Precision atomic spectroscopy provides a solid model independent bound on the existence of new dark forces among the atomic constituents. We focus on the keV-GeV region investigating the sensitivity to such dark sectors of the recent measurements on muonic atoms at PSI. To this end we develop for the first time, the effective field theory that describes the leading effect of a new (pseudo-)vector or a (pseudo-)scalar particle of any mass at atomic energies. We identify in the Lamb Shift measurement in muonic deuterium (μD) and the 2s Hyperfine Splitting (HFS) in muonic hydrogen (μH) the most promising measurements to probe respectively spin-independent and spin-dependent new forces. Furthermore, we evaluate the expression of the vector force HFS finding that a future measurement of the 2s HFS in regular hydrogen could provide the strongest atomic bound for such a force for masses above 100 MeV.

Italy

Claudia Frugiuele

Germany

Clara Peset

Muonic vs electronic dark forces: a complete EFT treatment for atomic spectroscopy

Positronium and muonium are purely leptonic atoms and hence free of an internal substructure. This qualifies them as potentially well suited systems to probe the existence of physics beyond the Standard Model. We hence carry out a comprehensive study of the sensitivity of current positronium and muonium precision spectroscopy to several new physics scenarios. By taking properly into account existing experimental and astrophysical probes, we define clear experimental targets to probe new physics via precise spectroscopy. For positronium we find that, in order for the spectroscopy bounds to reach a sensitivity comparable to the electron gyromagnetic factor, an improvement of roughly five orders of magnitude from state-of-the-art precision is required, which would be a challenge based on current technology. More promising is instead the potential reach of muonium spectroscopy: in the next few years experiments like Mu-MASS at PSI will probe new regions of the parameter space testing the existence of medium/short range (MeV and above) spin-dependent and spin-independent dark forces between electrons and muons.

Israel

Jesús Pérez-Ríos

Current and future perspectives of positronium and muonium spectroscopy as dark sectors probe

We compute the heavy quarkonium mass of <math><mi>l</mi><mo>≠</mo><mn>0</mn></math> (angular momentum) states, with otherwise arbitrary quantum numbers, with next-next-to-next-to-leading logarithmic (<math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LL</mi></mrow></math>) accuracy. This constitutes the first observable in heavy quarkonium for which two orders of the weak-coupling expansion sensitive to the ultrasoft scale are known and the resummation of ultrasoft logarithms is made. We also obtain, for the first time, resummed <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>3</mn></mrow></msup><mi>LL</mi></mrow></math> expressions for the different fine and hyperfine energy splittings of these states, which are not sensitive to the ultrasoft scale but still require resummation of (hard) logarithms. We do this analysis for the equal and non-equal mass cases. We also study an alternative computational scheme that treats the static potential exactly. We then perform a comprehensive phenomenological analysis: we apply these results to the <math><mi>n</mi><mo>=</mo><mn>2</mn></math>, <math><mi>l</mi><mo>=</mo><mn>1</mn></math> bottomonium, <math><msub><mi>B</mi><mi>c</mi></msub></math> and charmonium systems and study their convergence.

Spain

Antonio Pineda

Jorge Segovia

<math><mrow><mi>P</mi></mrow></math>-wave heavy quarkonium spectrum with next-to-next-to-next-to-leading logarithmic accuracy

We determine the charm and bottom quark masses using the N3LO perturbative expression of the ground state (pseudoscalar) energy of the bottomonium, charmonium and B  c  systems: the η  b , η  c  and B  c  masses. We work in the renormalon subtracted scheme, which allows us to control the divergence of the perturbation series due to the pole mass renormalon. Our result for the  <math><mover><mrow><mi>M</mi><mi>S</mi></mrow><mo>¯</mo></mover></math> $$ \overline{\mathrm{MS}} $$ masses reads  <math><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>c</mi></msub><mfenced><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>c</mi></msub></mfenced><mo>=</mo><mn>1223</mn><mfenced><mn>33</mn></mfenced></math> $$ {\overline{m}}_c\left({\overline{m}}_c\right)=1223(33) $$ MeV and  <math><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>b</mi></msub><mfenced><msub><mover><mi>m</mi><mo>¯</mo></mover><mi>b</mi></msub></mfenced><mo>=</mo><mn>4186</mn><mfenced><mn>37</mn></mfenced></math> $$ {\overline{m}}_b\left({\overline{m}}_b\right)=4186(37) $$ MeV. We also extract a value of α  s  from a renormalon-free combination of the η  b , η  c  and B  c  masses: α  s (M  z ) = 0.1195(53). We explore the applicability of the weak coupling approximation to bottomonium n = 2 states. Finally, we consider an alternative computational scheme that treats the static potential exactly and study its convergence properties.

The charm/bottom quark mass from heavy quarkonium at N3LO

We consider the $${\mathcal {O}}(1/m)$$ <math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>m</mi><mo>)</mo></mrow></math>  and the spin-independent momentum-dependent $${\mathcal {O}}(1/m^2)$$ <math><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow></math>  quasi-static energies of heavy quarkonium (with unequal masses). They are defined nonperturbatively in terms of Wilson loops. We determine their short-distance behavior through $$\mathcal{O}(\alpha ^3)$$ <math><mrow><mi>O</mi><mo>(</mo><msup><mi>α</mi><mn>3</mn></msup><mo>)</mo></mrow></math>  and $$\mathcal{O}(\alpha ^2)$$ <math><mrow><mi>O</mi><mo>(</mo><msup><mi>α</mi><mn>2</mn></msup><mo>)</mo></mrow></math> , respectively. In particular, we calculate the ultrasoft contributions to the quasi-static energies, which requires the resummation of potential interactions. Our results can be directly compared to lattice simulations. In addition, we also compare the available lattice data with the expectations from effective string models for the long-distance behavior of the quasi-static energies.

Maximilian Stahlhofen

Relativistic corrections to the static energy in terms of Wilson loops at weak coupling

We obtain a model-independent prediction for the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen. We use the relation of the Wilson coefficients of the spin-dependent dimension-six four-fermion operator of NRQED applied to the electron-proton and to the muon-proton sectors. Their difference can be reliably computed using chiral perturbation theory, whereas the Wilson coefficient of the electron-proton sector can be determined from the hyperfine splitting in hydrogen. This allows us to give a precise model-independent determination of the Wilson coefficient for the muon-proton sector, and consequently of the two-photon exchange contribution to the hyperfine splitting in muonic hydrogen, which reads δ E ¯ p μ , H F T P E n S = − 1 n 3 1.161 20 $$ \delta {\overline{E}}_{p\mu, \mathrm{H}\mathrm{F}}^{\mathrm{TPE}}(nS)=-\frac{1}{n^3}1.161(20) $$ meV. Together with the associated QED analysis, we obtain a prediction for the hyperfine splitting in muonic hydrogen that reads E pμ ,HF th (1 S ) = 182.623(27) meV and E pμ ,HF th (2 S ) = 22.8123(33) meV. The error is dominated by the two-photon exchange contribution.

Model-independent determination of the two-photon exchange contribution to hyperfine splitting in muonic hydrogen

We determine the 1 /m and 1 /m 2 spin-independent heavy quarkonium potentials in the unequal mass case with O $$ \mathcal{O} $$ ( α 3 ) and O $$ \mathcal{O} $$ ( α 2 ) accuracy, respectively. We discuss in detail different methods to calculate the potentials, and show the equivalence among them. In particular we obtain, for the first time, the manifestly gauge invariant 1 /m and 1 /m 2 potentials in terms of Wilson loops with next-to-leading order (NLO) precision. As an application of our results we derive the theoretical expression for the B c spectrum in the weak-coupling limit through next-to-next-to-next-to-leading order (N 3 LO).

Potential NRQCD for unequal masses and the B c spectrum at N 3 LO

We compute the spin-dependent and spin-independent structure functions of the forward virtual-photon Compton tensor of the proton at O(p3) using heavy baryon effective theory including the Delta particle. We compare with previous results when existing. Using these results we obtain the leading hadronic contributions, associated to the pion and Delta particles, to the Wilson coefficients of the lepton–proton four fermion operators in NRQED. The spin-independent coefficient yields a pure prediction for the two-photon exchange contribution to the muonic hydrogen Lamb shift, ΔETPE(π&amp;Δ)=34(13) μeV . We also compute the charge, 〈rn〉 , and Zemach, 〈rn〉(2) , moments for n≥3 . Finally, we discuss the spin-dependent case, for which we compute the difference between the four-fermion Wilson coefficients relevant for hydrogen and muonic hydrogen.

The two-photon exchange contribution to muonic hydrogen from chiral perturbation theory

We study the generation of neutrino masses and mixing in supersymmetric R-parity violating models containing two pairs of Higgs doublets. In these models, new RPV terms $ {{\widehat{H}}_{D1 }}{{\widehat{H}}_{D2 }}\widehat{E} $ arise in the superpotential, as well as new soft terms. Such terms give new contributions to neutrino masses. We identify the different parameters and suppression/enhancement factors that control each of these contributions. At tree level, just like in the MSSM, only one neutrino acquires a mass due to neutrino-neutralino mixing. There are no new one loop effects. We study the two loop contributions and find the conditions under which they can be important.