The recoil correction to the proton-finite-size contribution to the Lamb shift in muonic hydrogen

0.34 MB·English

by  Savely G. Karshenboim

The recoil correction to the proton-finite-size contribution to the Lamb shift in
muonic hydrogen
∗
Savely G. Karshenboim
Max-Planck-Institut fu¨r Quantenoptik, Garching, 85748, Germany and
Pulkovo Observatory, St.Petersburg, 196140, Russia
Evgeny Yu. Korzinin and Valery A. Shelyuto
D. I. Mendeleev Institute for Metrology, St.Petersburg, 190005, Russia
Vladimir G. Ivanov
Pulkovo Observatory, St.Petersburg, 196140, Russia
5
1 The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The
0 theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton
2 radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic
electron-protonscattering. Thatcreatesacertainproblemfortheinterpretationoftheresultsfrom
n
themuonic hydrogen Lamb shift. For thelatter we need also to takeinto account thetwo-photon-
a
J exchangecontributionwiththeprotonfinitesizeinvolved. Theonlywaytodescribeitreliesonthe
data from thescattering, which may producean internal inconsistency of theory.
6
Recentlytheleadingproton-finite-sizecontributiontothetwo-photonexchangewasfoundwithin
2
the external field approximation. The recoil part of the two-photon-exchange has not been con-

sidered. Here we revisit calculation of the external-field part and take the recoil correction to the
]
h finite-sizeeffects into account.
p
- PACSnumbers: 12.20.-m,13.40.Gp,31.30.J-,36.10.Ee,
p
e
h
I. INTRODUCTION The general problem in its calculation is that to find
[
it one has to use a certain description of the charge dis-

1 tribution (or G (q2)). There is no satisfactory ab initio
A few years ago the Lamb shift (2p 2s ) in E
v 1/2 − 1/2 model. So, one either could apply a certain variety of
muonic hydrogen was experimentally determined to a
9
models,whicharenotreallyrelatedtoexperimentaldata
great accuracy [1]. The related theoretical prediction
3
and arbitrary to a certain extent, and to estimate some-
5 is much less accurate, suffering from a bad knowledge
howthe uncertainty,or,alternatively,one couldtakead-
6 of the proton charge radius. Therefore, the experimen-
0 tal data and theoretical expression have been utilized to vantage of the existing experimental data. The first op-
1. determine an accurate value of the proton radius. Ac- tion does not really work for the proton. As for the sec-
ondoption,the experimentaldataby themselves arenot
0 tually, theory (see, e.g., [1–3]) disagreeswith experiment
accurate enough and one cannot use the data directly,
5 and therefore the muonic-hydrogen value of the proton
1 charge radius is inconsistent with results of hydrogen Instead one has to apply a fit. Unfortunately, and that
: is a real problem with the proton radius, the fits also
v spectroscopy [4] and evaluations of the elastic electron-
produce their value ofR and those values fromempiric
i proton scattering (see, e.g., [5–8]). E
X fits do not agree with the value from the muonic hydro-
ThetheoreticalpredictionfortheLambshiftinmuonic
r gen(see,e.g.,parametersoflowmomentumexpansionof
hydrogen is basically of the form
a
some empiric fits in Table I).
∆E =c +c R2 , (1) We have to clarify that the correction we discuss here
L 1 2 E
is a small one, while the contradiction is huge. The very
factandthe scaleof the discrepancydoes not dependon
where the first term, dominated by the Uehling correc-
thewaywetreatthehigher-orderproton-finite-sizeterm.
tionandother QEDeffects, does notdepend onthe pro-
Meanwhile, the eventual value of the muonic R does
ton rms electric charge radius R , and the second term E
E
depend on the way, we treat the higher-order finite-size
is dominatedbythe leading proton-finite-sizecorrection.
correction, at the level of the uncertainty of R . Here
Meanwhile, there is a higher-orderproton-finite-size cor- E
we revisit the correction.
rection, which does depend on the distribution of the
proton charge(or rather we prefer to discuss the electric The higher-order proton-finite-size correction can be
charge form factor of the proton G (q2)), but does not understood as a certain moment of the charge distribu-
E
have the required form of const R2. tion. Since the very mean radius of that distribution is
× E
under question because the results from the scattering
and the muonic hydrogen are different, we cannot di-
rectly apply the scattering results for these corrections
∗Electronicaddress: [email protected] (see, e.g., [9–12]).
2
Thus,thenext-to-leadingproton-finitesizecorrections formalism, such as the effective Dirac equation (EDE)
are sensitive to detail of the charge distribution. A self- [2, 19, 20]. Within such an approach the TPE diagrams
consistent consideration of those shape-sensitive contri- appear with certain subtractions (related to proton pole
butions was performed in [12]. The consideration was contributions) and on top of those subtracted two-body
done within the external field approximation, which de- TPEdiagramsone hastoaddaperturbativeseriesofan
livers the leading part of the shape-sensitive contribu- effective one-body theory, e.g., the EDE theory with a
tions. Here, we present a complete evaluation, including protononmassshell. ThelatterispresentedbytheFriar
recoileffects. Thoseinvolvenotonlythechargedistribu- contribution, while the former is related to two types of
tion, but also a distribution of the magnetic moment. theeffects. Thoseincludeelastic(withsubtractions)and
There are two main ingredients of our consideration. inelastic (polarizability) parts (see Figs. 2 and 3), which
We have to explain the method of a treatment of the at a certain stage are to be considered together.
higher-order finite-size correction within a theory of
muonic hydrogen, which would be consistent both with
the experimental data on the electron-proton scattering
and with the muonic value of the proton radius. The
method has been applied previously [12] within the ex-
ternal field approximation. Here we are to expand it to
accommodate the recoil finite-size corrections, which we FIG. 2: Two-photon-exchange: elastic contribution. Neces-
also need to consider. For that we follow [13, 14]. In sary subtractions, which follow from an EDE consideration,
both cases we start following cited papers, but modify are assumed. The circles are for the vertex with the appro-
priate form factors.
the approaches.
II. GENERAL EXPRESSION FOR THE
ELASTIC TWO-PHOTON-EXCHANGE
CONTRIBUTION
Let’s start with the expression for the proton-finite-
FIG. 3: Two-photon-exchange: inelastic contribution (the
size contribution including the recoil effects. They are
proton polarizability).
described by the two-photonexchange(TPE). The lead-
ing partofthe higher-orderproton-finite-sizetermis the
Indeed, once we have well-defined elastic and inelas-
so-called Friar term. That is a result derived within the
tic quantities, we can work on their calculation indepen-
external field approximation (see Fig. 1). The result for
dently. However,theirdefinitionis notastraightforward
the correction was first derived in the coordinate space
issue. As an example one can consider a calculation of
[15–17]andnext inthe momentum space[18] (see [2] for
the elastic part. It requires an appropriate on-shell ver-
details of the derivation)
tex for the proton. One may deal with the Dirac and
Pauli form factors F or with the Sachs ones G .
16(Zα)5m4 δ 1,2 E,M
∆E (nl) = r I l0 , (2) The difference betweenthose twoparametrizationsleads
Fr − π Frn3
to a contribution, which vanishes for the on-shellvertex,
∞
I = dq G (q2) 2 1 2G′ (0)q2 . but is not equal to zero for the TPE diagrams in Fig. 2,
Fr Z0 q4 h(cid:0) E (cid:1) − − E i welahsicthicipnavrotlvies nthoet wveerltl-edxeefisnoeffd-sbhyelilt.seIlnf.other words, the
Here and throughout the paper we apply the relativistic
Meanwhile, both elastic and inelastic contributions
units, in which ¯h = c = 1, m stands for the muon mass,
have their specifics and their calculation is very differ-
M is the proton mass and m is the reduced mass. The
r ent. Indeed, there are distinguished contributions into
momentum q is the Euclidean one.
each of them and those contributions can be studied in-
dependently of the definitions. One of them is the Friar
term, the dominant part of the elastic TPE and we in-
deed can study it independently of anything else. (As
we mention, the ambiguity in a possible definition of the
elastictermconcernsusuallyonlyrecoileffects,whilethe
Friar term is the result within the external field approx-
FIG.1: ThediagramfortheFriarcorrectiontoLambshiftin imation.) Indeed, the actual expression for the elastic
muonichydrogen. Theappropriatesubtractionsareassumed. part for the two-photon-exchange (eTPE) correction is
more complicated than the Friar contribution (2). Once
we intend to complete the calculation of the elastic part
If we like to consider the TPE correction with all re- and to add recoil corrections to the Friar term, we have
coil effects included, we have to use a certain two-body to check the definitions.
3
Therefore, we have to explicitly quote the definitions The first term apparently is the Friar term, which is
of the elastic and inelastic TPE and to be sure they are a non-recoil contribution. The other terms are for the
consistent. While calculationsofthe elasticandinelastic recoil. As it happens in the two-body systems (see, e.g.,
parts of TPE have been done in a single paper for a few variouscorrectionstotheLambshiftin[2])theexternal-
occasions [21–23], the first complete evaluation of TPE, field contributions and the recoil one are treated differ-
where consistency of their definitions was discussed, ap- entlyandtheyhavedifferentsubtractions. Thestructure
peared some time later in [13]. Unfortunately, it did not G2 1 is a result ofthe subtractionof the expressionfor
−
manage to produce consistent definitions of the elastic the point-like proton.
andinelasticpart. The consistencywasachievedlaterin
The explicit presentation of the f factors [13] is
[14].
The elastic part of the TPE contribution consists of
a QED result for a point-like proton and of a proton-
q γ (τ ) γ (τ ) 1
structure correction. We acknowledge that the point- f (m,M;q2) = 2 p 2 µ
E
likeparthasbeenalreadycalculatedanditissubtracted 2(M −m)(cid:20)(cid:18) √τp − √τµ (cid:19)1+τp
whilecalculatingtheeTPEterm. TheQEDTPEtermis 1 1
,
the well-known Salpeter contribution (see, e.g., [2, 20]). −(cid:18)√τp − √τµ(cid:19)(cid:21)
Itisfortheprotonwithoutanyanomalousmagneticmo-
qτ γ (τ ) γ (τ ) 1
ment, which satisfies the conditions F1(0)=1,F2(0)=0 fM(m,M;q2) = p 2 p 2 µ
or GE(0)=GM(0)=1. 2(M −m)(cid:20)(cid:18) √τp − √τµ (cid:19)1+τp
TheanalyticexpressionfortheeTPEcontributionwas γ (τ ) γ (τ )
1 p 1 µ
,
considered in [13, 14, 21, 22]. Here, we re-arrange the
expression for our purpose. The general structure is of −(cid:18) √τp − √τµ (cid:19)(cid:21)
M +m
the form [13, 14] (cf. [21, 22]) f (m,M;q2) = τ γ (τ ), (4)
F p 1 µ
m
16(Zα)5m4 δ
∆E (nl) = r I l0 ,
eTPE − π eTPEn3
where
I = I +I ,
eTPE Fr rec
∞
I = dq G (q2) 2 1 2G′ (0)q2 ,
Fr q4 E − − E q2
I = IZ0+I h+(cid:0) I , (cid:1) i τp = 4M2 ,
rec E M F
IE = ∞ dq4qfE(m,M;q2) GE(q2) 2−1 , τµ = 4qm22 ,
I = Z0∞ dqf (m,M;q2)h(cid:0)G (q2(cid:1)) 2 1i , γ1(τ) = (1−2τ) √1+τ −√τ +√τ ,
IM = Z0∞ dq4qfM(m,M;q2) h(cid:0)F M(q2) 2(cid:1) −1 i(.3) γ2(τ) = (1+τ)3/(cid:0)2−τ3/2− 23√(cid:1)τ . (5)
F q4 F 1 −
Z0 h(cid:0) (cid:1) i
The Dirac form factor We are going to treat electric and magnetic form fac-
tors somewhat different and we are to decompose the
F (q2)= GE(q2)+ 4qm22p GM(q2) Dirac form factor
1 1+ q2
4m2
p
1 2τ
F2 1 = (G2 1)+ p (G G 1)
ismorefundamentalthantheSachselectricandmagnetic 1 − (1+τ )2 E − (1+τ )2 M E −
p p
form factors G and G . However, just the latter are
E M τ2
used to be fitted in empiric-fit papers. + p (G2 1) (6)
Thelastterm,I ,wasexcludedin[13]fromtheelastic (1+τp)2 M −
F
partandattributedtothepolarizability. Lateron,itwas
re-established by [14] as a part of the elastic TPE. Most
into terms of the Sachs form factors. We also note that
oftheexistingresultsonthepolarizability[14,21,23–26]
are consistent with this definition1 or, as in case of [27], the eTPE contribution takes into account two structure
effects,whichareanon-zerovalueoftheanomalousmag-
can be re-adjusted to it.
netic moment of the proton F (0) = G (0) 1 = κ
2 M
− ≃
1.79285and the finite size of the proton. It is helpful to
disentanglethembecausetheformercanbefoundwitha
1 Actually,theverycalculationofthepolarizabilityin[13]iscon- very high accuracy, while the latter as we will show (see
also [12]) needs for its calculation additional experimen-
sistentwiththedefinitionoftheelasticpartincludingthisterm
aswasshownin[14]. tal data and is not free from uncertainties.
4
Eventually, we arrive at notnecessaryrelatedtothesizeofthecontribution. The
Friar term is not only the dominant term for the eTPE
Irec = Iκ+IEF+IM1+IM2 contribution,butitisalsothepartofeTPE,whichisthe
∞
dq mostsensitivetotheparametrizationoftheformfactors.
I = κ (2+κ)f +f ,
κ q4 M1 M2 The sensitivity of this term to the parametrization was
Z0
IEF = ∞ dq4qfE(cid:8)F(m,M;q2) GE((cid:9)q2) 2−1 , disTcuhsesesdituinat[i1o2n].with accuracy of the data for the in-
I = Z0∞ dqf G (q2)h2(cid:0) (1+κ(cid:1))2 ,i teesgtirmataitoinonissaprreemseandteedwiinthiFnigt.he5dtiapkoelne pfraormam[e1t2ri]z.atTiohne.
M1 q4 M1 M −
Z0∞ dq h(cid:0) (cid:1) i
I = f G (q2)G (q2) (1+κ) , (7)
M2 q4 M2 M E −
Z0 6
(cid:2) (cid:3)
where
4 GEHq2L
1 -1
f = f + f ,
EF E (1+τp)2 F 2 -2G’H0Lq2
τ2 q(cid:144)L
f = f + p f , 1 2 3 4 5
M1 M (1+τ )2 F
p -2
2τ
p
f = f . (8)
M2 (1+τ )2 F 1.0
p
0.8
Wecancomparetheintegrandsrelatedtovariouscon-
tributions to I as summarized in Fig. 4. The plot is
eTPE 0.6
prepared using the standard dipole approximation
0.4
Λ2 2
G (q2)= (9) 0.2
dip q2+Λ2
(cid:18) (cid:19)
q(cid:144)L
0 1 2 3 4 5
(with Λ2 = 0.71GeV2), which is good to estimate the
size of various contributions, but is not accurate enough
for real calculations. FIG. 5: Top: Fractional contributions to the integrand of
IFr in (2) as a function of q/Λ as follows from the dipole
model. The red dot-dashed line is for the subtraction term
20 with G′(0). The dashed line is for the subtraction term
with 1 and the blue solid line is for the G2 term, which
15
represents the data. Bottom: the integrand (in arbitrary
10 units) is presented with a dashed line and the integral
5 R0qdq/q4(cid:2)(GE(q2))2−1−2G′E(0)q2(cid:3) (in fractional units) in
presentedasafunctionofqwithasolidline. Theshadowbelt
qHGeVL is for the area in the q space from which 90% of the integral
0.2 0.4 0.6 0.8 1.0
-5 IFr come. See[12] for detail.
-10
Thegrayareaisforthearea,where90%oftheintegral
comes from. One sees that below 0.6Λ 0.5 GeV there
≃
is a massive cancellation between the data term (with
FIG. 4: Integrandsfor theindividual contributionsto IeTPE: GE(q2)) and the subtractions terms (with 1 and with
for the Friar term (2) (the black line), the κ term (the cyan G′(0)). The data on G (q2) roughly have accuracy at
E
dot-dot-dashed line), theEF term (thered dashed line), the the 1%levelandbelow this point the accuracyof the in-
M1term(thegreendottedline),andtheM2term(bluedot-
tegrand becomes worse than 1% and increases very fast.
dashed line).
Meanwhile, above 0.8Λ 0.65 GeV, the contribution of
≃
the data themselves (i.e. of the G (q2) term) is negligi-
E
′
ble, while the subtraction term with G (0) dominates.
E
Therefore, the integrand at low momentum is deter-
III. A PRELIMINARY CONSIDERATION mined by our knowledge of the curvature of (G (q2))2,
E
whileathighmomentumitis determinedbythe slopeof
As we see from the plot, the Friar term is indeed the (G (q2))2. Both values cannot be measured experimen-
E
dominant contribution. However, that is not only the tally, but can be extracted on base of a certain fitting
valueofthecontributionthatmatters. Theuncertaintyis procedure. Exceptfortheκterm,theintegrandofwhich
5
does not involve the data on the form factors, the other crepancyisthatlarge. Thescatteringdatafromdifferent
data arealsosensitive to the fittings. At lowmomentum experimentsareconsistentanditishardtoimaginethat
all the integrands are determined by the slope of the re- they are completely wrong.
lated G2 terms and thus are proportional to the electric The integrand involves massive cancellations at low q
and/ormagneticradiussquared. Indeed,anyfittingpro- domain, while for high q it is dominated by the contri-
duces a certain systematic uncertainty. bution, proportional to
Unfortunately, the determination of neither electric
1
nor magnetic radius is free of questions. Different ex- G′ (0)= R2 . (10)
tractionsfromdifferentkindsofdataleadtoinconsistent E −6 E
results. Wehavealreadymentionedastrongdiscrepancy
If one calculates the integral using scattering data, as
between muonic-hydrogenR [1] and the scattering val-
E it was done, e.g., in [13], the value of G′ (0) is indeed
ues (see, e.g., [5–8]). The results on R are not in a E
M taken from there and it is inconsistent with R from
perfect shape as well (see, e.g., [11, 28]). However, the E
the Lamb-shift. If one calculated the whole integralas a
discrepancyin determinationof R fromMAMI [5] and
M single thing, then there is no other choice because if we
global data [6] is not that important. The most impor-
′
takeavalueofG (0)inconsistentwiththeappliedfitfor
tant is that the determination of the charge radius R E
E G (q2), the integral is infrared divergent.
fromthescatteringandmuonichydrogenisnotinagree- E
Following[12],ourtreatmentoftheFriartermisbased
ment, which compromises the whole procedure. Mean-
on two important steps. First, following [9, 12], we split
while, the magnetic radius is even more sensitive to var-
the integration into two parts
ious systematic effects [29–31].
Aswementioned,wediscussarathersmallcorrection. ∞ q0 ∞
We anticipate that a different treatment of the higher- I = dq... I<+I> dq...+ dq.... (11)
≡ ≡
order proton-size contribution will somewhat shift the Z0 Z0 Zq0
value ofR frommuonic hydrogen,but notsignificantly
E Those two terms are to be treated differently.
(cf. [12]). So,weexpectthatthediscrepancywillremain
The separation disentangles the subtraction at low q
andwehavetofindawayhowtousethescatteringdata
and the asymptotic behavior at high q, which both in-
without a contradiction to the finally extracted value.
′
volve G (0), and, in particular, that allows to use for
We intend to use fits, rather than the data. Generally E
the high-q asymptotics the physical value of R . This
speaking, to find an appropriate fit is a tricky issue. A E
value is not fixed, being an adjustable parameter to be
pure empiric fit is the best suited to fit the data, how-
eventually found from a comparison of the theory and
ever, it ignores various theoretical constraints and con-
experiment on the muonic-hydrogen Lamb shift. In this
straintswhichfollowfromsomeotherscatteringchannels
area we integrate the G2 term and the subtractions sep-
(through theory). On the contrary, a fit, which follows
arately and while integrating G2 we rely on the empiric
all those constraint, either may disagree with the data
fits.
or may involve so complicated fit function that it would
Meanwhile,atthe low-q partofthe integralwerelyon
produceinstabilities. Therefore,weprefertouseempiric
the expansion of the form factor
fits, but we intend to apply them in a specific way.
onTeheevraeluaarteestwthoewhaaydsrtooniacpvpalycueummpiprioclafirtizs.atEio.ng.,cownhterin- GE(q2) 2 1 RE2 q2+Kq4 . (12)
≃ − 3
butiontotheanomalousmagneticmomentofmuon(see,
(cid:0) (cid:1)
e.g., [32]), there is no subtraction involved. The fit is Even a very bold estimation of the K coefficient allows
necessary only as a smooth approximating curve which at low q a better accuracy than the data by themselves
is consistent with all the data involved. Its parameters (see [12] and consideration below).
bythemselvesareirrelevant. Theotherstoryiswhenthe The final result takes the form
parameters of the fit are really applied. In the integral
in (2) they are. The very shape of an empiric fit has I =C R2 +I . (13)
R2 E
no physical sense. The parameters must be applied with E
a caution. A scenario, in which the result for the Friar Such a form of presentation is crubcial, allowing a self-
correctiondoesnotdependonthe low-q2 behaviorofthe consistent consideration. As the result, the theoretical
fit, was developed in [12]. expression for the Lamb shift in muonic hydrogen keeps
Themostplausiblescenariotoexplainthediscrepancy the desirableformas in (1) and the higher-orderproton-
between the scattering and muonic-hydrogen values of finite-sizeterm(2)contributestobothcoefficientsc and
1
R is that the scattering data are more or less correct, c . Meantime, we use the empiric fits only in the area
E 2
buttheaccuracyindeterminationoftheradiusisoveres- of the high-q part of the integrand and the result only
timated. The alternatives, such as a suggestion that the marginally depends on the details of their extrapolation
dataarenotcorrectorthemuonicvalueiswrong,arenot to q2 = 0 (see [12] and consideration below). As a final
an option here. If the muonic value is wrong,there is no importantstep,wehaveafreeparameterq whichweare
0
sense in a calculation of small corrections while the dis- to vary to minimize the uncertainty of the calculations.
6
IV. CALCULATION OF THE LEADING momentum contribution and its uncertainty
PROTON-FINITE-SIZE TPE CONTRIBUTION
q
(THE FRIAR TERM) 0
I =10(1 1) . (15)
Fr> ± Λ4
Let’s consider evaluation of the leading proton-finite-
For the low-q partwe haveexactly followed[12], while
size TPE contribution I in more details. To deal with
Fr for the high-momentumpartwe prefer a certainalterna-
a lower-momentum contribution
tion, whichwouldbe more suitable for the calculationof
I = q0 dq G (q2) 2 1 2G′ (0)q2 the recoil corrections. The higher-momentum part con-
Fr< q4 E − − E tains the data-related and radius-relatedcontributions
we expand thZe0form fha(cid:0)ctor (cid:1) i I = ∞ dq G (q2) 2 1 + 1RE2 . (16)
Fr> q4 E − 3q3 3 q
G (q2) 2 1 RE2 q2+Kdip(1 1)q4 , (14) Zq0 (cid:0) (cid:1) 0 0
E ≃ − 3 ± Combining the low-q2 and high-q2 contributions, we
(cid:0) (cid:1) find
and, following [12], we estimate the q4 coefficient with
1h9el.p8 GofeVth−e4)s.tanTdhaerddipdoipleoleformmoudleali(sKndoitp t=hat10b/aΛd4fo≃r IFr = q0 dq4q GE(q2) 2−1−2G′E(0)q2
tbhoeldveaslutiemfaotliloonwsi.ngE.fgr.o,mthethGe′m(0u)otnericmRsaErebydiffroeuregnhtlyfr7o%m b +Z0 ∞ dqh4q(cid:0) GE(q(cid:1)2) 2−1 , i
aspnedctfrroosmcotphyo’sseRfolloawndingfrofrmomschayttderroignegn’s-aRnd-dbeyutaerbiouumt 1Zq0 h(cid:0) (cid:1) i
E E C = , (17)
17%. The uncertainty of 100% for the curvature seems R2E:Fr 3q0
reasonable. The value is also consistent with the curva-
ture of various empiric fits (see, e.g., [33–38]; see also a which is necessary for the decomposition according to
discussion below). (13).
Thephysicalconditionontheexpansionisq <2m To calculate the centralvalue of the data-relatedterm
0 π
≃
0.28GeV. Wearetovaryq tominimize thefinaluncer- in(16),theintegrationin[12]wasdoneoverthefits [33–
0
tainty and we will check this constraint after we specify 37](seeAppendixA)andastheresulttherelatedmedian
the optimal value of q . value was taken. Here, we use the most recent empiric
0
The expansion (14) allows to easily find the low- fit from [37]
1+2.90966τ 1.11542229τ2+3.866171 10−2τ3
G (q2)= p− p × p , (18)
E 1+14.5187212τ +40.88333τ2+99.999998τ3+4.579 10−5τ4+10.3580447τ5
p p p × p p
whereτ isdefinedabovein(5). We estimateourknowl- comes from a relatively small area around the low limit
p
edge of the form factors as at the level of 1% and expect (cf. [12]). So, we expect the whole G2 contributions
that the dominant contribution to the integral for G2 known at the level of 2% and thus we arrive at
I = ∞ dq G (q2) 2 1 0.02 1 + 1RE2 , (19)
Fr> q4 E × ± − 3q3 3 q
Zq0 0 0
(cid:0) (cid:1) (cid:0) (cid:1)
where G (q2) is taken from (18). of the fits was estimated in [12] separately as
E
δI = 1 2δGE(q02) G (q2) 2 , (20)
Fr> 3q3 G (q2) dip 0
0 E 0
(cid:0) (cid:1)
Whilechoosingthefits,wehavelookedforthosewhich where we expect that δG (q2)/G (q2) 1%.
produce a smooth function for the all space-like q2 and Here, we apply a somewEha0t diffEeren0t e≃stimation of the
have a good value of the reduced χ2. We prefer ‘simple’ uncertainty. Itisestimatedas2%oftheG2 contribution
empiricfits. We believethatitistrickyto determinethe into I . The dominant part of the G2 integral comes
Fr>
accuracyfor the extrapolationto low q2 andfor the sub- fromthelowendoftheintegrationareaandasfarasthe
sequent differentiation (to find R ) and, most probably, low-end area dominates the estimations are close.
E
their inconsistency with the muonic value of the proton Aboveweexplainedthatitisnotconsistenttouseany
radiusisduetoanoverestimationoftheiraccuracy. Nev- fits. However,weneedto distinguishusingofthe central
ertheless, the form factor by itself (not its derivative) is value of the fits with a pre-estimated uncertainty, as we
fitted by those fits well. The accuracy of the application do,and the use ofthe parametersof the fits to find, e.g.,
7
some derivatives etc. If the fit over a large sample of the as we mentioned, it simplifies a comparison with [13].
data has the reduced χ2 about unity, that means that Note, that is not an improvement, but a simplification.
the fit is consistent with the data at the level of their The reason to check the median using various fits (some
uncertainty, which is about 1%. Such use of the fit does of which are partly out of date) was to check that the
not assume any model. Once one intends to take advan- result does not much depend on low-q2 behavior. The
tages of the large number of data points and to obtain a spread of the values of the radius was sufficient to prove
′
certainvalue,e.g.,G(0)withahighprecisionbecauseof such model independence, which was also confirmed by
statisticalaverage,the modelcomes outandthe system- the use a fit from [39], which, despite of unreasonable
aticuncertaintyduetoitsapplicationistobeestimated. low-q behavior,produces a quite reasonableresult. Only
We prefer to stay on the safe side and introduce the un- aftersuchtestsareperformed,theuseofthemostrecent
certainty at the level of the accuracy of the data which fit is possible.
grantsmodelindependence while applyingany fitwith a The extracted value of the proton radius depends on
good χ2 value. the separation parameter q in almost the same way as
0
The scatter due to choice of the parametrization (i.e. for [12]. The procedure (see [12] and sections below) al-
due to choice of the fits [33–37]) is smaller than the un- lowstovarythevalueoftheseparationparameterq and
0
certainty we introduced. The reasonfor the bigger value to find an optimal one. The results with different values
of the uncertainty is that all the fits basically fitted the ofq areconsistent. Thedeviationfromtheformertreat-
0
same data, which were only partly updated from a pub- mentoftheeTPE[13]islargeenough(atthelevelofthe
lication to a publication. The scatter does not reflect uncertainty) to be considered seriously. The dependence
the accuracy of the data, but rather the importance of R (q ) is shown in Fig. 6.
E 0
their updates. The uncertainties of I and I are
Fr< Fr>
considered as independent.
REHfmL
0.8415
Fit Ref. Type RE [fm] K [GeV−4]
0.8410
(A1) [34] chain fraction 0.90 34.3
(A3) [34] chain fraction 0.90 35.3
0.8405
(A5) [33] Pad´e approximation (q2) 0.86 28.0
(A7) [35] Pad´e approximation (q2) 0.88 31.1 0.8400
(A9) [36] Pad´e approximation (q2) 0.87 28.2
0.8395
(18) [37] Pad´e approximation (q2) 0.88 31.3
2mΠ q0HGeVL
0.1 0.2 0.3 0.4
TABLE I: The low-momentum expansion of the fits for the
electric form factor of the proton applied in [12]. The values
aregivenforcentralvaluesofthefitswithoutanyuncertainty.
Here: (cid:0)GE(q2)(cid:1)2 = 1−RE2q2/3+ Kq4 + .... The related FIG. 6: The value of the proton charge radius RE extracted
values for the standard dipole fit are RE = 0.811 fm and [12]frommeasurement[1]ofthemuonic-hydrogenLambshift.
K = 19.8 GeV−4. The spread of the central values of the The plot follows [12]. The horizontal belt is for the original
charge radius (from 0.86 to 0.90fm) for different fits (from value of the parameter. The other figure [of a more compli-
different sets of the data) is comparable with the spread of catedshape]isfortheradiusdeterminedin[12],presentedas
central values of the updated results on the proton radius, a function of the separation parameter q0. Every area corre-
obtained by different methods, the central values of which sponds to one sigma. The internal and darker part of each
are0.84fm(µH[1]),0.88fm(H&D[4]),0.895fm(globaldata figureisrelated tothepartial uncertaintyduetothecalcula-
without MAMI [6]—and the results of the other evaluations tion IFr, while the whole area is for the total uncertainty of
aresimilar)and0.88fm(MAMI[5]). Allthevalues,butfrom theradius. The final result of [12] RE =0.84022(56)fm was
muonic hydrogen, have uncertainty at the level of 1%. The obtained thereat theoptimal valueq0 ≃0.152GeV.
muonic oneis muchmore accurate.
Minimizing the uncertainty we arrive at
Tofacilitatefurthernumericalevaluationswithinclud-
ingtherecoileffects,wechoosehereinsteadofthemedian I = 1RE2 23.5(3.3)GeV−3 , (21)
Fr
the fit [37], which is the most recent one. This modifi- 3 q0 −
cation is not important because our estimation of the
uncertainty is substantially larger than the scatter. The where the numerical value corresponds to q0
≃
mostoftheintegral(fortheG2 term)comesfromthelow 0.147GeV. Itisimportantthatthenewoptimalvalueis
limit. Here,wechoosetoassignthe2%uncertaintytothe quitebelow2mπ whichistherequirementforthevalidity
whole G2 term. We have checked that this modification of the expansion (14).
is of marginal importance for the Friar term, however, While the decomposition R2 term and into the con-
8
stant I is somewhat different2, the related numerical re- cannot check it a priori because at low q2 we use the
sults at R = 0.84 fm are in a perfect agreement, being actual value of the proton radius. However, once the
E
17.6(3b.4) GeV−3 [12] and 17.7(3.3) GeV−3 (this work). complete calculation is performed, we find the value of
The approach applied here is a modification of the one the radius and can to check the continuity. Such an a
from [12] and the results for the radius are expected to posteriori test is plotted in Fig. 7. The red area is for
nearly coincide at R = 0.84 fm. The latter effectively thelow-q2expansionandthegreenoneisthefitfrom[37]
E
means that the extracted radius is to be approximately suggesting the uncertainty of 1%. They are consistent.
the same(with a marginaldeviation). (We widely use in
this paper the ‘reference’ value of various contributions G(cid:144)Gdip
1.02
at R = 0.84 fm—that is the approximate value consis-
E
tent with extractions from the muonic hydrogen [1, 12]
at the level of 10−3 which is completely sufficient taking 1.00
into account the uncertainty of our calculations of the
Friar and eTPE integrals at the level of 20%.) 0.98
Those numerical values should be compared with
the results which can be obtained if one directly ap- 0.96
plies empiric fits to (2). The related results are
I = 23.1(1.1)GeV−3 (after applying empiric fits [40]), qHGeVL
Fr 0.0 0.1 0.2 0.3 0.4 0.5
22.9(1.2)GeV−3 (afterapplying[12]empiricfits[35]and
[33]—thatissimilarlytoconsiderationin[13],whichwas
done for the complete I ) and I = 24.3(7) GeV−3 FIG.7: Thecontinuitytestfortheprotonelectricformfactor
eTPE Fr
(after applying empiric fits [10]). The uncertainty of our normalized against the standard dipole fit. The red area on
result is essentially larger than theirs and the the shift theleftwithfastincreasing(totheright)widthisforthelow
momentumapproximation,whilethebeltwithapproximately
from the quoted values to ours is larger than the un-
homogenous width on the right is for the empiric fit [37] we
certainty. We believe that the uncertainty of the cited
used,with the uncertaintyof 1%.
results was underestimated and suffered from an incon-
sistency(oncewe acceptthe muonicvalueforthe proton
The method suggested in [12] pretends that the re-
radius).
sult does not depend on the parametrization of the fit.
We havetorecallthegeometricalmeaningoftheFriar
To check that, a number of fits have been applied (see
term
Fig. 8). Those include the chain-fraction fits (in q2) [35]
I = π d3r d3r′ r r′ 3ρ(r)ρ(r′), (22) (the blue dashed line), the Pad´e-approximation (in q2)
Fr 48 | − | fits [33,35,36](the greendot-dashedlines)and[37](the
Z Z
green bold solid line), and a fit with the inverse polyno-
where in a non-relativistic approximation ρ(r), being a
mials in q [39] (the solid red line). The fits except for
Fourier transform of G (q), is the charge density in-
E [39] have reasonable low q behavior. They are analytic
side the proton. The muonic-hydrogen data assume a in q2 around q2 = 0 and the related value of the charge
smaller proton, thus the Friar term adjusted to those
radius varies from 0.86 to 0.90 fm, so the spread is com-
data should also be smaller (see also discussion on that
parablewith the distance to the muonic value of0.84 fm
in [10]). The radius is smaller by about 6% than the re- (see Table I). The high-q2 asymptotics is proportional
sultsfromtheempiricfitsandtheFriarterm(thecentral to q−2 for the chain fractions and to q−4 for the Pad´e
value) is smaller by about 15%, which seems consistent.
approximations. Therefore, the parameters and the gen-
Indeed, in a rigorous consideration, the Fourier trans-
eral behavior (far from the characteristic q area) vary
0
form of G is not the charge density in common sense.
E verymuch. Nevertheless,thespreadofthe valuesforthe
However, the presence of relativistic effects does not af-
integralI is smaller thanthe estimationofthe uncer-
Fr>
fectthefactthatthevalueoftheFriartermisdetermined
tainty [12]. The scatter of the results with inclusion of
to a certain extent by the proton size, it just makes this
the fit with unrealistic behavior [39] is somewhat larger
relation more complicated and involves corrections. The
than the uncertainty but still acceptable [12].
non-relativisticpartwithacleargeometricsenseremains
and it is dominant.
Onemaywonderwhethersplittingtheareaoftheinte- V. CALCULATION OF THE RECOIL
gration maintains the continuity of the form factor. We CORRECTIONS
AfterdiscussingtheleadingeTPEterm(withintheex-
ternal field approximation), we are prepared to consider
2 The result of [12] is IFr = R2E/(3q0)−22.2(3.4) GeV−3, where the eTPE recoil corrections to it. We start from (7). As
the numerical value uses the optimal value q0 ≃ 0.152 GeV. weseefromFig.4,the dominantproton-finite-sizeterms
(Indeed,sincetheuncertaintyofI> fortheFriartermisdefined
are the EF and the κ ones. The former is the proton-
here and in [12] differently, the optimal values, which minimize
theuncertainty, arealsosomewhatdifferent.) sizeterm,whilethelatterisnotrelatedtotheprotonsize
9
DG(cid:144)G directly. Domination of the EF term is due to q/M ex-
0.2 2mΠ 0.3 0.4 0.5qHGeVL pansion. Comparing the coefficient functions at the area
of m q M we find
-0.01 ≪ ≪
-0.02
-0.03
3 q
-0.04 fEF(m,M;q2) ,
≃ −8M
3 q 3
-0.05 f (m,M;q2) ,
M1
≃ −16 M
3 (cid:16)q 3(cid:17)
f (m,M;q2) , (23)
M2
FIG. 8: The fractional deviation of the fits (see in App. A ≃ 16 M
for details) and the data for the electric form factor GE(q2) (cid:16) (cid:17)
from the standard dipole form factor. The data are taken
from [35],wheretheprotonform factorswereextractedfrom
experimental data on the elastic electron-proton scattering.
The fits include the chain-fraction fits (A1) and (A3) (in q2) which explains why the EF term is dominating.
[35] (the blue dashed line), the Pad´e-approximation (in q2)
fits (A5), (A7), (A9), and (18) [33, 35, 36] (the green dot-
dashed lines) with the most recent one from [37] shown with An important area of the integration is related to q
a green bold solid line. An exceptional fit (A12) with an manditisusefultopresentamoredetailedasymptotic∼s
inverse polynomials in q [39] is shown with a solid red line.
at m M, q M (exactly in m/q)
≪ ≪
q4 4m2+q2 q +8m4 4m2+q2 2m 2m2q3
− − −
f = +... (24)
EF − (cid:16)p (cid:17) 16m(cid:16)4pM (cid:17)
Decomposing the EF term according (13) we find and in a close analytic form
κ M
I = q0 dq G (q2) 2 1 2G′ (0)q2 f (q2) Iκ = 16M2(M m) · −3(1+κ)ln m
EF q4 E − − E EF − (cid:20)
b Z+0 ∞ dhq(cid:0) G (q(cid:1)2) 2 1 f (q2),i +(1−κ) ln2− 14 +B· Mm 2 . (27)
1Zq0 q0q4dqh(cid:0) E (cid:1) − i EF wherethe value ofB (cid:18) (1). Ou(cid:19)r result(cid:16)for t(cid:17)hi(cid:21)s value is
C = f (q2). (25) ∼O
R2E:EF −3 q2 EF
Z0 κ 2 2M 2+11κ
B = − ln +
4 m 48
(cid:20) (cid:21)
That is very similar to the expression (17) for the Friar
κ 3 2M 43(1+κ) m 2
term. The evaluationofthe integralsisdone in asimilar + − ln +...
16 m − 384 M
way and we apply the same effective description for GE. (cid:20) (cid:21)(cid:16) (cid:17)
Theonlydifferenceisthatinformercasealltheintegrals As one sees (cf. (23)), this term is M−3 in contrast
exceptfortheG2termwerecalculatedanalytically,while to the leading part of the EF term∼, which is M−1.
here we have to do that numerically. ∼
However, the non-vanishing value of κ, which enters the
The other important term is the κ term. It is a combinations such as κ(1 + κ), produces a certain en-
proton-structure correction, but not a proton-finite-size hancement.
one. Since we decompose elastic TPE into the point- Let’s now consider the magnetic finite-size terms, I
M1
like Salpeter term and a proton-structure term, the sub- and I . At low-q we use expansion
M2
traction in the latter is due to a proton without any
anomalous magnetic moment. Therefore, there is a con- R2
G (q2) 2 =µ2 1 dipq2[1 0.4]+Kdipq4(1 1) ,
tribution to eTPE which does not vanish in the limit M p − 3 ± ± !
R = R = 0, which is the κ term. It does not involve
E M (cid:0) (cid:1) (28)
theparametrizationoftheformfactorsandmaybefound
where µ = 1+ κ 2.79285 is the proton magnetic
with a high accuracy. We obtained the result from both p p ≃
moment in the units of the nuclear magnetons, for the
numerically
magneticformfactor,similartothatforthe electricone.
However,thereisadifference. Whilefortheelectricform
I = 2.85023GeV−3 (26) factorthevalueofR istheadjustableparameterforthe
κ E
−
10
extraction of the charge radius from muonic hydrogen, but with a conservative uncertainty of 40% (to R2 ).
M
the rms magnetic radiusR should be a constraint. We
M As for the high-q, we apply the fits from [37], which
use here the dipole value of the radius
gives not only the fit for the electric form factor (18),
but also a fit for the magnetic one. The latter is of the
12
R2 = , form
dip Λ2
GM(q2) = 1−1.43573τp+1.19052066τp2+0.25455841τp3 . (29)
µ 1+9.70703681τ +3.7357 10−4τ2+6.0 10−8τ3+9.9527277τ4+12.7977739τ5
p p × p × p p p
q0 [GeV] CR2E [GeV−1] IbeTPE [GeV−3] IeTPE [GeV−3] ratherthanasthe rmssum. Thatleavesuswiththe EF
0.1 3.36220 −44.52 16.4(6.0) term as the main source of such an uncertainty since it
0.2 1.71465 −12.79 18.3(4.0) Contribution CR2E [GeV−1] IbeTPE [GeV−3] IeTPE [GeV−3]
Fr 2.271 −23.5(3.3) 17.68(3.31)
0.3 1.17525 −2.597 18.7(5.8)
κ 0 −2.850 −2.850
0.4 0.91153 2.527 19.1(7.6)
EF 0.039 1.26(4) 1.97(4)
0.5 0.75709 5.790 19.5(9.2)
M1 0 1.76(3) 1.76(3)
M2 −0.0006 −0.838(4) −0.849(4)
TABLE II: The results I and IeTPE for fits (18) and (29) at
b
variousq0. ThelastcolumnisforthenumericalvalueofIeTPE eTPE 2.309 −24.1(3.3) 17.71(3.27)
at RE =0.84fm.
TABLEIII:IndividualcontributionstotheIeTPE at optimal
q0 ≃ 0.147 GeV. The last column is for numerical value of
ThedecompositionintoI andC andfurthernumer-
R2E IeTPE at RE =0.84fm.
icalevaluationsofthosevaluesaresimilartothoseforthe
EF term. b
The numericalresultsforeTPEasa functionofdiffer-
isexpressedintermsofthesameparametersasthelead-
ent values of the separation parameter are summarized
ing term. The contribution to the integral by itself can
inTableII.The minimizationoftheuncertaintyleadsto
comefromavariousareaofq,however,theareasensitive
q 0.147GeV as the optimal value.
0 ≃ to the parametrization is around q0. The expansion at
The result is
low q (< q ) is the most uncertain at the largest ‘low q’
0
I = 24.14(3.27)+59.31r2 GeV−3 , (30) available,whiletheuncertaintyofthehigh-Qarea(>q0)
eTPE − p is mostly determined by the smallest ‘high q’ available.
where the err(cid:2)or of 2.90 GeV−3 corre(cid:3)sponds to I , Thecontributioncomesfromamorebroadareathanthat
eTPE< around q q 0.147GeV. In the limit m q M,
while the error of 1.53GeV−3 relates to IeTPE>. thesuppre∼ssio0n≃ofEF behavesasq0/M with≪sma0ll≪coeffi-
The individual contributions to the eTPE corbrection cients(<1)asfollowsfromtheasymptoticbehavior(23).
arepresentedin Table III, while the indivbidualcontribu- The optimal q is found to be q 0.147 GeV and thus
0 0
tions to the uncertainty are summarized in Fig. 9. We q < 2m . That leads to a suppr≃ession of the EF term
0 µ
seethattheindividualelectricandmagneticrecoilcontri- in respect to the Fr term by a factor of m/M ( 10%)
butions are at the level of 10-15%, however, they have a withsmallcoefficients(<1)partlybecause ofad∼ditional
trendforastrongcancellation. Thecancellationisrather softening of the integrands by factors q/2m < 1 and
µ
accidental and depends on the value of the anomalous softening of the character of the q-integration.
magnetic momentofthe protonκ ,the mass ratiom/M
p As an important test we check that the scatter due
and the proton radius. The terms differently depend on
to use of the empiric fits (except for those from [39]) is
the parametersandthe cancellationofthe centralvalues
below the uncertainty. The fits from [39] shift the result
doesnotproduceamassivecancellationoftheuncertain-
somewhatfarther,but the value is acceptable because of
ties.
the unreasonable behavior of the fits at low q.
However, the latter are much smaller, than the level
of the 10% of the Friar-term uncertainty. The size of
the contribution is not that important as its sensitivity
to the parametrization. It is also important, that any VI. CONCLUDING REMARKS
uncertaintiesatthe levelof10%ofthe Friar-termuncer-
tainty would be important only if it is correlated to the The extraction equation is of the similar form as for
main uncertainty and we have to sum them up linearly the Friar term in [12]