Polarization Transfer in Weak Pion Production off the Nucleon

Polarization transfer (PT) observables in the single pion production induced by the charged current interaction of the neutrino with the nucleon are examined. The polarization components of the final nucleon and the charged lepton are calculated within two models for the pion production. The predictions are made for neutrino energy of the order of 1~GeV as well as for the T2K energy distribution. It is demonstrated that the PT observables, the degree of polarization and the polarization components of outgoing fermions, are sensitive on assumptions about the nonresonant background model. In particular it is shown that the normal components of the polarization of the outgoing nucleon and the lepton are determined by the interference between the resonant (RES) and nonresonant (NB) amplitudes. Moreover the sign of the normal component of the polarization of the charged lepton is fixed by the relative sign between the RES and the NB amplitudes.


INTRODUCTION
The first theoretical studies of the single pion production (SPP) in neutrino-nucleon (νN ) scattering were performed more than forty years ago. A historical review of the topic can be found in [1]. Latest development of the accelerator neutrino oscillations experiments [2][3][4][5] has triggered off new interests in the SPP processes.
A further progress in the investigation of the fundamental properties of neutrinos: the oscillation phenomenon, the CP -violation in lepton sector, the mass hierarchy, requires an improvement of the experimental techniques for the measurement of the interactions of neutrinos with different nuclear targets [6] as well as a development of the theoretical models describing the neutrino-nucleus scattering [7].
The measurement of the neutrino oscillation parameters as well as extraction of the CP violation phase are made based on the analysis of the quasielastic (QE) neutrino-nucleus scattering, however, the SPP processes contribute to the background for the detection of the QElike events as well as the electron neutrinos in the far detector. Hence the SPP contribution can not be neglected in the analyses of neutrino scattering data. Additionally the investigation of the SPP in the νN interactions allows to study the weak excitation of the nucleon to the resonance states.
A first natural step in modeling the SPP in ν-nucleus scattering is the construction of the theoretical description for the ν-nucleon scattering. In this work we focus on the interactions of neutrinos with free nucleon target. Two mechanisms for the pion production in the νN * Electronic address: krzysztof.graczyk@uwr.edu.pl scattering can be distinguished: a resonant (RES) and a nonresonant. In the first the nucleon is excited to the resonance state, N * , which subsequently decays to πN system. In the other there is no N → N * transition. This contribution can be modeled by the so-called nonresonant background (NB) amplitudes allowed by the symmetries [8].
The choice of the degrees of freedom of the SPP model depends on the energy range in which it is applicable. If neutrino energy is relatively low, E ∼ 1 GeV, it is enough to consider the contribution to the scattering amplitude from the first resonance region. For larger energies one has to include also the resonances states from the second and the third resonance regions as well as higher order the NB terms. In this work we discuss neutrino energy of the order of 1 GeV, which is a kinematic domain typical for the long and short baseline experiments with accelerator source of neutrinos [2,3].
There are many phenomenological models describing the SPP in νN interactions [9][10][11][12][13][14][15][16][17][18][19][20], for more complete list see [7,21]. One of the main difficulties in modeling the pion production is proposing a consistent description of both the RES and the NB contributions. The analysis of the unpolarized cross section data does not allow to distinguish between the RES and the NB contributions. Moreover the main information about the SPP in the νN scattering is obtained from the analysis of the data collected by two bubble chamber experiments ANL [22] and BNL [23] in which the neutrino-deuteron scattering was investigated. There are new measurements of the SPP in the ν-nucleus scattering e.g. by Minerva [24] experiment. But in the analysis of these data the nuclear structure effects must be included which makes the studies complicated and model-dependent.
In this work we propose to study the polarization properties of the particles in the SPP processes. We shall show that the polarization transfer (PT) observables contain the non-trivial information about the resonance and the nonresonace transition. In particular we demonstrate that the investigation of the normal polarization components of the charged lepton and the final nucleon give a knowledge about the relations between the RES and the NB contributions. Indeed the normal polarizations are proportional to interference of the RES and NB amplitudes.
The PT observables have been studied experimentally and theoretically in electron scattering off the nucleon/nuclei for last years. More than forty years ago Akhiezer et al. [25,26] (see also [27][28][29]) showed that from the measurement of the PT observables in the elastic electron scattering off the proton the form factor ratio G p E /G p M (G p E/M is electric/magnetic proton form factor of the proton) can be obtained. This is an alternative method for the measurement of the elastic form factors to the famous Rosenbluth method. The ratio G p E /G p M obtained from the PT data turned out to be inconsistent with the Rosenbluth measurements. It triggered off more detailed experimental and theoretical studies of the elastic electron-nucleon scattering. A recent review of the topic can be found in [30]. The investigation of the polarization properties of final particles in the νN scattering is not a new idea. The PT observables in the QE νN scattering were discussed in [1]. Recently the problem has been refreshed in works [31][32][33][34]. Moreover, in Refs. [35][36][37] the polarization properties of the τ lepton produced in the QE and inelastic νN interactions were studied. Additionally in [38] the impact of the nuclear effects on the polarization of the τ lepton produced in the QE neutrino-nucleus scattering was investigated.
The polarization properties of the τ lepton in the SPP induced by the ν τ N scattering processes were studied by two groups: Hagiwara et al. [36] and Naumov et al. [35]. However, the discussed SPP models did not contain the NB contribution. In our work the NB contribution plays a central role. We show that the PT observables are sensitive on the NB contribution. We discuss the charged current interactions of the muon and the tau neutrinos with the nucleons. Eventually we investigate also the polarization properties of the final nucleon produced in the SPP process. This problem has been not studied before.
We show that the PT observables are sensitive on the various details of the SPP models, in particular the description of the NB contribution. In order to study the model-dependence of the predictions of the PT observables we consider two phenomenological approaches for the SPP: Hernandez-Nieves-Valverde (HNV) model as formulated in [8] and Fogli-Nardulli (FN) model as described in [11]. Both approaches are similar in the construction but it is demonstrated that small differences in the treatment of the NB contribution give rise to dispar- FIG. 1: Angular distribution of the particles, in the laboratory frame, in the process (1). The momenta k and k ′ denote the neutrino and the charged lepton respectively. The target is at rest while kπ and p ′ denote the pion and the outgoing nucleon momenta. The vectors ξ and ζ in green, blue and red denote longitudinal, normal, and transverse components of the charged lepton and the nucleon respectively.
ities in the predictions of the PT observables.
The paper is organized as follows: Section 2 introduces kinematics and the cross section formula, in Sec. 3 the polarization observables are given, Sec. 4 contains short review of the HNV and the FN models, while in Sec. 5 the numerical results are presented and discussed. We summarized in Section 6. Additionally we include three Appendixes.

KINEMATICS AND CROSS SECTION
We consider the SPP induced by the charged current (CC) νN interaction where l = µ, τ , k α = (E, k) and k ′ α = (E ′ , k ′ ) are the four-momenta 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. Notice that E x = x 2 + M 2 x . M , m and m π denote masses of the nucleon, charged lepton and the pion respectively.
The four-momentum transfer is defined as and the invariant hadronic mass W is given by Let us also define By θ ≡ ∠(k, k ′ ) the scattering angle between lepton momenta is denoted while by Ω the spherical angle (depending on θ) is denoted; φ π is the angle between the scattering plane (spanned by k and k ′ ) and the plane spanned by the pion and the final nucleon momenta, see Fig. 1.
The differential cross-section for the process (1) reads: Sum over the spins of the leptons and the nucleons is denoted by ′ and respectively. We systematically omit the spin notation in this section.
The scattering matrix, in the one-boson exchange approximation and for −q 2 ≪ M 2 W reads: where G F ≈ 1.16639× 10 −5 GeV −2 is the Fermi constant while θ C is the Cabibbo angle and cos θ C ≈ 0.97427. The expectation value of the CC weak hadronic current operator J µ reads Distinct phenomenological models can contain different numbers of diagrams contributing to the total scattering amplitude. Hence the total hadronic current can be given by the sum: where C a is the Clebsch-Gordan coefficient, see Table A. The total amplitude in the HNV model is described by seven Feynman diagrams, hence S HN V = {N P, CN P, CT, P P, P F, ∆P, C∆P }.
An explanation of the notation in the parentheses can be found in Fig. 2 and Ref. [8]. In the FN model the total amplitude is given by four diagrams The structure of the amplitudes in the HNV and FN models is the same, hence we introduce where R µ a = r µ a and R µ a =r µ for the HNV and the FN models respectively.
Eventually it is convenient to introduce the notation then FIG. 3: The differential cross section and its "pure" resonance (dotted), "pure" nonresonant (dashed-dotted) and the interference between resonant and nonresonant amplitudes (dashed-dotted-doted) contributions in the HNV (left figure) and the FN (right figure) models. The "pure" resonance contribution is given by |M∆P + MC∆P | 2 and |M∆P | 2 in the HNV and the FN models respectively. The neutrino energy E = 0.7 GeV and Q 2 = 0.1 GeV 2 .

POLARIZATION TRANSFER OBSERVABLES
We consider two types of the PT processes. In the first the polarization of the final lepton is examined, namely In the other the polarization of the final nucleon is a subject of studies Notice that ξ and ζ are the four-vector spins of the final lepton and the nucleon respectively. A spin four-vector s µ of a 1/2-spin fermion of mass M has three independent components. At rest it has the form In a frame in which a particle moves with the velocity p/E p the spin vector reads [39][40][41]: At any frame It is convenient to introduce: longitudinal (L), transverse (T ) and normal (N ) components of the spin where we choose the coordinates so that (see Fig. 1) and h l = ±1 as well as and h N = ±1.
For the process (14) the differential cross section reads while for the process (15) the cross section has the form: where | M f i | is summed over the spins. The four-vectors P l µ and P N µ describe the polarization of the charged lepton (l) and the final nucleon (N). The   components of polarization are given by the ratio [28,42]: where X = L (longitudinal), N (normal) and T (transverse) components of the polarization of the final fermion; Y = l, N and s = ξ, ζ.
Finally we define the degree of polarization of particle: Notice that 0 < P < 1. If P ≈ 0 than the particle is unpolarized, while for fully polarized particle P ≈ 1.

SPP MODELS
We consider the interactions of the neutrinos of energy of the order of 1 GeV with the free nucleon target. The dominant RES contribution is given by the weak N → ∆(1232) transition. In both the HNV and the FN models the N → ∆(1232) transition is described by the same formalism but in the FN model the weak vertex N W + ∆ is oversimplified. In phenomenological approaches the NB contribution is described by some number of diagrams allowed by a symmetry, however, in the FN model the number of diagrams is smaller. In next two subsections the main features of both descriptions are shortly reviewed.

HNV Model
We consider the HNV model as it is formulated in [8], however, we notice the latest developments of the approach in Refs. [43,44]. The NB contribution is described by five diagrams, see Fig. 2. They are determined from the non-linear sigma model. The main idea was to obtain, on the tree level, the axial and the vector currents from the non-linear sigma model and to associate them with their phenomenological counterparts. The resonance contribution is given by two diagrams describing the weak N → ∆(1232) transition. The ∆(1232) resonance is modeled by the 3/2-spin Rarita-Schwinger field [45].

Nonresonant Background
The NB contribution is generated by five hadron currents: where is the nucleon propagator. The electroweak nucleon vertex reads where F V k (q 2 ) is the vector nucleon form factor, see (B1); G A is the axial nucleon form factor (see: B2); g A = 1.26 is the axial nucleon coupling; the form factor F ρ is given by (B7).
The axial part of the vertex (41) reads [48] A νµ 3/2 (p, q) = The axial form factors are obtained from the analysis of the neutrino scattering data. However, the data are not enough accurate to get an information about four independent form factors [49]. Therefore the following simplifications are made: • C A 3 (Q 2 ) = 0, as it is suggested by simple quark model, see e.g. [50]; as it is supported by dispersion analyses [9]; • C A 6 is a function of C A 5 , namely: as it is obtained from the partially conserved vector current (PCAC) hypothesis.
For the numerical analyses we take the C A 5 fit from [47], see Appendix 1.

Fogli-Nardulli Model
The FN model 1 formulated in [11] describes the pion production in the first and the second resonance regions. But in this paper we consider only the first resonance region.

Nonresonant Background
The NB contribution is described by three diagrams: the pion in flight (P F ), the same as in the HNV model, and two nucleon pole diagrams: N P , and CN P . But in the latter the pseudoscalar pion-nucleon coupling was implemented, while in the HNV model the pseudovector pion-nucleon coupling is discussed. (48) where g N N π = √ 14.8 · 4π, see e.g. [52], F π form factor is given in the Appendix 1.

∆(1232) Contribution
In the FN model only one digram describes the SPP induced by the N → ∆(1232) transition. The structure of this current is the same as in the HNV model However, it is assumed that as well as The width of the ∆(1232) resonance is given by (C3).

Unpolarized Cross Sections
Most of the SPP models, including the HNV and the FN approaches, reproduce the cross section data for ν µ p → µ − pπ + process with reasonable accuracy. The agreement with the data is achieved by appropriate tuning the values of the parameters of the C A 5 axial form factor [11,16,47,51,53]. The main problem is to obtain coherent model predictions for all three CC channels. Indeed the ANL data for the νn → µ − nπ + process seem to be inconsistent with the other channels [47]. This can be caused by: oversimplified treatment of the deuteron structure effects in the analysis of the ANL data (see [54]) and/or incomplete description of the ∆(1232) resonance propagation (see [44]) and/or low quality of the data.
It seems that critical studies of the SPP models can be performed only if new more precise measurements of the interactions of the neutrinos with the free nucleon target will be delivered. Moreover the information about the SPP dynamics hidden in: the total and single differential cross sections (like dσ/dQ 2 ) and even double differential cross sections is limited because important features of an approach are integrated out. Hence the analysis of these cross section data does not allow to verify which    FIG. 14: The same caption as in Fig. 11 but T2K-flux averaged. model, among many in the market, is the closest, in the description, to the reality. It is illustrated in Fig. 3, where we plot d 2 σ/dW dQ 2 distribution and its partition into: (i) "pure" resonance contribution; (ii) interference between the RES and the NB amplitudes; (iii) "pure" NB contribution.
Although the RES and the NB contribute differently in both models the shape and the magnitude of d 2 σ/dW dQ 2 obtained within both approaches are very similar. The significant disparities between predictions of the both models manifest when the triple differential cross sections are examined. In Fig. 4 we plot d 3 σ/dE ′ dΩdE π and its partition into all possible interference terms calculated at particular kinematics. At low scattering angle the FN cross section increases rapidly while the HNV model predictions fall down. The low-θ behavior of the FN's cross section is determined by a |M N P | 2 contribution. In the HNV model the contribution from the N P diagram is smaller and it is suppressed by the contribution missing in the FN model, namely the interference terms: M C∆P M * N P , M CT M * P F and others. Certainly inspection of the triple differential cross sections, their dependence on φ π angle, may deliver a valuable information about the RES and the NB dynamics. Let us also mention that additional constraints on the NB contribution can be obtained from the combined analysis of various mass distributions. It is demonstrated in Ref. [16], where the detailed discussion of W (N π), W (µN ) and W (µπ) event distributions of the ANL and the BNL experiments were performed. However, in the next section we shall demonstrate that the PT observables contain an unique information about the dynamics of the SPP.

Polarization Transfer Observables
We start the presentation of the PT results from the discussion of the polarization properties of the charged lepton produced in process (14). In Figs. 5 and 6 the longitudinal, normal and transverse components as well as the degree of polarization of the µ − and µ + leptons are shown respectively. Additionally Fig. 7 includes the plots of µ − polarization components calculated for an averaged T2K [3] experiment neutrino energy, E = 0.6. While in Fig. 8 the polarization components of µ − lepton averaged over the T2K energy flux [55] are shown. The muon lepton is a light particle hence it should be almost polarized. However, we notice that for some kinematics, namely at low scattering angle, the µ ± lepton can be partially polarized.
A unique information about the SPP dynamics is hidden in the normal component of the polarization. Indeed this observable is dominated by the interference of the M ∆P and M C∆P amplitudes with the NB diagrams. Therefore the sign of the normal component is defined by the relative sign between the RES and the NB contributions. This property is illustrated in Fig. 9, where we plot the partition of the normal component into all interference terms. Now let as review the polarization properties the τ lepton produced in the ν τ N scattering. The mass of the τ is large hence it can be partially polarized. Indeed in Fig. 10 we plot the predictions of the degree of polarization. It varies from 0.3 to 1.0. Let us remark that the polarization vector of the τ lepton is one of the variables which describes the angular distribution of the products of its decay. The model dependence of the predictions of the polarization vector of the tau lepton on the SPP model assumptions seems to be stronger then in the case of muon lepton.
Interesting conclusions are obtained from the analysis of the polarization properties of the final nucleon produced in process (15). In Figs. 11 and 12 the angular dependence (in φ π angle) of the polarization components and the degree of polarization of the nucleon produced in ν µ N and ν µ N interactions are plotted. Similarly as for the discussion of the lepton polarization properties we show also the predictions of the polarization components: for the T2K neutrino averaged energy, Fig. 13, and T2K-flux averaged, Fig. 14. Notice that the resonance part of the normal, transverse and longitudinal polarization components have sinusoidal character which is distorted by the inclusion of the NB contribution. At some kinematics the normal polarization of the nucleon is large -it reaches the value 0.5. Similarly as in the lepton polarization case the interference of the RES with the NB diagrams gives sizeable contribution which for some kinematics becomes dominant. It is illustrated in Fig. 15.

SUMMARY
We discussed polarization properties of fermions produced in the SPP processes induced by the charged current neutrino-nucleon scattering. The components of the polarization vector of the outgoing charged lepton and the nucleon were calculated. In order to make the discussion more realistic we have made the predictions of the polarization properties for the T2K experiment. It turned out that the PT observables are very sensitive on the details of the SPP model. In order to investigate how a change in a model assumptions affects the predictions of the polarization components we considered two the SPP approaches. It was demonstrated that the most interesting information about the SPP dynamics is hidden in the normal component of the polarization of the outgoing lepton and the nucleon. In particular the sign of the normal polarization of the charged lepton is deter-mined by the relative sign between the NB and the RES amplitudes.
Eventually we conclude that the investigation of the polarization transfer observables in the SPP in νN scattering should deliver a complementary, to the spin averaged cross sections, knowledge about the resonance and the nonresonance contributions.
All algebraic calculations have been performed using FORM symbolic language [56].
The calculations have been carried out in Wroclaw Centre for Networking and Supercomputing (http://www.wcss.wroc.pl), grant No. 268. ton, F p k , form factors, namely F V k (q) = F p k (q) − F n k (q), k = 1, 2. (B1) We consider the same nucleon form factors as in [8].
The axial nucleon form factor reads where M A = 1.00 GeV, g A = 1.26.

HNV Model
The vector N → ∆(1232) transition form factors are given by [47]: where M V = 0.84 GeV, parameters K 1 , K 2 , A, B, C and D are in Table II.  The axial form factor C A 5 (Q 2 ) has the form: where M A∆ = 0.85 GeV and C A 5 (0) = 1.10 as obtained in Ref. [47].