Single Pion Production off Free Nucleons: Analysis of Photon, Electron, Pion and Neutrino Induced Processes
M. Kabirnezhad

                
                APS/123-QED
                
                
                
                
                
                  Imperial College London, Department of Physics,London SW7 2BZ, United Kingdom
                
                m.kabirnezhad@imperial.ac.uk
                
                
                
§ ABSTRACT 
    In this paper, I introduce a unified model for single-pion production across photo-, electro-, and neutrino-nucleon interactions, designed to be valid over a broad kinematic range that is crucial for accelerator-based neutrino experiments. This model includes vector and axial-vector nucleon transition form factors for all excited nucleons or resonances up to 2 GeV, as well as non-resonant backgrounds, within a meson dominance framework that adheres to QCD principles and ensures unitarity. This approach guarantees accurate asymptotic behaviour at high momentum transfer (Q^2) and effectively addresses the transition region. Additionally, the model employs the Conserved Vector Current (CVC) and Partially Conserved Axial Current (PCAC) relations to provide reliable predictions at very low Q^2, tackling challenges encountered by current neutrino experiments.

The unified model facilitates a comprehensive analysis by integrating all available data from electron, photon, pion, and neutrino scattering experiments. This integration enables a detailed investigation of nucleon structure within the resonance region and is particularly valuable for probing weak interactions, where neutrino-nucleon data are limited. The combined analysis allows for the simultaneous parameterisation and constraint of the model’s free parameters, while quantifying associated uncertainties, thus providing a robust and reliable framework for future neutrino measurements.     
        
               
            

        Single Pion Production off Free Nucleons: Analysis of Photon, Electron, Pion and Neutrino Induced Processes
    M. Kabirnezhad
    September 9, 2024
===========================================================================================================



§ INTRODUCTION

Neutrino interactions that result in the production of a single pion in the final state are critically important for accelerator-based neutrino experiments. Single pion production (SPP) channels constitute the largest fraction of the inclusive neutrino-nucleus cross-section in the 1-5 GeV energy region, which is relevant for all accelerator-based neutrino beams. Accurate models of SPP cross-section processes are required to precisely predict the number and kinematics of neutrino interactions. These models help establish the relationship between neutrino energy and energy deposition in a neutrino detector, which is crucial for interpreting experimental results and reducing systematic uncertainties. Therefore, it is essential that these models are valid within the kinematic region of the neutrino experiments.


Chiral perturbation theory (ChPT), the effective field theory of QCD, is adept at describing non-perturbative QCD processes, such as pion production, particularly at neutrino energies around 1 GeV. This makes ChPT highly useful for understanding interactions in this energy range. However, current and future neutrino experiments, such as NOvA and DUNE, operate in a wide energy beam that includes higher energies. These experiments explore a transition region where neutrino energies exceed E_ν = 1 GeV but remain below the energies where perturbative QCD becomes the dominant framework. This kinematic region, spanning both perturbative and non-perturbative regimes, features hadronic degrees of freedom that are a mixture of nucleons, resonances, and partons. This presents unique challenges in accurately modelling the interactions and particle productions observed in these experiments. This can explain current discrepancies in the first resonance region between MiniBooNE and MINERvA experiments, which have been discussed in Ref. <cit.>. 

Additionally, the transition region encompasses several processes where both resonant and non-resonant mechanisms play a role. Currently, 18 overlapping resonances with varying quantum numbers have been identified, each with resonance masses below 2 GeV. These overlapping resonances result in significant interference effects, not only among the resonances themselves but also between resonant and non-resonant interactions. Such complex interference patterns add to the difficulty of accurately understanding pion production. Therefore, a thorough understanding of these processes within the transition region is crucial for improving the precision of neutrino interaction predictions. Figure <ref> illustrates these complexities: the left panel shows how the hadron invariant mass W evolves, while the right panel depicts the corresponding Q^2 evolution, mapping out the regions explored by both current and future neutrino experiments.

In addition to model development and robust predictions, a crucial aspect of theoretical work in neutrino experiments is the quantification of uncertainties. While several SPP models exist
, most of them address only key processes, focusing primarily on the first and second resonance regions. However, none of these models systematically study or evaluate the systematic uncertainties inherent to the models themselves. Furthermore, existing SPP models are confined to the non-perturbative region, which does not encompass the broad kinematic range required for neutrino experiments. The primary goal of this work is to develop a comprehensive model that addresses these gaps, providing a robust framework to enhance the precision of neutrino measurements.

  
        
The valid region for the SPP model presented in this paper is defined by a hadron invariant mass (W) below 2 GeV , as illustrated in Fig. <ref>. This region is characterised by broad nucleon excitations superimposed on a smooth non-resonant background. For hadron invariant masses below 1.4 GeV, the target nucleon is predominantly excited to the Δ(1232) resonance, which then decays almost immediately into a single pion and a nucleon. In this lower mass range, the Δ resonance, with its significant contribution, is the sole resonance observed within the first resonance region (1.08GeV < W < 1.4 GeV).

The dynamics of the interactions become more intricate in the second resonance region (1.4 GeV < W < 1.6GeV), where the scattering processes are influenced by three isospin-1/2 resonances: the P_11(1440) Roper resonance, and two negative-parity states, D_13(1520) and S_11(1535). As W continues to rise into the third resonance region (1.6GeV < W < 2.0 GeV), the complexity further escalates with the contribution of an additional ten four-star resonances, as noted in Ref. <cit.>, significantly enriching the interaction dynamics.


A commonly used form factor in theoretical models, derived from elastic electron scattering data, is the dipole form factor. However, inelastic scattering data indicate that the dipole form factor is inadequate for accurately describing excited nucleons, particularly in the transition region. In contrast, the meson dominance (MD) form factor model, which accounts for the interaction between leptons and nucleons via meson exchange with analogous quantum properties, offers a more accurate representation. As emphasised in Ref. <cit.>, the MD model, consistent with QCD calculations, effectively captures the nucleon form factor across both perturbative and non-perturbative regimes, particularly addressing the hadron-quark transition region. The model incorporates several parameters that can be constrained by unitarity conditions and analytic models. However, precise quantification of these form factors requires accurate data across a broad kinematic range—a challenge compounded by the scarcity of available data for neutrino interactions in this context.


To address this challenge, this paper introduces a comprehensive unified model that extends its applicability to analogous processes involving not only neutrino beams but also electron, photon, and pion beams. The integration of these diverse probes is particularly advantageous, as electron, photon, and pion scattering experiments have yielded a wealth of data over the years, covering a broad range of kinematic conditions. By harnessing this extensive dataset, the model enables a more detailed and nuanced investigation of nucleon structure, offering insights that would be difficult to achieve using neutrino data alone.

The unified model capitalises on the strengths of each interaction type: electron scattering data provides precision and control for probing the vector current, photon interactions offer complementary insights into the vector current at very low Q^2, and pion scattering data, guided by the Partially Conserved Axial Current (PCAC) hypothesis, delivers crucial information on the axial-vector current at low Q^2. This comprehensive dataset is particularly valuable for addressing and potentially resolving discrepancies between single-pion production (SPP) models and neutrino data, especially in the low Q^2 region (Q^2 < 0.2 , GeV^2).

The advanced theoretical framework developed in this work is designed to incorporate these diverse datasets seamlessly, ensuring that the model remains consistent with fundamental QCD principles and preserves all symmetry relations. This consistency is crucial for making robust predictions about nucleon behaviour across various experimental contexts. Furthermore, the sophisticated analysis method employed in this paper allows for meticulous control over systematic uncertainties, which primarily arise from experimental data and the choice of model parametrisation. Robust predictions with quantified uncertainties are essential for precision neutrino measurements, paving the way for future discoveries.


        
          
        
    
    
   

            
 
 
                
            
            
            
            
            
         
                
        
                
                
        
            
               
        
               
        
                
        
       
        
        

§ MK MODEL (GENERAL DESCRIPTION) 

        
             Let us consider the weak, electro- and photo- pion production reactions:
        
    .[    ν_l; ν̅_̅l̅ ]}
                (k_1) + N(p_1)   →   .[  l; l̅ ]}
                (k_2) + N(p_2) + π(q),
    
                e(k_1) + N(p_1)   →    e(k_2) + N(p_2) + π(q),
    γ(k) + N(p_1)   →     N(p_2) + π(q)

        
        Here N represents a nucleon and l denotes the outgoing charged lepton or neutrino in charged current (CC) and neutral current (NC) interactions. The four-momentum of each particle is indicated in parentheses. In the weak and electro-production reactions, the momentum transfer is defined by 𝐤 =  𝐤_1 - 𝐤_2, and the invariant quantities Q^2 and W are given by:
        
    Q^2    =    -k^2 = -(k_1 - k_2)^2,
    
            W^2    =    (k+p_1)^2 = (p_2 + q)^2.

        Non-invariant quantities, such as pion angles, are defined in the centre-of-mass frame of the final pion and nucleon (the isobaric frame). This is represented by the equation:
        
    𝐪 + 𝐩_2 = 𝐤 + 𝐩_1 = 0.

        The pion angles in the isobaric frame are illustrated in Fig. <ref>. Variables in the laboratory frame are labeled by L.
        
        In this framework, the full kinematic differential cross section for single pion production in lepton-nucleon scattering is defined as:
        
    dσ(l N  → l'Nπ)/dk^2 dW dΩ_π =
     1/(2π)^4|q|/8M^2-k^2/(𝐤^𝐋)^2|ℳ|^2

        
         where ℳ, the matrix element for weak and electromagnetic interactions, can be expressed as follows:
         
    ℳ(ν_l N → l N π) =   G_F/√(2)a ϵ_ρ ⟨ Nπ| J_CC+^ρ |N  ⟩
    ℳ(ν̅_̅l̅ N →l̅ N π) =   G_F/√(2)a ϵ̅_ρ ⟨ Nπ| J_CC-^ρ |N  ⟩
    ℳ(e N → e' N π) =    e^2 ϵ_ρ^EM1/k^2⟨ Nπ| J_EM^ρ |N  ⟩

         
         
        where G_F is the Fermi coupling constant, and e is the electric charge. ϵ^ρ is the leptonic current and a is either the cosine of the Cabibbo angle for CC interactions or 1/2 for NC interactions. 
         
    ϵ_ρ   =   u̅_l(k_2) γ_ρ(1- γ_5) u_ν(k_1),
    ϵ̅_ρ   =   u̅_l̅(k_2) γ_ρ(1+ γ_5) u_ν̅(k_1),
    ϵ^EM_ρ   =   u̅_̅e̅(k_2) γ^ρ u_e(k_1),

        where u is the spinor of leptons.
        The lepton current (ϵ^ρ) can be interpreted as the intermediate gauge boson's polarisation vector:
        
    ϵ^ρ_λ = [C_L_λ e^ρ_L + C_R_λ e^ρ_R + C_λ e^ρ_λ],

        where 𝐞_L and 𝐞_R are the transverse polarisations (i.e., perpendicular to the momentum transfer), and 𝐞_λ is the longitudinal polarisation along the 𝐳 direction of the isobaric system. This gives:
        
    e^α_L   =   1/√(2)[  0  1 -i  0 ] ,
    
            e^α_R   =   1/√(2)[  0 -1 -i  0 ] , 
    
            e^α_λ   =   1/√(|(ϵ^0_λ)^2 - (ϵ^3_λ)^2 |)[ ϵ^0_λ     0     0 ϵ^3_λ ]

         and,
        
    C_L_λ   =   1/√(2)(ϵ_λ^1+ iϵ_λ^2) , 
    
            C_R_λ   =     - 1/√(2)(ϵ_λ^1- iϵ_λ^2) ,
    
            C_λ   =   √(|(ϵ^0_λ)^2 - (ϵ^3_λ)^2 |) .

        
        where λ= -(+) stands for the left (right)-handed helicity. In NC weak current and electron scattering lepton current, where the lepton mass is neglected, ϵ^ρ_+=0, therefore the gauge boson can be interpreted by three polarisations.

        
        The electromagnetic hadron current is given by
        
    J^ρ_EM = 1/2𝒱_Y^ρ + 𝒱_3^ρ.

        The weak hadron currents can be decomposed into vector current(𝒱):
        
    𝒱^ρ_CC   =   𝒱^ρ_1 + i𝒱^ρ_2 ,
    𝒱^ρ_NC   =    (1 - 2sinθ_W^2)𝒱^ρ_3 - sinθ_W^2)𝒱^ρ_Y - 1/2𝒱^ρ_S,

        and axial-vector (𝒜):
        
    𝒜^ρ_CC = 𝒜^ρ_1 + i𝒜^ρ_2
    𝒜^ρ_NC = 𝒜^ρ_3 - 1/2𝒜^ρ_S

        where 𝒱_1,2,3 (𝒜_1,2,3) are the component of the isovector part of the vector (axial-vector) current, 𝒱_Y and 𝒱_S (𝒜_S) are hypercharge and strange isoscalar current.
        
        Isospin symmetry establishes a connection between the matrix elements of the isovector components of the vector and axial-vector hadronic currents. Consequently, the matrix elements for antineutrino- and neutrino-induced reactions are proportional, given by:
        
    ⟨ pπ^-| J^CC-_ρ |p  ⟩   =   ⟨ nπ^+| J^CC+_ρ |n ⟩
    ⟨ nπ^-| J^CC-_ρ |n  ⟩   =   ⟨ pπ^+| J^CC+_ρ |p  ⟩
    ⟨ nπ^0| J^CC-_ρ |p  ⟩   =    -⟨ pπ^0| J^CC+_ρ |n  ⟩



 


        A real photon has two transverse polarisations.
        
        The matrix elements in Eq. <ref> can be written in terms of helicity amplitudes and the lepton coefficients in Eq. <ref>. The helicity amplitudes of the hadron current are defined with three indices: the helicity of the incident nucleon (λ_1), the helicity of the outgoing nucleon (λ_2), and the polarisation of the gauge boson (λ_k):
        
                
    F̃_λ_2, λ_1^λ_k = ⟨ Nπ| e^ρ_λ_k (1/2W) J^V_ρ |N  ⟩ ,
    G̃_λ_2, λ_1^λ_k = ⟨ Nπ| e^ρ_λ_k (1/2W) J^A_ρ |N  ⟩,

        Therefore, there are 16 helicity amplitudes to describe the vector current and 16 helicity amplitudes to describe the axial-vector current for weak CC interactions. For electron scattering and NC channels, the gauge boson has three polarisations; therefore, there are 12 helicity amplitudes for the electromagnetic current. For photon scattering, there are two polarisations and thus 8 helicity amplitudes.


        
        The hadronic currents and helicity amplitudes involve several form factors. The most sophisticated model to describe these form factors is the meson dominance (MD) model, based on the effective Lagrangian of quantum field theory as introduced by Sakurai and Gell-Mann in Refs. <cit.>. The MD model explains the interaction between leptons and nucleons through meson exchange with analogous quantum properties. The form factors can then be expressed in terms of the meson masses (m_j), the ratio of coupling strengths between the gauge boson and the meson, and between the meson and the nucleon (a_j), summing over all possible n mesons:              

    FF_N(k^2)= ∑_j=1^n a_j m^2_j/m_j^2 - k^2

        where a list of vector-mesons and axial-mesons is given in Table <ref>.
                
        
        There are two shortcoming from the form factor model; firstly, the FF_N in eq: <ref> does not have analytic properties and does not obey the unitarity condition. secondly, the asymptotic behaviour of the form factor differs from the asymptotic scaling behaviours predicted by QCD. However these properties can be imposed to the MD model.
        
         Imposing the conditions bring us to a complex system of equations for the coupling constant ratios, making it difficult to find solutions in the general case. However, in Ref. <cit.>, an equivalent system of equations is derived for the coupling constant ratios, with coefficients that are simply even powers of the corresponding vector-meson masses. In principle, one can find solutions to these equations even in the general case, referred to as linear superconvergence relations:
        
    ∑_j=1^n m^2_j a_j    =0,
    ∑_j=1^n m^4_j a_j    =0,
       ⋮
    ∑_j=1^n m^2(m-1)_j a_j    =0,

        where m≤ n shows the asymptotic behaviour of the form factors:
        
    FF_N(k^2)_|k^2| →∞∼ k^-2m

                
            
        


§ RESONANCE PRODUCTION

                



  
  The general form of helicity amplitudes was calculated in my previous papers <cit.>, and these are reproduced in Appendix <ref> and <ref>. The 16 vector helicity amplitudes (F̃_λ_2, λ_1^λ_k) are presented in <Ref>, and the 16 axial-vector helicity amplitudes (G̃_λ_2, λ_1^λ_k) are shown in Table <ref>. The definitions include several factors, such as Clebsch-Gordan coefficients for different SPP channels, branching ratios for each resonance decaying into a single pion, and Breit-Wigner parametrisation for the broad resonance states. These factors are common across all models. However, the key part of the definition lies in the resonance production amplitudes, which depend on the specific model chosen. The production amplitudes are calculated from the Feynman diagrams shown in Fig. <ref>. The goal of this section is to calculate these amplitudes, which are defined as follows:
        
    
    f^(V,A)_-3   =    -1/2W⟨ R, 3/2| e^ρ_R  J^(V,A)_ρ |N, 1/2  ⟩
    
                f^(V,A)_-1   =    -1/2W⟨ R, 1/2| e^ρ_R  J^(V,A)_ρ |N, -1/2  ⟩
    
                f^(V,A)(λ)_0+   =   1/2W√(-k^2)/|k|⟨ R, 1/2| e^ρ_R  J^(V,A)_ρ |N, 1/2  ⟩

Here f^(V,A) denotes the vector and axial-vector production amplitudes and J^(V,A) represent hadron currents. 


                

        
        

    

 §.§ The first and second resonance regions

     In the first and second resonance regions, the produced resonances correspond to the first four states listed in Table <ref>. These resonances have spins of either 1/2 or 3/2. The hadronic currents associated with these resonances can be expressed as follows:  

    ⟨ R(p)| J^ρ_3/2 |N(p_1)  ⟩   =   ψ̅_α(p)Γ^αρ_3/2u(p_1,s_z)
    ⟨ R(p)| J^ρ_1/2 |N(p_1)  ⟩   =   u̅(p)Γ^ρ_1/2u(p_1,s_z) .
   
          
In these equations, u(p) represents the Dirac spinor for spin-1/2 resonances, while ψ_α(p) denotes the Rarita–Schwinger spinor for spin-3/2 resonances. The operators Γ^αρ_3/2 and Γ^ρ_1/2 encapsulate the interaction dynamics and depend on the specific form of the current J^ρ, as well as the quantum numbers of the resonances involved. Given that the dynamics governing resonances with spin 1/2 and 3/2 differ significantly, I will discuss these cases separately to provide a clear understanding of their respective contributions:
         
       
       
        
        
        
         
       
        
      
  
        

  §.§.§ Resonances with spin 3/2 

        In the first and second resonance regions, there are two resonances with spin 3/2: the P_33 (Δ) resonance, which has positive parity, and the D_13(1520) resonance, which has negative parity. Their corresponding interactions are defined as follows:
        
    Γ^αρ_3/2(P_33)    =   [𝒱^αρ_3/2 - 𝒜^αρ_3/2]γ^5
    Γ^αρ_3/2(D_13)    =   [𝒱^αρ_3/2 - 𝒜^αρ_3/2]
 
        where the vector (𝒱^αρ_3/2) and axial-vector (𝒜^αρ_3/2) components are given by:
        
    𝒱^αρ_3/2   =   𝒞_3^V/M(g^αρk̸ - k^αγ^ρ) + 𝒞_4^V/M^2(g^αρ k.p - k^α p^ρ) 
            + 𝒞_5^V/M^2(g^αρk.p_1 - k^α p_1^ρ) ,
    𝒜^αρ_3/2   =   𝒞_3^A/M(g^αρk̸ - k^αγ^ρ) + 𝒞_4^A/M^2(g^αρ k.p - k^α p^ρ)
            + 𝒞_5^A g^αρ + 𝒞_6^A/M^2k^α k^ρ .

        
        The vector and axial-vector helicity amplitudes for the P_33 and D_13 resonances can be derived from Eqs. <ref> and <ref>. Here, we present the explicit forms of these amplitudes for P_33 resonance:
    

    
    f^V_-3 (P_33)   =    -|k|/W1/√(2𝒩)[C_3^V W_+/M 
              + C_4^V/M^2 Wk_0 +  C_5^V/M^2 (Wk_0 -k^2 )],
    
             f^V_-1 (P_33)   =    |k|/W1/√(2𝒩)[C_3^V/MW(k^2 - MW_+) 
              + C_4^V/M^2 Wk_0 +  C_5^V/M^2 (Wk_0 -k^2)],
    
            f^V(λ)_0+(P_33)    =    -|k|/W1/√(2𝒩)1/C_λ(|k|ϵ^0_λ - k_0ϵ^z_λ) [C_3^V/M  + C_4^V/M^2W  +  C_5^V/M^2 (W- k_0)]
 
         
    f^A_-3(P_33)    =    - √(𝒩/2) [C_3^A W_-/M + C_4^A/M^2 Wk_0 +  C_5^A ],
    
               f^A_-1 (P_33)   =    - √(𝒩/6) [C_3^A/M2|k| + k_0W_-/E_k + M - C_4^A/M^2 Wk_0 -  C_5^A ],
    
            f^A(λ)_0+(P_33)    =    -√(𝒩/3)1/C_λ[(C_3^A/M  + C_4^A/M^2W)(|kϵ^0_λ - k_0ϵ^z_λ)   -  C_5^A ϵ^z + C_6^A/M^2 |k|(|k|ϵ^0_λ - k_0ϵ^z_λ) ]
 
         and for D_13 resonance:
         
    f^V_-3 (D_13)   =    √(𝒩/2)[C_3^V/MW_- + C_4^V/M^2 Wk_0
              +  C_5^V/M^2(Wk_0 - k^2)],
    
               f^V_-1 (D_13)   =    √(𝒩/6)[C_3^V/M(W _-- 2k^2/E_k + M) 
             + C_4^V/M^2Wk_0  +  C_5^V/M^2(Wk_0 - k^2)  ],
    
            f^V(λ)_0+ (D_13)   =    √(𝒩/3)1/C_λ(|k|ϵ^0_λ - k_0ϵ^z_λ) [C_3^V/M  + C_4^V/M^2W  +   C_5^V/M^2 (W- k_0)]
 
        
    f^A_-3 (D_13)   =    -|k|/W1/√(2𝒩)[C_3^A W_+/M + C_4^A/M^2 Wk_0 +  C_5^A ],
    
             f^A_-1 (D_13)   =    |k|/W1/√(6𝒩) [C_3^A/M (W+M-2k_0) - C_4^A/M^2 Wk_0 -  C_5^A ],
    
            f^A(λ)_0+ (D_13)   =    |k|/W1/√(3𝒩)1/C_λ[(C_3^A/M  + C_4^A/M^2W)(|kϵ^0_λ - k_0ϵ^z_λ)   -  C_5^A ϵ^z + C_6^A/M^2 |k|(|k|ϵ^0_λ - k_0ϵ^3_λ) ]
 
        
        where 
        
    𝒩=√(M^2 + k ^2) + M/W.

        
         The coupling factors 𝒞^V_i (i= 3-5), represent either the CC vector form factors (C_i^V), the electromagnetic form factors (C_i^N, N=p,n) or the NC form factors (C̃_i^N, N=p,n) for the resonances with spin 3/2. Similarly, 𝒞^A_i, i= 3-6, represent either the CC axial-vector form factors (C_i^A) or NC form factors (C̃_i^A).
        
        

  §.§.§ Resonance with spin 1/2 

        In the first and second resonance regions, there are two resonances with spin 1/2: the P_11(1440) resonance, which has positive parity, and the S_11(1535) resonance, which has negative parity. Their corresponding interactions are defined as follows:
    
        
    Γ^ρ_1/2(P_11)    =   [𝒱^ρ_1/2 - 𝒜^ρ_1/2]
    Γ^αρ_1/2(S_11)    =   [𝒱^αρ_1/2 - 𝒜^αρ_1/2]γ^5
 
        where the vector (𝒱^αρ_1/2) and axial-vector (𝒜^αρ_1/2) components are given by:
        
    𝒱^ρ_1/2   =   ℱ_1^V/2M^2(k̸q^ρ - k^2 γ^ρ) + ℱ_2^V/2M(iσ^ραk_α) ,
    𝒜^ρ_1/2   =   ℱ_Aγ^ργ^5  + ℱ_P/M k^ργ^5 .

        The vector and axial-vector helicity amplitudes for the P_11 and S_11 resonances can be derived from Eqs. <ref> and <ref>. Here, we present the explicit forms of these amplitudes for P_11 resonance:
          
    f^V_-1 (P_11)   =    |k|/√(W(E_k +M))[ℱ_1^V - ℱ_2^V/W_+^2 k^2] 
    
            f^V(λ)_0+ (P_11)   =    -1/C_λ|k|/√(2W(E_k +M))(|k|ϵ^0_λ - k_0ϵ^3_λ) 
             1/W_+[ ℱ_1^V - ℱ_2^V]
 
         
    f^A_-1(p_11)    =     - ℱ_A √(E_k + M/W)
    
            f^A(λ)_0+ (P_11)   =    1/C_λ1/√(2W(E_k +M)) [ ℱ_A [(|k|ϵ^0_λ - k_0ϵ^3_λ) 
            + ϵ^3_λW_+ ] + ℱ_P|k|/M(|kϵ^0_λ - k_0ϵ^3_λ)  ]
 
        and for S_11 resonance:        
        
    f^V_-1 (S_11)   =    √((E_k + M)/W)[ℱ_1^V/W_+^2 k^2 - ℱ_2^V/W_+W_-] 
    
            f^V(λ)_0+ (S_11)   =    1/C^V_λ√(E_k + M/2W)(|k|ϵ^0_λ - k_0ϵ^3_λ)
           [-ℱ_1^V W_-/W_+^2 + ℱ_2^V/W_+]
 
         
    f^A_-1 (S_11)   =    -ℱ_A |k|/√(W(E_k +M))
    
                 f^A(λ)_0+(S_11)    =    -1/C_λ√(E_k +M/3W)[  ℱ_A( ϵ^0_λW_+)
            + ℱ_P/M(|k|ϵ^0_λ - k_0ϵ^3_λ) ]
 
         
    where ℱ^V_i (i= 1,2), represent either the CC vector form factors (F_i^V), the electromagnetic form factors (F_i^N, N=p,n) or the NC form factors (F̃_i^N, N=p,n) for the resonances with spin 1/2. Similarly, ℱ_A and ℱ_P represent for CC (NC) axial-vector form factors F_A, F_P (F^N_A, F^N_P).

                
    The vector form factors C_i^V and C_i^VN (F_i^V and F_i^VN) can be related to the electromagnetic form factors C_i^p,n (F_i^p,n) as is presented in Table <ref>.
                              

        

  §.§.§  Form factor parametrisation
 
       The Meson Dominance (MD) model, which aligns with Quantum Chromodynamics (QCD) calculations, effectively describes the nucleon form factor in both perturbative and non-perturbative regimes as highlighted in <cit.>. This model is particularly adept at addressing the hadron-quark transition region. In this work, we use a parametrization suggested by G. Vereshkov and N. Volchanskiy in Ref. <cit.>.
        
    C_α^(V,A)(k^2)   =   C_α ^(p)(0)/L_α(k^2)∑_k=1^Ka^(V,A)_α k m^2_k/m^2_k- k^2,   (α=3-5) 
    
                g_β^(V,A)(k^2)    =   g_β^(p)(0)/L_β(k^2)∑_k=1^Kb^(V,A)_β k m^2_k/m^2_k - k^2,   (β=1-2)
 
        Here, m^2_k = m^2_(ρ)k represents the ρ-meson mass for the Δ (P_33(1232)) resonance, while m^2_k = (m^2_(ω)k + m^2_(ρ)k)/2 pertains to the resonances in the second region, such as P_11(1440), D_13(1520), and S_11(1535). The values for m_(ρ)k and m_(ω)k are listed in Table <ref>.

The parameters a^(V,A)α k and b^(V,A)β k, which represent the coupling constant ratios of vector and axial-vector mesons to nucleons, are not free parameters. Instead, they are constrained by the linear superconvergence relations as expressed in Eq. <ref>:
        
                        
         For α=3:
                
    ∑_k=1^K    a_3 k =1,               ∑_k=1^K a_3 k  m^2_k=0,
       ∑_k=1^K    a_3 k  m^4_k=0.
 
                
                For α=4,5:
                
    ∑_k=1^K    a_α k =1,              ∑_k=1^K a_α k  m^2_k=0,
       ∑_k=1^K    a_α k  m^4_k=0,          ∑_k=1^K a_α k m^6_k=0.
 
                
                For β=1:
                
    ∑_k=1^K    b_1 k =1,               ∑_k=1^K b_3 k  m^2_k=0,
       ∑_k=1^K    b_1 k m^4_k=0.
 
                
                For β=2:
                
    ∑_k=1^K    b_2 k =1,              ∑_k=1^K b_2 k  m^2_k=0,
       ∑_k=1^K    b_2 k m^4_k=0,          ∑_k=1^K b_2 k  m^6_k=0.
 
                
        The logarithmic renormalisation L_α, β^(V,A)(k^2) is necessary to satisfy quark-hadron duality:

    L_α (β)^(A,V)(k^2) = 1 +  k_α (β)^(A,V)ln^n_α(β)(1 - k^2/Λ^2_QCD)
 
                where n_1 = n_3 ≃ 3, n_1>n_2 and n_5>n_3>n_4.
    This parametrisation provides a robust framework for analyzing the nucleon form factors, ensuring consistency with both theoretical predictions and experimental observations. The linear superconvergence relations play a crucial role in constraining the parameters, thereby enhancing the predictive power and reliability of the MD model in various resonance regions.         
                 
                
    

 §.§ The Third resonance regions

    The third resonance region is characterised by overlapping structures of multiple resonances, each contributing less significantly to the hadronic tensor compared to those in the earlier regions. Additionally, this region includes several resonances with spins greater than 3/2, which are inherently more complex to describe due to the increased number of degrees of freedom.

To avoid introducing an excessive number of parameters that would complicate the analysis of these overlapping resonances, I employ the quark harmonic oscillator model <cit.> to calculate the production amplitudes as described in Eq. <ref>. This approach simplifies the problem by reducing the complexity associated with higher-spin resonances.

In fact, these amplitudes have already been calculated in previous works <cit.>, where a simple dipole form factor was used for each resonance. The hadronic current for each resonance in this region is related to a dipole form factor, parameterised by two adjustable quantities, F_V(0) and M_V:

    F_V(k^2)    =    F_V(0)( 1- k^2/M_V^2)^-2(1 - k^2/M^2)^n/2
	

Here, F_V(k^2) represents the form factor as a function of the momentum transfer squared, k^2 . The parameter F_V(0) correspond to the form factor's value at zero momentum transfer. The factor (1 - k^2/M^2)^n/2 introduces additional flexibility in the form factor to account for the effects of higher resonances where n represents the number of oscillator excitations in the quark oscillator model.

By adopting this approach, the model remains manageable while still capturing the essential physics of the third resonance region. The use of a dipole form factor with adjustable parameters allows for a more practical and streamlined analysis, facilitating the study of this complex region without sacrificing accuracy.




§ NON-RESONANT BACKGROUND
 
    The helicity amplitudes of non-resonant background from the chiral-perturbation (ChPT) diagrams were calculated in the previous work <cit.> and are reproduced in Appendix <ref>. In this work, the dipole vector form factors of ChPT theory (low W) are changed to VMD form-factors as suggested by Lomon <cit.>. They are 
                
                  
    F_1^iv (k^2)   =    N/21.0317 + 0.0875(1 - k^2/0.3176)^-2/1 - k^2/0.5496 F_1^ρ(k^2) +  ℛ_ρ'm_ρ'^2/m_ρ'^2 - k^2 F_1^ρ(k^2) + (1 - 1.1192N/2 - ℛ_ρ') 	F_1^D(k^2),
    
                        F_2^iv (k^2)   =    N/25.7824 + 0.3907(1 - k^2/0.1422)^-1/1 - k^2/0.5362 F_2^ρ(k^2) + κ_ρ'ℛ_ρ'm_ρ'^2/m_ρ'^2 - k^2 F_2^ρ(k^2) + (κ_v - 6.1731N/2 - κ_ρ'ℛ_ρ') F_2^D(k^2),
    
                        F_1^is (k^2)   =    ℛ_ωm_ω^2/m_ω^2 - k^2 F_1^ω(k^2) +    ℛ_ϕm_ϕ^2/m_ϕ^2 - k^2 F_1^ϕ(k^2)  + (1 - ℛ_ω ) F_1^D(k^2),
    
                        F_2^is (k^2)   =    κ_ωℛ_ωm_ω^2/m_ω^2 - k^2 F_2^ω(k^2)  +  κ_ϕℛ_ϕm_ϕ^2/m_ϕ^2 - k^2 F_2^ϕ(k^2) (κ_s - κ_ωℛ_ω  - κ_ϕℛ_ϕ) F_2^D(k^2),
 
                
                where κ_v=3.793, κ_s=-0.12, m_ρ'=1.465 GeV, m_ω=0.78265 GeV and m_ϕ=1.020 GeV. The vector form-factors, F_1,2^ρ, F_1,2^ω and F_1,2^D, are given by:
                
    F_1^ℳ,D(k^2)   =   Λ^2_1,D/Λ^2_1,D - k̃^2Λ^2_2/Λ^2_2 - k̃^2, 
    
                      F_2^ℳ,D(k^2)   =   Λ^2_1,D/Λ^2_1,D - k̃^2(Λ^2_2/Λ^2_2 - k̃^2)^2,

                where ℳ= ρ and ω and Λ_1,D is Λ_1 for F_1,2^ℳ and Λ_D for F_1,2^D,
                
    F_1^ϕ(k^2)   =    F_1^ℳ(-k^2/Λ^2_1 - k^2)^3/2 ,     F^ϕ_1(0)=0, 
    
                      F_2^ϕ(k^2)   =    F_2^ℳ(μ^2_ϕ - Λ_1^2 k^2/μ^2_ϕΛ^2_1 - k^2)^3/2,

                where
                
    k̃^2 = k^2 ln[(Λ^2_D - k^2)/Λ_QCD^2 ]/ln(Λ^2_D/Λ^2_QCD)  .

                
                The non-resonant background of the MK model is extended to high W (W>1.4 GeV), using the Regge trajectory approach <cit.>. The propagators of the t-channel meson exchange ChPT diagrams are replaced by the corresponding Regge propagators (ReChi). The expressions of the Regge propagators for the meson trajectories of the π and ρ are the following:
                
    𝒫_π(t,s)    =    -α'_πΓ(- α_π(t) )(S/S^π_0)^α_π(t), 
    𝒫_ρ(t,s)    =    -α'_ρΓ(- α_ρ(t) )(S/S^ρ_0)^(α_ρ(t) - 1),

                where S^π_0 and S^ρ_0 are adjustable parameters. The Regge trajectory of π and ρ are as follows:
                
    α_π(t)    =    0.75(t- m_π^2),
    α_ρ(t)    =    0.53 + 0.85t.

                Following the Hybrid model's parametrisation <cit.>, the helicity amplitudes for non-resonant background will be:
                
    F̃ = cos^2 ϕ(W) F̃_ChPT + sin^2 ϕ(W)F̃_ReChi,

                where  F̃= F̃_λ_2, λ_1^λ_k and 
                
    ϕ(W) = π/2( 1- 1/exp (W-1.7)/0.1).

                F̃_ChPT are the helicity amplitudes for the ChPT background and F̃_ReChi are the modified helicity amplitudes with Regge propagators.

                
                This approach with the linear Regge trajectories (Eq. <ref>) is valid at high-energy and low-momentum transfer. To ensure that the model reproduces data at high momentum transfer, the trajectories are multiplied by a factor of (1 + 2.4Q^2/W^2)^-1, as suggested in Ref. <cit.>. The adjustable parameters in the Regge trajectory and VMD form factors are fit to the available exclusive electron scattering data in the next section.
 
                
                
                The non-resonant interactions follow the Hernandez et al. model <cit.> which is deduced from the chiral-perturbation (ChPT) theory Lagrangian of the pion-nucleon system, therefore, it is not reliable at high hadron invariant mass (W>1.4 GeV). However to  extend the model to high W (W>1.4 GeV), Regge trajectory approach <cit.> is applied, where the propagators of the t-channel meson exchange ChPT diagrams are replaced by the corresponding Regge propagators (ReChi). The expressions of the Regge propagators for the meson trajectories of the π and ρ are the following:
                
    𝒫_π(t,s)    =    -α'_πΓ(- α_π(t) )(S/S^π_0)^α_π(t), 
    𝒫_ρ(t,s)    =    -α'_ρΓ(- α_ρ(t) )(S/S^ρ_0)^(α_ρ(t) - 1),

                where S^π_0 and S^ρ_0 are adjustable parameters and the Regge trajectory of  the π and ρ are:
                
    α_π(t)    =    0.75(t- m_π^2),
    α_ρ(t)    =    0.53 + 0.85t.

                Following the Hybrid model's parametrisation <cit.>, the helicity amplitudes for non-resonant background will be:
                
    F̃ = cos^2 ϕ(W) F̃_ChPT + sin^2 ϕ(W)F̃_ReChi,

                where  F̃= F̃_λ_2, λ_1^λ_k and 
                
    ϕ(W) = π/2( 1- 1/exp (W-1.7)/0.1).

                F̃_ChPT is the helicity amplitudes for ChPT background and F̃_ReChi is the modified helicity amplitudes with Regge propagators.

                
                This approach with the linear Regge trajectories (equation <ref>) is valid at high Energy and low momentum transfer. To ensure that the model reproduces data at high momentum transfer, the trajectories are multiplied by a factor of (1 + 2.4Q^2/W^2)^-1, as suggested in reference <cit.>. The adjustable parameters in the Regge trajectory and VMD form factors are fit to the available exclusive electron scattering data in the next section.
 
                The helicity amplitudes of non-resonant background were calculated in the previous work <cit.> and they are given in Appendix <ref>. 
                
                
                
        

§ ANALYSIS OF EXCLUSIVE SPP DATA


Accurate determination of vector and axial-vector transition form factors, as well as non-resonant nucleon-pion production amplitudes, is essential for providing reliable predictions of SPP channels. These form factors involve several adjustable parameters, which can only be accurately defined through the use of experimental data.

        In this section, we conduct a comprehensive analysis of all available exclusive SPP data on proton and neutron targets. This joint analysis aims to determine the Q^2 dependence of all form factors described in the previous section, providing a detailed understanding of the underlying interaction dynamics. The analysis incorporates a wide range of data sources to ensure robust and accurate form factor determinations, considering various experimental conditions and measurements.
        
        The vector form factors in weak interactions are closely linked to the electromagnetic form factors. As a result, electron and photon scattering data on nucleon targets are invaluable for determining the vector form factors. Conversely, pion scattering data are essential for fitting the axial-vector form factors. By combining electron, photon, pion, and neutrino scattering data, we can achieve a comprehensive and precise extraction of both vector and axial-vector form factors. This integrated approach ensures coverage of the entire kinematic range relevant for Single Pion Production (SPP) processes. A list of resonances used in the MK model is given in <Ref>.
         
                


 §.§ Photo-Nucleon and Electro-Nucleon Single Pion Productions


Electron and photon scattering data are crucial for this analysis as they allow for probing the vector current operators using monochromatic incident beams across a wide range of kinematic regions. 
Given the limitations of theoretical predictions in accurately capturing the Q^2-dependence of form factors, particularly in the transition region, this experimental data is crucial for guiding the selection of optimal functions for various vector form factors, which are then applied to axial-vector form factors.  The interactions of electrons and photons with protons are described by the following reactions:
    
    e + p → e p  π^0     ,             e + p →  e n  π^+
    γ + p →  p  π^0     ,            γ + p →  n  π^+
       
Similarly, the interactions of electrons and photons with neutrons are:
    
    e + n →  e n  π^0     ,             
            e + n →  e p  π^-  
    γ + n →  n  π^0     ,            γ + n →  p  π^-.

        
        In previous work presented in Ref. <cit.>, the model for electron-proton interactions and the electromagnetic MD form factors were tested for their ability to describe excited protons within the transition region. In that study, the MD form factors were fitted to all available electron-proton scattering data in the two channels described by Eq. <ref>. The results demonstrated that the model could predict the data smoothly and accurately across these kinematic ranges.
        
        In the current study, we adopt a similar analysis methodology but extend it by including photo-proton scattering data in Eq. <ref>. This inclusion is particularly important for addressing the form factors at very low Q^2, a region that is less accessible through electron scattering alone. Photo-nucleon interactions provide complementary information that helps fill in these gaps, ensuring a more comprehensive understanding of the form factors across all relevant kinematic regions.

This study expands the investigation by incorporating data on surface plasmon polaritons (SPPs) induced by electrons and photons on neutron targets, as shown in Eq. <ref>. The recent analysis of π^- production with an electron beam has made this analysis possible for the first time. This advancement is crucial for determining the form factors of excited neutrons, a subject that has been less studied compared to protons.

The form factors obtained from interactions involving excited protons and neutrons are anticipated to be identical for isospin 3/2 resonances due to isospin symmetry considerations, as discussed by Lalakulich et al. <cit.>. However, for isospin 1/2 resonances, differences can emerge, reflecting the distinct internal dynamics of these states. Understanding these discrepancies is essential for a comprehensive understanding of nucleon resonance behavior.
  
  
        
        
                
                
               
                
                
                   
                
                
                

 §.§ Pion-Nucleon and Neutrino-Nucleon Single Pion Productions


        Neutrino interactions with weak current can provide information for both vector and axial-vector currents. The SPP neutrino interactions consist of CC and NC channels for neutrino and anti-neutrino beams.        
     The charged current SPP channels are:
        
    ν_l + p →l^- p  π^+     ,            ν̅_l + n →  l^+ n  π^-
    ν_l  +n → l^- n  π^+     ,            ν̅_l + p →  l^+ p  π^-
    ν_l + n → l^- p  π^0     ,            ν̅_l + p →  l^+ n  π^0
       
        and the neutral current SPP channels are:
        
    ν_l + p →ν_lp  π^0     ,            ν̅_l + p →ν̅_l p  π^0
    ν_l + p →ν_ln  π^+     ,            ν̅_l + p →ν̅_l n  π^+
    ν_l + n →ν_ln  π^0     ,            ν̅_l + n →ν̅_l n  π^0
    ν_l + n →ν_lp  π^-     ,            ν̅_l + n →ν̅_l p  π^-
  
Experimental knowledge of the neutrino-nucleon cross section for single pion production in the GeV neutrino energy range, which is crucial for current and planned neutrino oscillation experiments, remains limited. Most of the existing data originate from low-statistics bubble chamber experiments conducted with hydrogen or deuterium targets and involve muon (anti-)neutrino beams.

Unlike electron scattering experiments, where full kinematic measurements are typically available, neutrino data are generally reported as integrated cross sections or one-dimensional differential cross sections. The only data available on hydrogen targets are from CC neutrino and anti-neutrino channels measured by the BEBC experiment, which utilised very broad neutrino and anti-neutrino beams. Consequently, the most relevant weak interaction data for this study is the Q^2-differential cross section obtained from the BEBC experiment.    

There are more data on deuterium targets from the ANL and BNL bubble chamber experiments with neutrino beams in the few GeV range, which are more relevant for this study. In fact, all single pion production (SPP) models have used the ANL and BNL data to fit the axial form factor, assuming that nuclear effects on deuterium are negligible. However, recent measurements with an electron beam <cit.> indicate that the cross sections for hydrogen and deuterium differ by approximately 15%.

As a result, I am not using the direct cross-section measurements on deuterium targets, as the nuclear effects are not negligible. Nevertheless, I assume that the ratio of the cross-section data across different channels will effectively cancel out the nuclear effects. Consequently, the already limited neutrino data will become even more restricted.

Fortunately, pion scattering data on hydrogen can provide additional information for determining the axial form factor. The weak interaction, which includes both vector and axial vector currents, only has contributions from the axial vector current at very low 
 Q^2∼0. This phenomenon is explained by the conserved vector current (CVC) relation. In this region, the axial vector current is connected to elastic pion-nucleon interactions via the partially conserved axial current (PCAC) relation:

        
    dσ/dWdQ^2(ν_μ p →μ^- p π^+) |_Q^2=0   =   σ(π^+ p →π^+ p)
    dσ/dWdQ^2(ν̅_μ p →μ^+ p π^-) |_Q^2=0   =   σ(π^- p →π^- p)

        
Therefore, the axial form factors are fitted to the pion scattering data at very low Q^2. The Q^2-dependence of the axial form factors can only be determined using neutrino-nucleon single pion production (SPP) data, which is very scarce.
        
        The model includes 88 parameters to parametrise all the transition form factors for resonance production and non-resonant background. These parameters are fit to the experimental data by using the Minuit2 minimiser and the best fit result, with a reduced χ^2=3.05, are discussed below:
                

 



        
        

 
                
                

 §.§ VMD form-factors for spin-1/2 resonances

                There are two spin-1/2 resonances in the second resonance region i.e P_11(1440) and S_11(1535) with the form-factors in equation <ref>. Five mesons in the <Ref> are taken into account and therefore  K=5 for spin 1/2 resonances. As a result there will be two free parameters in b_1 k as they satisfy three superconvergence relations (equation <ref>) and one free parameter will be allowed for b_2 k due to the four superconvergence relations in equation <ref>. The QCD-scale in the logarithmic renormalisations (L^(p)_β) can vary between Λ∈ [0.19-0.24] GeV and n_β=1=3, n_β=2=2.
                All adjusted parameters related to spin-1/2 resonances are reported in <Ref>. 
                
                
                

 §.§ dipole form-factors for resonances in the third region

                The most characteristic feature of the third region is overlapping structures of several resonances with less overall contribution to the hadronic tensor. To avoid having many parameters (as we do with VMD form-factors) in a rather small phase space we use the Rein-Sehgal model with a simple dipole form-factor for each resonance. The hadron current of each resonance in the third resonance region will be related to a dipole form-factor with two adjustable parameters, F_V(0) and M_V:
                
    F_V(k^2)    =    F_V(0)( 1- k^2/M_V^2)^-2(1 - k^2/M^2)^n/2
	
                where n is the number of oscillators from the Rein-Sehgal model. The number of oscillators and parameters F_V(0) and M_V are reported in <Ref> for resonance in the third region.
                
                
                

 §.§ The VMD form-factors for non-resonant interactions

                The update for non-resonant interaction was presented in the previous section <ref>.
                The adjustable parameters for non-resonant form-factors are defined in equations (<ref>-<ref>) and the adjustable parameter for Regge propagator are defined in equation (<ref>). All parameters related to non-resonant interaction and their best fits are reported in <Ref>.
                
                
                
                
                

§ RESULT AND COMPARISON WITH EXPERIMENTAL DATA

                
                In this section we present results from the MK model and its comparison with electron scattering data. The experimental data is the measurements of the virtual photon cross section dσ_T/dΩ^∗ _π   + ϵdσ_L/dΩ^∗ _π[the virtual photon cross section is defined in the previous work <cit.>] <cit.>.
                The selected results are chosen to cover a broad range of Q^2 and W for two channels with pπ^0 and nπ^+ final states hadrons in <Ref> and <Ref> within 1σ error band. To have an accurate estimate for the systematic uncertainty, the correlations between parameters are included to avoid double counting or cancellation of the systematic effects. 

              
                
                 We also present results from other existing models:
                 
                
  * Unitary Isobar Model - MAID2007 <cit.> is the latest version of unitary isobar model for partial wave analysis on the world data of pion photo and electroproduction in the resonance region (Q^2<5.0   (GeV/c)^2). 
                
  * Dynamical coupled-channels (DCC) model <cit.>. The DCC analysis is on the limited pion photo and electroproduction data (Q^2<3.0   (GeV/c)^2).
                
  * Hybrid model <cit.> which is valid at low Q^2. The Rarita-Schwinger formalism plus Regge approach are used for the resonant and non-resonant interactions like the MK model, however the form-factors in both models are different.
                
                Note that the valid region for the above models is smaller than the MK model, therefore the missing results from the above models in the <Ref> is due to the model's limitations.

                
                <Ref> shows the cross-section for e p → e n π^+ channel and <Ref> shows the cross-section for e p → e p π^0 channel over a broad range of Q^2 (∈[0.16-6.00] (GeV/c)^2) in terms of the pion polar angle in the resonance rest frame. Plots at low invariant mass (W<1.15 GeV) are dominated by non-resonant background contributions and at higher W, the Δ resonance has a dominant contribution in plots with W ∈ [1.18- 1.28 GeV].  <Ref> show that all models have good agreement with data at low invariant mass (W<1.28 GeV) for both channels.

                
                 At still higher invariant mass (W ∈ [1.4- 1.6 GeV]) the next three resonances (P_11(1440), D_13(1520), and S_11(1535)) have the dominant effects in the second resonance region. The MAID results show a good agreement at low Q^2 (two top plots in <Ref>), however, at higher Q^2, it favours data from e p → e p π^0 channel over e p → e n π^+ channel. The second plot of <Ref> shows that the Hybrid model doesn't predict the backward pion well. Also, the model under-predict data in the dip region between the first and the second resonance region, i.e. W ∈ [1.28- 1.4 GeV], which is also visible in <Ref>.

                 
                 The rest of resonances in the <Ref> contribute to the third resonance region (W > 1.6 GeV) and only one measurement for e p → e n π^+ channel exists. This is presented in the third plot of <Ref> where the MAID result doesn't show good agreements with data.

                 
          
             
                Some of the datasets in <Ref> are over a large range of the hadron invariant mass (W) which will allow us to show the cross-section in different resonance regions as it is presented in <Ref> (<Ref>) for low (medium) Q^2. Comparing the two channels in each figure reveals the different W distribution which is mainly due to the different Clebsch-Gordan coefficients for isospin 3/2 and 1/2 resonances. The first resonance region is dominant in the e p → e p π^0 channel due to larger Clebsch-Gordan coefficient for the Δ (isospin 3/2) resonance. The second resonance region is more pronounced for the e p → e n π^+ channel because this region is populated mainly by isospin 1/2 resonances that have larger Clebsch-Gordan coefficients for this channel.

                
                All comments about the model comparisons in the first and the second resonance regions from the  <Ref> are visible in <Ref> (the third resonance region is not visible here). The DCC prediction for the e p → e n π^+ channel at low Q^2 (in <Ref>) and forward pion angle shows discrepancies with data in the second resonance region.  
                
            
                
                
                

§ CONCLUSION

         
The Single Pion Production (SPP) model presented in this work is meticulously designed to enhance our understanding and interpretation of neutrino measurements:


  * Comprehensive Coverage: The model encompasses all SPP channels and is applicable across the entire kinematic range relevant to the energy spans of all neutrino experiments.

  * NC Weak Interaction: By employing a unified model of electromagnetic and weak interactions along with isospin symmetry, the model delivers precise predictions for NC neutrino interactions, even in the absence of extensive data. This capability is particularly crucial for water Cherenkov detectors such as T2K and Hyper-K, where neutral current π^0 production constitutes a significant background.


  * Addressing Low Q^2: The model effectively addresses the low Q^2 region, where discrepancies with data currently exist. This is achieved through the use of photon scattering data and the Conserved Vector Current (CVC) hypothesis for the vector current, as well as pion scattering data and Partially Conserved Axial Current (PCAC) for the axial current.


  * Addressing High Q^2: Neutrino-nucleon-resonance vertices are parameterized with phenomenological form factors that possess analytic properties and adhere to the unitarity condition. These form factors are asymptotically consistent with QCD calculations, ensuring the model's validity at high Q^2.


  * Addressing High W: The model extends its validity to high W by utilizing Regge phenomenology, replacing t-channel Feynman propagators in non-resonant interactions with the corresponding Regge trajectories.


  * Integration with Event Generators: The model is designed for seamless integration into nuclear theory frameworks and neutrino event generators employed by neutrino experiments, ensuring accurate and consistent predictions.


  * Systematic Uncertainty Control: The calculations are conducted with stringent control over systematic uncertainties stemming from experimental errors, model dependencies, parameterization choices, and extrapolation to higher energies. This rigorous approach is crucial for the reliability and precision of neutrino measurements.


        
    By addressing these key areas, the SPP model provides a robust framework for enhancing the accuracy and reliability of neutrino and anti-neutrino  interaction studies, paving the way for more precise and insightful experimental results and CP discovery.  
                
                
              
               
        
               
               
               
                
                
                
                





§ HELICITY AMPLITUDES FOR THE RESONANT INTERACTIONS

    The helicity amplitudes of the vector and axial-vector currents are given in <Ref> and <Ref>, where f^V,A(R) is the production amplitudes and D(R) is the decay amplitude:

                    
    𝒟^j(R)=⟨ Nπ,λ_2|R λ_R ⟩   = σ^D C_Nπ^j√(χ_E)κ C_Nπ^I f_BW

                where f_BW(R) is the Breit-Wigner amplitude:
                
    f_BW(R) = √(Γ_R/2π)( 1/W- M_R + iΓ_R/2 )

                where 
    Γ_R = Γ_0 (|𝐪(W)|/|𝐪(M_R)|)^2l+1,

                and 
                
    κ= ( 2π^2 W^2/M^2 . 2/2j+1 1/|𝐪|)^1/2,

                Γ_0, M_R, σ^D and χ_E are given in <Ref> and C^I_Nπ are the isospin Clebsch-Gordan coefficients given in <Ref>.
                
                The explicit form of the d^j_λ,μ(θ) functions for j=l+1/2 are:
                
    d^j_1/21/2    = (l+1)^-1cosθ/2 (P'_l+1 - P'_l)
    
                    d^j_-1/21/2   = (l+1)^-1sinθ/2 (P'_l+1 + P'_l)
    
                    d^j_1/23/2    = (l+1)^-1sinθ/2 (√(l/l+2)P'_l+1 + √(l+2/l) P'_l)
    
                    d^j_-1/23/2   = (l+1)^-1cosθ/2 (-√(l/l+2)P'_l+1 + √(l+2/l) P'_l)

                where P_l are Legendre polynomials and P'_l= dP_l/dcosθ.
                
    

                
  
           The helicity amplitudes of the vector current are given in <Ref>, where 
                
    ℱ_i = K^V_i . F_i  (i=1, ...,6)

                where
                
    F_1   = V_1 + (V_3-V_4)(qk)/W_- +  V_4W_- - V_6 k^2/W_- ,
    
                    F_2   =-V_1 + (V_3-V_4)(qk)/W_+ +  V_4W_+ - V_6 k^2/W_+ ,
    
                    F_3    = V_3 - V_4 + V_25/W_+  , 
    
                    F_4    = V_3 - V_4 - V_25/W_-  , 
    
                    F_5    = V_1(W_+^2 - k^2)/2W - V_2(qk)(W_+^2 - k^2 + 2WW_-)/2W 
       + (V_3-V_4)(W_+q_0 - (qk)) + V_4(W_+^2 - k^2)W_-/2W 
       - V_5(qk)k_0 - V_6 (W_+^2 - k^2)W_-/2W + q_0 V_25 ,
    
                    F_6    =-V_1(W_-^2 - k^2)/2W + V_2(qk)(W_+^2 - k^2 + 2WW_-)/2W 
       + (V_3-V_4)(W_-q_0 - (qk))  + V_4(W_-^2 - k^2)W_+/2W 
       + V_5(qk)k_0 - V_6 (W_-^2 - k^2)W_+/2W - q_0 V_25 ,

                and V_i (i=1, ...,6) are presented in <Ref> where s, u, t are invariant Mandelstam variables:
                
    s = (p_2 + q)^2 = (p_1 + k)^2 = W^2 ,
    
                    t = (k-q)^2 ,   and   u=(q-p_1)^2 .

                 K^V_i are given in reference <cit.>:
                
    K_1^V    = W_- O_1+
    
                    K_2^V    = W_+ O_1-
    
                    K_3^V    = q^2 W_+ O_2-     K_4^V    = q^2 W_+ O_2-
    
                    K_5^V    = 1/O_2+
    
                    K_6^V    = 1/O_2-

                where
                
    O_1±   =    [(W^2_± - k^2)(W^2_± - m_π^2)]^1/2/2W
    
                    O_2±   =    [(W^2_± - k^2)/(W^2_± - m_π^2)]^1/2.

                
                  
              
                
                
       
       
       

 
 plainnat
               
                