One pion production in neutrino-nucleon scattering and the different parametrizations of the weak $N\rightarrow\Delta$ vertex

The $N \to \Delta$ weak vertex provides an important contribution to the one pion production in neutrino-nucleon and neutrino-nucleus scattering for $\pi N$ 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 parametrizations of the weak $N \to \Delta$ vertex and form factors (FF) and we study their effects on the determination of the axial coupling $C_5^A(0)$, the common normalization of the axial FF, which is predicted to hold 1.2 by using the PCAC hypothesis. Predictions for the $\nu_{\mu} p \to \mu^- p\pi^+$ total cross sections within the two approaches, which include the resonant $\Delta^{++}$ and other background contributions in a coherent way, are compared to experimental data.

• In ν μ → ν e appearance experiments (like MiniBooNE) one detects ν e in an (almost) pure ν μ beam. The neutral current reaction ν μ N → ν μ Nπ 0 , N = n, p (NC 1π 0 ) can become a source of background for the signal event ν e n → e − p when one of the photons in the π 0 → γ γ decay escapes detection leading to a misidentification of the electron and neutral pion [4].
Therefore, a precise knowledge of the cross sections of these elementary 1 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][6][7][8][9][10][11][12][13][14]. The scattering amplitude in all these models always contains a resonant term (R) in the π N system, described by the Δ(1232)-pole 1 We refer to neutrino-nucleon scattering as the elementary process that underlies neutrino-nucleus scattering.  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 π NΔ and W NΔ vertexes. The propagator can be written as [15] with the parameter b = A+1 2 A+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 π NΔ and W NΔ 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 FF's entering the W NΔ vertex can be fixed from the electromagnetic γ NΔ process by assuming the CVC hypothesis. No analogous symmetry allows to fix the axial-vector FF's. Among the axial FF's, the most relevant role is played by C A , depending on the assumed form for the axial vertex at zero momentum transfer. A reference value is provided by the PCAC hypothesis being C A [16]. The value C A 5 (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 ν p → 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:  [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 energydependent width) and the adopted convention for the FF's, the above mentioned models only differ in the way the W NΔ vertex is parameterized. In view of the different values obtained for C A 5 (0), it would be important to compare these parameterizations.
Let us consider here the amplitude for the elementary neutrinonucleon CC 1π production process (ν p → μ − pπ + , νn → μ − nπ + , νn → μ − pπ 0 ). From our Ref. [9], hereafter called BLM, we have the total amplitude being the set of 4-momenta of the initial nucleon, neutrino, muon, pion and final nucleon, respectively, and q = p μ − p ν ( Q 2 ≡ −q 2 ) 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 J λ i a vector-axial structure By assuming the CVC hypothesis in the vector sector, the axial FF at Q 2 = 0 can be fixed from the fit to the d σ /dQ 2 differential cross sections; the strong and other weak 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 m Δ → m Δ − iΓ Δ /2 everywhere the Δ mass appears in the propagator, with Γ Δ 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 The Q 2 -dependence of FF is assumed to be of the form given in Ref. [7], The Lorentz tensor structures are: Now, we want to express W V νμ in the so-called 'normal parity' (NP) decomposition. Using the non-trivial relation [25] and assuming a real Δ as in Ref. [23], and thus the validity of the Δ on-shell constrains (i.e.ψ 2 We have replaced q → −q in Ref. [21] and we have corrected a misprint (by adding a factor of 2 in the denominator of K M νμ ) in Refs. [9,10]. 3 The BD convention is used in Ref. [25].
Note that H νμ 5 tensor does not contribute to Eq. (5), but it will appear in forthcoming expressions. Eqs. (4) are independent of taking p = p Δ ± 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 νμαβ q α q β = 0. Thus, Eq. (5) is valid in both cases, but ) depends on the particular contribution to the amplitude. Now, if we set on the Δ-pole contribution and replace p = p Δ + q we can rewrite (5) as where we have introduced a new set of FF's 4 : being [26], m N = 0.940 GeV and the effective values G M (0) = 2.97 and G E (0) = 0.055 fixed from photoproduction reactions [22], we get 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 4 Since C V , in order to avoid kinematical singularities when where the j λ cc+ | i are given in Eq. (51) from HNV. Here the + sign corresponds to the pπ + and nπ + final state reactions and − to the pπ 0 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 don't do the same for the ω one.
Also, the expressions for the H νμ 3,4,5 tensors agree with those given in Eq. (6), but a different expression, H νμ 6 = m 2 N g νμ , is used for the remaining FF. In addition, they use the same Eq. (7) but with with m V = 0.84 GeV and In Eq. (9)  As it can be observed from Eqs. (9) and (12) 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 D 4 (Q 2 ) = 0 and again we use the approximation where the Δ is treated as real in the weak vertex, getting where the − sign corresponds to the weak vertex in Fig. 1(g) and + to that in Fig. 1(h). The Q 2 -dependence of the FF is [7] 5 C V i (Q 2 ) are obtained from photo and electroproduction data of Δ in terms of the multipole amplitudes E 1+ , M 1+ , and S 1+ . Recent data determine that [27]. where The normalization of the axial FF at Q 2 = 0 is fixed by comparing the non-relativistic limit ofū ν Δ W A νμ u in the Δ rest frame (p Δ = (m Δ , 0), p = (E N (q), −q)) with the non-relativistic QM [6,7]. We , and we can rewrite Comparison of Eq. (14) (for the plus sign) with (17) lead us to the following FF's (note that C A The corresponding expression from the HNV authors (by assuming C A Besides the different dependencies upon Q 2 through the F A (Q 2 ) 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 Q 2 = 0, we obtain which are close to the values obtained by HNV, namely Up to now, we have shown that a connection between the Sachs and NP parameterizations of the W NΔ 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 C A 5 (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 Q 2 -dependence of the FF's are shown in Fig. 2, where we compare the vector FF shown also for F V 5 (Q 2 ) FF. We also display for comparison, the  appreciated in the ratio for i = 4, 6 which are displayed in Fig. 3. As it can be observed, the Q 2 -dependence of these rations is not very strong and the departure from the unity comes essentially from differences in C A 4,6 (0). Since the effects of these FF are very suppressed in the cross section with respect to those due to C A 5 (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 ν p → 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 C A 5 (0) = 1.35 was previously obtained [9] by fitting the differential cross section d σ /dQ 2 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 γ 0 W V † μν γ 0 term (see Eq. (3)) appearing in the conjugated amplitude, changes its sign in the first term of Eq. (5). Taking into account that γ 0 (iH 3 γ 5 , i H 4,6 γ 5 ) † γ 0 = (−iH 3 γ 5 , i H 4,6 γ 5 ), the same result is obtained directly from Eq. (7). Now, the values obtained for C V 4,5 (Q 2 ) are not the same as the ones obtained previously for Fig. 1(h) graph owing to the change of sign for q · p Δ 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 , −iq μ q ν ) and the minus sign in Eq. (14). We get a different dependence on the C A

(Q 2 )
form factor and the sign of C A 6 (Q 2 ), but not in the value of C A 5 (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 p 2 Δ 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 C A 5 (0) will depend on the specific model used for the weak W NΔ vertex. Finally, we compare the calculations obtained within the BLM model (Sachs form for W NΔ 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 Γ Δ (p 2 Δ ), which would need to include energy-dependent vector FF's 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 C A 5 (0) = 1.35 in BLM case [9] and the value C A 5 (0) = 0.867 [8] is used for the HNV model (more recently a value C A 5 (0) = 1 was reported [14]).
In Fig. 5 we show results for the ν p → μ − pπ + total cross section as a function of the neutrino energy E ν ; 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 C A 5 (0) = 0.867 and 1.2 but with f π NΔ /m π × √ 2 (which duplicates the R cross section) within the HNV model, which are shown as "Δ × 2 1/2 ". The re-sults of these models are compared to experimental data from Ref. [18] (below an energy cutoff of 1.4 GeV in the π N 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 C A 5 (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 C A 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 ν p → μ − pπ + channel by adopting two different prescriptions for the W NΔ 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 C A 5 (0) that is fitted from data depends upon the specific parametrization of the weak vertex. In our model we use the Sachs parametrization, and make also a comparison with calculations adopting the Normal Parity form which get a very different value for C A 5 (0), trying to look for the origin of the differences in the weak pion production cross section results.