Hyperfine structure in hydrogen and helium ion

QED theory of the hyperfine splitting of the $1s$ and $2s$ state in hydrogen isotopes and helium-3 ion is considered. We develop an accurate theory of a specific difference 8E_{HFS}(2s)-E_{HFS}(1s). We take into account fourth order corrections and nuclear structure effects. The theoretical prediction is now of a higher accuracy than the experiment is. The study of the difference provides the most accurate test (on a level of a part in 10^8) of the QED theory of $1s$ HFS up to date. The theory agrees with most of the experimental data.

QED theory of the hyperfine splitting of the 1s and 2s state in hydrogen isotopes and helium-3 ion is considered. We develop an accurate theory of a specific difference 8EHFS(2s) − EHFS(1s). We take into account fourth order corrections and nuclear structure effects. The theoretical prediction is now of a higher accuracy than the experiment is. The study of the difference provides the most accurate test (on a level of a part in 10 8 ) of the QED theory of 1s HFS up to date. The theory agrees with most of the experimental data. The hyperfine structure (HFS) intervals of the ground state in a number of neutral atoms and singly charged ions can be measured with a high accuracy. However, theory even in the case of the simplest of them (such as hydrogen isotopes and the helium-3 ion) is essentially affected by nuclear structure effects which contribute from 30 to 200 ppm and cannot be calculated accurately. In contrast the 1s HFS interval in muonium is calculated with an uncertainty of about 0.1 ppm and can be used to accurately test the bound state Quantum Electrodynamics (QED). However, the muonium calculations involves precision values of the fundamental contants (α and m µ /m e ) and it would be important to test the QED calculations for the HFS without interferring with such a problem.
Study of a specific difference provides us with an opportunity to make a test of the QED theory on a level of accuracy essentially better than 1 ppm [1]. Such a high accuracy is possible because of an essential cancelation of nuclear contributions. We report here on new results for the difference in Eq. (1). We complete calculations of the fourth order corrections and find nuclear structure contributions which remain after cancelation of the leading effects. Some of the corrections obtained here are bigger than the experimental uncertainty [2] and must be taken into account. Our results are found to be in a fair agreement with most experimental data on hydrogen, deuterium and helium-3. We present a significant improvement of the theory for D 21 and demonstrate that the comparison of theory and experiment [2,3] for helium tests presently the QED theory of 1s and 2s HFS on the highest level, namely one part in 10 8 . That superceeds the muonium HFS by an order of magnitude.
The hyperfine splitting of an ns state in a hydrogenlike atom with a nuclear mass M and a nuclear spin I can be presented in the form Here Ry is the Rydberg constant, c is the speed of light, h is the Planck constant, µ B is Bohr's magneton and m is the electron mass. The nuclear magnetic moment µ in our notation can be negative (if its direction is opposite to the nuclear spin) and the Fermi energy E F , related to an energy splitting between the atomic state with total angular moment F = I +1/2 and I −1/2, can be negative as well. The QED correction for the HFS interval in the ground state is (see Ref. [4] where a e is the electrons anomalous magnetic moment. A comparison of the QED calculations with experimental values is summarized in Table I. To compute theoretical values we use fundamental and auxiliary constants from Refs. [5,6]. The QED expression above does not take into account any recoil effects. Recoil contributions involve high momentum transfer [16] and are essentially affected by the nuclear structure. In Table I we also present data for the 2s state, the theoretical expression for which is similar to Eq. (4) but some coefficients are different (see below).
One can see that the 1s hyperfine structure has been measured very accurately but any test of the QED calculations is limited by an essential contribution related to the nuclear structure which cannot be calculated precisely. In fact the uncertainty of the nuclear-structure contribution is at least 20% in hydrogen [18], and for deuterium the accuracy is not better [19]. In the case of tritium and helium-3 ion no results on the contribution of the nuclear effects has been obtained to the best of our knowledge. Thus, the pure QED theory is incomplete because of lack of the nuclear-structure contributions and a comparison of the QED theory with the experiment in Table I demonstrates how much it is incomplete. Our final target is a comparison of 1s and 2s HFS intervals and for this reason we do not try to correct the QED theory for the nuclear effects. Contrary, we compare the pure QED calculation and experimental data to "measure" the nuclear contribution.
For comparison we presented in Table I a theoretical result on 1s muonium HFS [17], which contains the recoil contributions and even small non-QED terms. Muonium, being a pure leptonic atom, is free of the nuclearstructure problem, however, the accuracy of any theoretical calculation is limited to 10 −7 by the uncertainty of experimental values for parameters needed to calculate the Fermi energy in Eq. (3). Those are the muon magnetic moment and the fine structure constant. Below we demonstrate that combining data for the 1s and 2s hyperfine structure in hydrogen and 3 He + we can go far beyond 1 ppm level [1] and hence develop a precision test of the QED theory for HFS compatible with the one related to muonium HFS.
The theory for the specific difference D 21 in Eq. (1) up to the third order in units of the Fermi energy was developed some time ago [20][21][22] The nuclear-structure corrections essentially shift the HFS value from its QED prediction. Three major nuclear effects contribute to the difference in Table I. Namely they are: • the nuclear charge and magnetic moment distribution (that is the biggest effect in the case of hydrogen); • a nuclear polarizability contribution (that is the biggest effect in the case of deuterium); • nuclear recoil contributions of order (Zα)(m/M )E F and higher.
There is also a correction to the Lamb shift caused by the nuclear structure where R E is the nuclear electric charge radius and relativistic units in whichh = c = 1 are used. When the contributions to HFS (6) and the Lamb shift (7) are determined, one can try to obtain a correction for difference D 21 . That is possible because most of the nuclearstructure corrections do not depend on the details of the atomic structure. Both, the leading contributions to the HFS and the Lamb shift are of a special factorized form The energy shift is the product of the nuclear-structure parameter A(Nucl) and the value of the wave function The leading correction to the difference in Eq. (1) must therefore vanish. The non-vanishing contributions can be expressed in terms of some effective δ-like potentials The coefficient A(Nucl) can be for various nuclear contributions calculated (see e.g. Eq. (7)) or determined from a comparison of experiment and a pure QED theory (see e.g. Eq. (6) where 1 + ζ = R 2 M /R 2 E is a ratio of quadratic magnetic and electric nuclear radii. We obtain the nuclear structure contribution to the 1s HFS interval from comparison in Eq. (6) and conservatively estimate the uncertainty as 10%. The Lamb shift contribution is taken from Eq. (7).
Partial results on the α(Zα) 3 contributions were found in Ref. [1]. They are related to effective non-relativistic potentials which lead to logarithmic contributions for the 1s state HFS [25]. The terms in the same order should also appear from potentials which contain some derivatives. A complete result on the self energy contribution was calculated after a suggestion by us in Ref. [27]. We report here the completion of the evaluation of the vacuum polarization effects. We derive an exact result for HFS of the 2s state and a contribution to D 21 is found via a comparison with the previously obtained result for the 1s state [28].
The fourth order contibutions are finally found to be The partial results for the constants C SE and C V P that are obtained in Ref. [1] contain some misprints. Being corrected, the partial results (C SE ≃ 2.5 and C V P ≃ 0.83) are found to be close to the complete results above.
That confirms an intuitive assumption that the potentials with derivatives lead to relatively small contributions. Smallness of terms with derivatives is important for our estimation of uncertainties of the nuclear-structure corrections. Let us discuss the uncertainty of the QED expression. The first two terms in Eq. (12) are found in the logarithmic approximation and we estimate the next-to-leading terms by a half-value of the leading contribution. However, in the case of the third term in Eq. (12) the situation is more complicated. First of all, the (Zα)(m/M )E F corrections to the 1s HFS contain a nuclear-structure dependence presented by ln(mR E ). Since we have not included them into the QED expression (4), they are effectively taken into account as a part of ∆E Nucl HFS . That means that an essential part of the (Zα) 3 (m/M )E F contribution into D 21 is effectively included into D 21 (Nucl) via Eq. (11). However, there are some contributions with loop momentum of about one electron mass and below which does not depend on the nuclear structure. They can be enhanced because of a relatively big magnetic moment (compared to the Dirac value) and we estimate the uncertainty of the last term in Eq. (12) as (µ/µ B )(Zα) 3 E F (cf. Eq. (5)).
All contributions to the difference D 21 in hydrogen, deuterium and helium-3 ion are summarized in Table II. Parameter ζ is known very badly, but it is not expected to be much larger than unity and hopefully the ζ-term is essentially below the uncertainties related to theory and experiment and thus may be excluded from further considerations.
An essential improvement of the theory is achieved. In previous papers related to third-order QED corrections [20,21] the uncertainty was not spelled out. We found here a number of corrections exceeding the experimental uncertainty. We state that after the examination presented here the theoretical predictions (Table II) D 21 (theor) = D are more accurate than the experiment. Five accurate measurements performed on three atomic systems are compared with our calculation in Table III. Four experimental results are in fair agreement with our theory, but a recent result for hydrogen [11] shows a 1.8σ discrepancy. The most important comparison is related to 3 He + : the 2s HFS was measured most accurately [2] and its value is also the most sensitive to higher order corrections (because of larger Z and larger nuclear contributions). Because of a fair agreement of our theory with the helium experiment we expect that in the case of hydrogen the discrepancy is related to a problem on the experimental side.
We consider comparison of theory and experiment for the difference D 21 as a test of a calculation of a statedependent part of corrections to E QED HFS (ns) and hence we present in the last column in Table III a standard deviation σ with respect to the Fermi energy E F , i. e. to a value directly related to the 1s HFS. That comparison demonstrates that study of D 21 in helium ion provides a more accurate test of QED than the study of the muonium HFS (σ/E F ≃ 0.1 ppm) and indeed of HFS in hydrogen and other atoms with a structured nucleus. The uncertainty for D 21 in hydrogen and deuterium is determined experimentally, while in the case of helium a value of σ contains an essential contribution from theory as well.
Most of so-called QED tests involve in part some other problems such as • verification of nuclear models and calculations of nuclear effects and hadronic contributions; • tests of consistency of data for fundamental constants (such as muon magnetic moment) or effective parameters (such as the proton charge radius) related to completely different experiments.
The D 21 theory is free of all these problems. No constants are really involved: an effective value of E Nucl HFS (1s) related to the nuclear effects arises from HFS theory. Its contribution being relatively small is under control as well as other nuclear contributions.
It is important to mention that presently there are three crucial higher-order QED contributions to hydrogenic energy levels: radiative recoil of the order α(Zα) 2 (m/M )E F , pure recoil of the order (Zα) 3 (m/M )E F and two-loop effects of the order α 2 (Zα) 6 m. The difference D 21 is sensitive to all of them and a progress in its calculation will therefore contribute into progress in theory of the hydrogen Lamb shift, muonium hyperfine structure and positronium energy levels. The most accurate measurement on this difference is related to an 25-years old experiment on helium ions [2] and we can hope that some experimental progress to improve the most precise test of QED theory for the hyperfine structure is possible.
An early part of this work was done during a short but fruitful stay of SGK at University of Notre Dame and he is very grateful to Jonathan Sapirstein for his hospitality, stimulating discussions and participation in the early stage of this project. We would like to thank Thomas Udem for useful discussions. The work was supported in part by RFBR grant 00-02-16718, Russian State Program "Fundamental Metrology" and DAAD.    (16), [14] 0.0 0.18