Computing neutrino cross sections from Euclidian responses

Abstract

Energy integrated neutrino cross sections are integrals of nuclear responses weighted with kinematic prefactors. We decompose the prefactors into a limited set of functions of energy transfer and show the relevant integrals are the moments of the responses, and integrals weighted with $1/(a+ω)^n$ with $n\leq 2$. These can be directly obtained from the Euclidean response, avoiding the need for inversion of the Laplace transform. As a proof of concept we study the procedure with toy-model responses for the quasielastic peak. We show that the different contributions can be straightforwardly organized in terms of relative importance, and how flux-averaged cross sections can be obtained. Using a realistic model for the response and numerical uncertainty we show that it is feasible to obtain the required integrals from the Euclidean response, with large uncertainties only for the third moment. Due to kinematic restrictions, the integrals contain contributions from the unphysical region for neutrino scattering, coming from high-momentum nucleons. We show that (in the absence of two-body currents) robust corrections for this contamination are obtained from the single-nucleon momentum distribution. These results present an opportunity to compute certain neutrino cross sections with ab-initio methods with controlled uncertainties.

Figures (14)

• Figure 1: Longitudinal ($L$), transverse ($T$), and transverse-interference ($T^\prime$) contributions to the single-differential cross section at fixed $q=300$ MeV (left), and $q=600$ MeV (right). Solid lines show the exact result. Dashed lines are the high-energy limits. Filled points show the first energy-dependent contribution. Open symbols show the next order contribution, equivalent to keeping only contributions up to and including the second moments. The bottom panels show the ratio of this approximation to the full results.
• Figure 2: Contributions to the energy-integrated cross section, obtained in $\omega$ space (lines), and directly from the Euclidean response (symbols). The left panel shows the cross section decomposed by contribution of the different responses. The right panel shows the contribution from terms proportional to integrals of responses weighted with $\omega^n$ and $\mathcal{E}_n$.
• Figure 3: Fits of the MiniBooNE flux with second order polynomials in windows of energy $E\in [E_0,E_0+q]$ for different values of $q$. For each $q$-value three $E_0$ windows are shown $E_0 \in \{q/2, 3q/4, q\}$, equally spaced in the physical region $E\in [q/2,q]$.
• Figure 4: MiniBooNE flux averaged cross sections. Symbols are obtained from the Euclidean response, lines are calculated from the responses in $\omega$-space. The left-hand panel shows the total cross section. Black line is the result with the exact MiniBooNE flux, the blue line is the quadratic approximation to the flux. The green line is the calculation with the quadratic flux where additionally the contribution of $\omega^2$ terms is dropped in the lepton prefactors. Green open symbols are the equivalent calculation directly from the Euclidean response. The right-hand panel shows the exact results for the different responses, compared to the result from the Euclidean response. The difference is due to dropping the $\omega^2$ term in the latter.
• Figure 5: Contribution from the unphysical region to the $n$-th moment of the responses, Eq. (\ref{['eq:approx_omega_unphys']}) for $q=500$. We scale all moments by $1/q^n$, and normalize to 1 for $n=0$. The filled circles denote the different nuclear responses, the solid line is the phase-space response $R_{PS}$ of Eq. (\ref{['eq:R_PS_SF']}). The ratio is shown in the bottom panel. The empty circles are the results when the responses are scaled by appropriate powers of $\omega$ from Eq. (\ref{['eq:R_w_scaling']}). The dashed line is the approximation for $R_{PS}$ from Eq. (\ref{['eq:R_app_ndist']}) determined from the momentum distribution.