Proton rms-radii from low-q power expansions?

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by  Ingo Sick

Proton rms-radii from low-q power expansions?
Ingo Sick and Dirk Trautmann
Dept. fu¨r Physik, Universita¨t Basel, CH4056 Basel, Switzerland∗
(Dated: January 10, 2017)
Several recent publications claim that the proton charge rms-radius resulting from the analysis
of electron scattering data restricted to low momentum transfer agrees with the radius determined
from muonic hydrogen, in contrast to the radius resulting from analyses of the full (e,e) data set
which is 0.04fm larger. Herewe show why thesepublications erroneously arrive at thelow radii.
7 PACSnumbers: 14.20Dh,21.10.Ft,25.30.Bf
1
0
2 Introduction. The determination of the rms-radius correspondingchargedensity — approximatelyan expo-
R of the proton charge distribution has recently at- nential — the moments r2n for n 2 grow unusually
n h i ≥
tracted much attention. While standard analyses of fast with increasing order n. In the form factor G(q)
a
J electron-protonscatteringdatayield0.879 0.009fm[1], the moments r2n are tightly coupled and give contri-
7 the Lamb shift measurement in muonic h±ydrogen gave butions of altehrnatiing signs. In an expansion with small
0.8409 0.0004fm [2]; this represents a 5σ discrep- n (n = 1,2) the values found for r2n depend on the
± ≈ h i
] ancy. The radii from electron scattering near 0.88fm maximum n and the value of the maximum momentum
x come from analyses that fit with excellent χ2 the world transfer q employed, andalways yield too small r2 .
e max h i
- cross section and polarization transfer data up to large This has recently been shown by Kraus et al.[13] who
l momentum transfer q, 5fm−1 to 12fm−1 [3–8]. Re- quantitatively demonstrate the pitfalls of fits with low
c
u cently, 3 publications [9–11] which restrict the analysis order power series by analyzing pseudo-data generated
n to the low-q data, with q = 0.7,0.9 and 1.6fm−1 re- with known R. They show that e.g. a linear fit in q2
max
[ spectively,findRinthe 0.84fmneighborhood,i.e. com- with q = 0.7fm−1 as employed in [9, 10] produces a
max
1 patible with the radius from muonic hydrogen. In this value of R which is low by 0.04fm.
v paper, we show why these analyses, which yield values This result of Kraus et al. can qualitatively be under-
9 of R 0.04fm lower than refs. [3–8], have led to erro- stood. When terminating the series eq.(1) with the q2-
0 neous≈ly low values. term, one implicitly posits r4 = 0. As r2 0.7fm2
8 Powerseriesexpansion. IntermsoftheelectricSachs this implies a charge densithy tihat is posithiveia≈t small r
1
form factor G (q) the proton charge rms-radius R is de- (charge proton +e), but has a negative tail at large r;
0 finedviathesleopeofG (q2)atq2 =0. Itthereforeseems due to the larger weight in the r4-term the tail can re-
. e
1 natural to parameterize G(q) in a power series duce r4 to 0. This negative tail of course also affects
0 r2 , handi leads to the systematically low values of R.
7 Ge(q)=1+q2a2+q4a4+q6a6+... (1) hThei samehappens mutatis mutandis withtruncationsat
1
higher order[13].
Xiv: tawhnhedecrahe6aR=rg2e−=dhren6−is/6it5ay02.4d0isNatrroienb-guriteviloeannt.ibvTyishttihecearlahltyiig,ohnaea4rlem=boehmhr4einni/dt1sa2on0f finTitheesizseeceoffnedc,to(FbvSiEou)s1,−−p1rGoeb(lqe)mdewcirtehasveesrlyikeloqwm2aqx:. tAhle-
ready at the q 0.8fm of maximal sensitivity of the
ar analysisrestrictedtodatawithlowmaximummomentum data to R (see≈below) the FSE q2R2/6 amounts to
transferq : atlowenoughq thetermsproportionalto ≈
max 0.09only. The smallnessofthe FSE emphasizesthat fits
q2n withn>1(orinsomecasesn>2)canbeneglected, used to extract R must reach the minimal χ2 achiev-
soalinear(quadratic)fitofthedataintermsofpowersof min
able, a visually good fit is not enough: a change of R
q2shouldsuffice. Loworder(oneparameter)fitsinterms
of 1% corresponds to a systematic change of G of only
ofderivedfunctionsase.g. adipole,G(q)=1/(1+q2b2)2, 0.0015 (0.17% of G ), a difference that is far beelow the
e
follow the same rationale, although these parameteriza-
resolution of typical plots of G (q) [9–11].
tionsdoimplicitlycontainhigherq2na2ncontributionsas ThesensitivityofthedatatoeRisshowninFig.1which
fixed by the analytical shape of the parameterization.
resultsfromanotchtestemployingSOGfitsoftheworld
Problems with expansions of the proton form factors
data (for recent reference to notch tests see [14]). When
in terms of q2n have been recognized earlier[12]. Due exploiting only part of the range of q 1.5fm−1, one
to the peculiar shape of the proton form factor — ap- ≤
looses part of the experimental information on R; anal-
proximately a dipole — and the peculiar shape of the yses which limit the data to e.g. 0.8 fm−1 as done in
refs.[9, 10] then ignore half of the data sensitive to R.
Restriction to a subset of the world data only amplifies
this problem.
∗ [email protected],[email protected]
2
FIG.1. (Coloronline)Sensitivity(arbitraryunits)tothe FIG. 2. The solid curves show the relative contribution
moments hr2i and hr4i obtained from fits of the world (in %) of the q2n terms to the finite size effect FSE in
data. Ge(q). Thedashedcurveshowstherelativecontribution
of an 0.2% uncertainty of the experimental Ge(q). For
comparison: the qmax of the fits linear in q2 (dipole) of
Contribution of higher moments. Foramoredetailed refs. [9, 10]([11]) amount to 0.72, 0.90 and 1.6fm−1,
discussionofthe problemswith eq.(1),we startfromthe respectively.
values of a2,a4,... determinedby Bernaueret al. [15] via
a power-seriesfit (with a χ2 as low as a spline fit) to the
Mainz data for q = 5fm−1. One might hope that,
max
of the data is 4-5 parameters (moments) which hardly
due to the large q and the high order 2n = 20 em-
max
can be represented correctly by a one-parameter form-
ployed,the values of the lowestmoments of interesthere
factorsuchas employedby Horbatsch+Hessels[11] (for a
should not be affected seriously by the above-mentioned
quantitative discussion see below).
problems [12]. Fig.2 shows the percent contribution of
Higher moments from world data. As was pointed
the a4 to a10 terms to the FSE. Also indicated is the
out in [12] and quantitatively demonstrated in [13] the
uncertainty in the FSE due to a (very optimistic) uncer-
determination of the lowest moments via a power-series
tainty of 0.2% in the experimental G (q).
e
fit is not veryreliable and for the higher n dependent on
This figure shows several features:
thecut-offinn. Wethereforehavemadeanindependent
1. At the q’s used in the ‘low-q fits’ referred to above,
with q =0.72 0.9fm−1, the contribution of the q4- determination.
max
term to the FSE− q2R2/6 amounts to 10–15% at the We use the world data up to the maximum momen-
≈ tumtransferavailableforG ,10fm−1 (notincludingthe
upper limit of the q-range where FSE is most sensitive e
data of ref.[15] which show systematic differences [3]).
to R. This shows immediately and without further cal-
This data set, which comprises 603 cross sections and
culation that neglecting this contribution in a linear fit
in terms of q2 must yield a value ofR2 which is low by a polarizationtransferpoints,iscorrectedfor2-photonex-
changeeffects [16] andfitted with a Fourier transformof
comparable percentage.
2. Eventhe contribution of the q6-term is not entirely Laguerrefunctionsoforder11forbothGe(q)andGm(q).
4 −1 Laguerre functions1 are particularly well suited as
negligible (15% of the q -term at q = 0.9fm ); when
attempting to determine a4 from a fit quadratic in q2 a –Theyprovideanorthonormalbasiswhichmakesmulti-
6 parameter fits very efficient (even if the polynomials are
wrong value results if the contribution of the q -term is
not strictly orthogonal over the limited q-range of the
not accounted for.
data).
3. Restriction of q to extremely low values, such
max
as to justifiably neglect the q4-term and maintain an ac- – They have a controlledbehavioratlargeradii r due to
curacy of 1% in R, would require q < 0.35fm−1. At the e−γr weight function, a consideration which is par-
max
ticularlyimportant[20]whenaddressinghighermoments
these values of q, the FSE is < 0.015, and the typical
(anaspectsharedwiththe parameterizationsoftheVec-
errorbars of G (q) would yield huge uncertainties in the
e
FSE contribution, hence R2 (see dashed curve). tor Dominance Model VDM).
–They providevaluesfor the momentsinsensitiveto the
Fig.2makesitobviousthatthelow-qfitsofrefs.[9,10],
4
which neglect the q -contribution, must find wrong val-
ues for R due to the omitted q4 term (for a quantitative
analysisseebelow). Fig.2alsoshows,withoutfurthercal-
culation, that for q 1.6fm−1 the information content 1 Forsimilarexpansionssee[17–19]
≤
3
cutoff in the number of terms employed; the moments rms-radius of 0.850 0.019fm and conclude that this
r2n are given by the lowest 2n+3 coefficients. value is consistent w±ith the muonic hydrogen result of
h Thie set of data can be reproduced with a χ2 of 542 0.84fm. Repeating their fit, but using the a4 deter-
with 548 degrees of freedom when the normalizations of mined much better from the high-q fit, yields a radius
the individual data sets are floated. When keeping the of 0.877 0.008fm, with lower χ2 and a significantly
±
normalizations at their measured values, and without in- smaller error bar. This result agrees with the 0.88fm-
creasing the error bars due to systematic error of the type results, and disagrees with the radius from muonic
normalizations, the χ2 amounts to 783 with 580 degrees hydrogen.
of freedom. These χ2 values are excellent given a set of Griffioen et al. also perform fits up to order q6, with
data measured over some 50 years. The resulting values a4,a6-values as given by simple models for the proton
for r4 are 2.01 0.05 (1.99)fm4. The quality of the fit chargedensity(uniform,exponential,gaussian)whichall
andhthie values of±the moments arevery close to the ones produce the same χ2; the resulting R-values are linearly
obtained using SOG [21] ( r4 = 2.03) or a VDM-type correlated with a4. Extrapolating these values linearly
4 h i
parameterization ( r = 2.01). We have verified that a to the value of a4 given by the fit to high-q data yields
variationofqmax bhetwieen7and12fm−1 anda variation R=0.876 0.008fm, again in agreement with the R’s
4 4 ±
ofnbetween10and13changes r by<0.03fm . Dis- in the 0.88fm region.
h i
tler et al.[22] obtained 2.59 0.19 0.04 from a mix of The bottom line: all the low-q fits of refs.[9, 10] yield
± ±
two form factor parametrizations fit separately to low-q radii in the 0.88fm region once the higher moments of
[15] and high-q [23] data. With these preliminaries we the chargedensity—whicharenon-zerobutignored(or
are in the position to quantitatively discuss the recent poorlyfixedinthe low-q fits due tothe truncationofthe
low-q fits. series in n of q ) — are properly accounted for.
max
Fits to very-low q data. Higinbotham et al. [10] per-
Fits to not-so-low q data. Horbatsch and Hessels
formalinearfitinq2toasubsetofthedataavailable,the
[11] employ the cross sections of ref.[7] up to a q of
max
form factors of Mainz80+Saskatoon74[24, 25]. For their 1.6fm−1. They parameterize the form factors via a 1-
highestq of0.9fm−1,whichyieldstheresultwiththe
max parameter dipole expression for both G and G . Their
smallest uncertainty, they find2 R = 0.844 0.014fm. fityieldsareducedχ2 of1.11,anda(chearge)rmms-radius
±
From this the authors conclude that R agrees with the
R=0.842 0.002fm. Fromthis,togetherwithotherfits
valueof0.84fmfrommuonichydrogen. Whenrepeating ±
whichyieldradiinear0.89fm,theauthorsconcludethat
exactly the same analysis, but adding in the q4 and q6
Risintherange0.84 0.89fm,i.e. couldbecompatible
contributions using the higher moments from the fit to −
with the radius from muonic hydrogen.
the high-q data, one finds a reduced χ2 (i.e. χ2 per de- Fig.2 showsthat forq =1.6fm−1 the moments up
max
gree of freedom) which is 11% smaller and a radius R of
to at least 2n=10 are important to get the full FSE. It
0.899fm. This R disagrees with the muonic value, and
ishighly unlikelythatthe one-parameterdipole contains
agrees with the above-cited R’s in the 0.88fm region. the mix of q2n-terms for 2n = 4...10 appropriate for the
Higinbotham et al. also perform a fit quadratic in q2,
proton. Indeed, expansionof the dipole in terms ofpow-
andfindaradiusof0.873±0.039fm. Thisagreeswiththe ers of q2 shows that the numerically largestdifference to
radiiinthe0.88fmregion,although,astheauthorswant
the power-series fit of [15] results from the contribution
toseeit,thevalueis“withinoneσ ofthemuonicresult”. ofthe r4 term. Thisdifferencein r4 alonewouldlead,
The uncertainty of ±0.039fm illustrates the large error at thehq =i 0.85fm−1 of maximal hsenisitivity to R, to a
bars resulting from the restriction of the analysis to a
difference ∆G of 0.0081 corresponding to 9.5% in the
fractionofthe q-regionsensitivetoR(see Fig.1)andthe FSE, hence Re2 (causing the systematic deviations just
largeuncertaintyof r4 duetothetruncationinq. When
using, instead of theh ri4 = 1.32 0.96 of Higinbotham visibleinFig.3of[11]). Thesameconsiderationappliesto
h i ± the parameterizationofG(q) asa(one-parameter)linear
et al., the value 2.01 0.05 we know from the fit to the
high-q data, the resu±lt for R becomes 0.901fm, with a afunndcttio=n1−q2c.zTwhiethlazck=in(g√fltcex−ibti−lit√ytocf)/t(h√etfict−futn+ct√iotnc),
smaller error bar of 0.010fm. −
causing systematic differences between data and fit and
Griffioen et al. [9] analyze part of the cross sections a χ2 larger than the one of already published fits, also
of [7] for q < 0.72fm−1 using eq.(1) including terms up
affects the results from the high-q fits of [9, 10].
to a4. They use a low-q parameterization for Gm/Ge ForthefitsofHorbatschandHesselsitisnotpractical
and take the shortcut of ignoring the free relative nor-
malizations of the individual data sets3. They find an to correct for the effect upon R of the incorrect higher
q2n-terms aswe did abovefor the analysesofrefs.[9,10];
too many terms 2n = 4...10 would contribute. In order
to demonstrate the importance of their effect we rather
quote the result of a Laguerre-function fit (4 terms each
2 Including Coulomb distortion would have increased R by
≈0.01fm[26] for Ge and Gm) to exactly the same data, yielding a
3 Correct treatment of the normalizations of the data sets of [7], lower reduced χ2 of 1.045 and a (charge) rms-radius
whichareindividuallyfloating,wouldhaveincreasedtheuncer- R = 0.884 0.016fm. Due to the lacking flexibility the
taintyofRbyafactor1.6. parameteri±zation of Horbatsch+Hessels has a χ2 that is
4
higher by 50! From such a “fit”, that is some 7 σ’s away n>1 are there, and they are known to be large. Ignor-
from a genuine best-fit, one obviously cannot get a sig- ing their strong correlationwith R [9–11] leads to wrong
nificant value for R. results for the proton rms-radius.
Conclusion. The moments r2n of the proton for
h i
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