Spin asymmetry in single pion production induced by weak interactions of neutrinos with polarized nucleons

The single pion production (SPP) in the charged-current neutrino (antineutrino) scattering off the polarized nucleon is discussed. The spin asymmetry is predicted within two approaches. The spin polarizations of the target nucleon that are longitudinal and perpendicular to the neutrino momentum are considered. It is shown, in several examples, that information about the SPP dynamics coming from the spin asymmetry is complementary to information obtained from measurements of spin averaged cross section. Indeed, the spin asymmetry is sensitive to the nonresonance background description of the SPP model. For the normal polarization of the target, the spin asymmetry is given by the interference between the resonance and the nonresonance contributions.


INTRODUCTION
The neutrino oscillation phenomenon has been investigated for several decades. The oscillation parameters are relatively well established [1], however, still two parameters, δ -CP -violation phase and θ 23 -mixing angle, are poorly known [2].
In the simplest two-flavor scenario the probability for the oscillation ν α → ν β reads P (ν α → ν β ) ≈ sin 2 (2θ) sin 2 (∆m 2 L/4E), where ∆m is neutrino mass differences, θ is a mixing angle, E is the neutrino energy, while L is a distance between the source of the neutrinos and the detector.
In the long baseline experiments, such as T2K [1] or Nova [3], the distance L is known. The neutrino beam, produced at accelerator, consists of mainly muon neutrinos of the energy of the order of 1 GeV. However, the beam is not monochromatic and its energy profile is obtained from the analysis of the interaction of the neutrinos with the target. Therefore the determination of the oscillation parameters depends on the accuracy in estimation of the neutrino energy.
Usually the neutrino energy is reconstructed from the analysis of the quasielastic (QE) neutrino-nucleus scattering. The reconstruction bases on the knowledge of the neutrino-nucleon and the neutrino-nucleus cross sections [4,5]. However, in the 1 GeV energy range a sizable fraction of the detected interactions is inelastic. In particularly, the so-called single pion production (SPP) processes are distinguished. The SPP events contribute to the background for the measurement of the QE scatter- * Electronic address: krzysztof.graczyk@uwr.edu.pl ing. Moreover, the neutral current π 0 production events can be mislead with the signal for ν µ → ν e oscillation.
Intense studies of the fundamental neutrino properties caused a new interest in the investigation of the neutrinonucleon and the neutrino-nucleus scattering. In this work we focus on the problem of the single pion production in the neutrino-nucleon scattering in the energy range characteristic for the long baseline neutrino oscillation experiments. This topic has been studied theoretically  and experimentally [27][28][29][30][31][32] for last fifty years.
The SPP scattering amplitude is dominated by the resonance (RES) contribution given by a weak nucleonresonance transition. However, a complete SPP model should include also the diagrams describing the so-called nonresonance background (NB) terms. The way the RES and NB contributions are treated gives rise to the differences between various theoretical approaches.
In order to test the SPP models their predictions must be confronted with the experimental measurements of the neutrino-nucleon and the neutrino-nucleus cross sections. As it is explained in our previous paper [33], the spin averaged cross sections contain only a part of the information about the dynamical structure of the SPP amplitudes. The complementary information can be obtained from the analysis of the polarization transfer (PT) observables.
The investigation of the PT in the neutrino-nucleon and the neutrino-nucleus scattering have been discussed since sixties [34][35][36][37][38][39][40][41][42][43][44][45][46]. Recently in [33,47] we reported the results of the discussion of the impact of the NB contribution on the PT observables. It was shown that the components of the polarizations of the charged lepton and the final nucleon contain a unique information about the relative phase between the RES and NB amplitudes which can be used to constrain theoretical models, in particulary, the description of the nonresonant background. In this report, instead of analyzing the polarizations of the final particles, the neutrino scattering off the polarized target is considered. We propose to investigate properties of a spin asymmetry observable. Similar quantity was discussed for the elastic electron-nucleon and the electron-nucleus scattering [48,49]. Indeed, the measurement of the asymmetry in the electron-nucleon scattering was proposed as an alternative technique, to the Rosenbluth method, for getting the electric and magnetic form factors of the nucleon 1 . In this work we calculate and analyze the spin asymmetry in the SPP induced by interactions of the neutrinos with the nucleons. We show that this observable is sensitive to the NB contribution. Hence, the spin asymmetry contain unique information about the SPP dynamics not accessible in the spin averaged cross section measurements.
Similarly as in [33], two different SPP approaches are 1 The first measurements of the spin asymmetry are reported in Ref. [50].
consider [8,51]. Our studies are restricted to the neutrinos of the energy of the order of 1 GeV. Therefore to model the RES contribution we consider only the weak N → ∆(1232) transition. The predictions are made for full models (RES and NB contributions) and the version of the models with resonance contribution only. The paper is organized as it follows: in Sec. 2 the necessary formalism is introduced, Sec. 3 presents the numerical results and their discussion, a summary is given in Sec. 4.

SPIN ASYMMETRY
Let us consider the SPP processes induced by the charged current muon netrino/antineutrino interactions with the polarized nucleon target, namely (2) where k α = (E, k) and k α = (E , k ) are the fourmomenta of the initial and the final leptons respectively, while p α = (E p , p), p α = (E p , p ) and k α π = (E π , k π ) denote the four-momenta of the incoming nucleon (N), the outgoing nucleon (N ) and the pion respectively. The calculations are made in the laboratory frame, hence the spin four-vector of the target reads where s 2 = 1. The four-momentum transfer is given by: where ω and q denote the transfer of the energy and the momentum respectively. Let us introduce the hadronic invariant mass and Eventually let Ω(θ, φ) denotes a solid angle depending on θ ≡ ∠(k, k ) and φ is a corresponding azimuth angle. We define a spin asymmetry by the ratio: where dσ is the differential cross section. The asymmetry is linear in s, namely A = s · a.
Notice that in the last variant the φ-dependence of dσ/dΩ can not be trivially integrated out. Indeed the rotational symmetry (along k) is broken by the choice of the direction of the target's spin. In order to perform calculations we choose the coordinates so that wheren is the normal vector of the scattering plane spanned by k and k .

Numerical implementation
Our main objective is to study the properties of the spin asymmetry, in particularly, its sensitivity to the NB contribution. To achieve this goal, similarly as in our previous work [33], in order to perform the calculations, two SPP approaches are considered: the model by Hernandez, Nieves, and Valverde (HNV), as described in [51], and the model by Fogli and Nardulli (FN), as given in [8]. In both descriptions the scattering amplitude is calculated in tree level approximation.
The predictions of the spin asymmetry for neutrino (antineutrino) scattering off longitudinally (8) and perpendicularly (9) polarized target are made for six charged-current SPP processes: The differential cross section, for given SPP processes, has the structure where D is the set of the diagrams, c a is the Clebsch-Gordan coefficient while M a is a matrix element for a diagram a.
The full amplitude of the HNV model consists of contributions from seven diagrams. The NB amplitudes are obtained from the nonlinear sigma model. The SPP contribution in the FN approach is given by five diagrams, where the NB contribution is motivated by the linear sigma model. All diagrams are plotted in Fig. 3. Our discussion is restricted to the first resonance region, hence, all calculations are preformed for W < 1.4 GeV.
In the HNV model the NB contribution is given by the following diagrams: nucleon-pole (NP), conjugate nucleon-pole (CNP), contact term (CT), pion in flight (PF) and pion-pole (PP). The ∆(1232) resonance contribution is described by two diagrams: ∆P -delta pole, and C∆P -conjugate delta pole.
The NB contribution in the FN model consists of three diagrams: pion in flight (P F ) and two nucleon pole diagrams: N P , and CN P . But in the latter two diagrams the pseudoscalar pion-nucleon coupling is implemented, in contrast to HNV model, where the pseudovector coupling is considered. The weak N → ∆(1232) transition is oversimplified. Indeed, there is only one resonance diagram and the N W − ∆ vertex is described by only two form factors.
More details about the implementation of both models, the choice of the transition form factors etc. can be found in our previous paper [33].

Spin asymmetry for longitudinal polarized target
The Fig. 4 presents the plots of the longitudinal spin asymmetry A (σ, E) calculated for the neutrino and the antineutrino scattering off the polarized target. The asymmetry varies from −0.5 to 0.5. Above E = 1 GeV A (σ, E) weakly depends on the neutrino energy.
For the channels (11) and (14) (related by the isospin symmetry), the NB contribution to A (σ, E) is negligible. Indeed in this case the resonance contribution, from ∆P , is dominant. Therefore the predictions of A (σ, E) within the HNV and the FN models are very similar. However, for the other channels the asymmetry is quite model-dependent. Indeed for the processes (12) and (15) (also related by the isospin symmetry) the asymmetry predicted within the HNV and FN models have com- pletely different functional dependence (different sign and magnitude). Moreover, in this case the NB contribution is large and it modifies significantly A (σ, E). To understand this property, we present the Fig. 5, where the contributions to the spin asymmetry from various diagrams are distinguished. It can be noticed that the deviations between the HNV and FN models are due to the presence of the diagrams C∆P and CT in the HNV model. Eventually, in both approaches the diagram N P gives rise to the difference between the full and the RES models predictions.
Similar observations, as above, can be made when A (dσ/dQ 2 , E) is examined, see Fig. 6.

Spin asymmetry for perpendicularly polarized target
The spin asymmetry is given by the scalar product s · a. In the case of the perpendicularly polarized target the components of s ⊥ are proportional to either sin(φ) or cos(φ). As the result the spin asymmetry can be written in the form: The A ⊥ (φ) is dominated by the sinusoidal part. It is shown in Fig. 7, where the asymmetry A ⊥ (dσ/dφ) is plotted. The sinusoidal character is maintained also when the asymmetry is calculated for the flux averaged cross sections, as it is illustrated in Fig. 8

φ-dependence of
calculated for the energy spectrum, Φ, of the T2K experiment [52] is plotted.
Similarly as in the case of the longitudinally polarized target for two channels (11) and (14) the asymmetry is dominated by resonance contribution of the ∆P diagram. Hence, in this case A ⊥ (φ) is insensitive to details of the NB model.
It is important to remind that the FN model does not contain the C∆P diagram, required by gauge invariance. Lack of this contribution leads to the deviation between predictions of the A ⊥ (φ) obtained for the FN and the HNV models for the channels (12) and (15), see Figs. 7 and 8 as well as Fig. 9. In the latter figure the decomposition of the asymmetry into contributions from various diagrams is shown.
The spin asymmetry (18) has contributions from a 2 sin(φ) and a 1 cos(φ), however, plots of Figs. 7 and 8 suggest that the a 2 is dominant. It is interesting to remark that the a 1 component, connected with cosine, is given by the interference between resonance and nonres-onance amplitudes. Hence, any deviation of A ⊥ (φ) from the sinusoidal dependence is induced by the NB contribution. If the spin vector s ⊥ is parallel to normal vector n then only a 1 component contributes to the spin asymmetry. In this case The above property is illustrated in Fig. 10 where we plot the decomposition of A ⊥ (φ = 0 • ) into contributions from various interferences between diagrams. In Fig. 11 (top panel) we plot A ⊥ (φ = 0 • ) as a function of energy. It is seen that the asymmetry is small but nonvanishing function of the energy. The asymmetry takes the largest values when s ⊥ is perpendicular (φ = 90 • ) to the normal vector n, see Fig. 11 (bottom panel).

SUMMARY
The single pion production in the neutrino/antineutrino scattering off the polarized target has been discussed. Two polarization of the target have been considered, namely, longitudinal and perpendicular to the neutrino beam. In both cases the spin asymmetry has been calculated within two different models for  spin asymmetry is sensitive to the nonresonant background. Moreover, it is shown that when polarization of the target is parallel to the normal of the scattering plane, the asymmetry is given by the interference between the resonance and the nonresonance diagrams. Summarizing: the spin symmetry contains additional, with respect to spin averaged cross sections measurements, information about the SPP dynamics, which can be used to constrain significantly the single pion production models.
The scattering amplitudes, cross sections have been calculated using symbolic programming language FORM [53].