2SHyperfine splitting of muonic hydrogen

Abstract

Corrections of orders ␣ 5 and ␣ 6 are calculated in the hyperfine splitting of the 2S state in the muonic hydrogen. The nuclear structure effects are taken into account in the one-and two-loop Feynman amplitudes by means of the proton electromagnetic form factors. The total numerical value of the 2S state hyperfine splitting in the muonic hydrogen is 22.8148 meV. This value can be considered as a reliable estimate in conducting a corresponding experiment with an accuracy 10 −5 . The value of the Sternheim hyperfine splitting interval ͓8⌬E HFS ͑2S͒ − ⌬E HFS ͑1S͔͒ is obtained with an accuracy 10 −6 .

Laboratory tests of Lorentz andCPTsymmetry with muons

“…Also, only coefficients with even n contribute to Eq. (45), as those with odd n come only with even j and hence cancel via the geometrical factor (43). For example, we find 32 independent observable combinations can contribute for d ≤ 6, and they are denoted as H exists only for n ≤ 2.…”

“…The theory relating the hyperfine splitting to the Zemach radius [18,42,43] shows that the shift δE HF can be understood as a change δr Z given as δr Z ðfmÞ ≃ −6.2 × 10 12 δE HF ðGeVÞ: ð27Þ…”

The size of the proton from the Lamb shift in muonic hydrogen

“…A treatment of possible solutions to this puzzle can be found in [35]. Assuming the theory for the Lamb shift in μp [37][38][39][40][41][42] and for the H atom [37,43] to be correct, and using the precisely measured 1S-2S interval in H [6,8], the 1S-and 2S-Lamb shifts in H calculated with our r p value, and the most recent value for the fine structure constant α [44], we obtain a new value for the Rydberg constant R ∞ = 10,973,731.568160 (16) m −1 (relative accuracy 1.5 × 10 12 ). This result is 4.6 times more precise than the CODATA-value [36], but −110 kHz/c or 4.9 σ away.…”

Breit equation with form factors in the hydrogen atom

“…This includes the proton structure corrections to the standard Coulomb potential plus terms such as the Darwin term (which takes into account the relativistic effects), the spin-spin and spin-orbit interaction terms (corresponding to fine and hyperfine structure) and retarded potential terms. We focused in particular on certain terms involving the spin-spin and spin-orbit interaction with the aim of studying the proton form factor effects (or proton finite size effects) in the hyperfine splitting in electronic and Our FSC of -0.11234 meV to the 2S hyperfine splitting in muonic hydrogen is close to the order α 5 corrections of -0.183 meV [28] and -0.1518 meV [35] using other approaches.…”

“…In fact, the energy of the 2S splitting is 4 E hf s (2S f =1 1/2 ). In the evaluation of the proton radius in [19], the values of the hyperfine splittings were taken from [35], where the FSC for the 2S were evaluated using the Zemach method and those for the 2P case were not taken into account. Their FSC (taken from HF S = 3.392570 meV.…”