The finite nuclear mass $(Z\,α)^2\,m/M\,E_F$ correction to the hyperfine splitting in hydrogenic systems is calculated using a combined relativistic heavy particle and nonrelativistic quantum electrodynamics. The obtained results are in disagreement with previous calculations by Bodwin and Yennie [Phys. Rev. D {\bf 37}, 498 (1988)]. The comparison of improved theoretical predictions with the corresponding measurements in hydrogen reveals $5σ$ discrepancy, which indicates problems with the proton structure corrections.
Nuclear recoil (finite nuclear mass) effects in atomic systems are quite difficult to calculate, even for simple systems like hydrogen-like ions. Although the reduced mass accounts for the finite nuclear mass on the level of the Schrödinger equation, it is not the case for a Dirac equation. In fact, one has to refer to a full quantum electrodynamic (QED) theory to account for relativistic nuclear recoil effects. There exist many formulations of two-body bound-state theory in the literature, mostly based on the Bethe-Salpeter approach [1]. In all these formulations, the kernel of the integro-differential equation can only be constructed perturbatively in powers of the fine structure constant α. Alternative formulation, introduced by Caswell and Lepage [2], is the nonrelativistic QED (NRQED), where one constructs perturbatively an effective Lagrangian, by matching with a full QED theory. This was a very fruitful formulation, which, combined with dimensional regularization allowed for many important results for simple atomic systems, like the recent one in Ref. [3].
Another development in bound-state QED was initiated by V. Shabaev [4,5], who formulated in 1985 the exact nonperturbative formula for the first order in the electron nuclear mass ratio correction to atomic energy levels. A few years later this correction was independently rederived in Ref. [6], and since than it has been used for the calculation of the Lamb shift in hydrogen-like ions [7,8]. Very recently this original formula has been extended to finite size nuclei [9], to the hyperfine splitting [10], and to higher powers in the m/M mass ratio [11] under the name heavy particle QED (HPQED).
From the other side, measurement of the Lamb shift and hyperfine splitting in hydrogen, hydrogen-like ions, and other simple atomic systems are accurate enough to determine fundamental constants [12] and in general to test fundamental interaction theory. Particularly accurate is measurement of HFS of the ground state of the hydrogen atom, which after averaging over several results [13] is E exp hfs = 1 420 405.751 768(1) kHz, However, it has the least accuracy of theoretical predictions among all transitions in the hydrogen atom. In spite of the fact that the proton size is 5 orders of magnitude smaller than the atomic hydrogen, it gives a significant -33 ppm correction to the hydrogen HFS. To verify this proton structure effect and to test the standard model of fundamental interactions, all other quantum electrodynamics effects shall be rigorously calculated. Since the original calculation by Bodwin and Yennie [14] of relativistic nuclear recoil effects, theoret-ical predictions have been in a few σ disagreement with this measurement. The dubious correction was the one related to the proton structure, namely the distribution of the magnetic moment and the proton polarizability. Despite many further investigations of the proton structure effects, this discrepancy persisted.
In this work we recalculate relativistic nuclear recoil effects of order (Z α) 2 m/M in a system consisting of a lepton of mass m and an arbitrary nucleus of mass M , using complementarly NRQED and HPQED approaches, and obtain a result in disagreement with the previous calculations by Bodwin and Yennie in Ref. [14]. Moreover, the discrepancy between theoretical predictions and the measured hyperfine splitting in hydrogen is even increased, for which we do not have any obvious explanation. Most probably, the analogous measurement of the hyperfine splitting in µH [15,16] will shed light on this discrepancy. Since the muon is about 200 times heavier, the proton structure effects are much more significant there. At the same time, the muon proton mass ratio ∼ 0.11 is not very small, and thus nuclear recoil effects are more important. In fact, our result for the relativistic recoil effect significantly improves the accuracy of theoretical predictions not only for H, but also for µH HFS.
Before moving to the main calculations, let us introduce the hyperfine splitting and the basis notation. In a relativistic framework the interaction between the static point nuclear magnetic moment ⃗ µ
and an electron leads to the hyperfine splitting of atomic energy levels given by the expectation value of the magnetic interaction E hfs = ⟨V hfs ⟩ with the Dirac wave functions. This formula is valid for the infinite nuclear mass only, and it is not obvious how to account for the finite nuclear mass in this relativistic framework. In fact, later in this work we present a recently derived [10] exact relativistic formula for the leading recoil correction to HFS in mass ratio m/M . Nevertheless, it is more convenient here to perform a nonrelativistic expansion in Z α, where finite nuclear mass effects can easily be accounted for. And so, the leading hyperfine splitting for S states is given by
where µ is the reduced mass, g is the nuclear g-factor defined by
and where q = -Z e and e is the electron charge.
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