Weak pion production off the nucleon in covariant chiral perturbation theory

Weak pion production off the nucleon at low energies has been systematically investigated in manifestly relativistic baryon chiral perturbation theory with explicit inclusion of the $\Delta$(1232) resonance. Most of the involved low-energy constants have been previously determined in other processes such as pion-nucleon elastic scattering and electromagnetic pion production off the nucleon. For numerical estimates, the few remaining constants are set to be of natural size. As a result, the total cross sections for single pion production on neutrons and protons, induced either by neutrino or antineutrino, are predicted. Our results are consistent with the scarce existing experimental data except in the $\nu_\mu n\to \mu^-n\pi^+$ channel, where higher-order contributions might still be significant. The $\Delta$ resonance mechanisms lead to sizeable contributions in all channels, especially in $\nu_\mu p\to \mu^- p\pi^+$, even though the considered energies are close to the production threshold. The present study provides a well founded low-energy benchmark for phenomenological models aimed at the description of weak pion production processes in the broad kinematic range of interest for current and future neutrino-oscillation experiments.


I. INTRODUCTION
Neutrino interactions with matter are at the heart of many relevant phenomena in astrophysics, nuclear and particle physics. Among them, neutrino oscillations have revealed that neutrinos are massive, providing evidence of physics beyond the Standard Model. Precision studies of neutrino-oscillation parameters demand a good understanding and accurate modeling of neutrino interactions with nucleons and nuclei [1][2][3]. In this context, weak pion production has been actively investigated.
Single-pion production amounts to one of the leading contributions to the inclusive (anti)neutrino-nucleus cross section in the energy range of interest for ongoing and future oscillation experiments. As such, it can be part of the signal or a background that should be precisely constrained. Single charged pion production in charged-current interactions is a source of events that can be misidentified as quasielastic [ν l (ν l )N → l ∓ N ] ones if the pion is not identified, introducing a bias in the kinematic neutrino energy reconstruction 1 . Furthermore neutral-current π 0 production events in Cherenkov detectors contribute to the electron-like background in ν µ → ν e measurements. In spite of the progress, 20-30% errors are currently taken for single-pion production in oscillation analyses due to conflicts between data sets and models [4].
It was early acknowledged that, at low and intermediate energies, weak pion production should proceed predominantly through the excitation of the ∆(1232)3/2 + resonance (see Ref. [5] and references therein). Isobar models accounting for heavier nucleon resonances were subsequently developed [6,7]. The nucleon-to-resonance transitions were parametrized in terms of real form factors obtained from quark models [7][8][9][10] or phenomenology. In the later case, owing to the symmetry of the conserved vector current under isospin rotations, vector transition form factors can be related to electromagnetic ones extracted from electron scattering data while the partial conservation of the axial current (PCAC) allows to derive the off-diagonal Goldberger-Treiman (GT) relation for the leading axial couplings [6,[11][12][13]. Additional, but rather limited, information on the transition axial form factors can be obtained from available weak pion-production bubble-chamber data on hydrogen and deuterium [14][15][16]. Non-resonant mechanisms were added to the resonant ones in Refs. [6,17,18], and further extended to fulfill chiral symmetry constraints at threshold in Ref. [19]. In these studies, the range of applicability of the Born terms is expanded by the introduction of form factors. In the approach of Ref. [19] (denoted as HNV from now on), a good agreement with bubble chamber data was achieved at the price of introducing tensions in the value of the leading N − ∆(1232) axial coupling, C A 5 in the notation of Ref. [5], with respect to the GT value at a 2σ level [15] 2 . The two values could be reconciled by imposing Watson's theorem in the dominant partial wave [22]. The importance of a consistent treatment of the ∆(1232) was stressed in Refs. [23,24], also accounted for in the HNV model by the introduction of new contact terms that absorb the unphysical spin-1/2 components in the ∆ propagator [25]. Extensions of the HNV model to higher energies have been developed 1 Neutrino fluxes are not monochromatic. Therefore, the neutrino energy, on which oscillation probabilities depend, is not known on an event-by-event basis but can be approximately reconstructed from the finallepton kinematics in quasielastic events. 2 Deviations from the N − ∆(1232) off-diagonal GT relation are expected only at the few-% level, as they arise from chiral symmetry breaking. Systematic studies of the corrections to this GT relation using chiral perturbation theory have been reported in Refs. [20,21].

A. Kinematics, Lorentz and isospin decompositions
Charged-current weak pion production off the nucleon consists of processes of the type induced either by neutrinos ν or antineutrinosν ; see Ref. [40] for a classic review of electroweak pion production. This reaction is described by the Lorentz-invariant amplitude T f i , which is defined by In the antineutrino case, one replaces ν →ν and − → + in the above definition. The amplitude T f i is a function of the following six Mandelstam variables, which fulfill the constraint where the neutrino mass has been approximated to zero. We work in the isospin limit so the mass of all nucleons (pions) has been set to m N (M π ). Henceforth, t is always given in terms of the other five invariants. In the limit |t 1 | M 2 W , where M W is the vector W-boson mass, the scattering amplitude T f i can be written as where the leptonic and hadronic currents, denoted by L µ and H µ , respectively, are given by in terms of the isovector vector and axial-vector currents V a µ and A a µ ; H µ depends only on variables s 2 , t 1 and t. Its isospin structure has the form where χ i and χ f are isospinors of the initial and final nucleon states, respectively. Furthermore, the Lorentz decomposition reads with the Lorentz axial-vector operators 3 and Lorentz vector operators The set of vector operators is complete but they are not independent if the conservation of the vector current is imposed. To be specific, there exist two constraints on V i : with k ≡ k 1 − k 2 . Eventually, once the functions H ± µ are determined, the hadronic transition amplitudes for the various physical weak pion production processes can be readily obtained through

B. Cross section
Unless otherwise stated, the energies and momenta are defined in the center-of-mass frame of the initial (anti)neutrino and nucleon. The directions of pion and lepton three-momenta directions are specified in the reference frame depicted in Fig. 1. By construction, Oxz is the lepton scattering plane.
The total cross section reads where x 1 = cos θ 1 and φ 12 is the angle between the Oxz plane and the one spanned by k 2 and q. Here, the limits for the lepton energy ω l are given by and the ones for the pion energy ω π are In the above, m denotes the outgoing-lepton mass. The invariant amplitude squared can be written as in terms of the conventional leptonic and hadronic tensors. From Eq. (6) the leptonic tensor for a neutrino-induced process is given by with 0123 = +1. For the corresponding antineutrino reaction, the term proportional to the fully anti-symmetric tensor gets a minus sign. On the other hand, the hadronic tensor H µν reads whereH µ = γ 0 H † µ γ 0 . The hadronic transition amplitudes H µ are those introduced in Eq. (13).
The total cross section is a function of only s, so that the other four Mandelstam variables should be expressed in terms of s and the integration variables: where x i ≡ cos θ i and the moduli of the three momenta are with The invariant s can be related to the energy of the neutrino in the laboratory frame, E ν , by so that the total cross section can expressed as a function of E ν .

III. SYSTEMATIC ANALYSIS OF THE HADRONIC TENSOR IN CHPT
In this section, the different ingredients required to obtain the hadronic current in ChPT are presented.

A. Power counting
As an expansion in powers of momenta and light-quark masses, ChPT relies on a hierarchy of the contributions (diagrams) known as power counting. The presence of matter fields as explicit degrees of freedom introduces new scales that do not vanish in the chiral limit, causing the presence of power counting breaking (PCB) terms [41] in the diagrams with loops. To remedy this problem, various approaches have been proposed in the past thirty years: e.g., the heavy baryon (HB) formalism [42,43], the infrared regularization (IR) prescription [44,45], and the extended-on-mass-shell (EOMS) scheme [46][47][48]. 4 For ChPT in the one-baryon sector, denoted in short as BChPT, the EOMS scheme has proven to be a very effective tool. It is covariant and preserves the analytic structure of the calculated physical quantities with correct power counting. When the proper limits are taken, EOMS reproduces the results obtained using the HB or the IR formalisms but usually offers a faster chiral convergence. Due to the above-mentioned facts, the EOMS scheme is gaining a widespread acceptance and has been applied to many relevant processes, e.g. pionnucleon scattering [50][51][52][53] and pion photoproduction [54][55][56], among others. Furthermore, the EOMS scheme has also been extended beyond the low-energy region [57] and to heavylight systems [58][59][60].
The explicit inclusion in BChPT of baryon states heavier than the nucleon, such as the ∆ resonance, is not trivial. The ∆(1232) excitation is the lightest baryon resonance, located only ∼ 200 MeV above the πN threshold, and hence crucial for a good description of the πN physics even at low energies. In BChPT with ∆(1232), apart from the external momenta p and the pion mass M π , an additional small parameter appears, namely the mass difference δ = m ∆ − m N ∼ 300 MeV. Different assumptions about the expansion parameters lead to different power-counting rules. In the small scale expansion (SSE) scheme proposed in Refs. [61,62], both δ and M π are counted as O(p). Instead, in the so-called δ-counting, developed in Ref. [63], a different counting, δ ∼ O(p 1 2 ), is introduced in order to preserve the hierarchy p/Λ χSB ∼ M π /Λ χSB ∼ (δ/Λ χSB ) 2 , with Λ χSB ∼ 1 GeV being the chiral symmetry breaking scale.
In the present work, we are interested in the energy range from the production threshold E thr.
ν +M π ( 415 MeV for = µ), where the invariant mass of the final hadronic πN system, denoted as √ s 2 , is well below the ∆-resonance peak.
Hence, we prefer to employ the δ-counting rule. Specifically, for a given Feynman diagram with L loops, V (k) vertices of O(p k ), I π internal pions, I N nucleon propagators and I ∆ ∆-propagators, its chiral dimension D is obtained according to the rule Here, we aim to perform a calculation of the hadronic transition amplitude up to the chiral

B. Chiral effective Lagrangians
Given our working accuracy and according to the power counting rule (24), the following chiral Lagrangians are needed for our calculation, where superscripts represent chiral orders while subscripts denote the relevant degrees of freedom. For clarity, the effective Lagrangian is classified in three parts: the purely pionic sector, the pion-nucleon sector and the one involving ∆ resonances.

Pionic interactions
The required terms in the purely pionic sector are given by [32,41] where F L µν = ∂ µ l ν −∂ ν l µ −i[l µ , l ν ] is the left-handed field-strength tensor; l µ = −g W V ud l a µ τ a /2 is the left-handed external field and τ a (a = 1, 2, 3) are the Pauli matrices. 5 Here χ = diag{M 2 , M 2 } is the mass matrix with M being the pion mass in the isospin limit. Tr[· · · ] denotes the trace in flavor space. Furthermore, F is the pion decay constant in the chiral limit and 3,4,6 are mesonic low-energy constants (LECs). The Goldstone pion fields are collected in the 2 × 2 matrix U where the corresponding covariant derivative has also been defined.

Interactions with nucleons
The relevant terms describing the interactions between pions, or external fields l µ , and nucleons read [64] with the nucleon doublet Ψ N = (p, n) T . Here, m and g are the nucleon mass and axial charge in the chiral limit. The LECs c i and d j have units of GeV −1 and GeV −2 , respectively. The involved chiral blocks are given by 5 We identify l 1 µ = W 1 µ , l 2 µ = W 2 µ and l 3 µ = 0, to which the physical weak-boson fields W ± µ are related via Note that, to be consistent with Eq. (5), we always factorize out the combination −g W /(2 √ 2) V ud from the hadronic transition amplitude H µ calculated in subsection III C . Furthermore, the factor g W /(2 √ 2), together with an identical one from the lepton sector, is absorbed in the Fermi where M W denotes the mass of the vector W boson.
In practice, the Levi-Civita tensor can be expressed in terms of Dirac gamma matrices: In such a manner, the Lorentz structure of the hadronic transition amplitude can be readily expressed in terms of the operators given in Eqs. (10) and (11).

Interactions with ∆
The ∆-resonance is a state of spin-3/2, which can be represented by a vector-spinor Ψ µ in the Rarita-Schwinger formalism [65]. It is also a field of isospin-3/2, thus, it can be described by a vector-spinor isovector-isospinor field Ψ i µ , with µ and i being the Lorentz vector and isovector indices, respectively. We refer the reader to Ref. [62] for the so-called isospurion formulation where the relations between the field Ψ i µ and the physical ∆(1232) states, ∆ ++ , ∆ + , ∆ 0 and ∆ − , are presented. The interactions of ∆ resonances with pions read where m ∆ is the ∆ bare mass and a 1 a bare coupling constant; the covariant derivative is defined by Furthermore, ξ 3 2 ij = δ ij − 1 3 τ i τ j is the isospin-3/2 projection operator; the Dirac matrices with multiple Lorentz indices are defined as Finally, the effective Lagrangian for pion-nucleon-∆ interaction has the form [61, 62, 66] where h A denotes the LO axial coupling constant, b k are NLO LECs, and the chiral blocks with isovector index i are defined as In fact, as pointed out in Ref. [67], the b 2 and b 7 terms can be eliminated thanks to the identity, . Furthermore, the b 3 and b 8 terms are redundant too [67,68], which has been explicitly checked in πN scattering [52], showing that their contributions can be absorbed in the LO ∆-exchange and contact terms. Therefore, for L (2) πN ∆ , we only need to take the b 1 term into consideration.

C. Hadronic transition amplitudes
The tree-level diagrams relevant to our calculation up to O(p 3 ) are depicted in Fig. 2. They are labeled according to the scheme shown in Table II in Appendix A. Therein, the chiral order of each tree-level contribution is specified, as well, for convenience. The explicit expressions for the corresponding amplitudes are listed diagram by diagram in this appendix. In Fig. 2, the diagrams with mass insertions in the internal pion, nucleon and ∆ propagators are not shown. Such amplitudes with mass insertions in internal nucleon and ∆ lines, which are generated by terms proportional to c 1 term in L (2) πN and the a 1 term in L (2) π∆ , can be taken into account by the following replacement in the nucleon and ∆ propagators: On the other hand, the insertions in pion propagators, generated by the l 3 and l 4 terms in L ππ , contain momentum-dependent pieces. Hence, their contribution can not be incorporated as in the nucleon and ∆ cases. Instead, the contribution of a diagram with one insertion in a pion line results from the substitution with where q π is the momentum transferred in the pion propagator. Note that, up to the order we are working in, the pion-insertions for diagrams T D 12 , T E 112 , T F 12 and T G 112 need to be taken into consideration only once, since ξ(q 2 π ) is of order O(p 2 ). For the calculation of loop contributions, we need all the diagrams generated from the topologies shown in Fig. 3. In total, there are 89 diagrams. An example of how to generate  them from topology (b) of Fig.3 is shown in Fig. 4. The calculation of these one-loop amplitudes is straightforward but yields lengthy analytical expressions, which we do not show explicitly here. Finally, the contributions of diagrams corresponding to loop corrections on the external legs are included through wave function renormalization, which is discussed in the next section.

D. Renormalization
In the above subsection, we have described the calculation of the hadronic transition amplitudes up to O(p 3 ), corresponding to the Feynman diagrams excluding corrections at external pion and nucleon legs. In fact, the sum of all their contributions yields the amputated amplitude,Ĥ µ , for which the superscripts '±' are suppressed for brevity. According to the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula [69], the full amplitude is related to the amputated one through where Z π and Z N are wave function renormalization functions of the pion and nucleon fields, respectively. Their explicit expressions are given in Appendix C.
In the full amplitude, the loop contributions are evaluated using dimensional regularization. The ultraviolet (UV) divergences stemming from the loops are subtracted using the modified minimal subtraction (MS-1) scheme and absorbed by the LECs appearing in the counter terms generated by the effective Lagrangian. That is, we split the bare LECs in the following way, where , d the number of space-time dimension, and γ E the Euler constant. We refer to the effective Lagrangians, in Eqs. (30), (31) and (27), for the values of the indices i, j, k. Furthermore, β X are beta functions. As already mentioned in the beginning of this section, there exist PCB terms due to the appearance of nucleon internal lines in the loop diagrams. To restore the power counting, we apply the EOMS scheme. Therefore, after the cancellation of the UV divergences, one has to perform additional finite shifts for the O(p) and O(p 2 ) UV renormalized LECs as withβ X being the beta functions for this finite renormalization. The verification of the cancellation of UV divergences and PCB terms is delicate. The vector and axial-vector operators given in Eqs (10) and (11) are not well suited to perform a chiral expansion, due to the fact that sometimes the chiral order of their combination is underestimated. For instance, the chiral orders of O A µ,5 and O A µ,6 are both assigned to be O(p). Consequently, the combination O A µ,5 − O A µ,6 is naively counted as O(p). However, its actual chiral order should be O(p 2 ), since (p 1,µ − p 2,µ ) gives an additional contribution of O(p). Therefore, to overcome such issues during renormalization, we have chosen a chiralexpansion-suited (CES) basis, see Eq. (B1) and Eq. (B2) in Appendix B. Another advantage of the CES basis is that vector current conservation is automatically implemented. With the help of the CES basis, we remove the UV divergences and PCB terms order by order in the chiral expansion, and obtain the explicit expressions for the β functions, namely β X andβ X in Eqs (45) and (46), which are relegated to Appendix C.
All the parameters in the renormalized full amplitude are UV finite. For practical convenience, we write F , M ,m andg in terms of their corresponding physical values, F π , M π , m N and g A by using the relations specified in Eq. (C7) and Eq. (C9). The terms of O(p 4 ) and higher orders generated by the above substitutions, as well as by the wave function renormalization in Eq. (44) are neglected.

A. Low energy constants
The available data for neutrino-induced charged-current single pion production on nucleons at low energies are very scarce. In fact, they are limited to the early experimental measurements at the ANL [70,71] and the BNL [72,73] hydrogen-and deuterium-filled bubble chambers. Muon neutrino beams were used for both ANL and BNL with average energies around 1 GeV and 1.6 GeV, respectively. Although events for all allowed channels induced by muon neutrinos were detected, almost all the data are beyond the energy region where ChPT is expected to be valid. This is also the case for the data on muon antineutrinoinduced processes measured at CERN-PS [74]. Therefore, the task of fixing the unknown LECs present in the hadronic transition amplitudes calculated above by fitting the above mentioned ν −ν data is unattainable. Nonetheless, most of the required LECs are known, as they have been obtained in the analysis of other processes or physical quantities. We take their values from the studies of πN scattering [50][51][52] and the axial radius of the nucleon [75], which used the EOMS scheme as in the present calculation.
For the parameters appearing in the LO Lagrangians, i.e., L ππ , L where h A is determined from the strong decay width of ∆ → πN (Γ str. ∆ = 118 ± 2 MeV [76]). See Ref. [77] for details. In the higher-order effective Lagrangians relevant to our calculation, there are in total 22 LECs. Three of them, r 3 , r 4 and d r 16 , become irrelevant after the procedure of renormalization and replacement of the LO parameters by their physical ones as discussed in the previous section. Furthermore, as shown in Table I, most of them are pinned down in processes other than weak pion production. The so-called scale-independent parameter¯ 6 was extracted from the electromagnetic charge radius of the pion r 2 π at O(p 4 ) in Ref. [32]. The value of 6 at the renormalization scale µ, denoted by r 6 in Eq. (45), can be obtained through the following renormalization group equation [32] with where B 0 is a constant related to the quark condensate and β 6 = −1/6 as can be seen from Eq. (C11). As usual in BChPT, we set µ = m N , which yields LECsc i 's and d r j 's displayed in Table I, exceptc 6 and d r 22 , have been fixed in pion-nucleon scattering, calculated up to O(p 3 ) using the δ-counting within the EOMS scheme [50]. This is exactly the same approach employed in the present study. The model was fitted to the experimental phase shifts from Ref. [79]. On the other hand, the value ofc 6 has been obtained in Ref. [78] by adjusting the corresponding chiral results for the magnetic moments of protons and neutrons, µ p and µ n , to their empirical values from Ref. [76]. There are two determinations of this parameter in Ref. [78]: one is obtained without and the other with explicit ∆'s which are present only in loops. We have chosen the former determination as the central value forc 6 , since in the adopted power-counting rule loops with internal ∆'s are of higher order and beyond our consideration. The difference between the two determinations is then assigned to the error ofc 6 . Specifically, we havec 6 = (1.35 ± 0.04) GeV −1 in the end 6 . As for d r 22 , it is pinned down in the extraction of the nucleon axial charge and radius from lattice QCD results in Ref. [75]. Similarly toc 6 , the ∆ resonance is involved in the axial form factor only at loop level, hence we employ its value from the ∆-less fit therein.
Finally, as demonstrated in Ref. [77], the electromagnetic width of the ∆ resonance can be expressed in terms of the NLO πN ∆ coupling b 1 . Given that Γ em /(Γ em +Γ str ) = 0.55−0.65% with Γ str. ∆ = 118 ± 2 MeV [76], the value of b 1 is fixed to be b 1 = (4.98 ± 0.27)/m N . As mentioned in Ref. [77], the sign of b 1 remains undetermined, but here we have chosen a positive sign as further discussed in the next subsection.
Apart from the known parameters discussed above, there are still 7 values, while the errors are propagated in quadrature. 6 In Ref. [78], the ρ meson is explicitly included in the calculation and the combinationc 6 = c 6 + c ρ 6 is determined, where c ρ 6 = −G ρ /(2g ρ ) is the part saturated by the ρ, given in terms of parameters G ρ , g ρ , related to ρ interactions. In our case, without explicit ρ meson, this ρ contribution is absorbed by the LEC. Therefore, we identify ourc 6 withc 6 rather than c 6 . 7 Some of these LECs, d 8 , d 9 , d 20 and d 21 , also appear in pion electroproduction on the nucleon. Their values have been determined in the analysis of that process in Ref. [37]. Although Ref. [37] uses the EOMS scheme, we cannot use their results directly because the ∆ is not included in the calculation.

B. Total cross sections
Once the parameters in the hadronic transition amplitudes have been specified, we are now in the position to make predictions for experimental observables. First, the (anti)neutrino-induced pion-production cross sections are calculated up to O(p 3 ). The convergence properties of our results are then discussed. We consider the muon flavor, for which the available measurements have been performed. As previously explained in subsection III A, we expect our model to be reliable up to energies E ν 415 MeV, so that we are relatively far from the ∆ pole and the δ counting is appropriate.
Only ∆ Only N Full N &∆ HNV model ANL In the left (right) column of Fig. 5, the results are shown for neutrino (antineutrino)induced pion production, respectively. The plots are displayed up to E ν = 450 MeV, slightly above our validity limit, to better show the trends of the curves. The ∆-width effect is taken into account as well by means of Eq (A4), though its contribution is of higher order. Furthermore, its effects are really minor in the energy region we are concerned with. Its implementation enables us to eventually extend our results smoothly to higher energies, even passing the ∆-peak. Due to the nearby existence of the ∆ pole, the ∆ contribution (black dash-dotted line) increases rapidly in the region above E ν 415 MeV, as can be observed especially from the plot for the reaction ν µ p → µ − pπ + . Meanwhile, except for this latter channel, the nucleonic contribution (blue dashed line) grows steadily and dominates the total cross sections in the region below E ν 415 MeV. The bands in the plots show the uncertainty associated to the error estimations of the LECs discussed in the previous section.
In the considered energy region, there is only one experimental data point from the ANL measurements [70,71] for each neutrino-induced reaction channel. As can be seen in Fig. 5, our full chiral predictions (red lines with bands), at E ν ∼ 400 MeV, are in good agreement with the ANL data in the channels of ν µ p → µ − pπ + and ν µ n → µ − pπ 0 . However, the theoretical cross section for the ν µ n → µ − nπ + reaction is smaller than the central value of the experiment. Nevertheless, the chiral calculation for this latter channel is still consistent with data due to the large experimental uncertainties. Unfortunately, for the antineutrino processes, so far there are no available data at low energies, preventing us from assessing our predictions.
We also compare our results with those of the HNV model [19], which allows for a simple but meaningful comparison: the HNV phenomenological model, gives a good description of the weak pion production process for a wider range of neutrino energies well above 1 GeV. This model incorporates both the contributions from the ∆ pole mechanisms and nonresonant terms constrained by chiral symmetry and given by the tree diagrams of Fig. 2 at their lowest order. The counterparts of those diagrams in ChPT are represented by the LO ∆-less tree diagrams of O(p) and the ∆-exchange ones of O(p 3/2+5/2 ). In particular, for the ∆ contribution, we find the following correspondence where C 5A and C 3V are two of the Adler N → ∆ axial and vector form factors that are conventionally used in the literature [5,80] including the HNV model. Imposing the values of h A and b 1 specified in the above subsection, we obtain C 5A (0) 1.17 ± 0.02 and C 3V (0) 2.01 ± 0.11, which are comparable to C 5A (0) = 1.2, as obtained with an offdiagonal Goldberger-Treiman relation, and C 3V (0) = 2.13 from pion photoproduction. This observation also supports the choice of a positive b 1 . Note, that while HNV does not obey a systematic power counting neither includes loop diagrams, some higher-order corrections are implemented through phenomenological form factors for the vertices in the axial and vector weak currents 8 . Our results are systematically larger than the HNV ones. This is mainly due to the inclusion of the O(p 3 ) terms coming both from tree and loop diagrams. The enhancement improves the agreement with data though the large error bars preclude any strong claim. Particularly interesting is the ν µ n → µ − nπ + channel where there is a large contribution of the O(p 3 ) terms but the results are still below data.
In Fig. 6, we display the total cross sections for the neutrino reactions order by order, in order to show the convergence properties of the chiral series 9 . For all the channels, a calculation with a higher chiral order brings the predictions closer to the experimental data. Moreover, the resulting contribution when stepping from O(p 2 ) up to O(p 3 ) is quite significant in the improvement of the predictions. On the other hand, it seems clear that next order effects could still be relevant. In fact, it has been shown for ν µ n → µ − nπ + , that the failure on the description of the ANL data might be cured by partially restoring unitarity [22]. This can be approximately done by imposing Watson's theorem for the dominant vector and axial multipoles [22]. In a systematic ChPT calculation, this corresponds to the inclusion of higher-order loops: especially those whose internal pion and nucleon lines can be put on shell. Another possible solution has been suggested in Ref. [25] and amounts to the need of extra higher-order contact terms.

V. SUMMARY AND OUTLOOK
Charged current (anti)neutrino-induced pion production off the nucleon at low energies has been systematically studied for the first time within the framework of manifestly relativistic baryon chiral perturbation theory up to O(p 3 ), (NNLO), for the low-energy chiral representation of the hadronic-transition amplitude. The ∆(1232) resonance has been included explicitly using the δ-counting rule. To tackle the power-counting violation of the nucleon loops we have performed the renormalization in the EOMS approach [46][47][48] in which the power counting is restored by means of finite shifts of the LEC values in the chiral effective Lagrangians after the conventional UV subtraction in the MS-1 scheme.
Remarkably, at this order, most of the involved LECs (15 out of 22) have been previously determined in other processes such as pion-nucleon scattering. Furthermore, another 4 of the remaining unknown LECs in the O(p 3 ) πN Lagrangian may be obtained in the future from available pion electroproduction data. For numerical estimates, the unknown LECs have been assumed to be of natural size. Consequently, we have predicted the total cross sections in all the physical reaction channels, both for neutrino-and antineutrinoinduced pion production. We have also estimated the theoretical uncertainties due to the limited knowledge of some LECs. Our results are expected to be reliable up to the neutrino laboratory energy of E ν = 415 MeV, which is relatively close to the threshold and well below the ∆ peak. Hence, the energy range is well suited for the adopted δ-counting. Nonetheless, mechanisms involving the ∆ resonance contribute significantly to all production channels, especially to the ν µ p → µ − pπ + one.
It has been found that our predictions are consistent with the few existing experimental ANL data for the neutrino-induced processes except ν µ n → µ − nπ + . This might indicate that higher-order contributions are still relevant for this channel as suggested by the more phenomenological study of Ref. [25]. Lacking a full calculation, such higher-order contributions might be approximated by unitarity corrections or some extra contact counterterms. So far there are no low-energy experimental data for antineutrino-induced pion production on nucleons. Our results for these processes provide a set of theoretical predictions that fully rely on ChPT.
Finally, our chiral representation of weak pion production can be applied to study various low-energy theorems in the future. It can also be adapted to analyze pion photo-and electro-production by further incorporating the isoscalar vector part of the hadronic currents. Most importantly, the present study provides a well founded low-energy benchmark for phenomenological models aimed at the description of weak pion production processes in the broad kinematic range of interest for current and future neutrino-oscillation experiments.

Appendix A: Chiral hadronic amplitudes at tree level
In what follows, all the tree amplitudes corresponding to the diagrams specified in Table II are listed. We use the abbreviations: with x ∈ {s 2 , u}. The Mandelstam variable u is defined as u ≡ (p 1 − q) 2 , and hence can be written in terms of the variables in Eq. (3) via u = 2m 2 N − s 2 − t − t 1 . Hereafter, the Lorentz indices of the axial and vector operators are suppressed. Furthermore, we shall use the shorthands: As for the ∆-exchange diagrams, the ∆-width effect can be included through the following substitution: with the energy-dependent width given by [81] being λ(a, b, c) ≡ a 2 +b 2 +c 2 −2ab−2ac−2bc the Källén function and θ(x) the step function.
(A7) 11 : • Diagram T D 12 : (A9) • Diagram T E 112 : (A10) • Diagram T F 12 : • Diagram T G 112 : • Diagram T B 2 : • Diagram T C 12 : (A15) • Diagram T F 22 : • Diagram T C 13 : (A20) • Diagram T C 31 : • Diagram T D 32 : . (A23) • Diagram T E 114 +T E 312 +T E 132 : • Diagram T F 14 : . (A25) • Diagram T F 32 : 4. At O(p 3/2 ): • Diagram T A∆ 11 : (A28) • Diagram T C∆ 11 : • Diagram T E∆ 112 : (A30) • Diagram T G∆ 112 : • Diagram T A∆ 21 : • Diagram T C∆ 12 : Appendix B: Chiral-expansion-suited operators As mentioned in Section III D, the vector and axial-vector operators given in Eqs. (10) and (11) are not suited to perform a chiral expansion. In practice, we prefer to use the following axial-vector operators with P µ ≡ (p µ 1 + p µ 2 )/2. For vector operators, we follow the basis proposed in Ref. [40]: for which the vector-conservation assumption is automatically implemented. The axialvector structure functions in the new basis can be obtained through while the vector structure functions arẽ , In the chiral expansion of ∆-less amplitudes, we treat The UV divergence is contained in the quantity R = 2/(d − 4) + γ E − 1 − ln(4π), being γ E the Euler constant. We denote A 0 and B 0 loop integrals with removed UV-divergent parts (multiples of R) byĀ 0 andB 0 , respectively. To proceed, the nucleon and pion wave function renormalization constants can be written as respectively, where the O(p 2 ) parts are The relations between the renormalized (or chiral limit) masses and the physical ones read Likewise, for the leading couplings g A and F π , one has