Theory of QED radiative corrections to neutrino scattering at accelerator energies

Control over quantum electrodynamics (QED) radiative corrections is critical for precise determination of neutrino oscillation probabilities from observed (anti)neutrino detection rates. It is particularly important to understand any difference between such corrections for different flavors of (anti)neutrinos in charged-current interactions. We provide theoretical foundations for calculating these corrections. Using effective field theory, the corrections are shown to factorize into soft, collinear, and hard functions. The soft and collinear functions contain large logarithms in perturbation theory but are computable from QED. The hard function parametrizes hadronic structure but is free from large logarithms. Using a simple model for the hard function, we investigate the numerical impact of QED corrections in charged-current (anti)neutrino-nucleon elastic cross sections and cross-section ratios at GeV energies. We consider the implications of mass singularity theorems that govern the lepton-mass dependence of cross sections for sufficiently inclusive observables and demonstrate the cancellation of leading hadronic and nuclear corrections in phenomenologically relevant observables.

Figure 1

Kinematics of charged-current elastic scattering for neutrino (left) and antineutrino (right).

Figure 2

Ratio of one-loop cross section to the tree-level result in the static limit, for scattering of electron (anti)neutrinos (left plot) and muon (anti)neutrinos (right plot). The observable including only the radiation of one soft photon of energy below $\mathrm{\Delta }E=20\mathrm{MeV}$ is shown by the blue solid curve. The correction including also radiation of one collinear hard photon is represented by the red dashed line, with cone sizes $\mathrm{\Delta }\theta =10°$ and 2° appropriate for electrons and muons, respectively. Inclusive cross sections are shown by the green dotted line. All curves are shown with the choice of the scale corresponding to the physical nucleon mass $\mu =M$.

Figure 3

Decomposition of full theory diagram into hard, soft and collinear momentum regions.

Figure 4

Differential cross section $\mathrm{d}{\sigma }^{\gamma }$ for one radiated photon of energy above $\mathrm{\Delta }E=20\mathrm{MeV}$ within the angle $\mathrm{\Delta }\theta$ to the lepton direction, divided by the tree-level charged-current elastic cross section $\mathrm{d}{\sigma }_{\mathrm{LO}}$. The ratio $\mathrm{d}{\sigma }^{\gamma }/\mathrm{d}{\sigma }_{\mathrm{LO}}$ is computed using Eq. (36) for the scattering of electron (anti)neutrinos (left plot) and muon (anti)neutrinos (right plot). The bottom, middle, and top curves correspond to electromagnetic jet energy 600 MeV, 2 GeV, and 6 GeV, respectively.

Figure 5

Derivative of $\mathrm{d}{\sigma }^{\gamma }$ in Fig. 4 with respect to $\mathrm{\Delta }\theta$, divided by the tree-level charged-current elastic cross section $\mathrm{d}{\sigma }_{\mathrm{LO}}$.

Figure 6

Hadronic model for hard matching at one-loop level. The photon is exchanged between the charged lepton and nucleon lines.

Figure 7

Uncertainties from the hard function on the ratio of the differential cross section, with only one soft photon radiated, to the tree-level result in ${\nu }_{e}n\to e^{-}p$ (left plots) and ${\overline{\nu }}_{e}p\to e^{+}n$ (right plots). For initial (anti)neutrino energies $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots), the soft-photon energy cutoff is $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively. The uncertainty propagated from the tree-level form factors is shown by the red dashed line. The difference obtained by varying nucleon electromagnetic form-factor parameter ${\mathrm{\Lambda }}^2$ as ${\mathrm{\Lambda }}^2\to 1.1{\mathrm{\Lambda }}^2$ in the loop diagram is shown by the green dash-dotted line. As discussed in the text, the total contribution of $G_E^n$ and $G_M^n$ at the neutron electromagnetic vertex and the variation of the proton electromagnetic vertex obtained by adding $G_E^p\to G_E^p+G_E^n$ and $G_M^p\to G_M^p+G_M^n$ represent an estimate for neglected contributions in the hadronic model. These variations are shown by the black dotted line and by the black fine-dotted line, respectively. The sum of all variations in quadrature is shown by the blue solid line.

Figure 8

The same as Fig. 7 but for the muon flavor.

Figure 9

Ratio of the differential cross section, with only soft-photon radiation, to the tree-level result in ${\nu }_{e}n\to e^{-}p$ (left plots) and ${\overline{\nu }}_{e}p\to e^{+}n$ (right plots). For incident (anti)neutrino energies $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots), the soft-photon energy cutoff is $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively. The resummed result including all corrections of $\mathcal{O}(\alpha )$ is shown by the blue solid lines. It is compared to the resummation result including only the leading logarithms, shown by the green dotted lines, and to the result including subleading logarithms of order $\mathcal{O}(\sqrt{\alpha })$, shown by the red dashed lines. To illustrate the net effect of resummation, we also present the fixed-order result using the same hadronic model by the black dash-dotted lines. Hadronic uncertainty, described in Sec. 4e2, is displayed for the fixed-order calculation. Only perturbative uncertainty, estimated by scale variation as described in the text, is presented for the first three lines.

Figure 10

Differential cross-section ratio to the tree-level result for electron (anti)neutrino charged-current elastic scattering on the nucleon accompanied by soft radiation, as a function of $Q^2$ for a few different soft-photon energy cutoffs $\mathrm{\Delta }E$. Left and right plots are for neutrino and antineutrino scattering, respectively. Upper and lower plots are for $E_{\nu }=600\mathrm{MeV}$ and $E_{\nu }=2\mathrm{GeV}$, respectively. The curves show the results for $\mathrm{\Delta }E=10$, 20, and 40 MeV.

Figure 11

The same as Fig. 10 but for the muon flavor.

Figure 12

Ratio of the differential cross section to the tree-level result for ${\nu }_e$ (${\overline{\nu }}_e$) including only soft or both soft and collinear photon emission. Left and right plots are for neutrino and antineutrino scattering, respectively. Upper and lower plots are for $E_{\nu }=600\mathrm{MeV}$ and $E_{\nu }=2\mathrm{GeV}$, with soft-photon energy cutoff $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively. The lower blue dash-dotted band corresponds to curves in Fig. 10. The upper pink dashed band includes also real photons within a cone of size $\mathrm{\Delta }\theta =10°$. The difference between the green dashed line and unity represents the uncertainty of the tree-level cross section itself; this uncertainty largely cancels in the displayed cross-section ratios.

Figure 13

Ratio of unpolarized radiative charged-current-elastic-like ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$ differential cross sections to the tree-level result with collinear photon energy above the threshold for final muon to be misidentified as electron, as described in the text. Reference scenarios with $E_{\gamma }>E_{\mu }-m_{\mu }$ and $E_{\gamma }>200\mathrm{MeV}$ are shown for comparison. Plots are for (anti)neutrino beam energies $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots). The calculation based on our default electroweak vertex is labeled as default model. The calculation when the electroweak vertex is taken as in diagrams with local interactions is labeled as local model.

Figure 14

Ratio of unpolarized radiative charged-current-elastic-like ${\nu }_e$ and ${\overline{\nu }}_e$ differential cross sections to the tree-level result with hard photons outside the jet cone. Curves show $E_{\gamma }>20$, 40, and 80 MeV, for the angle between the electron and photon $\theta >10°$. Plots are for (anti)neutrino beam energies $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots). The calculation based on our default electroweak vertex is labeled as default model. The calculation when the electroweak vertex is taken as in diagrams with local interactions is labeled as local model.

Figure 15

The same as Fig. 14 but for the muon flavor charged-current-elastic-like cross section, with $\theta >2°$.

Figure 16

Ratio of the inclusive differential cross section including noncollinear emission of one hard photon to the tree-level result is presented as a function of $Q^2$. Electron neutrino-neutron scattering is shown on left plots and antineutrino-proton scattering on right plots. For comparison, the result including only soft photon radiation with energy below $\mathrm{\Delta }E$ is shown by the black dash-dotted lines and light blue band. The $E_{\ell }+E_{\gamma }$ energy spectrum is shown by the red dashed lines; the spectrum of events reconstructed from the energy within the $\mathrm{\Delta }\theta =10°$ cone is shown by the dark-blue dashed double-dotted lines and blue band; and the energy spectrum reconstructed from the electron energy is shown by the green dotted lines. (Anti)neutrino beam energies are taken as $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots), and the soft-photon energy cutoff is $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively. $Q^2$ is defined according to the spectrum in figures. The displayed $Q^2$ range for 600 MeV (anti)neutrino beam corresponds to the maximum for tree-level elastic kinematics. For 2 GeV (anti)neutrinos, the displayed $Q^2$ range is truncated to focus on the phenomenologically relevant regime. Central value cross sections for the entire $Q^2$ region are displayed in Fig. 18.

Figure 17

The same as Fig. 16 but for the muon flavor. The cone observable is not presented since the radiation inside the cone is negligible for the muon flavor.

Figure 18

Tree-level differential cross section for the (anti)neutrino beam energies $E_{\nu }=600\mathrm{MeV}$ and $E_{\nu }=2\mathrm{GeV}$ is compared to the spectra of radiated events, with photons above the energy $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively, when $Q^2$ is defined from the lepton energy $E_{\ell }$ or from the sum of lepton and photon energies $E_{\ell }+E_{\gamma }$. Results are shown for electron and muon flavors, for neutrino-neutron scattering on the left plot and for antineutrino-proton scattering on the right plot. The flavor difference of tree-level cross sections is unresolved on this figure over the wide range of allowed kinematics; see upper curves on all plots. We also present the spectrum of events reconstructed from the energy within the cone of angle $\mathrm{\Delta }\theta =10°$ for the electron flavor. $Q^2$ and its range are defined according to the spectrum in figures. Integrals over all $Q^2$ values for $E_{\ell }$, energy in 10° cone, and $E_{\ell }+E_{\gamma }$ spectra are the same.

Figure 19

Electron over muon flavor ratios of unpolarized differential cross section in neutrino-neutron (left plots) and antineutrino-proton (right plots). Blue solid lines and blue band, tree level; black dash-dotted lines and light blue band, including radiation of soft photons with energy below $\mathrm{\Delta }E$; red dashed lines and pink band, including radiation of soft photons with energy below $\mathrm{\Delta }E$ and including radiation of collinear photons for the electron flavor as described in Sec. 6b but not for the muon flavor. For (anti)neutrino beam energies $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots), the soft-photon energy cutoff is $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively.

Figure 20

Antineutrino-proton over neutrino-neutron differential cross-section double ratios to the tree-level results with electron (left plots) and muon (right plots) flavors including the radiation of soft photons with energy below $\mathrm{\Delta }E$, shown by the black dash-dotted lines and light blue band, and accounting also for the radiation of collinear photons in case of electron flavor as described in Sec. 6b, shown by the red dashed lines and pink band. For (anti)neutrino beam energies $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots), the soft-photon energy cutoff is $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively. Differences between the double ratios with and without collinear photons arise from subleading effects and are indistinguishable in the figure (see text). The relative uncertainty of the tree-level antineutrino over neutrino cross-section ratio is represented by the green dotted line as a deviation from unity.

Figure 21

(Inclusive) Electron over muon flavor ratios of unpolarized differential cross sections in neutrino-neutron (left plots) and antineutrino-proton (right plots) scattering at tree level, shown by the blue solid lines; the ratio of $E_{\ell }$ spectra is shown by the green dotted lines; the ratio of spectrum of events reconstructed from the energy within the 10° cone for the electron flavor to the $E_{\ell }$ spectrum for the muon flavor is shown by the dark-blue dashed double-dotted lines and blue band; and the ratio of $E_{\ell }+E_{\gamma }$ energy spectra is shown by the red dashed lines. (Anti)neutrino beam energies are taken as $E_{\nu }=600\mathrm{MeV}$ (upper plots) and $E_{\nu }=2\mathrm{GeV}$ (lower plots); the soft-photon energy cutoff for jet events with electron (anti)neutrinos is $\mathrm{\Delta }E=25\mathrm{MeV}$ and $\mathrm{\Delta }E=10\mathrm{MeV}$, respectively. $Q^2$ is defined according to the spectrum in figures.

Figure 22

(Inclusive) The same as Fig. 20 but for inclusive observables as specified in the caption to Fig. 21.

Figure 23

Estimated cross-section ratio to the tree-level result for the $\mathrm{CC}0\pi$ process in T2K, NOvA, DUNE, and MINERvA (LE and ME flux configurations) experiments after averaging over the corresponding neutrino flux, as a function of $Q^2=-{(p_{\nu }-p_{\mu })}^2$. See text for details.

Figure 24

The effect of radiative corrections to the tree-level prediction for the $\cos {\theta }_{\mu }=0.94–0.98$ bin of the T2K ND280 ${\nu }_{\mu }$ (left) and ${\overline{\nu }}_{\mu }$ (right) CCQE-like data [219] is shown for the cross section per nucleon of a CH target.

Figure 25

The effect of radiative corrections to the tree-level prediction for the $\cos {\theta }_{\mu }=0.60–0.70$ bin of the T2K ND280 ${\nu }_{\mu }$ (left) and ${\overline{\nu }}_{\mu }$ (right) CCQE-like data [219] is shown for the cross section per nucleon of the CH target.

Figure 26

The effect of radiative corrections to the tree-level prediction for one slice of the MINERvA LE ${\nu }_{\mu }$ CCQE-like data [214] (left) and ${\overline{\nu }}_{\mu }$ CCQE-like data [226] (right) is shown for the cross section per nucleon of the CH target.

Figure 27

The effect of radiative corrections to the tree-level prediction for the same slice of the MINERvA ME ${\nu }_{\mu }$ [217] (left) and ${\overline{\nu }}_{\mu }$ [227, 228] (right) CCQE-like data is shown for the cross section per nucleon of the CH target.