Modelindependent determination of the magnetic radius of the proton from spectroscopy of ordinary and muonic hydrogen
Abstract
To date the magnetic radius of the proton has been determined only by means of electronproton scattering, which is not free of controversies. Any existing atomic determinations are irrelevant because they are strongly modeldependent. We consider a socalled Zemach contribution to the hyperfine interval in ordinary and muonic hydrogen and derive a selfconsistent modelindependent value of the magnetic radius of the proton. More accurately, we constrain not a value of the magnetic radius by itself, but its certain combination with the electriccharge radius of the proton, namely, . The result from the ordinary hydrogen is found to be , while the derived muonic value is . That allows us to constrain the value of the magnetic radius of proton at the 10% level.
pacs:
12.20.m, 13.40.Gp, 31.30.J, 32.10.Fn 36.10.GvI Introduction
While a discrepancy between results on determination of the electric charge radius of the proton has lately attracted attention of theoreticians and experimentalists, a controversy in determination of the magnetic radius is rather in shadow. The situation is summarized in Fig. 1. The proton charge radius has already been discussed in (1), which is referred here as paper I. This paper is a direct continuation of paper I and we do not reproduce here any plots or equations from there.
A stronger interest to the situation with the electric charge radius is due to a broader variety of the data and more important applications, such as determination of the Rydberg constant. While in the case of the magnetic radius there is a discrepancy between two scattering results (3); (2), the set of results for also includes the spectroscopic data on hydrogen and deuterium (7) and on muonic hydrogen (8); (9).
The contradiction between different values of is rather serious, while reading the published results literally. To certain extent it is expected that the discrepancy is partly due to different treatment of the proton polarizability contribution (10); (11). However, that may remove only a part of the discrepancy.
The spectroscopic data and, in particular, results on the hyperfinestructure (HFS) interval in hydrogen () and muonic hydrogen () (9), may present a source for an independent extraction of , however, at present there is no modelindependent constraints on from the HFS interval in muonic or ordinary hydrogen. Values published from time to time are deduced from models of the proton form factors, but there has been no realistic model of the proton developed to date.
Any comparison of ‘pure’ QED theory with the experimental results on hydrogen has been ‘contaminated’ for decades by the presence of certain proton finitesize and polarizability contributions. While the experimental value of the interval in hydrogen had been for a while among the most accurately measured physical quantities, the most sensitive QED tests for the HFSinterval theory has been performed not with the interval in hydrogen, but with quantities free of the influence of the nuclear structure. Such quantities have been provided by a study of leptonic atoms, such as muonium or positronium. Another opportunity is a comparison of the and HFS intervals measured with the same atom. Details of those QED tests with the HFS can be found, e.g., in review (12).
Here we explore a related question. The main purpose of this note is to estimate the constraints on the magnetic radius of the proton, , from the hyperfine splitting in muonic and ordinary hydrogen. If the proton polarizability contributions are known with a sufficient accuracy, we can experimentally determine the value of the proton finitesize contribution by a comparison of the theory and experiment. Such a contribution must be sensitive to the distribution of both electric charge and magnetic moment inside the proton. Considering that contribution in an appropriate way, we intend to extract a constraint on a certain combination of and .
While the QED effects are well understood (see, e.g., (12)), the total theoretical accuracy for the HFS interval in both muonic and ordinary hydrogen is completely determined by the protonstructure terms, namely, by the elastic twophoton contribution and by the proton polarizability correction. In case of hydrogen the experimental uncertainty is negligible, while for H it is compatible with and somewhat higher than the theoretical one.
As for calculation of the elastic term, its dominant part can be found in the external field approximation. We have to deal with integral
(1) 
which determines the dominant protonfinitesize contributions into the HFS interval in ordinary and muonic hydrogen
(2) 
where is the socalled Fermi energy, is the reduced mass
of a bound electron (in hydrogen) or muon (in muonic hydrogen) and
is the proton magnetic moment in units of the
nuclear magnetons. For available experimental data for
ordinary hydrogen
(3) 
We have no direct experimental knowledge on the integrand in (1), which consists of the subtracted form factors of the proton, . In particular, the accurate data fail at low momenta, which essentially contribute to the integral. Everything used in the integrand was a result of certain fitting rather than direct measurements. (We in part explore here ideas presented previously in (15) and developed in paper I.)
The situation with the integrand in (1) is illustrated in Fig. 2, where various fractional contributions to the integrand are estimated from the dipole model and presented as a function of . The red dotdashed line is for the subtraction term with unity. The blue solid line is for the term, which is related to the data. The integral is fast convergent at high . At low , say below , the data contribution produces a large uncertainty and any successful result for the Zemach contribution obtained previously was based on a certain, sometimes unrealistic, model.
We are going to split the integration into two parts:
(4) 
For higher momenta, we will use direct experimental data (or rather their realistic approximation). The accuracy of the form factors is roughly 1%. The integral over the direct data is indeed singular at , because the experimental values of and are not equal to unity exactly — they are only consistent with unity within the uncertainty, which produces the singularity. The smaller is the larger is the uncertainty of the related integral.
On the other hand, we can expand the form factors at low momentum
Some contributions into vanish because of the subtraction and the uncertainty comes from the remaining terms. The smaller is the smaller is the uncertainty. Here, and . The contribution to
(5) 
is to be treated separately. That is the ‘signal’ that we use to constrain . The leading remaining term is the term, which is responsible for the uncertainty.
The idea is to apply a certain model to estimate the uncertainties and to find a value of , which corresponds to the smallest uncertainty possible (cf. paper I).
Concluding on the model to estimate the uncertainty, we note that the dipole form factor is a reasonable estimation for the form factors as far as we discuss general features, but not any accurate particular value. So, we can, e.g., set for
(6) 
and estimate the coefficient as (cf. (1)). Here, we use for various preliminary estimations the standard dipole model
and apply for numerical evaluations , which corresponds to fm.
Ii Consideration within the dipole model
Let us perform an evaluation of following the consideration of in paper I.
The complete dipole value useful for further estimation of the fractional uncertainties is
(7)  
Iii Splitting the integral into parts
As we intend to split the integral into two parts, let us start with the highermomentum part
(8)  
Its uncertainty is estimated, by considering the part of the integral, singular at the limit . The result is
(9)  
or
where and we suggest for our estimations that both electric and magnetic form factors roughly follow the standard dipole fit and we experimentally know both of them within 1% uncertainty
Here we apply the dipole values for estimation of absolute and fractional uncertainties (cf. paper I).
Meanwhile, at low momenta, we find
(10)  
where was defined above.
Iv The extraction: a general consideration
Combining an experimental value, QED contributions and a polarizability correction we obtain
(11)  
where we noted that with accuracy sufficient for the denominator. Alternatively, we can write
(12) 
Indeed, there are also some higherorder protonstructure corrections, such as a recoil part of the twophoton exchange. We assume that they are included if necessary in the QED or polarizability term.
Often in some papers, they present rather than . Some ‘experimental’ values of are summarized in Table 1. Any ‘experimental’ value is a result of an extraction procedure that deeply involves theory and, in particular, a calculation of the proton polarizability, which dominates in the uncertainty budget for hydrogen and produces an uncertainty comparable with the measurement uncertainty for muonic hydrogen.
Atom  State  Ref.  

H  1.047(16) fm  1.5%  (16)  
H  1.037(16) fm  1.5%  (17)  
H  1.082(37) fm  3.4%  (9) 
Meantime, according to our theoretical consideration
(13)  
and thus we arrive at
V The extraction: the uncertainty budget and its optimization
It may be useful to introduce the fractional uncertainty of
Since roughly , a somewhat different value
(14) 
is roughly equal to , but easier to handle. It is sufficient to minimize the uncertainty of determination of .
It is equal to the rms sum of partial uncertainties, which are
With
and
we obtain
To find , we have to utilize the results from Table 1
where for ordinary hydrogen we use the result from (16).
With the uncertainty determined, let us consider behavior of the uncertainty as a function of . All the partial uncertainties as well as the total one as a function of are plotted in Fig. 3 both for ordinary (top) and muonic (bottom) hydrogen.
The optimal values, which minimize the uncertainty, and the partial contributions to the total uncertainty for those values are collected in Table 2.
Atom  Best  Best  Scat  Scat  

H  0.278  0.234 GeV  9.4%  4.8%  6.4%  4.8%  1.3%  3.2% 
H  0.312  0.263 GeV  13.3%  9.9%  8.1%  3.5%  0.9%  2.0% 
Vi The fits of the proton form factors
Now we are to find by integration over the data. As in paper I, we use for that a certain set of fits.
As an approximation we utilize the fits for the proton form factors and from Arrington and Sick, 2007 (18), Kelly, 2004 (19), Arrington et al., 2007 (20), Alberico et al., 2009 (21), Venkat et al., 2011 (22), and from Bosted, 1995 (23). The details of the fits for the electric form factor are presented in (1), while for the magnetic one they are summarized in Appendix A.
Two of the fits for are with socalled chain fractions, five are with Padé approximations with polynomials in , and one is a Padé approximation with polynomials in .
As well as in case of a pure electric integral in (1), the fit (24) of Bosted (23) is perfect for the tests. It is a Padé approximation with polynomials in , not in . It definitely has a lowmomentum behavior strongly different from others.
The fits are quite close to one another and to the standard dipole parametrization in the area of interest. They are more or less consistent to each other and to the dipole one (see Fig. 4). Comparison of the fits for the magnetic form factor to the related fit for the electric form factor is also presented (see, Fig. 5).
The lowmomentum behavior of the fits is summarized in Table 3. Indeed, the fit (24) from (23) is excluded.
Ref.  Fit  Type  

[fm]  [fm]  [GeV]  
(18)  (17)  Chain fraction  0.90  0.86  31.2 
(18)  (18)  Chain fraction  0.90  0.87  32.2 
(19)  (19)  Padé approximation ()  0.86  0.85  26.8 
(20)  (20)  Padé approximation ()  0.88  0.86  29.2 
(21)  (21)  Padé approximation ()  0.87  0.87  28.7 
(21)  (22)  Padé approximation ()  0.87  0.86  28.2 
(22)  (A)  Padé approximation ()  0.88  0.86  29.5 
Vii Integration over the fits
Integrating over the fits for the optimal GeV for hydrogen, we find that varies from GeV to GeV if we exclude Padé approximation in (23) or from GeV if we include it. For detail see Table 4. Here, we accept
as the mean value (excluding (24)), that leads to
The uncertainty of integral above does not include scattering in the calculation of because we estimated the uncertainty of this term in a more conservative way as explained above.
Fit  Type  

(18)  Chain fraction  
(18)  Chain fraction  
(19)  Padé approximation ()  
(20)  Padé approximation ()  
(21)  Padé approximation ()  
(21)  Padé approximation ()  
(22)  Padé approximation ()  
(23)  Padé approximation () 
Eventually, we obtain a constraint on the magnetic radius from the HFS interval in hydrogen
(15) 
and we remind that for the standard dipole parametrization .
The related fractional scatter is 0.013 if we exclude (24) from the consideration and it is 0.032 if we include it. The result for the combination of the proton electric and magnetic radius is consistent with the value from the standard dipole model within its 9% uncertainty.
The same evaluation can be performed for various and the results are summarized in Table 5. All the results are consistent. The scatter is below the uncertainty except for very low , where behavior of the fits becomes model dependent.
Scatter  Scatter  

0.20  0.169 GeV  0.99(13)  0.03  0.10 
0.25  0.211 GeV  1.02(9)  0.02  0.05 
0.30  0.253 GeV  1.03(10)  0.01  0.02 
0.35  0.295 GeV  1.04(11)  0.007  0.01 
0.40  0.337 GeV  1.05(14)  0.004  0.007 
0.50  0.421 GeV  1.08(21)  0.002  0.002 
Similar treatment for muonic hydrogen produces GeV as the optimized value. The results of integration over the fits for vary from GeV excluding (24) and from GeV including it to GeV (see Table 6). We consider
as the mean value that leads to
The uncertainty of integral above does not include scattering in a calculation of because we estimated the uncertainty of this term in a more conservative way as explained above.
Fit  Type  

(18)  Chain fraction  
(18)  Chain fraction  
(19)  Padé approximation ()  
(20)  Padé approximation ()  
(21)  Padé approximation ()  
(21)  Padé approximation ()  
(22)  Padé approximation ()  
(23)  Padé approximation () 
The constraint from the HFS interval in muonic hydrogen is found to be
(16) 
The fractional scatter is 0.009 (excluding (24)) or 0.020 (including (24)). The result is consistent with the value from the standard dipole moment within the uncertainty of 13%.
The results obtained at various values of the separation parameter are consistent to each other (see Table 7 for details).
scatter  scatter  

0.20  0.169 GeV  1.14(19)  0.03  0.10 
0.25  0.211 GeV  1.13(14)  0.02  0.05 
0.30  0.253 GeV  1.13(13)  0.01  0.02 
0.35  0.295 GeV  1.13(14)  0.007  0.01 
0.40  0.337 GeV  1.13(16)  0.004  0.007 
0.50  0.421 GeV  1.14(22)  0.002  0.002 
Viii Conclusions
Our strategy to evaluate was dictated by our purpose, which is to determine (cf. (15)). For a different purpose the strategy would be different.
To obtain a constraint on the magnetic radius of the proton , the compilation of all constraints on the electromagnetic radii of the proton is presented in Fig. 6. We plot there constraints from Fig. 1 and also present two constraints on derived here from a study of the HFS intervals in ordinary and muonic hydrogen.
That is a general picture. It already has certain features similar to those considered in paper I. The overall accuracy of spectroscopic extractions of the magnetic radius looks comparable with scattering results—not with their claimed uncertainty, but with their discrepancy.
The preliminary results on the proton magnetic radius from atomic spectroscopy are presented in Table 8. It involves all possible combinations of spectroscopic constraints on (from HFS) and on (from the Lamb shift). Note that , assuming that is known with a good accuracy and that roughly .
Transition/atom  Lamb (H)  Lamb (H) 

HFS (H)  0.80(8) fm  0.76(8) fm 
HFS (H)  0.88(11) fm  0.85(11) fm 
If we accept the value of the proton charge radius as
which seems a reasonable choice until the controversy in its determination is not resolved, then we arrive at
as to the best constraint on the magnetic radius from spectroscopy.
We have performed a number of consistency checks described above such as consideration of various values of the separation parameter. All the results are consistent. The estimation of the term is consistent with all fits with reasonable behavior at low discussed in Appendix (as well as with the fits from MAMI (see (24) for detail)). In case of the fits considered above the only fit with unreasonable behavior is that from (23), which in particular produces infinite values of the charge and mangetic radii. It is important that all fits but (24) agree with each other within at 1% level in the region where the separation parameter was chosen in Tables 5 and 7 as seen in Fig. 7.
The fit of (23) with incorrect behavior at low momentum transfer is responsible for a scatter of values of bigger than (see Fig. 7) (cf. (1)). However, taking into account its unrealistic behavior, that is acceptable. The other fits agree with each other within 1% in the region crucial for a choice of . A similar situation with the electric radius (see Fig. 6 of paper I).
A value of the magnetic radius extracted from electronproton scattering strongly depends on treatment of the proton polarizability (10); (11). To apply the Rosenbluth separation, one has to rely on a certain model for the proton polarizability. We note, however, that the electric form factor at low momentum transfer is less sensitive to the model and as far as we are not going to go too low, one may use experimental results on from recoil polarimetry (see, e.g., (4)), which are free of the polarizability problem.
Similarly to the case of examination for in (1), we conclude that the estimation of the uncertainty of the form factors at the level of 1% for applied is validated by the behavior of the fits and by the scale of the scatter. Nevertheless, a direct investigation of the problem would be useful.
The author is grateful to S. Eidelman and V. Ivanov for useful discussions. This work was supported in part by DFG under grant HA 1457/91.
Appendix A Fits for applied in the paper
The fits for applied in the papers fall into three classes.
1). Two fits deal with chain fractions. Those are from from
Arrington and Sick, 2007, (18). One is from analysis
of (18) alone
(17) 
while the other exploit the evaluation of the twophoton effects from (25)
(18) 
and
Five fits are Padé approximations with polynomials in . Those include fits from Kelly, 2004, (19)
(19) 
where
from Arrington et al., 2007, (20)
(20) 
from Alberico et al., 2009, (21)
(21)  
(22) 
and two fits Venkat et al., 2011, (22)
(23)  
The remaining fit from Bosted, 1995, (23)
(24)  
is a Padé approximation with polynomials in . That is a phenomenological fit designed to be used for medium and high . It is not expected to be appropriate at low . Providing a reasonably good approximation at medium momentum transfer, the fit apparently has incorrect low behavior and incorrect analytic properties such as a branch point at .
Footnotes
 The HFS interval in hydrogen is also well measured (14). The experimental accuracy is worse than for the , however, it still supersedes the theoretical accuracy. The and data are consistent and a separate consideration of the HFS interval would not add any new information on the proton structure. Meanwhile, comparison of the and results allows a sensitive test of QED (see, e.g., (12)).
 Here, is the numerical value for momentum transfer in GeV.
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