]>PLB29863S0370-2693(13)00981-710.1016/j.physletb.2013.12.006The AuthorsTheoryFig. 1Background ((a)–(g)) and resonant (h) contributions to the scattering amplitude.Fig. 2Comparison of the Q2-dependencies of the vector (V) and axial (A) FFʼs, between the BLM [9] and HNV [8] models. The FF in C5V(Q2) for the HNV model, is also shown for comparison.Fig. 3The Q2 dependency of the double ratio CiA(Q2)/C5A(Q2) (i=4,6) for BLM and HNV models.Fig. 4Total cross section of νp→μ−pπ+ within the BLM model described in the text. Results using Sachs (thick lines) and Normal Parity (thin lines) decompositions, are shown for the B, R and B+R contributions to the scattering amplitude.Fig. 5The R and R+B contributions to the cross section in the BLM [9] and HNV [8] models for different values of the axial constant C5A(0). Also results for the HNV model where Δ×21/2 means that for the πNΔ strong coupling constant we use fπNΔ/mπ×2, are shown both for the same C5A(0)=0.87 and for C5A(0)=1.2.One pion production in neutrino–nucleon scattering and the different parameterizations of the weak N→Δ vertexC.BarberoabG.López CastrocA.Marianoab⁎aDepartamento de Física, Universidad Nacional de La Plata, C. C. 67, 1900 La Plata, ArgentinaDepartamento de FísicaUniversidad Nacional de La PlataC. C. 67La Plata1900ArgentinabInstituto de Física La Plata, CONICET, 1900 La Plata, ArgentinaInstituto de Física La PlataCONICETLa Plata1900ArgentinacDepartamento de Física, Centro de Investigación y de Estudios Avanzados, Apdo. Postal 14-740, 07000 México, DF, MexicoDepartamento de FísicaCentro de Investigación y de Estudios AvanzadosApdo. Postal 14-740México, DF07000Mexico⁎Corresponding author.Editor: W. HaxtonAbstractThe N→Δ weak vertex provides an important contribution to the one pion production in neutrino–nucleon and neutrino–nucleus scattering for πN invariant masses below 1.4 GeV. Beyond its interest as a tool in neutrino detection and their background analyses, one pion production in neutrino–nucleon scattering is useful to test predictions based on the quark model and other internal symmetries of strong interactions. Here we try to establish a connection between two commonly used parameterizations of the weak N→Δ vertex and form factors (FF) and we study their effects on the determination of the axial coupling C5A(0), the common normalization of the axial FF, which is predicted to hold 1.2 by using the PCAC hypothesis. Predictions for the νμp→μ−pπ+ total cross sections within the two approaches, which include the resonant Δ++ and other background contributions in a coherent way, are compared to experimental data.KeywordsNeutrino scatteringΔ resonanceEffective modelsNeutrino oscillation experiments search a distortion in the neutrino flux at a detector positioned far away (L) from the source. The comparison of near and far neutrino energy spectra, leads to information about the oscillation probability P(νi→νj)=sin22θijsin2Δmi,j2L2Eν, and then about the θij mixing angles and Δmi,j2 mass squared differences. Currently, new high quality data are available from MiniBoone [1], SciBoone [2] and new data are expected from Minerva [3] experiment, which is fully devoted to cross sections measurements of neutrino–nucleus interactions.The charged current quasielastic scattering (CCQE) νln→l−p reaction, with the nucleon bounded in the nucleus target, is usually used as signal event. Although the neutrino energy is not directly measurable, it can be reconstructed from the reaction products through two body kinematics (exact only for free nucleons). However, competition with other processes could lead to a possible misidentification of the arriving neutrinos. In fact:•Disappearance searching experiments νμ→νx (like SciBoone) use νμn→μ−p CCQE reaction to detect an arriving neutrino and reconstruct its energy. However, the determination of the neutrino energy Eν could be wrong due to a fraction of background events νμp→μ−pπ+ (CC 1π+) that can mimic a CCQE signal if the pion is absorbed in the target and/or is not detected.•In νμ→νe appearance experiments (like MiniBooNE) one detects νe in an (almost) pure νμ beam. The neutral current reaction νμN→νμNπ0, N=n,p (NC 1π0) can become a source of background for the signal event νen→e−p when one of the photons in the π0→γγ decay escapes detection leading to a misidentification of the electron and neutral pion [4]. Therefore, a precise knowledge of the cross sections of these elementary11We refer to neutrino–nucleon scattering as the elementary process that underlies neutrino–nucleus scattering. 1π processes in charged (CC) and neutral current (NC) neutrino–nucleon scattering is a prerequisite for the proper interpretation of the experimental data. This will allow to make simulations in event generators to eliminate fake events coming from 1π processes to get more realistic countings of quasielastic (QE) events. We will focus in this work on the CC 1π production, which is the channel that enables to fit the axial form factor of our interest.Several models have been developed over the last thirty years to evaluate the corresponding elementary cross sections [5–14]. The scattering amplitude in all these models always contains a resonant term (R) in the πN system, described by the Δ(1232)-pole contribution in Fig. 1(h) and (in some cases) by higher mass intermediate resonances, plus a background (B) term describing other processes, as shown in Fig. 1(a)–(f) (the cross-Δ contribution in Fig. 1(g) can also be included in this background) leading to πN final states. Therefore, the scattering amplitude can be written as: M=MB+MR. Since we are including only the Δ(1232) as the main resonance contribution, we will compare with data by applying a cut in the πN invariant energies at 1.4 GeV.The difference between all these models stem mainly from the treatment of the vertexes and the propagator used to describe the Δ resonance and from the consideration (or not) of the background and its interference with the resonant contribution. In order to compare the Δ baryon contribution (both to B and R amplitudes) between different approaches we need to carefully analyze both, the Δ propagator and the πNΔ and WNΔ vertexes. The propagator can be written as [15](1)Gαβ(pΔ)=p̸Δ+mΔpΔ2−mΔ2{−gαβ+13γαγβ+23mΔ2pΔαpΔβ−13mΔ(pΔαγβ−γαpβ)−b(p̸−mΔ)3mΔ2[γαpΔβ−(b−1)γβpΔα+(b2p̸Δ+(b−1)mΔ)γαγβ]}, with the parameter b=A+12A+1, where A is an arbitrary parameter related with the contact transformations upon the Δ field. Since the physical amplitude should be independent of A, the strong and weak vertexes involving the Δ in Fig. 1(h) should also depend on the A-parameter in order to cancel the A-dependence of the corresponding amplitude. In this case both the πNΔ and WNΔ vertexes should fulfill these requirements and thus a set of A-independent reduced Feynman rules can be obtained [15]. Equivalently, one may choose a common value for A in the Feynman rules involving the Δ particle to built the amplitude. In Ref. [9] the value A=−1/3 was assumed, coinciding the rules with those in Ref. [15]. However, a common mistake is to use the value A=−1/3 which simplifies the vertices simultaneously with A=−1, which simplifies the propagator. This procedure is inconsistent, leading to non-physical expression for the amplitude.The vector FFʼs entering the WNΔ vertex can be fixed from the electromagnetic γNΔ process by assuming the CVC hypothesis. No analogous symmetry allows to fix the axial-vector FFʼs. Among the axial FFʼs, the most relevant role is played by C5A(0) or, equivalently, D1(0)=3C5A(0), depending on the assumed form for the axial vertex at zero momentum transfer. A reference value is provided by the PCAC hypothesis being C5A(0)=1.2 [16]. The value C5A(0)∼1 [6] is obtained within quark models (QM); however, it is well known that it corresponds to a ‘bare’ estimate that should be dressed by the pion cloud contribution. This dressing can be done dynamically as in [7] where the QM value is enlarged around 35%, or in an effective way by fitting the experimental data for the νp→μ−pπ+ differential cross section [6]. Data on weak pion production on nucleons are scarce and not much precise being the most used those obtained by experiments at Argonne National Laboratory (ANL) [17] and/or Brookhaven National Laboratory (BNL) [18]. The different values assumed or obtained are: C5A(0)=1.20 [5], 1.38 [6,7], 0.867 [8,11], 1.35 [9,10], 1.17 [12,13], 1.00 [14]. These different yields depend upon the treatment of the Δ(R+B) contributions to the amplitude as modeled by different authors. For example in Ref. [5] the total amplitude is built at tree level by using a complex pole only in the denominator of the Δ propagator, which is inconsistent with the choice of the W or πNΔ vertexes, as it was mentioned above; at the same time, the contributions of Figs. 1(d)–1(g) were not included in the background. In Ref. [7] the inclusion of pion cloud dynamical effects (PCE) is achieved through a T-matrix approach and all terms are included in the B amplitude, but the same vertex-propagator consistency problems for the Δ are present. In Refs. [8,11,14] the model of Ref. [5] is extended by adding terms in the B amplitude guided by the effective SU(2) σ-model Lagrangian, but consistency problems (A=−1/3 in the vertexes and A=−1 for the propagator) persist; a value for C5A(0) close to the QM and below the PCAC one is obtained in this case. In Refs. [9,10] the problem of consistency of the Δ vertex-propagator is solved together with the question of including the Δ finite width effects, and the value obtained for C5A(0) is close to the one corresponding to PCE dressed effects. Finally, in Refs. [12,13], where the production and decay of the Δ resonance are separated in the amplitude, a value close to PCAC is obtained.Apart from the consistency problems in treating the Δ resonance, the treatment of the Δ instability (constant or energy-dependent width) and the adopted convention for the FFʼs, the above mentioned models only differ in the way the WNΔ vertex is parameterized. In view of the different values obtained for C5A(0), it would be important to compare these parameterizations. Let us consider here the amplitude for the elementary neutrino–nucleon CC 1π production process (νp→μ−pπ+, νn→μ−nπ+, νn→μ−pπ0). From our Ref. [9], hereafter called BLM, we have the total amplitude(2)Mi=−GFVud2u¯(pμ)γλ(1−γ5)u(pν)u¯(p′)Oiλ(p,p′,q)u(p),i=B,R with GF=1.16637×10−5 GeV−2, |Vud|=0.9740 (p, pν, pμ, k, p′) being the set of 4-momenta of the initial nucleon, neutrino, muon, pion and final nucleon, respectively, and q=pμ−pν (Q2≡−q2) being the momentum transferred from leptons to hadrons. We adopt here the metric and conventions of Bjorken and Drell (BD) [19] and for the hadronic currents Jiλ a vector-axial structure (Jiλ≡Viλ−Aiλ). By assuming the CVC hypothesis in the vector sector, the axial FF at Q2=0 can be fixed from the fit to the d〈σ〉/dQ2 differential cross sections; the strong and other weak couplings involved in OBλ(p,p′,q) and ORλ(p,p′,q) are those of the BLM approach. Here we introduce the unstable character of the Δ by through the complex mass scheme (CMS) [20] consisting in the replacement mΔ→mΔ−iΓΔ/2 everywhere the Δ mass appears in the propagator, with ΓΔ a constant. This procedure avoids the inclusion of ad-hoc corrections to the vertices in order to restore gauge invariance (which occurs if the CMS is adopted only for the denominator of the propagator) in processes where a photon is radiated from the Δ resonance [15].Next, we compare the WNΔ vertex, defined below as Wνμ≡WνμV+WνμA, in different prescriptions. Previously, in BLM and [7,10,21–23] a covariant multipole decomposition analogous to the Sachs choice [24] of nucleon FF for WV was adopted, namely22We have replaced q→−q in Ref. [21] and we have corrected a misprint (by adding a factor of 2 in the denominator of KνμM) in Refs. [9,10].:(3)WνμV(pΔ,q,p)=2[(GM(Q2)−GE(Q2))KνμM+GE(Q2)KνμE+GC(Q2)KνμC]. The Q2-dependence of FF is assumed to be of the form given in Ref. [7], Gi(Q2)=Gi(0)(1+Q2/MV2)−2(1+aQ2)e−bQ2≡Gi(0)GV(Q2) with MV=0.82 GeV, a=0.154/(GeV/c)2, b=0.166/(GeV/c)2. The Lorentz tensor structures are:(4)KνμM=−KM(Q2)ϵνμαβ(p+pΔ)2αqβ,KνμE=4(mΔ−mN)2+Q2KM(Q2)ϵνλαβ(p+pΔ)α2qβϵμγδλpΔγqδiγ5,KνμC=2(mΔ−mN)2+Q2KM(Q2)qν[Q2(p+pΔ)μ2+q⋅p+pΔ2qμ]iγ5 with KM(Q2)=3(mN+mΔ)2mN[(mN+mΔ)2+Q2].Now, we want to express WνμV in the so-called ‘normal parity’ (NP) decomposition. Using the non-trivial relation [25]33The BD convention is used in Ref. [25].−iϵαβμνaμbνγ5=(a̸b̸−a⋅b)iσαβ+b̸(γαaβ−γβaα)−a̸(γαbβ−γβbα)+(aαbβ−aβbα), and assuming a real Δ as in Ref. [23], and thus the validity of the Δ on-shell constrains (i.e. ψ¯Δμγμ≃0, ψ¯ΔμpΔ,μ≃0, pΔ2≃mΔ2 being ψΔμ de Δ field) we get a simplified version(5)WνμV(pΔ,q,p)=2i{−(GM(Q2)−GE(Q2))mΔKM(Q2)H3νμ+[GM(Q2)−GE(Q2)+22GE(Q2)(q⋅pΔ)−GC(Q2)Q2(mΔ−mN)2+Q2]KM(Q2)H4νμ−[22GE(Q2)mΔ2+(pΔ⋅q)GC(Q2)(mΔ−mN)2+Q2]KM(Q2)H6νμ}γ5, where(6)H3νμ(p,pΔ,q)=gνμq̸−qνγμ,H4νμ(p,pΔ,q)=gνμq.pΔ−qνpΔμ,H5νμ(p,pΔ,q)=gνμq.p−qνpμ,H6νμ(p,pΔ,q)=gνμq2−qνqμ. Note that H5νμ tensor does not contribute to Eq. (5), but it will appear in forthcoming expressions. Eqs. (4) are independent of taking p=pΔ±q (here the + sign corresponds to the Δ-pole contribution (Fig. 1(h)) and − sign to the cross-Δ term (Fig. 1(g))) which is clear since ϵνμαβqαqβ=0. Thus, Eq. (5) is valid in both cases, but the specific value of q⋅pΔ(=±mN2+Q2−mΔ22) depends on the particular contribution to the amplitude. Now, if we set on the Δ-pole contribution and replace p=pΔ+q we can rewrite (5) as(7)WνμV(pΔ,q,p=pΔ+q)=iΓνμV(pΔ,q),ΓνμV(pΔ,q)=3[−C3V(Q2)mNH3νμ−C4V(Q2)mN2H4νμ−C5V(Q2)mN2H5νμ+C6V(Q2)mN2H6νμ]γ5, where we have introduced a new set of FFʼs44Since C6V(Q2)∼−4GE(Q2)mΔ2+GC(Q2)(mN2+Q2−mΔ2)(mΔ−mN)2+Q2 we adopt GC((mΔ−mN)2)=2mΔ/(mΔ−m)GE((mΔ−mN)2), in order to avoid kinematical singularities when Q2→−(mΔ−mN)2 [23]. As (mΔ−mN)2≅(0.04GeV/c)2 we assume GC(0)≅2mΔmΔ−mNGE(0).:(8)C3V(Q2)=mΔmNRM[GM(0)−GE(0)]FV(Q2),C4V(Q2)=−RM[GM(0)−3mΔmΔ−mNGE(0)]FV(Q2),C5V(Q2)=0,C6V(Q2)=−RM2mΔmΔ−mNGE(0)FV(Q2), being RM=32mNmN+mΔ and FV(Q2)=(1+Q2(mN+mΔ)2)−1GV(Q2).Using mΔ=1.211 GeV [26], mN=0.940 GeV and the effective values GM(0)=2.97 and GE(0)=0.055 fixed from photoproduction reactions [22], we get(9)C3V(0)=2.02,C4V(0)=−1.24,C5V(0)=0,C6V(0)=−0.24.In order to make a numerical comparison with other calculations that use the NP parametrization, we consider Refs. [8] (hereafter denoted as HNV) and [11], which both use the same model. Our hadronic weak vertices defined in Eq. (2) are related with those used in [8,11] (where the W boson is considered as an incoming particle) as(10)OB(a,b,c,d,e,f)λ(q)=±i[jcc+λ|NP(−q)+jcc+λ|CNP(−q)+jcc+λ|CT(−q)+jcc+λ|PP(−q)+jcc+λ|PF(−q)],OB(g)λ(q)=±ijcc+λ|CΔP(−q),ORλ(p,p′,q)=±ijcc+λ|ΔP(−q), where the jcc+λ|i are given in Eq. (51) from HNV. Here the + sign corresponds to the pπ+ and nπ+ final state reactions and − to the pπ0 one, since for the latter the isospin matrix elements accounts a minus sign with respect to ours. Let us remark that the authors in HNV include the ρ meson contribution through a modification in the contact term but donʼt do the same for the ω one. Also, the expressions for the H3,4,5νμ tensors agree with those given in Eq. (6), but a different expression, H6νμ=mN2gνμ, is used for the remaining FF. In addition, they use the same Eq. (7) but with CiV(Q2)=CiV(0)FiV(Q2) being (Q2 in units of GeV2)(11)F3V(Q2)=F4V(Q2)=1(1+Q2/mV2)21(1+Q2/4×mV2)2,F5V(Q2)=1(1+Q2/mV2)21(1+Q2/0.776×mV2)2, with mV=0.84 GeV and(12)C3V(0)=2.13,C4V(0)=−1.51,C5V(0)=0.48,C6V(0)=0. In Eq. (9) we get C5V(0)=0 as assumed in the M1 dominance model55CiV(Q2) are obtained from photo and electroproduction data of Δ in terms of the multipole amplitudes E1+,M1+, and S1+. Recent data determine that E(S)1+/M1+∼−2.5%, and this ‘dominance’ of M1+ leads to C5V(0)=0 and the relation C4V(0)=−mNmΔC3V(0) [27]. and C6V(0)≠0 since our H6νμ satisfies current conservation condition (qνH6νμ=0) as demanded by the CVC hypothesis. As it can be observed from Eqs. (9) and (12), our values for C3,4V(0) are consistent with each other.Now, let us consider the axial-vector contribution WνμA. Within the BLM model, the axial vertex is taken as in Refs. [6,7], which can be obtained after multiplying WνμV by −γ5. It reads(13)WνμA(pΔ,q,p)=i[D1(Q2)gνμ−D2(Q2)mN2(p+pΔ)α(gνμqα−qνgαμ)+D3(Q2)mN2pνqμ−iD4(Q2)mN2ϵμναβ(p+pΔ)αqβγ5]. The last term in Eq. (13) will be dropped since we will not take into account the contribution of the Δ deformation to the axial current, i.e., we set D4(Q2)=0 and again we use the approximation where the Δ is treated as real in the weak vertex, getting(14)WνμA(pΔ,q)=i[(D1(Q2)±D2(Q2)Q2mN2)gνμ−2D2(Q2)mN2H4νμ±D3(Q2)+D2(Q2)mN2qνqμ], where the − sign corresponds to the weak vertex in Fig. 1(g) and + to that in Fig. 1(h). The Q2-dependence of the FF is [7](15)Di(Q2)=Di(0)FA(Q2),fori=1,2,D3(Q2)=2mN3(mN+mΔ)(Q2+mπ2)D1(0)FA(Q2), where FA(Q2)=(1+Q2MA2)−2(1+aQ2)e−bQ2 with MA=1.02 GeV. The normalization of the axial FF at Q2=0 is fixed by comparing the non-relativistic limit of u¯ΔνWνμAu in the Δ rest frame (pΔ=(mΔ,0), p=(EN(q),−q)) with the non-relativistic QM [6,7]. We have(16)D1(0)=6gA5mN+mΔ2mNFA(−(mΔ−mN)2),D2(0)=−D1(0)mN2(mN+mΔ)2, and we can rewrite(17)WνμA(pΔ,q,p=pΔ+q)=iΓνμA(pΔ,q),ΓνμA(pΔ,q)=3[C5A(Q2)gνμ−C4A(Q2)mN2H4νμ+C6A(Q2)mN2qνqμ]. Comparison of Eq. (14) (for the plus sign) with (17) lead us to the following FFʼs (note that C5A(0)=D1(0)3)(18)C4A(Q2)=−2mN2(mN+mΔ)2C5A(Q2)[1−Q2(mN+mΔ)2]−1,C5A(Q2)=D1(0)3FA(Q2)[1−Q2(mN+mΔ)2],C6A(Q2)=2mN3(mN+mΔ)(Q2+mπ2)C5A(Q2)[(1−Q2+mπ2mN(mN+mΔ))1−Q2(mN+mΔ)2]. The corresponding expression from the HNV authors (by assuming C3A=0) are(19)C4A(Q2)=−14C5A(Q2),C5A(Q2)=C5A(0)FA(Q2),C6A(Q2)=C5A(Q2)mN2Q2+mπ2, with FA(Q2)=(1+Q2/mA2)−2(1+Q2/3×mA2)−2 and mA=1.05 GeV. Besides the different dependencies upon Q2 through the FA(Q2) functions used in Eqs. (18) and (19), we observe further differences coming from the contributions of terms between square brackets in (18). Note that, at Q2=0, we obtain(20)C4A(0)=−0.38C5A(0),C6A(0)=0.87C5A(0)mN2mπ2, which are close to the values obtained by HNV, namely(21)C4A(0)=−0.25C5A(0),C6A(0)=C5A(0)mN2mπ2.Up to now, we have shown that a connection between the Sachs and NP parameterizations of the WNΔ vertexes can be established, and that the structure of the FF under the approximations assumed are consistent. Nevertheless, to make complete the comparison, both models should be confronted within a numerical calculation where also the fitting of C5A(0) enter into the game. We are going to achieve this by using results previously obtained within the BLM [9] and HNV [8] models. The effects of adopting different parameterizations for the Q2-dependence of the FFʼs are shown in Fig. 2, where we compare the vector FF FV(Q2) from BLM with F3V(Q2)=F4V(Q2) from HNV; the Q2-dependence is shown also for F5V(Q2) FF. We also display for comparison, the axial FF in C5A(Q2) for BLM, FA(Q2)[1−Q2/(mN+mΔ)2], and the corresponding one in HNV model. As it can be appreciated, we do not expect important differences in the cross sections coming from the different Q2-dependencies of the FFʼs. Despite the fact that C5V(0)=0 in the BLM model while C5V(0)=0.48 in the HNV one, the magnitude and quick drop of F5V(Q2) seems to indicate a small contribution of this form factor.Now, we focus on the effect of the additional Q2-dependent terms appearing in the C4,6A(Q2) FFʼs in the BLM (Eq. (18)) but not in the HNV (Eq. (19)) model, and also in the normalization conditions, Eqs. (20) and (21) at Q2=0. These effects are better appreciated in the ratio CiA(Q2)C5A(Q2) for i=4,6 which are displayed in Fig. 3. As it can be observed, the Q2-dependence of these rations is not very strong and the departure from the unity comes essentially from differences in C4,6A(0). Since the effects of these FF are very suppressed in the cross section with respect to those due to C5A(0), we do not expect important differences between both approaches due to these contributions.Next, we compare calculations for the total cross section of the most relevant νp→μ−pπ+ reaction, using alternatively the Sachs (Eqs. (3), (4), (13), (15) and (16)) and NP (Eqs. (7), (8), (17) and (18)) vertex, within the BLM model. We remark here that, within this model, a value C5A(0)=1.35 was previously obtained [9] by fitting the differential cross section d〈σ〉/dQ2 using a Sachs decomposition for the weak vertex.Before we discuss the results, let us mention that the contribution of Fig. 1(g) to the γ0WμνV†γ0 term (see Eq. (3)) appearing in the conjugated amplitude, changes its sign in the first term of Eq. (5). Taking into account that γ0(iH3γ5,iH4,6γ5)†γ0=(−iH3γ5,iH4,6γ5), the same result is obtained directly from Eq. (7). Now, the values obtained for C4,5V(Q2) are not the same as the ones obtained previously for Fig. 1(h) graph owing to the change of sign for q⋅pΔ in (5) for the cross-Δ channel. In this sense, the representation given in Eq. (3), apart from the assumed approximations, is not totally equivalent to that given in Eq. (7). For the axial part of the cross-Δ contribution we take into account that γ0(igμν,iH4,iqμqν)†γ0=(−igμν,−iH4,−iqμqν) and the minus sign in Eq. (14). We get a different dependence on the C5A(Q2) form factor and the sign of C6A(Q2), but not in the value of C5A(0). Again the result will not be the same as taking directly the conjugate of Eq. (17).As it can be observed in Fig. 4, results for the resonant R cross section using the NP vertex are slightly below the one obtained by using the Sachs vertex for the values of the constants and FF in correspondence. This can be understood considering that moving from Eq. (3) to (7) we have assumed the Δ to be on-shell (real Δ), which changes the momentum dependence of the vertex, and its coupling to the propagator (1) that has components behaving differently as pΔ2 increases. As far as the background contribution B (which includes the graph Fig. 1(g)) is concerned, the effect is opposite and is mainly due to the same approximation, and the effect of the conjugation mentioned above is of minor importance. As a consequence, the R−B interference will be different in both models and the cross section obtained within the NP model will have a value that is below the results obtained using the Sachs parametrization. This indicates that the fitted value of C5A(0) will depend on the specific model used for the weak WNΔ vertex.Finally, we compare the calculations obtained within the BLM model (Sachs form for WNΔ vertex) with the corresponding ones from HNV (NP form). The main difference between both models, apart from the specific parameters and FF, is the form adopted for the Δ propagator in Eq. (1): we use a value A=−1/3 consistent with the adopted for the vertex, and HNV take A=−1 which is equivalent to dropping the second term in Eq. (1). Second, the authors in HNV use an energy-dependent width ΓΔ(pΔ2), which would need to include energy-dependent vector FFʼs induced from vertex corrections as it is required by gauge invariance in the case that the corresponding radiative scattering is considered [15,26]. We have adopted the value C5A(0)=1.35 in BLM case [9] and the value C5A(0)=0.867 [8] is used for the HNV model (more recently a value C5A(0)=1 was reported [14]).In Fig. 5 we show results for the νp→μ−pπ+ total cross section as a function of the neutrino energy Eν; the R and R+B contributions are plotted separately. As it can be observed, the results for the resonant R contribution to the cross section in the HNV model (thin dashed lines) roughly account one-half of the cross section in the BLM model (thin full lines). By this reason, we probe with results obtained by using C5A(0)=0.867 and 1.2 but with fπNΔ/mπ×2 (which duplicates the R cross section) within the HNV model, which are shown as “Δ×21/2”. The results of these models are compared to experimental data from Ref. [18] (below an energy cutoff of 1.4 GeV in the πN invariant mass). As it can be observed, the results of the BLM model agree with data (see also Ref. [9]); the results from HNV model using C5A(0)=0.867 agrees with data only if the resonance Δ contribution to the cross section is multiplied by a factor of two. Results corresponding to C5A(0)=0.867 in the HNV model are well below data, and this cannot be attributed to the different parameterizations of the weak vertex (Sachs and NP) since; as we have seen before, these differences are much smaller if the same value of C5A(0) are used. Note also that the results corresponding to C5A(0)=1.2 and fπNΔ/mπ×2 agree very well with those reported in HNV [8] for this value of the axial constant.In summary, in this work we have compared calculations for the total cross section of the νp→μ−pπ+ channel by adopting two different prescriptions for the WNΔ weak vertex. Important differences are observed, showing that the momentum behavior of the Sachs parametrization for the vertex is not the same as the one assumed for the Normal Parity case. As a consequence, the value of C5A(0) that is fitted from data depends upon the specific parametrization of the weak vertex. 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