Wayne W. Repko
The method proposed by Kinoshita and Nio to compute higher order vacuum polarization contributions to the Coulomb potential in muonic hydrogen is generalized to obtain relativistic corrections to their results.
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We calculate the corrections of orders alpha^3, alpha^4 and alpha^5 to the Lamb shift of the 1S and 2S energy levels of muonic hydrogen (mu p) and muonic deuterium (mu d). The nuclear structure effects are taken into account in terms of the proton r_p and deuteron r_d charge radii for the one-photon interaction and by means of the proton and deuteron electromagnetic form factors in the case of one-loop amplitudes. The obtained numerical value of the isotope shift (mu d) - (mu p) for the splitting (1S-2S) 101003.3495 meV can be considered as a reliable estimation for corresponding experiment wi…
The contribution of hadronic vacuum polarization to the Lamb shift in muonic hydrogen is evaluated with the account of modern experimental data on the cross section of $e^+e^-$ annihilation into hadrons. The numerical value of this contribution to the (2P-2S) shift in muonic hydrogen is equal to 10.77 $μeV$.
We show that the value of the proton mass depends on each bound state of muonic or electronic hydrogen atom. The charged particle bound to the proton produces magnetic field inside the proton. This makes a change to the amount of chiral condensate inside the proton. The change gives rise to the shift in the value of the proton mass. Numerically, the shift in the $2S$ state of the muonic hydrogen atom can be of the order of $0.1$ meV. The effect may solve the puzzle of the proton radius.
The proton structure and proton polarizability corrections to the Lamb shift of electronic hydrogen and muonic hydrogen were evaluated on the basis of modern experimental data on deep inelastic structure functions. Numerical value of proton polarizability contribution to (2P-2S) Lamb shift is equal to 4.4 GHz.
The contribution of hadronic vacuum polarization to the hyperfine splitting of the muonic hydrogen ground state is calculated with the account of experimental data on the cross section of e^+e^-\to annihilation into hadrons and the dipole parameterization of electromagnetic proton form factors.
The contribution of the proton polarizability to the ground state hyperfine splitting in muonic hydrogen is evaluated on the basis of modern experimental and theoretical results on the proton polarized structure functions. The value of this correction is equal to 4.6(8)\cdot 10^{-4} times the Fermi splitting E_F.
The maximal acceleration corrections to the Lamb shift of muonic hydrogen are calculated by using the relativistic Dirac wave functions. The correction for the $2S-2P$ transition is $\sim 0.38$ meV and is higher than the accuracy of present QED calculations and of the expected accuracy of experiments in preparation.
Corrections of orders $α^5$ and $α^6$ are calculated in the fine structure interval $ΔE^{fs}=E(2P_{3/2})-E(2P_{1/2})$ and in the hyperfine structure of the energy levels $2P_{1/2}$ and $2P_{3/2}$ in muonic hydrogen. The obtained numerical values $ΔE^{fs}= 8352.08 μeV$, $Δ\tilde E^{hfs}(2P_{1/2})=7964.36 μeV$, $Δ\tilde E^{hfs}(2P_{3/2})=3392.59 μeV$ can be considered as a reliable estimate for the comparison with corresponding experimental data and for the extraction of the experimental value of the Lamb shift $(2P-2S)$ in muonic hydrogen.
The contribution of the $α^3$ single electron-loop vacuum-polarization diagrams to the Lamb shift of the muonic hydrogen has been evaluated recently by two independent methods. One uses the exact parametric representation of the vacuum-polarization function while the other relies on the Padé approximation method. High precision of these values offers an opportunity to examine the reliability of the Monte-Carlo integration as well as that of the Padé method. Our examination covers both muonic hydrogen atom and muon $g-2$. We tested them further for the cases involving two-loop vacuum polarizati…
A detailed discussion of the long-range one-pion exchange (Yukawa potential) contribution to the 2S hyperfine splitting in muonic hydrogen which had, until recently, been disregarded is presented. We evaluate the relevant vertex amplitudes, in particular $π^0μ^+μ^-$, combining low energy chiral expansions together with experimental data on $π^0$ and $η$ decays into two leptons. A value of $Δ{E}^π_{HFS}= -(0.09 \pm 0.06)\ μ\rm{eV}$ is obtained for this contribution.
The sixth-order electron-loop vacuum-polarization contribution to the $2P_{1/2} - 2S_{1/2}$ Lamb shift of the muonic hydrogen ($μ^{-} p^+$ bound state) has been evaluated numerically. Our result is 0.007608(1) meV. This eliminates the largest uncertainty in the theoretical calculation. Combined with the proposed precision measurement of the Lamb shift it will lead to a very precise determination of the proton charge radius.
Corrections of orders alpha^5, alpha^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. Total numerical value of the 2S state hyperfine splitting 22.8148 meV in the (μp) can be considered as reliable estimation for the corresponding experiment with the accuracy 10^{-5}. The value of the Sternheim's hyperfine splitting interval [8ΔE^{HFS}(2S)-ΔE^{HFS}(1S)] is obtained with the accuracy 10^{-6}.
Currently, both the g factor measurement of the muon as well as the Lamb shift 2S-2P measurement in muonic hydrogen are in disagreement with theory. Here, we investigate possible theoretical explanations, including proton structure effects and small modifications of the vacuum polarization potential. In particular, we investigate a conceivable small modification of the spectral function of vacuum polarization in between the electron and muon energy scales due to a virtual millicharged particle and due to an unstable vector boson originating from a hidden sector of an extended standard model. W…
We obtain a model independent expression for the muonic hydrogen Lamb shift up to $\mathcal O (m_μα^6, m_μα^5 \frac{m_μ^2}{m_ρ^2})$. The hadronic effects are controlled by the chiral theory, which allows for their model independent determination. We give their complete expression including the pion and Delta particles. Out of this analysis and the experimental measurement of the muonic hydrogen Lamb shift we determine the electromagnetic proton radius: $r_p$=0.8412(15) fm. This number is at 6.8$σ$ variance with respect to the CODATA value. The parametric control of the uncertainties allows us …
Corrections of orders alpha^5, alpha^6 are calculated in the hyperfine splitting of the muonic hydrogen ground state. 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 modification of the hyperfine splitting part of the Breit potential due to the electron vacuum polarization is considered. Total numerical value of the 1S state hyperfine splitting 182.638 meV in the (mu p) can play the role of proper estimation for the corresponding experiment with the accuracy 30 ppm.
In the framework of the quasipotential method in quantum electrodynamics we calculate the contribution of light pseudoscalar (PS) and axial-vector (AV) mesons to the interaction operator of a muon and a proton in muonic hydrogen atom. The coupling of mesons with the muon is via two-photon intermediate state. The parametrization of the transition form factor of two photons into PS and AV mesons, based on the experimental data on the transition form factors and QCD asymptotics, is used. Numerical estimates of the contributions to the hyperfine structure of the spectrum of the S and P levels are …
The reduced-mass dependence of relativistic and radiative effects in simple muonic bound systems is investigated. The spin-dependent nuclear recoil correction of order (Zalpha)^4 mu^3/m_N^2 is evaluated for muonic hydrogen and deuterium, and muonic helium ions (mu is the reduced mass and m_N is the nuclear mass). Relativistic corrections to vacuum polarization of order alpha (Zalpha)^4 mu are calculated, with a full account of the reduced-mass dependence. The results shift theoretical predictions. The radiative-recoil correction to vacuum polarization of order alpha (Zα)^5 ln^2(Zalpha) mu^2/m_…
From the comparison between the experimental and theoretical results for the energy difference between 2s1/2 and 2p1/2 states in hydrogen and hydrogen-like ions with different values of Z, we estimate bounds for the presence of an extra fractional dimension. These bounds are higher when we work with muonic hydrogen and muonic deuterium, which open the possibility that future measurements of the Lamb shift in muonic atoms with one electron and higher values of Z may show evidence of the extra dimension.
The contribution of virtual excitations to the energy levels of electronic and muonic hydrogen is investigated combining a model-independent approach for the main part with quark model predictions for the remaining corrections. Precise values for the polarization shifts are obtained in the long-wavelength dipole approximation by numerically integrating over measured total photoabsorption cross sections. These unretarded results are considerably reduced by including retardation effects in an approximate way since the average momentum transfer (together with the mean excitation energy) turns out…
A bound-state field theory approach to muonic hydrogen is set up using a variant of the Furry representation in which the lowest-order Hamiltonian describes a muon in the presence of a point Coulomb field, but the origin of the binding field is taken to be three charged quarks in the proton which are modeled as Dirac particles that move freely within a spherical well. Bound-state field theory techniques are used to evaluate one- and two-photon effects. Particular attention is paid to two-photon exchange diagrams, which include the effect of proton polarizability. In addition the modification o…
In view of the recently observed discrepancy of theory and experiment for muonic hydrogen [R. Pohl et al., Nature vol. 466, p. 213 (2010)], we reexamine the theory on which the quantum electrodynamic (QED) predictions are based. In particular, we update the theory of the 2P-2S Lamb shift, by calculating the self-energy of the bound muon in the full Coulomb+vacuum polarization (Uehling) potential. We also investigate the relativistic two-body corrections to the vacuum polarization shift, and we analyze the influence of the shape of the nuclear charge distribution on the proton radius determinat…
The contribution from the axial-vector meson exchange to the potential of the muon-proton interaction in muonic hydrogen induced by anomalous axial-vector meson coupling to two photon state is calculated. It is shown that such contribution to the hyperfine splitting in muonic hydrogen is large and important for a comparison with precise experimental data. In the light of our result, the proton radius "puzzle" is discussed.
The $n=2$ and $n=3$ levels for muonic Helium are calculated using a potential that includes all one-loop and recoil effects. Electronic vacuum polarization corrections are calculated using an extension of Kinoshita and Nio method. For $n=2$, the results are $2p_{1/2}-2s_{1/2}=1375.05\pm 1.4$ meV and $2p_{3/2}-2s_{1/2}=1521.65\pm 1.4$ meV, essentially in agreement with the latest summary of the current calculations. The $n=3$ results are summarized in tabular form and give $3p_{1/2}-3s_{1/2}=394.76\pm 0.43$ meV and $3d_{3/2}-3p_{3/2}=111.40$ meV.
Comparing electronic Hydrogen with muonic Hydrogen shows that the discrepancy in measurement of the Lamb shift in the both systems are relatively of order of $(\frac{m_μ}{m_e})^{4-5}$. We explore the spectrum of Hydrogen atom in noncommutative $QED$ to compare the noncommutative effects on the both bound states. We show that in the Lorentz violating noncommutative QED the ratio of NC-corrections is $(\frac{m_μ}{m_e})^3$ while in the Lorentz conserving NCQED is $(\frac{m_μ}{m_e})^5$. An uncertainty about $1 \,Hz\ll 3\,kHz$ in the Lamb shift of Hydrogen atom leads to an NC correction about $10 \…
We present the leading-order prediction of baryon chiral perturbation theory for the proton polarizability contribution to the 2S hyperfine splitting in muonic hydrogen, and compare with the results of dispersive calculations.
We present a high-precision calculation of the recoil--finite-size correction to the hyperfine splitting (HFS) in muonic and electronic hydrogen based on nucleon electromagnetic form factors obtained from dispersion theory. This will help guide the upcoming searches of the HFS transition in muonic hydrogen, and will allow a precise determination of the polarizability and Zemach radius contributions when this transition is found.
We reexamine the structure of the $n=2$ levels of muonic hydrogen using a two-body potential that includes all relativistic, recoil and one loop corrections. The potential was originally derived from QED to describe the muonium atom and accounts for all contributions to order $α^5$. Since one loop corrections are included, the anomalous magnetic moment contributions of the muon can be identified and replaced by the proton anomalous magnetic moment to describe muonic hydrogen with a point-like proton. This serves as a convenient starting point to include the dominant electron vacuum polarizatio…
On the basis of quasipotential method in quantum electrodynamics we calculate nuclear finite size radiative corrections of order $α(Zα)^5$ to the hyperfine structure of S-wave energy levels in muonic hydrogen and muonic deuterium. For the construction of the particle interaction operator we employ the projection operators on the particle bound states with definite spins. The calculation is performed in the infrared safe Fried-Yennie gauge. Modern experimental data on the electromagnetic form factors of the proton and deuteron are used.
We constrain the possibility of a new pseudoscalar coupling between the muon and proton using a recent measurement of the 2S hyperfine splitting in muonic hydrogen.
Reexamination of vacuum-polarization corrections to the self-energy in muonic bound systems
本文重新检验了μ子束缚系统中自能的真空极化修正，发现联合效应在Z=1-6系统中可造成百分之几水平的扰动，并更新了μ氢、μ氘和μ氦的修正处理。
We explore the possibility that a new interaction between muons and protons is responsible for the discrepancy between the CODATA value of the proton radius and the value deduced from the measurement of the Lamb shift in muonic hydrogen. We show that a new force carrier with roughly MeV mass can account for the observed energy shift as well as the discrepancy in the muon anomalous magnetic moment. However, measurements in other systems constrain the couplings to electrons and neutrons to be suppressed relative to the couplings to muons and protons, which seems challenging from a theoretical po…
Recently the muonic hydrogen lamb shift has been measured with unprecedented accuracy, allowing for a precise determination of the proton radius. This determination is 5 sigma away from the previous CODATA value obtained from (mainly) the hydrogen lamb shift and the electron-proton scattering. Within an effective field theory formalism, I will define the proton radius and briefly review some aspects of the theoretical prediction for the muonic hydrogen lamb shift, studying both the pure QED-like computation and the hadronic effects.
The Lamb shift in muonic hydrogen was measured some time ago to a high accuracy. The theoretical prediction of this value is very sensitive to the proton-finite-size effects. The proton radius extracted from muonic hydrogen is in contradiction with the results extracted from elastic electron-proton scattering. That creates a certain problem for the interpretation of the results from the muonic hydrogen Lamb shift. For the latter we need also to take into account the two-photon-exchange contribution with the proton finite size involved. The only way to describe it relies on the data from the sc…
In the framework of the quasipotential method in quantum electrodynamics we calculate the contribution of pseudoscalar mesons to the interaction operator of a muon and a proton in muonic hydrogen atom. The parametrization of the transition form factor of two photons into $π$, $η$ mesons, based on the experimental data on the transition form factors and QCD asymptotics is used. Numerical estimates of the contributions to the hyperfine structure of the spectrum of the S and P levels are presented.
Contributions to the energy levels in light muonic atoms and, in particular, to the Lamb shift fall into a few well-distinguished classes. The related diagrams are calculated using different approaches. In particular, there is a specific kind of non-relativistic contributions. Here we consider such corrections to the Lamb shift in order $α^5m_μ$. These contributions are due to free vacuum polarization loops as well as to various effects of light-by-light scattering. The closed loop in the related diagrams is an electronic one, which allows a non-relativistic consideration of the muon. Both kin…
We provide an accurate evaluation of the two-photon exchange correction to the hyperfine splitting of S energy levels in muonic hydrogen exploiting the corresponding measurements in electronic hydrogen. The proton structure uncertainty in the calculation of $α^5$ contribution is sizably reduced.
Recently a high-precision measurement of the Lamb shift in muonic hydrogen has been performed. An accurate value of the proton charge radius can be extracted from this datum with a high accuracy. To do that a sufficient accuracy should be achieved also on the theoretical side, including an appropriate treatment of higher-order proton-structure effects. Here we consider a higher-order contribution of the finite size of the proton to the Lamb shift in muonic hydrogen. Only model-dependent results for this correction have been known up to date. Meantime, the involved models are not consistent eit…
Based on the simple phenomenological $σππ$, $σN N$ and $ππγ$ interactions, we estimate the $σ$ exchange effect in the Lamb shift of muonic hydrogen. We at first calculate the effective couplings of $σμμ$ by two photons and two pions exchange, then calculate the corresponding corrections to the energy shift of the 2S/2P states of muonic hydrogen. We find the correction to the energy shift of 2S state is about -14$μ$eV by the current used parameters, which is about $44\%$ of the usual two photons exchange contribution, and is larger than the current experimental precision, and should be consider…
We reevaluate the Zemach, recoil and polarizability corrections to the hyperfine splitting in muonic hydrogen expressing them through the low-energy proton structure constants and obtain the precise values of the Zemach radius and two-photon exchange (TPE) contribution. The uncertainty of TPE correction to S energy levels in muonic hydrogen of 105 ppm exceeds the ppm accuracy level of the forthcoming 1S hyperfine splitting measurements at PSI, J-PARC and RIKEN-RAL.
We investigate the influence of the spatial extent of the proton magnetization and charge densities on the 2S hyperfine splitting in muonic hydrogen. The use of a non-perturbative relativistic Dirac approach leads to corrections of 15% to values obtained from the perturbative treatment encapsulated by the Zemach radius, which surpass the next-leading order contribution in the perturbation series by an order of magnitude.
To date the magnetic radius of the proton has been determined only by means of electron-proton scattering, which is not free of controversies. Any existing atomic determinations are irrelevant because they are strongly model-dependent. We consider a so-called Zemach contribution to the hyperfine interval in ordinary and muonic hydrogen and derive a self-consistent model-independent 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 electric-charge radius of the proton, namely, R_E^2+R_M^2.…
On the basis of quasipotential approach in quantum electrodynamics we calculate vacuum polarization and quadrupole corrections in first and second orders of perturbation theory in hyperfine structure of P-states in muonic deuterium. All corrections are presented in integral form and evaluated analytically and numerically. The obtained results can be used for the improvement of the transition frequencies between levels 2P and 2S.
We present a precise determination of the polarizability and other proton structure dependent contributions to the hydrogen hyperfine splitting, based heavily on the most recent published data on proton spin dependent structure functions from the EG1 experiment at the Jefferson Laboratory. As a result, the total calculated hyperfine splitting now has a standard deviation slightly under 1 part-per-million, and is about 1 standard deviation away from the measured value. We also present results for muonic hydrogen hyperfine splitting, taking care to ensure the compatibility of the recoil and pola…
We present the derivation of the formulas for the proton structure-dependent terms in the hyperfine splitting of muonic hydrogen. We use compatible conventions throughout the calculations to derive a consistent set of formulas that reconcile differences between our results and some specific terms in earlier work. Convention conversion corrections are explicitly presented, which reduce the calculated hyperfine splitting by about 46 ppm. We also note that using only modern fits to the proton elastic form factors gives a smaller than historical spread of Zemach radii and leads to a reduced uncert…
The energy interval $ (3S-1S) $ in muonic hydrogen is calculated on the basis of quasipotential approach in quantum electrodynamics. We take into account different corrections of orders $α^3÷α^6 $, which are determined by relativistic effects, the effects of vacuum polarization, nuclear structure and recoil, as well as combined corrections including the above. Nuclear structure effects are expressed in terms of the charge radius of the proton in the case of one-photon interaction and the proton electromagnetic form factors in the case of two-photon exchange interaction. The value of the energy…
For muonic hydrogen, positronium, and ordinary hydrogen, we show that the existence of a relativistic two-body wave equation whose energy levels are physically accurate to order $(Zα)^4$ (where $Zα$ is the binding coupling constant) implies a previously unknown two-body Sommerfeld energy formula which can be used to predict energy terms to order $(Zα)^6$ using simple algebra. Two such terms are verified to be physically correct by earlier $(Zα)^6$ calculations for positronium. For muonic hydrogen and ordinary hydrogen, these terms are predictions.
We study the hydrogen and muonic hydrogen within an effective field theory framework. We perform the matching between heavy baryon effective theory coupled to photons and leptons and the relevant effective field theory at atomic scales. This matching can be performed in a perturbative expansion in alpha, 1/m_p and the chiral counting. We then compute the O(m_{l_i}^3 alpha^5/m_p^2 x logarithms) contribution (including the leading chiral logarithms) to the Hyperfine splitting and compare with experiment. They can explain about 2/3 of the difference between experiment and the pure QED prediction …
The one-loop nuclear structure corrections of order (Z alpha)^5 to the Lamb shift and hyperfine splitting of the deuterium are calculated. The contribution of the deuteron structure effects to the isotope shift (ep)-(ed), (mu p)-(mu d) in the interval (1S - 2S) is obtained on the basis of modern experimental data on the deuteron electromagnetic form factors. The comparison with the similar contributions to the Lamb shift for electronic and muonic hydrogen shows, that the relative contribution due to the nucleus structure increases when passing from the hydrogen to the deuterium.
We demonstrate that criticisms concerning our work nucl-th/9704043, made in the recent preprint hep-ph/9903352, are unfounded. The improvements over our results claimed in hep-ph/9903352 are in fact spurious, being based mainly on misunderstandings and on dubious theoretical models.
The current discrepancy of theory and experiment observed recently in muonic hydrogen necessitates a reinvestigation of all corrections to contribute to the Lamb shift in muonic hydrogen muH, muonic deuterium muD, the muonic 3He ion, as well as in the muonic 4He ion. Here, we choose a semi-analytic approach and evaluate a number of higher-order corrections to vacuum polarization (VP) semi-analytically, while remaining integrals over the spectral density of VP are performed numerically. We obtain semi-analytic results for the second-order correction, and for the relativistic correction to VP. T…
Resonant states of muonic three-particle systems with lithium, helium and hydrogen nuclei
该研究结合指数与高斯基变分法及复坐标旋转（CCR）方法，计算含氦、锂、氢核的四类缪子三粒子系统的共振态能量。
In the framework of the quasipotential method in quantum electrodynamics we calculate corrections to the nuclear structure proportional to $r_N^2$ from two-photon exchange amplitudes in the fine and hyperfine structure of P-states in muonic hydrogen, as well as the photon-photon interaction amplitudes, leading to the exchange of the axial vector meson. In constructing the quasipotential of the muon-nucleus interaction, we use the method of projection operators on states of two particles with a definite spin and total angular momentum. Analytical calculation of the matrix elements is performed …
We illustrate how to apply modern effective field theory techniques and dimensional regularization to factorise the various scales which appear in QED bound states at finite temperature. We focus here on the muonic hydrogen atom. Vacuum polarization effects make the physics of this atom at finite temperature very close to that of heavy quarkonium states. We comment on the implications of our results for these states in the quark gluon plasma. In particular, we estimate the effects of a finite charm quark mass in the dissociation temperature of bottomonium.
Relativistic and Recoil Corrections to Vacuum polarization in muonic systems: Three--photon exchange, gauge invariance and numerical values
本文研究了μ子束缚系统中真空极化相对论与反冲修正的规范不变性，通过包含三光子交换图并基于NRQED$_μ$框架证明该不变性，给出了核电荷数Z=1,2,6的单μ子离子的规范无关修正更新值。
The extremely precise extraction of the proton radius by Pohl et al. from the measured energy difference between the 2P and 2S states of muonic hydrogen disagrees significantly with that extracted from electronic hydrogen or elastic electron-proton scattering. This is the proton radius puzzle. The origins of the puzzle and the reasons for believing it to be very significant are explained. Various possible solutions of the puzzle are identified, and future work needed to resolve the puzzle is discussed.
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 th…
We compute the spin-dependent and spin-independent structure functions of the forward virtual-photon Compton tensor of the proton at one loop 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, $ΔE_{\rm TPE}(π\&Δ)…
The recent conundrum with the proton charge radius inspires reconsideration of the corrections that enter into determinations of the proton size. We study the two-photon proton-structure corrections, with special consideration of the non-pole subtraction term in the dispersion relation, and using fits to modern data to evaluate the energy contributions. We find that individual contributions change more than the total, and present results with error estimates.
We develop a method for lattice QCD calculation of the two-photon exchange (TPE) contribution to the muonic-hydrogen Lamb shift. To demonstrate the feasibility of this method, we also present an exploratory study with a gauge ensemble at $m_π= 142$ MeV. By adopting the infinite-volume reconstruction (IVR) method along with an optimized subtraction scheme, we obtain a preliminary result of the TPE contribution which agrees well with previous calculation using other methods and one magnitude smaller compare to the large $\sim300~μ$eV discrepancy for the proton radius puzzle.
We develop a method for lattice QCD calculation of the two-photon exchange contribution to the muonic-hydrogen Lamb shift. To demonstrate its feasibility, we present the first lattice calculation with a gauge ensemble at $m_π= 142$ MeV. By adopting the infinite-volume reconstruction method along with an optimized subtraction scheme, we obtain $ΔE_{\text{TPE}} = -28.9(4.9)~μ\text{eV} + 93.72~μ\text{eV}/\text{fm}^2 \cdot\langle r_p^2 \rangle$, or $ΔE_{\text{TPE}} = 37.4(4.9)~μ$eV, which is consistent with the previous theoretical results in a range of 20-50 $μ$eV.
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