Neutrino-induced coherent $\pi^{+}$ production in C, CH, Fe and Pb at $\langle E_{\nu}\rangle \sim 6$ GeV

MINERvA has measured the $\nu_{\mu}$-induced coherent $\pi^{+}$ cross section simultaneously in hydrocarbon (CH), graphite (C), iron (Fe) and lead (Pb) targets using neutrinos from 2 to 20 GeV. The measurements exceed the predictions of the Rein-Sehgal and Berger-Sehgal PCAC based models at multi-GeV $\nu_{\mu}$ energies and at produced $\pi^{+}$ energies and angles, $E_{\pi}>1$ GeV and $\theta_{\pi}<10^{\circ}$. Measurements of the cross-section ratios of Fe and Pb relative to CH reveal the effective $A$-scaling to increase from an approximate $A^{1/3}$ scaling at few GeV to an $A^{2/3}$ scaling for $E_{\nu}>10$ GeV.

In neutrino-induced coherent pion production the nucleons in the nucleus recoil in phase under the impact of an incident neutrino. The nucleus remains in its initial quantum state and recoils with an energy below the detection threshold of most neutrino detectors. A π meson and a lepton are created, both with relatively small angles with respect to the incoming neutrino. Both charged (CC) and neutral current (NC) interactions can occur, induced by a neutrino or anti-neutrino of any flavor, according to ν l + A → l + π + A, where ν l is a neutrino of flavor l, A is the nucleus, and l and π, a lepton and a pion of the proper charge, respectively. The four-momentum transfer to the nucleus, must be between |t min | Q 2 + m 2 π /2E π 2 [1], and for the interaction to happen, where p ν , p l and p π are the neutrino, lepton and pion fourmomenta, respectively; p T and p L are the lepton's or pion's transverse and longitudinal momenta, respectively; E is the lepton's or pion's total energy, Q 2 is the square of the four-momentum transferred by the neu-trino, m π is the pion mass, and R N is the nuclear radius.
Historically, most experiments [2-17] used the Rein-Sehgal model (R-S) [18] to simulate coherent π production. It is based on Adler's Partially Conserved Axial Current (PCAC) theorem [19], which relates the neutrino-nucleus inelastic cross section to the pionnucleus elastic cross section, assuming the incoming neutrino and the outgoing lepton are parallel (when Q 2 = 0), and neglecting the lepton mass. The CC channel differential cross section is where y = ν/E ν = (E ν − E µ ) /E ν ≈ E π /E ν , f 2 π is the pion decay constant, E ν is the neutrino energy, and dσ π ± A /d|t| is the pion-nucleus elastic cross section. The model extrapolates Eq. (2) to Q 2 > 0 with a form factor m 2 A / m 2 A +Q 2 2 , where m A ≈ 1 GeV is the axialvector mass.
CC coherent pion production is an important background for CC quasi-elastic interactions in ν µdisappearance measurements [20], when the π + is misreconstructed as a proton. It is also a significant fraction of MINERvA's own CC1π + sample [21]. Both are very valuable for upcoming neutrino oscillation analyses in the few-GeV region [22,23].
By using |t| to isolate signal-like events, MINERvA was the first experiment to observe the CC coherent π ± in that energy region, using ν µ and ν µ beams on a hydrocarbon (CH) target [24,25]. These and two later publications [26,27] used an improved version of the R-S model that includes the lepton mass [28,29].
These measurements are obtained using the NuMI beam line at the Fermi National Accelerator Laboratory [31] where 120-GeV protons colliding on a graphite target, create hadrons which are focused using a pair of magnetic horns, and sent to a decay pipe where they create a beam of muon-neutrinos, with E ν ∼6.0 GeV [30], made of ∼95% ν µ , and ∼5% of ν µ , ν e and ν e [32]. The neutrino beam is simulated with a Geant4 model [33,34].
The MINERvA detector consists of an inner detector made of an upstream "nuclear target" and a downstream "tracker" region, and an outer detector composed of electromagnetic (ECAL) and hadronic (HCAL) calorimeters [35]. The nuclear target region is ∼1.4 m long with five different passive materials: solid C, Fe, and Pb; and liquid He and H 2 O, all installed in seven targets. Following the beam direction, solid targets are labeled from 1 to 5. Targets 1, 2 and 5 had segments of Fe and Pb, and thickness of ∼2.6 cm in targets 1 and 2, and ∼1.3 cm in target 5. Target 3 had C, Fe, and Pb segments, with thickness of ∼7.6 cm, ∼2.9 cm and ∼2.6 cm, respectively. Target 4 was made of Pb with a thickness of ∼0.8 cm. Eight planes of tracking plastic scintillator (CH) were placed between the targets (only four between targets 4 and 5). Different target positions and thicknesses tried to equalize mass, acceptance and particle containment; maximize event rates, vertex and track resolution; and minimize the energy threshold of particles exiting the passive materials. The tracker region is ∼2.7 m long and made of 120 scintillator planes. Planes consist of 127 triangular prism scintillator strips with 33-mm base, 17-mm height, and varying length to form an hexagonal plane. Planes are rotated by 60 • with respect to adjacent ones, enabling three-dimensional reconstruction. The detector's single hit position resolution is ∼3 mm and the time resolution is 3 ns [35]. The ECAL surrounds the inner detector, and the HCAL surrounds the ECAL. The former (latter) consists of planes of lead (iron) and scintillator to contain and track electromagnetically (strongly) interacting particles. Located 2 m downstream of MINERvA, the MINOS near detector [36,37] served as a magnetized spectrometer to determine muon charge and momentum.
The signal process is CC coherent interactions on C, Fe and Pb, induced by a ν µ from 2 to 20 GeV. Events with pion angle larger than 70 degrees cannot be tracked and have zero efficiency. Events in the nuclear target (tracker) region with muon angle larger than 13 degrees have an efficiency of ∼1% (∼4%) due to MINOS acceptance. The percentage of simulated signal events in these categories is ∼5% for CH and C, and ∼2% for Fe and Pb.
Neutrino interactions are simulated using a modified version of the GENIE event generator v2.12.6 [38,39]. The signal's cross section is given by the R-S model with the lepton mass correction. Background processes ( Fig.  1) in increasing hadronic invariant mass W , are: CC quasielastic (QE), correlated pairs of nucleons (2p2h), resonant π + production (Non-QE, W < 1.4 or RES), inelastic scattering (1.4 < W < 2.0 or INE) and deep inelastic scattering (W > 2.0 or DIS). Quasielastic scattering is simulated using the Llewellyn-Smith model [40] with an axial-vector form factor from a z-expansion fit to deuterium data [41] and a correction from the Valencia Random Phase Approximation (RPA) [42]. The 2p2h process is simulated with the Valencia model [43][44][45] and modified according to a "low recoil" fit by MINERvA [46]. Resonant pion production uses the Rein-Sehgal model [47] with its normalization increased 15% based on fits from a deuterium data analysis [48], plus an additional ad hoc suppression for Q 2 < 0.7 [GeV/c] 2 due to collective nuclear effects [49]. Inelastic interactions use a tuned model of discrete baryon resonances [47], and the Bodek-Yang model for the transition region to DIS, as well as non-resonant pion production across the full W range [50], that was reduced by 43% based on a tune to the same deuterium data [48]. These tunes to GENIE are labeled as the MINERvA tune v4.4.1 [51].
Final state particles coming from the GENIE simulation are propagated through the detector using a Geant4 simulation of the detector's geometry and material composition, light yield and energy deposition of the particles in the scintillator, and their hadronic and electromagnetic interactions [52,53]. The detector's energy scale was established by making sure that simulated throughgoing muons agreed with data in both light yield and reconstructed energy deposition. The detector's simulated response to different particles is validated in a test beam measurement [54], and the effects of accidental activity, electronics charge and time resolution were also included [35].
Scintillator strips with deposited energy greater than 1 MeV are grouped per plane according to their position and time, into "clusters". These are grouped with clusters in adjacent planes to form tracks. Backwardsprojected tracks find interaction vertices. Angles are measured between the simulated beam direction and the direction of the track in its first planes downstream of the vertex.
This analysis isolates events with two tracks from a common vertex. The reconstructed momentum of the muon candidate is the addition of the momentum determined by range inside MINERvA plus its momentum determined by range or curvature inside MINOS. The pion candidate has to be fully contained inside MINERvA, so |t| can be measured. The pion total energy is reconstructed calorimetrically from all the energy not associated with the muon, given the assumption E ν ≈ E π + E µ from Eq. (1), where E µ is the muon's total energy.
The reconstructed interaction vertex is defined as the upstream end of the muon track, and it is required to be inside the fiducial volume under study. The CH fiducial volume is 108 planes long (∼2.4 m) centered in the tracker region with the area of a 0.85-m apothem hexagon [55]. The fiducial volume in the passive targets, is the area times the thickness of the segment of interest. For the passive materials, the vertex is projected into the z center of the target, where the (x , y) coordinate determines the segment (material). Events from different targets but same material, are combined into a single sample.
The reconstructed neutrino energy must be between 2 and 20 GeV to remove events with mis-reconstructed muon energy [34]. To reject protons from quasielastic and resonance production backgrounds, dE/dx-based χ 2 compared to pion and proton hypotheses of the pion candidate track are built. A log likelihood ratio [56] between the hypotheses removes (keeps) ∼70% (∼87%) of protons (pions) according to the simulation.
The energy of the vertex region (E vtx ), defined as a 200-mm radius, 7−plane height cylinder centered at the interaction vertex, must be consistent with the energy deposited by one minimum-ionizing charged pion and one muon. The E vtx distribution of simulated signal events is fit to a Gaussian function, and events within ±1σ of the mean are selected. Due to different target thickness, E vtx is target-dependent, varying from ∼60 to ∼95 MeV. This cut removes (keeps) ∼86% (∼60%) of the background (signal).
Due to their proximity to tracking scintillator planes, the C, Fe and Pb samples have contamination from events occurring in scintillator upstream and downstream of the passive material. These events are considered background, and are tuned using the plastic regions between passive targets as sidebands. There is an "upstream" and a "downstream" plastic sideband for each passive material. The tuned plastic backgrounds represent ∼13%, ∼14% and ∼21% of the C, Fe and Pb selected samples, respectively.
After removing events with high E vtx , and subtracting the plastic background, all samples in Fig. 1 show a signal dominance at low |t|. For heavier nuclei, the signal shrinks to a lower |t| region as R N increases. The C distribution has a significant excess of RES and INE events from ∼0.025 to ∼0.5 [GeV/c] 2 compared to CH, despite both being interactions on Carbon. This is due to the ∼7.6-cm thickness of the C segment, where one or more pions from those backgrounds are absorbed inside the passive material, which allows the event to pass the E vtx cut. A high |t| sideband (0.2 < |t| < 0.7 [GeV/c] 2 ) is used to tune the QE, RES (Non-QE, W < 1.4), INE (1.4 < W < 2.0) and DIS (W > 2.0) backgrounds. Due to their small content, "Coherent" and "Other Interactions" (NC-, ν µ -or (−) ν e -induced) are not tuned, and 2p2h is considered QE during the tuning. Because the C target has limited statistics, two modifications were made to the fit for that target only: RES and INE were combined, and the QE and DIS scale factors were replaced by their CH counterparts. The scale factors for each of the backgrounds are in the supplemental material.
An iterative unfolding approach [57] was used to correct the background-subtracted distributions for resolution effects. The unfolded distributions were then efficiency-corrected. The cross sections were extracted according to the expression σ = N DAT A ef f / (ΦT ), where N DAT A ef f is the background-subtracted, unfolded and efficiency-corrected data, Φ is the incident neutrino flux, and T the number of C, Fe or Pb nuclei. The largest sources of inefficiency come predominantly from well-understood random processes, which supports the assumption that the non-detected events have the same relative background composition.
The extracted cross sections are compared to the R-S model (GENIE v2.12.6) and to the Berger-Sehgal (B-S) model (GENIE v3.0.6) [58][59][60]. The latter is also PCAC-based, and also includes the muon mass correction, but uses pion-carbon data [1] to model the elastic pion-nucleus cross section, instead of pion-deuterium data as the R-S model. Figure 2 shows the total cross section as a function of E ν , with the flux integrated per bin, where both models under-predict the reaction rate at high neutrino energies in the four materials. Inner (outer) error bars are the statistical (statistical+systematic) uncertainties. The differential cross sections with respect to E π and θ π , are flux-averaged from 2 < E ν < 20 GeV. In dσ/dE π (Fig.  3) there is a clear disagreement between the models and the data of the two heavier nuclei, for low (high) E π in iron (lead). Figure 4 shows that the models also underpredict the dσ/dθ π cross section at very forward angles in all materials. Notably, forward pion production in the heavier nuclei is enhanced relative to scattering on carbon, where for lead, the cross section becomes negligible for θ π > 30 • .
The simultaneous neutrino exposure of the various targets enables precise measurement of cross section ratios thanks to the same beam configuration in all targets at any given time. Figure 5 shows the cross section ratios as a function of E ν : σ C /σ CH , σ F e /σ CH and σ P b /σ CH .  63], respectively. The PCAC-based Belkov-Kopeliovich (B-K) model predicts a scaling close to A 1/3 at low pion energy but close to A 2/3 at high pion energy [64,65]. In terms of neutrino energy, the B-K model predicts a scaling of ∼A 1/3 (∼A 2/3 ) at neutrino energies below (above) ∼10 GeV [66]. Cross section ratios as function of Eν : σC /σCH , σF e/σCH , and σ P b /σCH , in reading order. The upper (lower) dashed line is the ratio predicted by an A 2/3 (A 1/3 ) scaling. The slope is the best A-scaling fit. The 2-3 GeV bin is not included in the σ P b /σCH fit due to the null cross section in lead in that bin (Fig. 2.) The measured σ F e /σ CH resembles the trend predicted by B-K, where below ∼8 GeV there is a clear agreement with the A 1/3 scaling, and a better agreement with the A 2/3 scaling above ∼10 GeV, with a constant increase in between. A similar trend occurs for the measured σ P b /σ CH but with an A-scaling larger than predicted below 10 GeV.
The statistical uncertainty of the total cross section dominates in the three passive materials (Fig. 6). The largest systematic uncertainties are related to the detector's geometry and particles interacting in it (Detector Model), like the muon energy deposition in MINERvA and MINOS [67]. Uncertainties associated with the "Interaction Model", come from GENIE and the uncertainties from the MINERvA tune v.4.4.1. The "Physics Sideband" is the uncertainty on the backgrounds scale factors, plus a "per-bin" uncertainty covering for the remaining disagreement between data and the simulation in the high |t| sideband.
The "Flux" uncertainty comes from the uncertainty on the beam line parameters, and hadron interactions [34].
It was further constrained from 7.6% to 3.9% using a neutrino-electron scattering measurement [68].
Other sources of uncertainty are the discrepancy in the detector mass; modifications to the QE-like background (Low Recoil and RPA); low Q 2 suppression of resonant pion production; and the uncertainty on the plastic background scale factors (Plastic Sideband). They contribute less than ∼5% (∼15%) to the total cross section uncertainty in CH (C, Fe and Pb). The CH sample provides the most precise measurement of the interaction so far, reducing the total uncertainty from ∼25% to ∼15% compared to the previous MINERvA measurement [25]. Cross section ratios have a further reduction of some systematic uncertainties, in particular the flux, reduced by ∼75% of itself (Fig. 6). The measurements in this letter represent the first simultaneous measurement of the interaction in multiple materials and the first measurement in nuclei with A > 40 ( 56 Fe and 207 Pb), from which cross section ratios with respect to CH are measured. The data indicates that the R-S and B-S PCAC models do not accurately describe the angular dependence on θ π , the energy-dependence on E π , or the A-dependence. While the σ F e /σ CH qualitatively agrees with the B-K model's energy-dependent A-scaling, σ P b /σ CH does not, at least at low E ν .
The estimate of the cross sections A-scaling provided in this letter could be used to extrapolate to materials where measurements do not exist or the statistics are limited, like H 2 O for Hyper-K or Ar for DUNE. For the latter, pion production will make up around three quarters of the detected neutrino-induced events.
This document was prepared by members of the MIN-ERvA Collaboration using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.

Ratio to CH
FIG. 14. Ratios of the differential cross section as a function of θµ. C, Fe and Pb with respect to CH, in reading order. Data is compared to the Rein-Sehgal (red) and Berger-Sehgal (blue) models.