Flux-Integrated Cross Section Measurement

• Flux-integrated cross sections are measures that normalize neutrino interactions over the full energy spectrum to reduce nuclear model uncertainties.
• They utilize computational frameworks like RDWIA and RFGM to extract key parameters such as the axial mass by comparing predictions with experimental data.
• This method mitigates systematic biases and serves as a benchmark for enhancing Monte Carlo simulations and theoretical models in neutrino physics.

A flux-integrated cross section measurement is a central methodology in modern experimental nuclear and neutrino physics that provides robust, model-independent observables for testing interaction models and constraining fundamental parameters. In the context of neutrino experiments, a flux-integrated cross section represents a convolution of the interaction probability with the entire neutrino energy spectrum of the beam, normalized by the total incident flux, and thus avoids several sources of nuclear model dependence that afflict traditional flux-averaged cross sections. This paradigm underpins precision comparisons with theoretical predictions, facilitates accurate extraction of quantities such as the axial mass, and supports rigorous tests of nuclear effects across energy and angular kinematic domains.

1. Definition and Theoretical Foundation

The flux-integrated cross section, $\langle\sigma\rangle_\Phi$, is defined as the observable rate of a given process normalized by the number of targets and by the total integrated neutrino flux, encompassing the full spectrum of incident neutrino energies. In the case of neutrino-nucleus scattering, this is operationalized as

$$\langle \frac{d\sigma}{d\Omega} \rangle_\Phi = \frac{1}{N_T \Phi_\mathrm{total}}\int_0^\infty \frac{d^2N}{d\Omega dE} dE,$$

where $N_T$ is the number of target nuclei, and $\Phi_\mathrm{total} = \int I_\nu(E) dE$ is the integrated flux. Critically, in a flux-integrated measurement, the normalization is to a fixed, experimentally determined total flux, rather than a shape-weighted average over energy intervals that depend on kinematic or nuclear model thresholds.

The weight function used in integration is shown for the Booster Neutrino Beamline (BNB) as

$$\widetilde{W}_{(\nu, \bar{\nu})}(T, \cos\theta, \varepsilon) = \frac{I_{(\nu,\bar{\nu})}(\varepsilon)}{\Phi_\mathrm{BNB}},$$

where $I_{(\nu, \bar{\nu})}(\varepsilon)$ is the flux at energy $\varepsilon$, and $\Phi_\mathrm{BNB}$ is the total integrated flux [(Butkevich, 2010), Eqs. (15)-(20)].

2. Nuclear Model Independence of Flux-Integrated Cross Sections

A key advantage of flux-integrated cross section reporting is nuclear model independence. Traditional flux-averaged differential cross sections require integration only over kinematically allowed ranges $$[\varepsilon_\mathrm{min}, \varepsilon_\mathrm{max}]$$, which are dependent on the chosen nuclear model, particularly in how Fermi motion, binding energies, and final state interactions restrict the accessible neutrino energy for a given outgoing lepton kinematic bin. In contrast, the flux-integrated approach uses the full incident spectrum, $\Phi_\mathrm{BNB}$, as the normalization denominator. This normalization does not depend on nuclear model-specific definitions of the energy limits and hence does not inherit their biases or uncertainties (Butkevich, 2010). The model independence is particularly strict at the level of single and double differential cross sections, making these observables ideal for benchmarking theoretical treatments of nuclear effects.

3. Computational Frameworks: RDWIA and RFGM Methods

The evaluation and prediction of flux-integrated cross sections generally proceed via two principal theoretical frameworks:

• Relativistic Distorted Wave Impulse Approximation (RDWIA): The RDWIA takes the impulse approximation as its baseline, in which the neutrino interacts with a single bound nucleon whose wave function is obtained from relativistic mean-field theory. The ejected nucleon is described by a distorted wave function accounting for final state interactions (FSI) using a complex optical potential; for inclusive observables where all final states are summed, only the real part of this potential is used. The transition amplitude is formally given by:

$$\langle p, B | J^\mu |A \rangle = \int d^3r \, e^{i\mathbf{q}\cdot \mathbf{r}}\, \overline\Psi^{(-)}(\mathbf{r})\, \Gamma^\mu\, \Phi(\mathbf{r}),$$

where $\Phi(\mathbf{r})$ is the bound-nucleon wave function, and $\Psi^{(-)}$ describes the outgoing nucleon with FSI.

• Relativistic Fermi Gas Model (RFGM): The RFGM models the nucleus as a Fermi gas of non-interacting nucleons up to momentum $p_F$ and with a fixed binding energy $\epsilon_b$. The outgoing nucleon is described by a plane wave, and FSI are incorporated at a basic level with Pauli blocking. The axial form factor is parameterized as a dipole:

$F_A(Q^2) = \frac{g_A}{(1 + Q^2/M_A^2)^2},$

where $M_A$ is the axial mass parameter to be extracted from data fits.

These two approaches, while anchored to the same one-body current, differ mainly in the treatment of nuclear binding, FSI, and many-body effects. For flux-integrated cross section calculation, both are applied across the full neutrino energy spectrum of the beam, using a consistent total flux normalization.

4. Experimental Implementation and Data Comparison

Flux-integrated cross section measurements have been realized most notably in MiniBooNE quasi-elastic (QE) neutrino scattering on $^{12}$C and similar targets (Butkevich, 2010), as well as in T2K and MicroBooNE.

• Single Differential Cross Sections: For measured $d\sigma/dQ^2$, the RDWIA predictions (with $M_A \approx 1.37$ GeV/$c^2$) agree with the experimental results over the full $Q^2 \lesssim 1$ (GeV/$c$)$^2$ range. The RFGM overpredicts the cross section at low $Q^2 \lesssim 0.2$ (GeV/$c$)$^2$, where Fermi motion and nuclear correlations begin to dominate.
• Double Differential Cross Sections: The most model-independent benchmark is $d^2\sigma/dT d\cos\theta$ for the outgoing muon. The RDWIA reproduces the MiniBooNE data across most kinematic bins, but both RDWIA and RFGM fall short at low muon energies ($T \lesssim 0.3$ GeV) and very forward angles ($\cos\theta > 0.9$). This region exposes limitations in FSI or multinucleon effects.

The overall normalization and the kinematic dependence, particularly in the double-differential distributions, critically test the completeness and accuracy of nuclear modeling.

5. Axial Mass Extraction from Flux-Integrated Cross Sections

A principal outcome of flux-integrated cross section analysis is the extraction of the effective axial mass, $M_A$, which governs the $Q^2$-dependence of the weak current axial form factor. The analysis performs a shape-only fit to the measured $d\sigma/dQ^2$ (or energy-dependent bins) using

$F_A(Q^2) = \frac{g_A}{(1 + Q^2/M_A^2)^2},$

with the model predictions binned identically as the data: $$\langle \frac{d\sigma}{dQ^2} \rangle_i = \frac{1}{\Delta Q^2} \int_{Q^2_i}^{Q^2_{i+1}} [d\sigma/dQ^2(Q^2)]^\text{int} dQ^2,$$ (Eq. (21), (Butkevich, 2010))

The fit is performed over all $Q^2$ bins to determine the best-fit $M_A$. Both RDWIA and RFGM approaches yield $M_A \simeq 1.36$–$1.37$ GeV/$c^2$ in agreement with the MiniBooNE value. This approach is robust to the nuclear model, due to the flux-integrated normalization.

6. Statistical and Systematic Considerations

By design, flux-integrated cross section measurements minimize model-related systematic uncertainties, especially those tied to the accessible kinematic phase space of the neutrino energy. The principal contributors to residual uncertainties are as follows:

• Flux Normalization: Determined from beam simulations, hadron production data (e.g., HARP), and in-situ monitoring.
• Detector Effects: Acceptance, resolution, and efficiency corrections, typically addressed with detailed Monte Carlo and control data (embedding the same flux model).
• Background Subtraction: For example, in MiniBooNE, it includes pion absorption, dirt events, and other neutrino species.
• Interaction Modeling: While the cross section normalization above is model-independent, predictions for specific differential cross sections, especially in regions sensitive to FSI and multinucleon processes, are subject to theoretical uncertainties in the input nuclear models.

Statistical uncertainties are determined by Poisson fluctuations in event counts, propagated through efficiency correction and unfolding.

7. Implications and Future Directions

The use of flux-integrated (as opposed to flux-averaged) cross section measurements is now standard for reporting high-precision, model-independent neutrino-scattering results across experiments (including MiniBooNE, T2K, MicroBooNE, and LHC-based measurements such as FASER (Collaboration et al., 4 Dec 2024)). This methodology provides stringent constraints for refining nuclear models, especially those needed to accurately interpret oscillation experiments where the neutrino energy reconstruction depends crucially on the quality of nuclear effect modeling.

In addition, flux-integrated double differential cross sections serve as benchmarks for developing improved Monte Carlo event generators that incorporate realistic treatment of FSI, multinucleon correlations, and nuclear-medium modifications, all of which are essential in the neutrino energy range of a few hundred MeV to several GeV. The accurate extraction of the axial mass, $M_A$, and identification of discrepancies in particular kinematic regions (such as low-muon energy and forward scattering) highlight where further theoretical development or new physics contributions may be relevant.

This comprehensive framework for flux-integrated cross section measurement not only mitigates biases associated with energy-dependent selection and model-dependent unfolding but also enables consistent and rigorous comparisons across different experimental setups and theoretical predictions. The approach underpins both the present and future program of neutrino and nuclear cross section physics.