The CCQE Workhorse: How Long-Baseline Experiments Reconstruct Neutrino Energy

When a muon neutrino at a few hundred MeV meets an atomic nucleus, several things can happen. It can scatter elastically off the whole nucleus through the neutral current. It can excite a baryon resonance and produce a pion. It can break a nucleon out, producing a hadronic shower with multiple particles in the final state. But the single dominant exclusive channel — the one most commonly used to identify neutrino events and reconstruct neutrino energies in long-baseline oscillation experiments — is charged-current quasi-elastic scattering, abbreviated CCQE.

The process is

${\nu }_{\mu }+n\to {\mu }^{-}+p$

in which the incoming muon neutrino converts a bound neutron into a proton by emitting a muon. The two final-state charged particles — muon and proton — leave clean tracks in a typical detector, the muon’s curvature in a magnetic field gives its momentum, and the angle between the muon and the neutrino beam direction can be measured precisely. Under the assumption that the struck neutron was at rest and the kinematics is purely two-body, the outgoing lepton’s energy and angle alone determine the incident neutrino energy. This is what makes CCQE such an attractive event topology for oscillation physics: you can reconstruct the neutrino energy event by event from quantities you can measure cleanly.

That is also where the trouble starts. The actual nucleus is not a collection of free neutrons. The struck nucleon may rescatter inside the nucleus before exiting; or the neutrino may have interacted with a pair of correlated nucleons rather than a single one; or the resulting hadronic system may include particles whose absorption inside the nucleus removes them from view. Each of these effects biases the energy reconstruction in ways that have, over the past fifteen years, become major systematics in long-baseline oscillation measurements.

This post is about how CCQE works as a tool for oscillation experiments, what the dominant complications are, and why neutrino-nucleus interaction modelling has become one of the most active areas of precision neutrino physics.

The naive picture

In the free-nucleon approximation, the CCQE cross-section is calculable from the V-A structure of the weak interaction and standard nucleon form factors. For a struck neutron at rest, energy and momentum conservation give

$E_{\nu }^{\mathrm{rec}}=\frac{m_p^2-m_n^2-m_{\mu }^2+2m_n{E}_{\mu }}{2(m_n-E_{\mu }+p_{\mu }\cos {\theta }_{\mu })}$

where $E_{\mu }$ and $p_{\mu }$ are the outgoing muon’s energy and momentum, and ${\theta }_{\mu }$ is its angle with respect to the neutrino direction. Measure those three lepton observables and you have the neutrino energy — exactly what an oscillation analysis needs.

The cross-section itself depends on the nucleon’s vector and axial form factors. The vector form factors are tightly constrained by electron-scattering measurements. The axial form factor is parameterized by the dipole form $F_A(Q^2)=g_A/(1+Q^2/M_A^2{)}^2$ with $g_A\approx 1.27$ from neutron beta decay and $M_A$ the axial mass, an empirical parameter around 1 GeV. The leading-order CCQE cross-section at the relevant energies is then a known function of $E_{\nu }$ and the form factor parameters.

This level of treatment was the standard approach through the early 2000s. It works adequately for many purposes, but it is not what real experiments measure.

Why the nucleus is more than a sum of nucleons

The real target is an oxygen, carbon, or argon nucleus, depending on the experiment. The struck nucleon is bound in the nuclear potential and moves with Fermi momentum of order 250 MeV/c. Both effects modify the CCQE kinematics.

The Fermi motion smears the relation between lepton kinematics and neutrino energy: even for true CCQE on a single bound nucleon, the reconstructed $E_{\nu }^{\mathrm{rec}}$ from the formula above deviates from the true $E_{\nu }$ by an amount of order tens of MeV. This is calculable from the nuclear momentum distribution and is the smallest of the nuclear effects.

The binding energy reduces the outgoing nucleon’s kinetic energy by about 25–30 MeV, again calculably. Combined with the Fermi motion, these effects are typically handled with a spectral function approach that accounts for the nucleon’s momentum and binding distribution simultaneously.

These corrections, while real, are tractable and modeled with reasonable accuracy. The harder problems are two phenomena that became apparent only in the mid-2010s with high-statistics neutrino-nucleus data: two-particle-two-hole excitations and final-state interactions.

Two-particle-two-hole excitations

A neutrino can interact not with a single nucleon but with a correlated pair. Inside a nucleus, two-nucleon correlations are common: at short distances, neutron-proton pairs are tightly correlated by the nuclear force and have momenta significantly larger than the Fermi level. When a neutrino’s $W$ boson is absorbed by such a pair, both nucleons are knocked out, producing a two-proton final state (for a $W^{+}$ absorbed on an $np$ pair) plus the muon.

The signature in a detector — a muon and one or more protons — overlaps with the CCQE signature, but the kinematics is different: the energy balance now involves two recoiling nucleons rather than one. Applying the CCQE energy-reconstruction formula to a 2p2h event gives an $E_{\nu }^{\mathrm{rec}}$ that is systematically below the true neutrino energy by 100-200 MeV.

The 2p2h fraction depends on nuclear physics inputs and is uncertain at the 30% level. It contributes 15-25% of “CCQE-like” events in the sub-GeV range relevant to T2K and a similar fraction at NOvA energies. The resulting bias in extracted oscillation parameters is sufficient to shift $\Delta m_{31}^2$ by 1-2% — not catastrophic, but at the threshold of precision targets for the next generation.

Final-state interactions

The proton ejected from the nucleus in a true CCQE event does not always emerge cleanly. It may scatter off other nucleons on the way out, losing energy to additional protons or neutrons or pions, or being absorbed entirely. Final-state interactions, or FSI, transform a CCQE event at the elementary vertex into a more complex final state at the level of the detector.

The reverse can also happen: a non-CCQE event with a pion in the final state can have its pion absorbed by the nucleus before exiting, leaving only a muon-plus-proton signature that mimics CCQE.

Modelling FSI requires a transport simulation that propagates each hadronic final-state particle through the residual nucleus, accounting for nucleon-nucleon scattering and absorption cross-sections. Several FSI models exist — GENIE, NEUT, NuWro — and they disagree at the 10-20% level on the rates of various processes. The choice of FSI model is one of the main systematic uncertainties in modern oscillation analyses.

Cross-section measurements and their puzzles

Direct measurements of the CCQE-like cross-section have produced some surprising results. The MiniBooNE collaboration’s analysis of mineral-oil data in 2010 found that the effective axial mass needed to fit their cross-section data was about 1.35 GeV — significantly higher than the world-average value of 1.0 GeV from low-energy deuterium-bubble-chamber data. The discrepancy was eventually understood not as an actual axial mass shift but as a manifestation of the 2p2h contribution: at the time, CCQE Monte Carlo predictions did not include 2p2h, so the missing strength was absorbed by inflating the axial mass.

The Martini and Nieves models, both incorporating 2p2h based on different microscopic nuclear-physics frameworks, reconciled the discrepancy and have since been adopted as the standard CCQE-event-generator inputs. But the two models still disagree by 20-30% on the 2p2h cross-section, and several modern measurements — MINERvA, T2K near-detector, NOvA — have placed constraints that the two models bracket rather than agree on.

Implications for oscillation analyses

Modern oscillation experiments deal with these uncertainties by combining several strategies.

The first is near-far cancellation: an oscillation measurement compares the near-detector flux with the far-detector flux, and many cross-section uncertainties cancel between the two. The cancellation is imperfect because the near and far detectors sample different parts of the neutrino flux due to off-axis or shaped-beam techniques, but it substantially reduces the impact of cross-section systematics.

The second is constrained joint fits: rather than treating cross-section parameters as nuisance variables with prior uncertainties, modern fits include the near-detector data itself in the likelihood and let the cross-section parameters be fit alongside the oscillation parameters. This gives the data direct constraining power on the cross-section model.

The third is dedicated cross-section measurements: experiments like MINERvA at Fermilab, T2K’s near detectors, and the upcoming SBND focus specifically on measuring neutrino-nucleus cross-sections, including the inclusive rate, the CCQE-like sub-sample, and 2p2h-like topologies. The resulting measurements feed back into the cross-section model inputs.

Even with all this, the cross-section systematic remains one of the dominant sources of uncertainty for DUNE and Hyper-Kamiokande, and substantial theoretical and experimental effort continues to bring it down. Argon-target cross-section measurements at SBND are particularly important for DUNE’s far detector, since the LArTPC physics involves a heavier and more complex nuclear target than the water in Super-K or the scintillator in NOvA.

Summary

Charged-current quasi-elastic scattering, ${\nu }_{\mu }+n\to {\mu }^{-}+p$, is the dominant exclusive channel for sub-GeV neutrino-nucleus interactions and the working tool of long-baseline oscillation experiments at the GeV scale. Reconstructing the neutrino energy from the outgoing lepton’s kinematics, under the quasi-elastic assumption, is straightforward in principle — but two effects make it more complex in practice. Two-particle-two-hole excitations of correlated nucleon pairs produce a CCQE-like signature with systematically reduced reconstructed energy. Final-state interactions inside the nucleus transform clean CCQE events into multi-nucleon topologies and vice versa. Both effects are at the 15-25% level for the relevant energy range, and both shift extracted oscillation parameters by amounts comparable to the experimental precision. Resolving them through near-far cancellation, joint fits with near-detector data, and dedicated cross-section measurements is one of the central activities of modern long-baseline neutrino physics, and remains a dominant systematic for the precision goals of DUNE and Hyper-Kamiokande.