A Relations between electromagnetic form factors and helicity amplitudes

Photon emission in neutral current interactions at intermediate energies


Neutral current photon emission reactions with nucleons and nuclei are studied. These processes are important backgrounds for appearance oscillation experiments where electromagnetic showers instigated by electrons (positrons) and photons are not distinguishable. At intermediate energies, these reactions are dominated by the weak excitation of the resonance and its subsequent decay into . There are also non-resonant contributions that, close to threshold, are fully determined by the effective chiral Lagrangian of strong interactions. In addition, we have also included mechanisms mediated by nucleon excitations () from the second resonance region above the . From these states, the contribution of the turns out to be sizable for (anti)neutrino energies above 1.5 GeV. We have extended the model to nuclear targets taking, into account Pauli blocking, Fermi motion and the in-medium resonance broadening. We present our predictions for both the incoherent and coherent channels, showing the relevance of the nuclear corrections. We also discuss the target mass dependence of the cross sections. This study is important in order to reduce systematic effects in neutrino oscillation experiments.


I Introduction

A good understanding of (anti)neutrino cross sections is crucial to reduce the systematic uncertainties in oscillation experiments aiming at a precise determination of neutrino properties Formaggio and Zeller (2012). Our present knowledge of neutrino-nucleus interactions has been significantly improved by a new generation of oscillation and cross section experiments. Quasielastic (QE) scattering measurements have been published by MiniBooNE Aguilar-Arevalo et al. (2010a, b, 2013a) at neutrino energies  GeV, by MINERFields et al. (2013); Fiorentini et al. (2013) at  GeV and by NOMAD at high (3-100 GeV) energies Lyubushkin et al. (2009). Detailed single pion production data have become available from MiniBooNE Aguilar-Arevalo et al. (2010c, 2011a, 2011b) for different reaction channels including the coherent one, which has also been studied by SciBooNE Hiraide et al. (2008); Kurimoto et al. (2010) at  GeV and NOMAD Kullenberg et al. (2009). Finally, new inclusive cross section results have been reported by T2K Abe et al. (2013a), SciBooNE Nakajima et al. (2011), MINOS Adamson et al. (2010) and NOMAD Wu et al. (2008) Collaborations. These results challenge our understanding of neutrino interactions with matter and have triggered a renewed theoretical interest Morfin et al. (2012). Quasielastic scattering has been investigated with a local Fermi gas Graczyk and Sobczyk (2003); Nieves et al. (2004); Athar et al. (2005); Martini et al. (2009), realistic spectral functions Benhar et al. (2005); Ankowski and Sobczyk (2008), different models to describe the interaction of the knocked-out nucleon with the residual nucleus Martinez et al. (2006); Butkevich and Kulagin (2007); Meucci et al. (2011) and using the information from electron scattering data encoded in the scaling function Caballero et al. (2005). The importance of two-nucleon contributions for the proper understanding of QE-like and inclusive cross sections has emerged in different studies Martini et al. (2009); Amaro et al. (2011); Nieves et al. (2011), and their impact in the kinematic neutrino-energy reconstruction has been stressed Martini et al. (2012); Nieves et al. (2012a); Lalakulich and Mosel (2012). Incoherent pion production has also been scrutinized using microscopic models for the reaction mechanism on the nucleon Sato et al. (2003); Hernandez et al. (2007, 2010a); Leitner et al. (2009a); Serot and Zhang (2012); Hernández et al. (2013), with special attention paid to pion final state interactions in nuclei Ahmad et al. (2006); Leitner et al. (2009b); Golan et al. (2012); Lalakulich and Mosel (2013); Hernández et al. (2013). New microscopic models have been developed for coherent pion production Singh et al. (2006); Alvarez-Ruso et al. (2007a); Amaro et al. (2009); Nakamura et al. (2010); Zhang and Serot (2012a) while traditional ones, based on the partial conservation of the axial current (PCAC), have been updated Paschos et al. (2006); Berger and Sehgal (2009); Hernandez et al. (2009); Kopeliovich et al. (2012).

One of the possible reaction channels is photon emission induced by neutral current (NC) interactions (NC), which can occur on single nucleons and on nuclear targets. Weak photon emission has a small cross section compared, for example, with pion production, the most important inelastic mechanism. In spite of this, NC photon emission turns out to be one of the largest backgrounds in oscillation experiments where electromagnetic showers instigated by electrons (positrons) and photons are not distinguishable. Thus, NC events producing single photons become an irreducible background to the charge-current (CC) QE signatures of () appearance. This is precisely the case of the MiniBooNE experiment that was designed to test an earlier indication of a oscillation signal observed at LSND Athanassopoulos et al. (1996, 1998). The MiniBooNE experiment finds an excess of events with respect to the predicted background in both and modes. In the mode, the data are found to be consistent with oscillations and have some overlap with the LSND result Aguilar-Arevalo et al. (2013b). MiniBooNE data for appearance in the mode show a clear () excess of signal-like events at low reconstructed neutrino energies ( MeV) Aguilar-Arevalo et al. (2007, 2013b). However, the distribution of the events is only marginally compatible with a simple two-neutrino oscillation model Aguilar-Arevalo et al. (2013b). While several exotic explanations for this excess have been proposed, it could be related to unknown systematics or poorly understood backgrounds in the experimental analysis. In a similar way, NC is a source of misidentified electron-like events in the appearance measurements at T2K Abe et al. (2013b). Even if the NC contribution to the background is relatively small, it can be critical in measurements of the CP-violating phase. It is therefore very important to have a robust theoretical understanding of the NC photon emission reaction, which cannot be unambiguously constrained by data. This is the goal of the present work.

The first step forwards a realistic description of NC photon emission on nuclear targets of neutrino detectors is the study of the corresponding process on the nucleon. Theoretical models for the reaction have been presented in Refs. Hill (2010); Serot and Zhang (2012). They start from Lorentz-covariant effective field theories with nucleon, pion, but also scalar () and vector (, ) mesons as the relevant degrees of freedom, and exhibit a nonlinear realization of (approximate) chiral symmetry. The single mechanism of excitation followed by its decay was considered in Ref. Barbero and Mariano (2012), where a consistent treatment of the vertices and propagator is adopted. Several features of the previous studies, in particular the approximate chiral symmetry and the dominance of the mediated mechanism are common to the model derived in our work. In Ref. Serot and Zhang (2012), a special attention is paid to the power counting, which is shown to be valid for neutrino energies below 550 MeV. However, the neutrino fluxes of most neutrino experiments span to considerably higher energies. Thus, in Ref. Zhang and Serot (2013), the power counting scheme was abandoned, and the model of Serot and Zhang (2012) was phenomenologically extended to the intermediate energies ( GeV) relevant for the MiniBooNE flux, by including phenomenological form factors. Though the extension proposed for the and the nucleon Compton-like mechanisms seems reasonable, the one for the contact terms notably increases the cross section above GeV (they are more significant for neutrinos than for antineutrinos). Since the contact terms and the associated form factors are not well understood so far, the model predictions for  GeV should be taken cautiously, as explicitly acknowledged in Ref. Zhang and Serot (2013).

In nuclear targets, the reaction can be incoherent when the final nucleus is excited (and fragmented) or coherent, when it remains in the ground state. It is also possible that, after nucleon knockout, the residual excited nucleus decays emitting low-energy rays. This mechanism has been recently investigated Ankowski et al. (2012) and shall not be discussed here. The model of Ref Hill (2010) has been applied to incoherent photon production in an impulse approximation that ignores nuclear corrections Hill (2011). These are also neglected in the coherent case, which is calculated by treating the nucleus as a scalar particle and introducing a form factor to ensure that the coherence is restricted to low-momentum transfers Hill (2010). More robust is the approach of Refs. Zhang and Serot (2012b, a) based on a chiral effective field theory for nuclei, again extended phenomenologically to higher energies Zhang and Serot (2013). In addition to Pauli blocking and Fermi motion, the resonance broadening in the nucleus, is also taken into account. The latter correction causes a very strong reduction of the resonant contribution to the cross section, in variance with our results, as will be shown below. The ratio of the to photon and to decay rates is enhanced in the nuclear medium by an amount that depends on the resonance invariant mass, momentum and also production position inside the nucleus, as estimated with a transport model Leitner et al. (2008); Leitner (2009). The coherent channel has also been studied in Refs. Gershtein et al. (1981); Rein and Sehgal (1981) at high energies. A discussion about these works can be found in Section V.E of Ref. Hill (2010).

It is worth mentioning that both the models of Ref. Hill (2010) and Refs Serot and Zhang (2012); Zhang and Serot (2012b, a, 2013) have been used to calculate the NC events at MiniBooNE with contradicting conclusions Hill (2011); Zhang and Serot (2013). While in Ref. Hill (2011) the number of these events were calculated to be twice as many as expected from the MiniBooNE in situ estimate, much closer values were predicted in Ref. Zhang and Serot (2013). The result that NC events give a significant contribution to the MiniBooNE low-energy excess Hill (2010) could have its origin in the lack of nuclear effects and rather strong detection efficiency correction.

Here we present a realistic model for NC photon emission in the  GeV region that extends and improves certain relevant aspects of the existing descriptions. The model is developed in the line of previous work on weak pion production on nucleons Hernandez et al. (2007) and nuclei for both incoherent Hernández et al. (2013) and coherent Alvarez-Ruso et al. (2007b); Amaro et al. (2009) processes. The model for free nucleons satisfies the approximate chiral symmetry incorporated in the non-resonant terms and includes the dominant excitation mechanism, with couplings and form factors taken from the available phenomenology. Moreover, we have extended the validity of the approach to higher energies by including intermediate excited states from the second resonance region [, and ]. Among them, we have found a considerable contribution of the for (anti)neutrino energies above 1.5 GeV. When the reaction takes place inside the nucleus, we have applied a series of standard medium corrections that have been extensively confronted with experiment in similar processes such as pion Oset et al. (1982); Nieves et al. (1993a), photon Carrasco and Oset (1992) and electron Gil et al. (1997a, b) scattering with nuclei, or coherent pion photo Carrasco et al. (1993) and electroproduction Hirenzaki et al. (1993).

This paper is organized as follows. The model for NC production of photons off nucleons is described in Sec. II. After discussing the relevant kinematics, we evaluate the different amplitudes in Subsec. II.2. In the first place, the dominant and non-resonant contributions are studied (Subsec. II.2.1). Next, we examine the contributions driven by resonances from the second resonance region (Subsec. II.2.2). The relations between vector form factors and helicity amplitudes, and the off-diagonal Goldberger-Treiman (GT) relations are discussed in Appendices A and B, respectively. NC reactions in nuclei are studied in Sec. III. First, in Subsec. III.1, we pay attention to the incoherent channel driven by one particle–one hole (1p1h) nuclear excitations. Next, in Subsec. III.2, the coherent channel is studied. We present our results in Sec. IV, where we also compare some of our predictions with the corresponding ones from Refs. Hill (2010); Zhang and Serot (2013). This Section is split in two Subsections, where the results for NC on single nucleons (Subsec. IV.1) and on nuclei (Subsec. IV.2) are discussed. Predictions for nuclear incoherent and coherent reactions are presented in Subsecs. IV.2.1 and IV.2.2, respectively. Finally the main conclusions of this work are summarized in Sec. V.

Ii Neutral current photon emission off nucleons

In this section, we describe the model for NC production of photons off nucleons,


ii.1 Kinematics and general definitions

Figure 1: Representation of the different LAB kinematical variables used through this work.

The unpolarized differential cross section with respect to the photon kinematical variables (kinematics is sketched in Fig. 1) is given in the Laboratory (LAB) frame by


As we neglect the neutrino masses, , and , where , and are the incoming neutrino, outgoing neutrino and outgoing photon momenta in LAB, in this order; MeV is the Fermi constant, while and stand for the leptonic and hadronic tensors, respectively. The leptonic tensor1


is orthogonal to the four momentum transfer , with . The hadronic tensor includes the non-leptonic vertices and reads


with the nucleon mass2 and the energy of the outgoing nucleon. The bar over the sum of initial and final spins denotes the average over the initial ones. The one particle states are normalized as . Then, the matrix element is dimensionless. For the sake of completeness, we notice that the NC, and electromagnetic (EM), currents at the quark level are given by


where , and are the quark fields and the weak angle (). The zeroth spherical component of the isovector operator is equal to the third component of the isospin Pauli matrices .

By construction, the hadronic tensor accomplishes


in terms of its real symmetric, , and antisymmetric, , parts. Both lepton and hadron tensors are independent of the neutrino flavor and, therefore, the cross section for the reaction of Eq. (1) is the same for electron, muon or tau incident (anti)neutrinos.

Let us define the amputated amplitudes , as


where the spin dependence of the Dirac spinors (normalized such that ) for the nucleons is understood, and is the polarization vector of the outgoing photon. To keep the notation simple we do not specify the type of nucleon ( or ) in . In terms of these amputated amplitudes, and after performing the average (sum) over the initial (final) spin states, we find


After performing the integration, there is still a left in the hadronic tensor, which can be used to perform the integration over in Eq. (2).

ii.2 Evaluation of the amputated amplitudes

The contribution, chiral symmetry and non-resonant terms

Just as in pion production Hernandez et al. (2007), one expects the NC reaction to be dominated by the excitation of the supplemented with a non-resonant background. In our case, the leading non-resonant contributions are nucleon-pole terms built out of and vertices that respect chiral symmetry. The dependence of the amplitudes is introduced via phenomenological form factors. We also take into account the subleading mechanism originated from the anomalous vertex, that involves a pion exchange in the channel. Thus, in a first stage we consider the five diagrams depicted in Fig. 2. The corresponding amputated amplitudes are


with the electron charge, such that , MeV the pion decay constant and the axial nucleon charge; and are the pion and masses, respectively.

Figure 2: Model for photon emission off the nucleon; direct and crossed nucleon-pole terms (a,b), direct and crossed -pole terms (c,d) and the anomalous channel pion exchange term (e). Throughout this work, we denote these contributions as , , , and , respectively.

As it will be clear in the following, each of the building blocks of the model is gauge invariant by construction . The vector parts of these amplitudes are also conserved (CVC) .

and amplitudes: The nucleon NC and EM currents are given by


where and are the NC vector and axial form factors3 while are the EM ones. These form factors take different values for protons and neutrons. For , we have that




where , GeV, , , and  Krutov and Troitsky (2003).

The NC vector form factors can be referred to the EM ones thanks to isospin symmetry relationships,


where are the strange EM form factors. Furthermore, in the axial sector one has that


where is the axial form factor that appears in CCQE interactions, for which we adopt a conventional dipole parametrization


with an axial mass  GeV Bodek et al. (2008); is the strange axial form factor. At present, the best determinations of the strange form factors are consistent with zero Pate and Trujillo (2013), thus they have been neglected in the present study.

amplitudes: The channel pion exchange contribution arises from the anomalous () Lagrangian Hill (2010)


together with the leading order interaction term


where , , , are the nucleon, neutral pion, photon and boson fields, respectively. Besides, is related to the Fermi constant and the -boson mass as ; is the number of colors. The Lagrangian of Eq. (21) arises from the Wess-Zumino-Witten term Wess and Zumino (1971); Witten (1983), which accounts for the axial anomaly of QCD.

and amplitudes: In the driven amplitudes of Eq. (12), is the spin 3/2 projection operator, which reads


is the resonance width in its rest frame, given by


with , the coupling obtained from the empirical decay width (see Table 1); , and is the step function.

The weak NC and EM currents for the nucleon to transition are the same for protons or neutrons and are given by


where and ; , and are the NC vector, NC axial4 and EM transition form factors, respectively. As in the nucleon case, the NC vector form factors are related to the EM ones


according to the isovector character of the transition. These EM form factors (and couplings) can be constrained using experimental results on pion photo and electroproduction in the resonance region. In particular, they can be related to the helicity amplitudes , and  Lalakulich et al. (2006); Leitner et al. (2009a) commonly extracted in the analyses of meson electroproduction data. The explicit expressions are given in Appendix A. For the helicity amplitudes and their dependence we have taken the parametrizations of the MAID analysis Drechsel et al. (2007); MAID ()5 In the axial sector, we adopt the Adler model Adler (1968); Bijtebier (1970)


for the subleading (in a expansion) form factors and assume a standard dipole for the dominant


with and  GeV fixed in a fit to BNL and ANL data Hernandez et al. (2010a).

The second resonance region

Here, we extend the formalism to the second resonance region, which includes three isospin 1/2 baryon resonances , and (see Table 1). In this way, we extend the validity of the model to higher energies. A basic problem that has to be faced with resonances is the determination of the transition form factors (coupling constants and dependence). As for the , we obtain vector form factors from the helicity amplitudes parametrized in Ref. Drechsel et al. (2007). The equations relating helicity amplitudes and form factors are compiled in Appendix A. Our knowledge of the axial transition form factors is much poorer. Some constraints can be imposed from PCAC and the pion-pole dominance of the pseudoscalar form factors. These allow to derive off-diagonal Goldberger-Treiman (GT) relations between the leading axial couplings and the partial decay widths (see Table 1 and Appendix B for more details).

[MeV]   [MeV] or
1232 3/2 3/2 + 117 100%  6
1440 1/2 1/2 + 300 65% 0.47
1520 3/2 1/2 115 60% 2.14
1535 1/2 1/2 150 45% 0.21
Table 1: Properties of the resonances included in our model Beringer et al. (2012). For each state, we list the Breit-Wigner mass () , spin (), isospin (), parity (), total decay width (), and axial coupling (denoted for spin 1/2 states and for spin 3/2 states).

For each of the three , and states, we have considered the contribution of direct () and crossed () resonance pole terms as depicted in Fig. 3.

Figure 3: Direct (a) and crossed (b) pole contributions to the NC photon emission process. We have considered the three resonances right above the

and : The structure of the contribution of these two resonances to the amputated amplitudes is similar to the one of the nucleon [Eq. (10)]. We have


the resonance masses are listed in Table 1 while the widths are discussed in Appendix C. The EM and NC currents read


for the and


for the 7 As in the nucleon case, isospin symmetry implies that


with expressed in terms of the corresponding helicity amplitudes (see Appendix A). The NC axial form factors are


The couplings are obtained from the GT corresponding relations and have values given in Table 1. The dependence of these form factors is unknown so we have assumed a dipole ansatz with a natural value of GeV for the axial mass. No information is available about the strange form factors but they are likely to be small and to have a negligible impact on the observables, so we set them to zero.

: In this case, the structure of the contribution of this resonance to the amputated amplitudes is similar to that of the , differing just in the definition of the appropriate form factors and the isospin dependence. Thus, we have


where the resonance mass is given in Table 1 and the width is discussed in Appendix C; is the spin 3/2 projection operator given also by Eq. (23), with the obvious replacement of by . Besides, the EM and NC transition currents are given by


where and ; , and are the NC vector, NC axial and EM form factors, respectively. The NC vector form factors are related to the EM ones in the same way as for the other isospin states considered above, namely


where are obtained from the helicity amplitudes using Eqs. (87-89). For the axial form factors, one again has that


We take a standard dipole form for the dominant axial NC form factor


with from the corresponding off diagonal GT relation (see Appendix B and Table 1), and set GeV as for the other . The other axial form factors are less important because their contribution to the amplitude squared is proportional to . We neglect them together with the unknown strange vector and axial form factors.

Iii Neutral current photon emission in nuclei

In this section we outline the framework followed to describe NC photon emission off nuclei. Both incoherent and coherent reaction channels are considered.

iii.1 Incoherent neutral current photon emission

To study the incoherent reactions


we pursue the many body scheme derived in Refs. Nieves et al. (2004, 2006, 2011) for the neutrino propagation in nuclear matter and adapted to (semi)inclusive reactions on finite nuclei by means of the local density approximation. With this formalism, the photon emission cross section is


in terms of the leptonic tensor of Eq. (3) and the hadronic tensor , which is determined by the contributions to the selfenergy with a photon in the intermediate state


In the density expansion proposed in Ref. Nieves et al. (2004), the lowest order contribution to is depicted in Fig. 4. The black dots stand for any of the eleven terms (, , , , with , , , ) of the elementary amplitude derived in Sec. II. The solid upwards and downwards oriented lines represent nucleon particle and hole states in the Fermi sea.

Figure 4: Diagrammatic representation of the one-particle-one-hole-photon (1p1h) contributions to the self-energy in nuclear matter. The black dots represent amplitudes.

This selfenergy diagram (actually 121 diagrams) is readily evaluated as 8


where and is the amputated amplitude for the process


The nucleon propagator in the medium reads