Elastic lepton-proton two-photon exchange scattering: An exact HB$χ$PT analysis including hadronic effects at NNLO

TL;DR

This work delivers a fully analytic evaluation of two-photon exchange (TPE) corrections to elastic lepton–proton scattering within heavy-baryon chiral perturbation theory up to NNLO. It extends prior exact NLO results by incorporating kinematical NNLO recoil corrections and dynamical NNLO contributions from proton form-factor insertions, including pion-loop induced hadronic structure effects via $F_1^p$ and $F_2^p$. The analysis demonstrates good perturbative convergence, with NNLO corrections being small and negative, and reveals non-vanishing proton-structure dependent ${\mathcal O}(\alpha/M^2)$ terms not captured by soft-photon approximations. IR divergences cancel consistently with bremsstrahlung, but remaining limitations include irreducible two-loop pionic TPE and inelastic intermediate states; the results highlight important discrepancies with SPA-based approaches and offer precise input for low-energy experiments like MUSE.

Abstract

We present an exact analytical evaluation of the two-photon exchange (TPE) correction to the elastic lepton-proton differential scattering cross section at low-energies within the framework of heavy-baryon chiral perturbation theory. Our analysis focuses on the kinematic regime relevant to the ongoing MUSE experiment, and we therefore restrict the intermediate states to the dominant elastic channel. All loop integrals are evaluated analytically without approximations. Radiative and chiral recoil contributions of the proton are included, retaining kinematical and dynamical TPE corrections to the cross section through next-to-next-to-leading order [i.e., ${\mathcal O}(α/M^2)$] accuracy in the recoil expansion where $M$ is the proton mass. At this chiral order, pion-loop contributions demonstrate that structure-dependent TPE effects arise through the proton form factors. Our analytical results for the scattering cross section reveal non-vanishing residual proton structure effects of ${\mathcal O}(α/M^2)$, despite substantial cancellations between TPE box and crossed-box contributions. Such effects were entirely absent at this accuracy in our earlier analysis based on the soft-photon approximation. Although the next-to-leading-order contributions are numerically sizable, the next-to-next-to-leading-order TPE corrections are found to be small, thereby indicating that the chiral expansion exhibits reasonably good perturbative convergence.

Figures (6)

• Figure 1: The LO [i.e., $\mathcal{O}(\alpha^2)$] and NLO [i.e., $\mathcal{O}(\alpha^2/M)$] TPE diagrams contributing to the $\mathcal{O}(\alpha^3/M)$ lepton-proton elastic differential cross-section, are shown. The thick, thin, and wiggly lines denote the proton, lepton, and photon propagators. The blobs and crosses denote the insertions of the NLO proton-photon vertices and ${\mathcal{O}}(1/M)$ proton propagators, respectively (figure reproduced from Ref. Goswami:2025zoe).
• Figure 2: The box and crossed-box pairs of finite fractional kinematical NNLO corrections arising from the LO and NLO TPE diagrams (a) - (h), are shown together with the contributions arising from the NLO seagull diagram (i) and the interference of the NLO Born (OPE) amplitude ${\mathcal{M}}^{(1)}_\gamma$ with the LO TPE diagrams (a) and (b), entering the ${\mathcal{O}}(\alpha/M^2)$ component $\delta^{\rm (LO+NLO;2)}_{\gamma\gamma}$ [see Eq. \ref{['eq:LO_NLO_TPE_delta2']} ] of the corrections to the elastic lepton–proton differential cross section. Note that in the figure, we present the combined O($\alpha/M^2$) contribution from the LO pair of TPE diagrams (a) and (b), i.e., $\overline{\delta^{(ab;1/M^2)}_{\gamma\gamma}}+\overline{\delta^{(a_1b_1;1/M^2)}_{\gamma\gamma}}$, which is labeled as "ab+${\rm a}_1{\rm b}_1$ (Exact)". For comparison, the corresponding SPA results of Goswami et al.Goswami:2025zoe are also displayed. It is noteworthy that the tiny seagull contribution (inset panel) has no SPA counterpart. The results for e–p ($\mu$–p) elastic scattering are shown in the left (right) panel as functions of the squared four-momentum transfer $|Q^2|$, for three MUSE incident lepton beam momenta: 210 MeV/c, 153 MeV/c, and 115 MeV/c. All fractional corrections are given relative to the LO OPE differential cross section.
• Figure 3: The one-loop NNLO [i.e., $\mathcal{O}(\alpha^2/M^2)$] TPE diagrams contributing to the $\mathcal{O}(\alpha^3/M^2)$ lepton-proton elastic differential cross section, are shown. The thick, thin, and wiggly lines denote the proton, lepton and photon propagators. The small solid circles and square green boxes denote the insertions of NLO ($\nu=1$) and NNLO ($\nu=2$) proton-photon interaction vertices, respectively. The crosses and green-square boxes represent the insertions of the ${\mathcal{O}}(1/M)$ and ${\mathcal{O}}(1/M^2)$ proton propagator components. Panel (A) contains the true TPE one-loop diagrams, while panel (B) shows the reducible two-loop TPE diagrams involving pion-loops and counter terms (not explicitly shown above, instead see Fig. 1 of Ref. Das:2025jfh) at the proton-photon vertices, as depicted by the large green blobs. These pion-loop insertions renormalize the diagrams (p) – (s) with NNLO proton–photon vertex corrections, effectively reducing them to one-loop graphs dressed with proton form factors.
• Figure 4: The box and crossed-box pairs of finite fractional dynamical NNLO corrections arising from the NNLO TPE diagrams, (j) – (${\rm s}_\pi$), are shown together with the interference of the NNLO OPE amplitude ${\mathcal{M}}^{(2)}_\gamma$ with the LO TPE diagrams (a) and (b), entering the ${\mathcal{O}}(\alpha/M^2)$ component $\delta^{\rm (NNLO;2)}_{\gamma\gamma}$ [Eq. \ref{['eq:NNLO_TPE_delta2']}] of the electron-proton (left panel) and muon-proton (right panel) elastic differential cross sections. For comparison the corresponding SPA results of Goswami et al.Goswami:2025zoe are also shown. Within the SPA evaluation the only non-vanishing NNLO TPE contribution arises from diagrams (n) and (o). The result is identical to the one obtained in our exact calculation, as indicated by the label “no (SPA = Exact)” for the curves. The purple and red shaded bands reflect the uncertainties associated with varying the proton rms radius across the range spanned by the CREMA determination [$r_p=0.84087(39)$ fm] Antognini:2013txnPohl:2013yb and the MAMI-ISR measurement [$r_p=0.87\pm (0.014){\rm stat.}\pm (0.024){\rm syst.}\pm (0.003)_{\rm mod.}$ fm] Mihovilovic:2019jiz. The central curve in each band corresponds to result obtained using the mean value, $\overline{r_p}\sim 0.855$ fm. All fractional corrections are given relative to the LO OPE differential cross section relevant to MUSE kinematical range.
• Figure 5: The finite fractional NLO [i.e., ${\mathcal{O}}(\alpha/M)$] corrections $\delta^{\rm (1)}_{\gamma\gamma}$ [see Eq. \ref{['eq:TPE_delta1']}] arising from relevant box and crossed-box pairs of TPE diagrams, contributing to the elastic lepton-proton differential cross section up-to-and-including NLO [i.e., ${\mathcal{O}}(\alpha^3/M)$] accuracy, is presented. For comparison the NLO SPA results of Goswami et al.Goswami:2025zoe are also displayed. The results for e–p ($\mu$–p) elastic scattering are shown in the left (right) panel as functions of the squared four-momentum transfer $|Q^2|$, for three MUSE incident lepton beam momenta: 210 MeV/c, 153 MeV/c, and 115 MeV/c. All corrections are given relative to the LO OPE differential cross section.