Coherency and incoherency in neutrino-nucleus elastic and inelastic scattering

Neutrino-nucleus scattering $\nu A\to \nu A$, in which the nucleus conserves its integrity, is considered. Our consideration follows a microscopic description of the nucleus as a bound state of its constituent nucleons described by a multiparticle wave function of a general form. We show that elastic interactions keeping the nucleus in the same quantum state lead to a quadratic enhancement of the corresponding cross section in terms of the number of nucleons. Meanwhile, the cross section of inelastic processes in which the quantum state of the nucleus is changed, essentially has a linear dependence on the number of nucleons. These two classes of processes are referred to as coherent and incoherent, respectively. Accounting for all possible initial and final internal states of the nucleus leads to a general conclusion independent of the nuclear model. The coherent and incoherent cross sections are driven by factors $|F_{p/n}{|}^2$ and $(1-|F_{p/n}{|}^2)$, where $|F_{p/n}{|}^2$ is a proton/neutron form factor of the nucleus, averaged over its initial states. Therefore, our assessment suggests a smooth transition between regimes of coherent and incoherent neutrino-nucleus scattering. In general, both regimes contribute to experimental observables. The coherent cross-section formula used in the literature is revised and corrections depending on kinematics are estimated. Consideration of only those matrix elements which correspond to the same initial and final spin states of the nucleus and accounting for a nonzero momentum of the target nucleon are two main sources of the corrections. As an illustration of the importance of the incoherent channel, we considered three experimental setups with different nuclei. As an example, for ${}^{133}\mathrm{Cs}$ and neutrino energies of 30–50 MeV, the incoherent cross section is about 10%–20% of the coherent contribution if the experimental detection threshold is accounted for. Experiments attempting to measure coherent neutrino scattering by solely detecting the recoiling nucleus, as is typical, might be including an incoherent background that is indistinguishable from the signal if the excitation gamma eludes its detection. However, as is shown here, the incoherent component can be measured directly by searching for photons released by the excited nuclei inherent to the incoherent channel. For a beam experiment these gammas should be correlated in time with the beam, and their higher energies make the corresponding signal easily detectable at a rate governed by the ratio of incoherent to coherent cross sections. The detection of signals due to the nuclear recoil and excitation $\gamma \mathrm{s}$ provides a more sensitive instrument in studies of nuclear structure and possible signs of new physics.

Figure 1

Left panel: Front of incoming neutrino plane-wave (solid vertical line) scatters on nucleons at fixed positions, ${\mathit{x}}_j$ and ${\mathit{x}}_k$, respectively. Nonzero scattering angle $\theta$ develops the phase difference $\mathrm{\Delta }\phi =\mathit{q}({\mathit{x}}_j-{\mathit{x}}_k)$ of two fronts of scattered neutrino plane-waves (dashed lines) which leads to a loss of coherence. Right panel: Neutrino scatters off a $k$th or $j$th nucleon described by a wave function exemplified here as a Gaussian profile. The outgoing neutrino wave, as for any nucleon target, is a superposition of waves $e^{i\mathit{q}{\mathit{x}}_k}$ weighted by $|{\psi }_n({\mathit{x}}_1\dots {\mathit{x}}_A){|}^2$.

Figure 2

Expected kinetic energy of nucleus ${}^{133}\mathrm{Cs}$ scattered in an elastic interaction with a neutrino as a function of its energy. The upper plot corresponds to $\mathrm{\Delta }{ϵ}_{mn}=0$ and four different values of $\cos \theta$, where $\theta$ is the neutrino scattering angle. The lower plot illustrates the impact of nonzero $\mathrm{\Delta }{ϵ}_{mn}$ for a fixed value $\cos \theta =-1$.

Figure 3

A qualitative picture of a coherent neutrino-nucleus interaction. A neutrino interacts with a nucleon initially having a particular momentum $\mathit{p}=\alpha \mathit{q}$ aligned along $\mathit{q}$ and given by Eq. (31). Since the nucleus initially is at rest, all the nucleons except the target one have a momentum $-\mathit{p}$ shown by line dashed. The final momentum $\mathit{p}+\mathit{q}=(1+\alpha )\mathit{q}$ of the target nucleon is also aligned along $\mathit{q}$. In the figure, an angle between the $\mathit{p}$ and $\mathit{p}+\mathit{q}$ vectors differs from $\pi$ for visual clarity. After the interaction the increased energy of the target nucleon and acquired three-momentum $\mathit{q}$ are transferred to the entire nucleus, leaving the internal quantum state of the latter unchanged. A $Z$-boson having a wavelength comparable to the size of the nucleus produces a coherent enhancement of scattering amplitudes.

Figure 4

Longitudinal component $p_L$ of the nucleon momentum corresponding to the energy-momentum conservation in neutrino-nucleus scattering.

Figure 5

The Helm form factor $F^{\text{Helm}}$ [80] as a function of the absolute value of three-momentum transfer $|\mathit{q}|$ (bottom horizontal axis). The upper horizontal axis corresponds to the kinetic energy of ${}^{133}\mathrm{Cs}$ nucleus.

Figure 6

Differential cross sections $\frac{d\sigma }{d{T}_A}$ for coherent (solid lines) and incoherent (dashed lines) neutrino-nucleus scattering for ${}^{74}\mathrm{Ge}$ (top), ${}^{133}\mathrm{Cs}$ (middle) and ${}^{40}\mathrm{Ar}$ (bottom) nuclei and different values of neutrino energies. Vertical lines correspond to experimental energy thresholds.

Figure 7

Integral cross sections $\sigma$ for coherent (solid lines) and incoherent (dashed lines) neutrino-nucleus scattering for ${}^{74}\mathrm{Ge}$ (top), ${}^{133}\mathrm{Cs}$ (middle), and ${}^{40}\mathrm{Ar}$ (bottom) nuclei and different values of neutrino energies. The integrals are calculated for idealistic threshold-less ($T_A^{\min }=0$, blue lines) experimental setups and accounting for state-of-the-art thresholds $T_A^{\min }$ (red lines) achieved by three considered experimental setups.

Figure 8

Ratio ${\sigma }_{\text{incoh}}/{\sigma }_{\mathrm{coh}}$ for neutrino scattering off of a ${}^{133}\mathrm{Cs}$ nucleus as a function of $E_{\nu }$. The two curves correspond to a $T_A^{\min }=0(5)\mathrm{keV}$ detection threshold.

Figure 9

Ratio $R$ of differential coherent cross section $d\sigma /d{T}_A$ calculated in this work in Eq. (48) to that used by the COHERENT Collaboration and reproduced in Eq. (54). Both cross sections are averaged over the nucleus spin, assuming neutrino scattering off of a ${}^{133}\mathrm{Cs}$ nucleus. The ratio is shown as a function of kinetic energy $T_A$ of the nucleus.

Figure 10

The left pictogram corresponds to a fermion current $h_{++}^{\eta }$ with positive projections of its spin (shown by double arrowed line) on the given axis (shown by dashed line) in both initial and final states. This current is decomposed into a sum of the current with negative helicity in both the initial and final states $h_{--}^{\chi }$ and of the current in which the initially negative helicity is flipped $h_{+-}^{\chi }$ weighted with $\sin \frac{\theta }{2}$ and $\cos \frac{\theta }{2}{e}^{-i\phi }$, respectively, as illustrated by the right pictogram.

Figure 11

Both balls are initially at rest when the right ball is hit with a momentum $\mathit{q}$ (left). After some time their center-of-mass moves with momentum $\mathit{q}$ and half of the transferred energy is accumulated in the potential energy of the spring, shown by its tension and extension (right).

Figure 12

Balls move towards and away each other with maximal momenta $\pm \mathit{q}/4$ while their center-of-mass is at rest, when the right ball having momentum $-\mathit{q}/4$ is hit and acquired the momentum $\mathit{q}$ (left). After a while, their center-of-mass moves with momentum $\mathit{q}$, while the spring still has the same potential energy, remaining in the same state. The balls have the same maximal momenta $\pm \mathit{q}/4$.