Angular distributions in electroweak pion production off nucleons: Odd parity hadron terms, strong relative phases, and model dependence

The study of pion production in nuclei is important for signal and background determinations in current and future neutrino oscillation experiments. The first step, however, is to understand the pion production reactions at the free nucleon level. We present an exhaustive study of the charged-current and neutral-current neutrino and antineutrino pion production off nucleons, paying special attention to the angular distributions of the outgoing pion. We show, using general arguments, that parity violation and time-reversal odd correlations in the weak differential cross sections are generated from the interference between different contributions to the hadronic current that are not relatively real. Next, we present a detailed comparison of three state-of-the-art, microscopic models for electroweak pion production off nucleons, and we also confront their predictions with polarized electron data, as a test of the vector content of these models. We also illustrate the importance of carrying out a comprehensive test at the level of outgoing pion angular distributions, going beyond comparisons done for partially integrated cross sections, where model differences cancel to a certain extent. Finally, we observe that all charged and neutral current distributions show sizable anisotropies, and identify channels for which parity-violating effects are clearly visible. Based on the above results, we conclude that the use of isotropic distributions for the pions in the center of mass of the final pion-nucleon system, as assumed by some of the Monte Carlo event generators, needs to be improved by incorporating the findings of microscopic calculations.

Figure 1

Model for the $W^{+}N\to N^{\prime }\pi$ reaction as introduced in Ref. [10]. It contains the Delta ($\mathrm{\Delta }P$) and crossed Delta pole ($C\mathrm{\Delta }P$) terms (first row), the nucleon ($NP$) and crossed nucleon pole ($CNP$) terms (second row), the contact $CT$ and pion pole ($PP$) terms (third row), and the pion in flight ($PF$) term (fourth row).

Figure 2

$D_{13}(1520)$ contributions to $W^{+}N\to N^{\prime }\pi$ introduced in Ref. [20]. Both $D_{13}$ ($DP$) and crossed $D_{13}$ pole ($CDP$) terms are considered.

Figure 3

Definition of the scattering and reaction planes. The $X^{*}{Y}^{*}{Z}^{*}$ coordinate axes move along with the CM system of the final pion-nucleon and their orientation has been chosen in such a way that the lepton momenta lie in the $O^{*}{X}^{*}{Z}^{*}$ plane with the positive $Z^{*}$ axis chosen along $\overrightarrow{q}$ and the positive $Y^{*}$ axis chosen along $\vec{k}\wedge {\vec{k}}^{\prime }$.

Figure 4

${\nu }_{\mu }p\to {\mu }^{-}p{\pi }^{+}$ total cross section as a function of the neutrino energy. In the left panel a kinematical cut $W_{\pi N}<1.4\mathrm{GeV}$ has been included. The corresponding experimental data have been taken from the reanalysis done in Ref. [24] of old ANL (crosses) and BNL (open squares) data, where the $W_{\pi N}<1.4\mathrm{GeV}$ cut is also implemented. In the right panel a kinematical cut $W_{\pi N}<2\mathrm{GeV}$ has also been applied to the theoretical calculation. The data have been taken from the reanalysis done in Ref. [33] of old ANL (crosses) and BNL (open squares) data, without any cut on $W_{\pi N}$.

Figure 5

${\nu }_{\mu }n\to {\mu }^{-}n{\pi }^{+}$ (left panels) and ${\nu }_{\mu }n\to {\mu }^{-}p{\pi }^0$ (right panels) total cross sections as a function of the neutrino energy. In the upper panels, the kinematical cut $W_{\pi N}<1.4\mathrm{GeV}$ has been included in the data points, taken from the reanalysis of the experimental cross sections carried out in [24], and both in the DCC and HNV theoretical predictions. For the DCC model, we also show the results for $W_{\pi N}<2\mathrm{GeV}$ in the bottom panels. The corresponding experimental data have also been taken from the reanalysis carried out in Ref. [24] of the old ANL (crosses) and BNL (open squares) data that does not incorporate any cut in the available phase space.

Figure 6

${\nu }_{\mu }n\to {\nu }_{\mu }p{\pi }^{-}$ total cross section as a function of the neutrino energy. The corresponding experimental data have been taken from Ref. [34] where no kinematical cut was implemented. A kinematical cut $W_{\pi N}<1.4\mathrm{GeV}$ has been, however, imposed for the HNV model (it has a moderate effect in this energy range). We present the DCC results both with $W_{\pi N}<1.4\mathrm{GeV}$ and $W_{\pi N}<2\mathrm{GeV}$ cuts.

Figure 7

CC total cross sections as a function of the neutrino energy from different theoretical models. A kinematical cut $W_{\pi N}<1.4\mathrm{GeV}$ on the final pion-nucleon invariant mass has been included.

Figure 8

NC total cross sections as a function of the neutrino energy from different theoretical models. A kinematical cut $W_{\pi N}<1.4\mathrm{GeV}$ on the final pion-nucleon invariant mass has been included.

Figure 9

$\mathrm{CC}–d\sigma /(d{Q}^{2}d{W}_{\pi N})$ differential cross sections as a function of $W_{\pi N}$, for fixed $E_{\nu }=1\mathrm{GeV}$ and $Q^2=0.1{\mathrm{GeV}}^2/c^2$.

Figure 10

$\mathrm{NC}–d\sigma /(d{Q}^{2}d{W}_{\pi N})$ differential cross sections as a function of $W_{\pi N}$, for fixed $E_{\nu }=1\mathrm{GeV}$ and $Q^2=0.1{\mathrm{GeV}}^2/c^2$.

Figure 11

CC-neutrino $A^{*},B^{*},C^{*},D^{*}$ and $E^{*}$ structure functions, as defined in Eq. (31), represented as a function of $\cos {\theta }_{\pi }^{*}$ for fixed $E_{\nu }=1\mathrm{GeV}$, $W_{\pi N}=1.23\mathrm{GeV}$ and $Q^2=0.1{\mathrm{GeV}}^2/c^2$.

Figure 12

Same as Fig. 11 for CC-antineutrino reactions.

Figure 13

Semisums and semidifferences (in MeV units) of the neutrino and antineutrino $A^{*}$ structure functions displayed in Figs. 11 and 12.

Figure 14

Neutrino $\mathrm{CC}–d\sigma /(d{Q}^{2}d{W}_{\pi N}d{\mathrm{\Omega }}_{\pi }^{*})$ differential cross section in units of ${10}^{-38}{\mathrm{cm}}^2{\mathrm{c}}^2/{\mathrm{GeV}}^3$, as a function of ${\varphi }_{\pi }^{*}$ and ${\theta }_{\pi }^{*}$, evaluated for fixed $E_{\nu }=1\mathrm{GeV}$, $Q^2=0.1{\mathrm{GeV}}^2/c^2$ and $W_{\pi N}=1.23\mathrm{GeV}$ values.

Figure 15

The same as in Fig. 14 for antineutrino $\mathrm{CC}–d\sigma /(d{Q}^{2}d{W}_{\pi N}d{\mathrm{\Omega }}_{\pi }^{*})$.

Figure 16

The same as in Fig. 14 for neutrino $\mathrm{NC}–d\sigma /(d{Q}^{2}d{W}_{\pi N}d{\mathrm{\Omega }}_{\pi }^{*})$.

Figure 17

The same as in Fig. 14 for antineutrino $\mathrm{NC}–d\sigma /(d{Q}^{2}d{W}_{\pi N}d{\mathrm{\Omega }}_{\pi }^{*})$.

Figure 18

$d\sigma /d{\mathrm{\Omega }}_{\pi }^{*}$ differential cross section in units of ${10}^{-38}{\mathrm{cm}}^2$, as a function of ${\varphi }_{\pi }^{*}$ and ${\theta }_{\pi }^{*}$, evaluated at $E_{\nu }=1\mathrm{GeV}$ and with a $W_{\pi N}<1.4\mathrm{GeV}$ cut.

Figure 19

$d\sigma /d\cos {\theta }_{\pi }^{*}$ differential cross section in units of ${10}^{-38}{\mathrm{cm}}^2$ for $E_{\nu }=1\mathrm{GeV}$, and with a $W_{\pi N}<1.4\mathrm{GeV}$ cut.

Figure 20

$d\sigma /d{\varphi }_{\pi }^{*}$ differential cross section in units of ${10}^{-38}{\mathrm{cm}}^2$ for $E_{\nu }=1\mathrm{GeV}$, and with a $W_{\pi N}<1.4\mathrm{GeV}$ cut.

Figure 21

Shape comparison of the theoretical $d\sigma /d\cos {\theta }_{\pi }^{*}$ (left panels) and $d\sigma /d{\varphi }_{\pi }^{*}$ (right panels) differential cross sections with unnormalized ANL [35] and BNL [36] data. A cut $W_{\pi N}<1.4\mathrm{GeV}$, in the final pion-nucleon invariant mass, is imposed in both data and the theoretical results. Theoretical results have been obtained averaging over the neutrino flux for neutrino energies in the [0.5, 6] GeV interval, setting their overall size to reproduce the areas under the experimental data. Predictions from two previous versions of the HNV model are also displayed: HNV1 stands for the HNV model without the $\mathrm{\Delta }$ propagator modification of Eq. (9), while to compute the HNV2 results, the implementation of the Watson theorem has been further suppressed.

Figure 22

Inclusive $d\sigma /(d{\mathrm{\Omega }}^{\prime }d{E}^{\prime })$ cross section off protons (sum of the differential distributions for the $e^{-}p\to e^{-}p{\pi }^0$ and $e^{-}p\to e^{-}n{\pi }^{+}$ reactions), as a function of the invariant mass $W_{\pi N}$ and for fixed ${\theta }^{\prime }=37.1°$. The four-momentum transfer square $Q^2$ varies in the interval $[0.18,0.04]{\mathrm{GeV}}^2/c^2$, when $W_{\pi N}\in [1,1.4]\mathrm{GeV}$. Data are taken from Ref. [37]. In the right panel, predictions from the HNV and two previous versions of that model are displayed: HNV1 stands for the HNV model without the $\mathrm{\Delta }$ propagator modification of Eq. (9), while to compute the HNV2 results, the implementation of the Watson theorem has been further suppressed.

Figure 23

Data and theoretical predictions for the ${\tilde{\sigma }}_T=\int {\sigma }_{T}d{\mathrm{\Omega }}_{\pi }^{*}$ and ${\tilde{\sigma }}_L=\int {\sigma }_{L}d{\mathrm{\Omega }}_{\pi }^{*}$ inclusive cross sections off protons ($p{\pi }^0+n{\pi }^{+}$), as a function of the $\pi N$ invariant mass, and for two fixed values of $Q^2=0.2{\mathrm{GeV}}^2$ (left panel) and $Q^2=0.5{\mathrm{GeV}}^2$ (right panel). Data are taken from Ref. [38].

Figure 24

Comparison of the ${\sigma }_T+ϵ{\sigma }_L$, ${\sigma }_L$, ${\sigma }_{TT}$, ${\sigma }_{LT}$, ${\sigma }_{L{T}^{\prime }}$ pion polar angular distributions obtained using the DCC, SL and HNV models for the $e^{-}p\to e^{-}p{\pi }^0$ (left panels) and $e^{-}p\to e^{-}n{\pi }^{+}$ (right panels) channels. The kinematics correspond to $Q^2=0.06{\mathrm{GeV}}^2/c^2$, $W_{\pi N}=1.221\mathrm{GeV}$ and an incoming electron energy of 0.855 GeV. For $e^{-}p\to e^{-}p{\pi }^0$, the ${\sigma }_L$ contribution is negligible so that ${\sigma }_T+ϵ{\sigma }_L\approx {\sigma }_T$, while for $e^{-}p\to e^{-}n{\pi }^{+}$ we also show ${\sigma }_T$ in the first panel. Data from Ref. [39] are available only for the $p{\pi }^0$ channel.

Figure 25

Comparison of the ${\sigma }_T+ϵ{\sigma }_L$, ${\sigma }_{TT}$ and ${\sigma }_{LT}$ pion polar angular distributions obtained using the DCC, SL and HNV models for $e^{-}p\to e^{-}p{\pi }^0$ (left panels) and $e^{-}p\to e^{-}n{\pi }^{+}$ (right panels) at $Q^2=0.4{\mathrm{GeV}}^2/c^2$ and $W_{\pi N}=1.22\mathrm{GeV}$ or 1.23 GeV, respectively. Data from Refs. [41, 42] for $e^{-}p\to e^{-}p{\pi }^0$ and $e^{-}p\to e^{-}n{\pi }^{+}$, respectively, are displayed as well. In the two bottom panels, we also show the $p{\pi }^0$ and $n{\pi }^{+}$ measurements of Refs. [43] (left) and [40] (right) at $Q^2=0.4{\mathrm{GeV}}^2/c^2$ and $W_{\pi N}=1.22\mathrm{GeV}$, together with the theoretical predictions, of the ${\sigma }_{L{T}^{\prime }}$ distribution. Finally in all panels, the HNV1 curves stand for the results obtained within the HNV model, when the propagator modification of Eq. (9) is not considered, while to obtain the HNV2 predictions, the implementation of the Watson theorem is further suppressed.

Figure 26

Same as Fig. 25 ($Q^2=0.4{\mathrm{GeV}}^2/c^2$), but for higher $\pi N$ invariant masses, $W_{\pi N}=1.30\mathrm{GeV}$ and 1.29 GeV for $e^{-}p\to e^{-}p{\pi }^0$ and $e^{-}p\to e^{-}n{\pi }^{+}$, respectively.