Incoherent Neutrinoproduction of Photons and Pions in a Chiral Effective Field Theory for Nuclei
We study the incoherent neutrinoproduction of photons and pions with neutrino energy . These processes are relevant to the background analysis in neutrino-oscillation experiments [for example, MiniBooNE; A. A. Aquilar-Arevalo et al. (MiniBooNE Collaboration), Phys. Rev. Lett. 100, 032301 (2008)]. The calculations are carried out using a Lorentz-covariant effective field theory (EFT), which contains nucleons, pions, the Delta (1232) (), isoscalar scalar () and vector () fields, and isovector vector () fields, and has chiral symmetry realized nonlinearly. The contributions of one-body currents are studied in the local Fermi gas approximation. The current form factors are generated by meson dominance in the EFT Lagrangian. The conservation of the vector current and the partial conservation of the axial current are satisfied automatically, which is crucial for photon production. The dynamics in nuclei, as a key component in the study, is explored. Introduced -meson couplings explain the spin-orbit coupling in nuclei, and this leads to interesting constraints on the theory. Meanwhile a phenomenological approach is applied to parametrize the width. To benchmark our approximations, we calculate the differential cross sections for quasi-elastic scattering and incoherent electroproduction of pions without a final state interaction (FSI). The FSI can be ignored for photon production.
pacs:25.30.Pt; 24.10.Jv; 11.30.Rd; 12.15.Ji
This paper is a continuing work of bookchapter (); 1stpaper (), focusing on neutrinoproduction of photons and pions from nuclei with neutrino energy . In Refs. bookchapter (); 1stpaper (), we introduced the resonance as a manifest degrees of freedom to the effective field theory (EFT), known as quantum hadrodynamics or QHD SW86 (); SW97 (); Furnstahl9798 (); FSp00 (); FSL00 (); EvRev00 (); LNP641 (); EMQHD07 (). (The motivation for this EFT and some calculated results are discussed in Refs. SW97 (); Furnstahl9798 (); HUERTAS02 (); HUERTASwk (); HUERTAS04 (); MCINTIRE04 (); MCINTIRE05 (); JDW04 (); MCINTIRE07 (); HU07 (); MCINTIRE08 (); BDS10 ().) To calibrate the reaction mechanism on the nucleon level, we studied the productions from nucleons 1stpaper (). The calculations are motivated by the fact that the neutrinoproductions of and photons from nuclei (and nucleons) are potential backgrounds in neutrino-oscillation experiments (e.g., MiniBooNE MiniBN2007 (); MiniBN2009 (); MiniBN2010 ()). Currently, it is still a question whether the neutral current (NC) photon production might explain the excess events seen at low reconstructed neutrino energies, which the MicroBooNE experiment plans to answer MicroBN2011 (). Moreover, the authors of Refs. Harvey07 (); Harvey08 (); Hill10 (); Gershtein81 () point out the possible role of anomalous interaction vertices involving , , and the photon in NC photon production. So it is necessary to calculate the cross sections for these processes. Here by using the QHD EFT, we study incoherent production, in which the nucleus is excited. Coherent production with the nucleus being intact is a topic of future work. 111Recently, a unified framework for handling both coherent and incoherent production has been proposed in Martini09 (). We will discuss the power-counting 222In an EFT, there are an infinite number of interaction terms allowed by various constraints. To organize them, we can associate power-counting to each vertex and diagram. The calculation can be done in a perturbative way by summing diagrams up to some particular power . See Refs.Furnstahl9798 (); MCINTIRE07 (); HU07 (); MCINTIRE08 (); BDS10 (); bookchapter (); 1stpaper () for detailed discussions about power-counting in QHD EFT. of the calculations through which we will show that the contributions of the anomalous interactions are small in the incoherent NC production of photons (where they contribute at next-to-next-to-leading-order). To benchmark the approximation scheme, we study electron scattering in both quasi-elastic and pion production channels.
There have been several experiments measuring the weak response of nuclei across the quasi-elastic region to the excitation peak. In most experiments k2k05plb (); k2k05prl (); k2k06prd (); k2k08prd (); Sciboone08prd (); Miniboone08prl (); Miniboone08plb (); Miniboone09prl (), which have and as the primary target nuclei, the mean energy of the beam is around . As emphasized in 1stpaper (), we expect our theory to work up to GeV, so we do not rely on these experiments to constrain the theory at this stage. On the theoretical side, much work has been done (e.g., in Martini09 (); Butkevich08 (); Butkevich09 (); Singh92 (); Singh93 (); Meucci04_1 (); Meucci04_2 (); Nieves04 (); Martinez06 (); GiBUU2006cc (); GiBUU2006nc (); GiBUU2009 (); Kartavtsev06 (); Praet09 (); GiBUUprc79_057601 (); Singh98 (); Sato03 (); Szczerbinska07 (); Amaro05 (); Caballero05 (); Amaro07 (); Martini07 (); Amaro07_prc (); Ivanov08 ()). Most of these papers are based on the global or local Fermi gas approximation and include contributions from one-body currents, with improved treatment for final-state interaction (FSI) and dynamics in the medium. The same approach has also been applied in electron scattering (e.g., in Gil97 ()). In Amaro05 (); Caballero05 (); Amaro07 (); Martini07 (); Amaro07_prc (); Ivanov08 (), scaling approaches are used to address quasi-elastic scattering. Moreover, the contribution from two-body currents was studied nonrelativistically, for example, in Alberico84 (). In most of these calculations, the dynamics in nuclei is based on the work of oset87 (), in which the self-energy has been studied using a nonrelativistic model. Parallel to the nonrelativistic studies, some work has been initiated in the relativistic framework, QHD EFT, using the local Fermi gas (LFG) approximation and including one-body currents Rosenfelder80 (); wehrgerger89 (); wehrgerger90 (); wehrgerger92 (); wehrgerger93 (). The two-body current was investigated relativistically in Dekker94 (); DePace03 (). These works mainly focus on electron scattering. But the handling of the resonance in these papers is somewhat phenomenological. Moreover, in both nonrelativistic and relativistic studies, photon production is rarely investigated.
In this paper, we also apply the LFG approximation Rosenfelder80 () to study the one-body current contribution. As shown in bookchapter (); 1stpaper (), we make use of meson dominance to generate form factors for various currents. Because of the built in symmetries in the Lagrangian, conservation of vector current and the partial conservation of axial current are satisfied. These properties are well preserved in the LFG approximation. Especially for photon production, vector current conservation is crucial. The dynamics, as a key component in this work, is explored to some extent. We introduce interactions between and non-Goldstone meson fields to generate the spin-orbit (S-L) coupling that has been introduced in phenomenological models horikawa80 (); Nakamura10 (). On the other hand, phenomenological knowledge about S-L coupling puts constraints on these couplings. Moreover, the decay width increases in the nucleus, because more decay channels are opened up and this effect overcomes the reduction of pion decay phase space. Here we follow the phenomenological studies and separate the width to the pion decay width and anything else parametrized by the imaginary part of the spreading potential. As a result of opening new decay channels, the flux having excited a resonance can be transferred to channels that do not involve pion or photon production. Moreover, Pauli blocking can reduce the pion and photon production cross section further, because of the reduction of the final particle’s phase space. 333The binding effect should be important when the neutrino energy is close to threshold, where the simple approximations used here are not feasible. But this is clearly not important around 0.5 GeV. In this paper, we explore how both and nonresonant contributions are reduced compared to those in free nucleon scattering. However, we do not include FSI effects for pions and knocked out nucleons. The simple treatment can be found in Adler79 (); paschos00 (), while the complete treatment is implemented in various event generators of experiments (e.g., NUANCE nuance ()), and the GiBUU model GiBUU2006cc (); GiBUU2006nc (). Hence we only compare our predictions with the output of NUANCE without FSI. 444The predictions from NUANCE shown throughout this paper are obtained from the NUANCE v3 event generator nuance (). Multiple resonances are considered in NUANCE, but the dominates. The axial mass GeV is used which is the same as that used by the MiniBooNE experiment for their baseline calculations MBCCQE (). However, the actual backgrounds used in their final analyses were scaled to data in a separate exercise.
The paper is organized as follows. In Sec. II, we first discuss the LFG approximation and then apply it to electron quasi-elastic scattering, which serves as a benchmark. In Sec. III, the calculation scheme for pion (photon) production is briefly introduced. Then the dynamics is studied with emphasis on the connection between -meson interactions and S-L coupling. The modification of the width is also discussed. After that, electron scattering at the peak is studied, and results are compared with data with explanation of the missing strength. The cross sections of neutrinoproduction of pions are also shown and compared to NUANCE’s output. Sec. IV is dedicated to NC photon production. Finally Sec. V contains a short summary. In the appendices, we show detailed kinematic analyses for both quasi-elastic scattering and pion production.
Ii Quasi-elastic scattering in the LFG approximation
This section serves as an illustration of the LFG approximation used for quasi-elastic scattering and for photon and pion production. (See Ref. 1stpaper () for discussion on the free nucleon interaction amplitude in all these processes.) Here we make use of the mean-field approximation to calculate the nuclear ground state. The relevant leading order Lagrangian is
(where the full Lagrangian can be found in bookchapter (); Furnstahl9798 () for example). The mean-field approximation is presented simply as follows. Inside nuclear matter, vector and , and scalar fields develop nonzero expectation values. In the laboratory frame of the matter, only two fields ( and ) have nonzero values (but in the isospin asymmetric case, can also develop a nonzero value). As a result, the nucleon’s mass is modified: . At the lowest order, the spectrum of nucleons is . Inside a finite nucleus, due to different boundary conditions, the mean-field expectation value is space dependent and can be calculated numerically. By using this approximation, we can calculate the bulk properties of the nucleus, the details of which can be found in Ref. Furnstahl9798 () for example.
Following Furnstahl9798 (), we calculate the local density and field expectation value in (the major nucleus in the MiniBooNE’s detector). Figs. 1 and 2 show the results based on and parameter sets in Furnstahl9798 (). We will explore the difference due to the two sets in electron quasi-elastic scattering.
To calculate the electroweak response of nuclei, we use the LFG approximation. This approach has been applied in Rosenfelder80 () to study electron quasi-elastic scattering. First, by assuming the impulse approximation (IA), the interaction happens every time between probe and each individual nucleon This only holds when the transferred momentum is high enough that the interference between different nucleons is reduced due to the big recoil. Second, the response of the nucleus is the incoherent sum of the response of the fermion gas in different regions. This works when the probe’s wave length is small enough compared to a characteristic length scale of the nucleus density profile. The discussion can be summarized in the following equation:
In this equation, and are the incoming and outgoing lepton momenta, respectively, is the momentum transfer, and are the scattered nucleon’s initial and final momenta and spin projection, and is the one-body interaction amplitude. The kinematic configuration is shown in Fig. 3. The integration over initial and final nucleon momenta depends on the space dependent Fermi momentum. A detailed discussion about this equation can be found in Appendix A.
The interaction amplitude in Eq. (2) can be expressed in terms of various current matrix elements (where , , and are, respectively, the isovector vector current, the isovector axial-vector current and the baryon current, bookchapter (); 1stpaper ()). For electron quasi-elastic scattering,
where and in the state are nucleon isospin. For charged current (CC) quasi-elastic scattering,
where , is the Fermi constant and is the and quark mixing element in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. For NC quasi-elastic scattering,
where is the weak mixing angle. The electroweak currents of leptons are well known, and can be found in 1stpaper (). But we need to include the nucleon spectrum modification to the results of 1stpaper (), which is straightforward to complete in the LFG approximation.
A short discussion on FSI is in order here. The picture is the following: the interaction channels are opened in the initial interacting vertex, and then these channels would couple to each other when particles are traveling through the nucleus. The flux among all the initial channels are redistributed due to FSI. This picture is adopted in the GiBUU model calculations for example for CC and NC processes GiBUU2006cc () GiBUU2006nc (). From conservation of probability, assuming the picture mentioned above is valid, we should expect the sum of these channels in the initial vertex to match the inclusive data. Moreover, Coulomb distortion of the electron is not included in this calculation.
In the upper panels of Figs. 4, 5, and 6, we present differential cross sections for electron scattering off at given electron energies and scattering angles. In this section we only focus on the so-called quasi-elastic peak at the lower energy region, which is believed to be dominated by one nucleon knock out. The higher energy peak will be discussed in Sec. III. In Fig. 4, the electron energy is , and the scattering angle is . The plots “G1” and “G2” are the calculations done with G1 and G2 parameter sets Furnstahl9798 (). The difference between the two is small. The data are from Ref. Barreau83 (). The validity of the form factors realized by meson dominance needs to be discussed here. In this figure, and at the peak. Below the peak, is slightly higher than , and above the peak, . As discussed in 1stpaper (), meson dominance works when , and hence it can be applied here. This is also true for Figs. 5 and 6. In Fig. 5, the electron energy is , and the scattering angle is . In Fig. 6, the electron energy is , and the scattering angle is . The data are from Barreau83 () and Connell87 (). Again, we see only small differences between the “G1” and “G2” parameter sets.
Iii pion production
iii.1 Approximation scheme and dynamics in the nuclear medium
By using the LFG approximation detailed before, the formula for the cross section can be written as
The details can be found in Appendix B. The notations for various momenta are explained in Fig. 7. All the integrations except the volume integration depend on the space coordinate through the space dependent Fermi momentum. The amplitude in Eq. (6) is similar to those in Eqs. (3), (4), and (5), except that the hadronic currents should be changed to those relevant to pion production:
Several Feynman diagrams contribute here, as shown in Fig. 8, including diagrams with the [(a) and (b)] and all the rest which we define as nonresonant diagrams. See Ref. 1stpaper () for details about them. Among the medium-modifications of the matrix elements, the behavior of the needs to be singled out. First, let us focus on the real part of the self-energy. We start from the following Lagrangian (and a similar Lagrangian can be found in wehrgerger93 ()):
Here the field is given by the Rarita-Schwinger representation, and are isospin indices bookchapter (). At the normal nuclear density, the is not stable in the nuclear medium. So the expectation values of meson fields are not changed in normal nuclei at the mean-field level. Similar to the nucleon case, the spectrum in nuclear matter (without the -pion interaction) is given by
The effect of introducing and couplings on the equation of state (EOS) was analyzed in wehrgerger89 (); boguta81 (); kosov98 (). Some constraints on the couplings, and , were calculated in wehrgerger89 (); kosov98 (). Here we resort to the scattering problem to find other constraints. In pion-nucleus scattering studies horikawa80 (); Nakamura10 (), S-L coupling of the inside the nucleus was introduced by hand, although its origin is not clear in the nonrelativistic model. In this model, a mechanism similar to the generation of the nucleon’s S-L coupling is used to generate ’s. Following discussions in Furnsthal98 () and using the Lagrangian in Eq. (7), we can estimate the S-L coupling of the as
In Fig. 9, we compare our estimates of defined in Eq. (8) with two different phenomenological fits for . We can see that our estimates based on three different parameter sets, , are consistent with the “Nakamura” result in Nakamura10 (), while the “Horikawa” result in horikawa80 () is significantly larger than the “Nakamura” result when . Meanwhile, all the couplings are consistent with the “naturalness” assumption, which also motivates our choice of . We do not show the consequence of , since there is no S-L coupling generated in this case.
Second, we turn to the imaginary part of the self-energy. It is known that Pauli blocking effects decrease the width due to reduction of the pion-decay phase space, while the collision channels, for example, increase the width. The two competing processes have been investigated in nonrelativistic models. At normal nuclear density, the net result is to increase the width oset87 (). In phenomenological fits hirata79 () horikawa80 (), this increase is taken into account by introducing a density-dependent complex spreading potential for the . Here we follow this approach. Above the pion threshold,
is the pion-decay width oset87 (); wehrgerger90 (). 555In the calculation, only Pauli-blocking is considered. Modifications of the real part of the nucleon and self-energies are not included. () is the width in other channels, and it has been fitted in horikawa80 (); Nakamura10 (). Below the pion threshold (which is useful in photon production),
This procedure turns out to work qualitatively, as will be shown later. Furthermore, in Gil97 (), the self-energy calculated in oset87 () is used for inclusive electron scattering off nuclei. In Sec. III.2, we will compare our results using Eqs. (9) and (10) with those using Eq. (11) and the width in Gil97 (); oset87 ().
iii.2 Pion electroproduction
Here we focus on the region beyond quasi-elastic scattering in the upper panels of Figs. 4, 5, and 6. It is believed that the second peak mainly comes from the excitation inside the nucleus. In the upper panels of these figures, we provide our pion-production results (without FSI) due to six different calculations. We include the full set of Feynman diagrams in the first five calculations, and diagrams with the in and channels in the sixth. The difference among the first three calculations is the choice of parameter sets: , , and . In these three, the width shown in Eqs. (9) and (10) is applied. In the fourth calculation, we set and apply the constant shift of the width as shown in Eq. (11). The fifth calculation is done by using the self-energy as calculated in oset87 (); Nieves93 (), which is essentially repeating the calculations in Gil97 (). The sixth calculation has and uses the same width as used in the first three.
First, let us discuss the location of the -peak along the axis. The different choice of and indicates different binding potentials for . For , the real part of the self-energy is the same as in vacuum without any binding. For , has the same attractive potential as the nucleon. The vector part tuned by provides a repulsive potential. So we can see that has a deeper binding potential than . Hence, is less bound than and is less bound than . In Figs. 4, 5, and 6, the location of the peak in first three calculations indeed follows this argument. (We can estimate the location of the peak in a global Fermi gas model. 666Following Rosenfelder80 (), for , we assume a global Fermi gas, with the nucleon effective mass , , (in the laboratory frame), and . Meanwhile in the channel, the momentum is . It is easy to calculate the -location of the peak by setting . For and (see Fig. 4), if and if .) The fourth calculation with and constant width does not give the correct peak position in the three figures: It underestimates (overestimates) ’s contribution on the left (right) side of the peak, because the constant width assumption overestimates (underestimates) ’s width on the left (right) side. The fifth calculation by using modification calculated in oset87 (); Nieves93 () gives the correct location of the peak. Comparing the second with the sixth calculation, we can see the significance of nonresonant contributions (they use the same set of parameters and the width).
However, the pion production channel could not explain the full strength of the peak. Meanwhile in the “dip” region between the quasi-elastic scattering and the peak, the calculations also miss strength. This indicates we miss other channels from dip region to the peak. Missing strength at the peak position can be qualitatively explained by considering the fourth calculation in the upper panels, whose simple treatment of the width makes analysis transparent. According to Eqs. (9) and (11), we estimate in the sense of averaging over , and hence . The comparable width of other decay channels shows their importance to the inclusive data. Moreover, there are contributions from two-body currents without as an intermediate state. In the lower panels of Figs. 4, 5, and 6, we add up three different channels: quasi-elastic, pion production, and two-body current contributions [labeled as meson-exchange-current (MEC) in the plots]. The label “” for pion production applies under the assumption that and that no new channel takes away the flux from the pion production. The MEC-contributions are from DePace03 () donnelly2012 (). Here the total strength matches well with the inclusive data. However a detailed study of different channels in the QHD EFT framework is needed to address this issue conclusively.
The difference between our calculations and those in wehrgerger89 () where the QHD model is also applied should be mentioned here. Ours are strictly based on the field theory, while in wehrgerger89 () the is introduced by hand (where they were convoluted with the cross section based on a “stable” theory with a Lorentzian weight function). Moreover, we take into account the contribution from other diagrams, which are not considered in wehrgerger89 (). The two results are different somewhat, but our choice is consistent with the analysis in wehrgerger89 ().
Moreover, we can see that the differences in cross sections obtained using are not significant, which indicates that the total cross sections of neutrinoproduction processes are not sensitive to them either. This will be confirmed by the results in Sec. III.3.
iii.3 CC and NC pion production
Fig. 10 shows the total cross section averaged over proton or neutron number for CC pion production in (anti)neutrino– scattering. We also compare our result with NUANCE’s output without FSI. In each figure, our calculations including different diagrams and using different and are shown. The “only ” calculation only takes into account diagrams. In the others, all the diagrams up to are included. Systematically in all the channels, our “only ” calculation is close to the NUANCE output. But other diagrams contained in “” calculations are not negligible in all the channels around the resonance region, especially when the -channel contribution is suppressed by the small Clebsch-Gordan coefficients (for example, ). In the very low energy region away from the resonance, nonresonant diagrams dominate. See 1stpaper () for the power counting of diagrams. Moreover, we check that the contributions of higher order () diagrams are tiny. We also can see that below 0.5 GeV, the results are bigger than the results and the results are bigger than the results. Following the discussions in Sec. III.2 about the location of the peak in pion electroproduction, we expect that, at a given energy, excitation occupies more phase space in than in and more in than in . So the pattern among the three different calculations is consistent with the qualitative analysis. Here is presented simply for the purpose of comparison, and its conflict with the S-L coupling is presented in Sec. III.1.
One question needs to be raised: Do we have -dominance in the nuclear scattering around GeV ? If we compare the “” calculations with “only ” calculations in every channel, the answer is no. It turns out that the contribution is strongly reduced due to the broadening of its width, compared to its contribution in free nucleon scattering. Meanwhile nonresonant contributions are reduced by Pauli blocking. To see this qualitatively, compare our results here with the cross sections shown in 1stpaper () for production from free nucleons. In 1stpaper (), two different calculations can be found, including “Only ” and “”. We just show the total cross sections at GeV in Tab. 1 for neutrino scattering. For example, “” indicates the channel . “(f)” and “(b)” correspond to scattering from free nucleons and from bound nucleons in respectively. In both “only (b)” and “ (b),” (and note that calculations with only and are not shown in the figures). “Nonresonant (b)” is the difference between the two, and can be viewed qualitatively as the contributions of the nonresonant diagrams. 777In principle, there are interferences between contributions from and other diagrams. At GeV, we can assume in most of phase space that the is “on shell” while contributions from other diagrams are real, and hence the interferences are small. The labeling for free nucleon scattering is the same. We can see that the contribution in nuclear scattering has been reduced systematically by around 50% in all channels, compared to its contribution in nucleon scattering; the nonresonant contributions are also strongly reduced. Clearly, the nonresonant contributions are not negligible in both nucleon and nuclei scattering. The same situation occurs in the antineutrino scattering channels and hence are not shown explicitly. This underscores the importance of including nonresonant contributions in CC pion production.
|only (f)||(f)||Nonresonant (f)||Only (b)||(b)||Nonresonant (b)|
In Fig. 11, we show the total cross section for NC pion production from . The categorization of the different calculations are the same as those for CC scattering. Again the NUANCE output is close to our “only ” calculation. Among the first three calculations in each channel, at fixed (anti)neutrino energy, gives a larger cross section than and gives a larger cross section than . This is the same as in the CC production, which has been explained in terms of kinematics. Moreover, we can see how -dominance is violated in the NC case, as shown in Tab. 2 (in which the labelings are the same as those in Tab. 1, and free nucleon scattering results are from 1stpaper ()). The same is true for antineutrino–nucleus scattering.
|only (f)||(f)||Nonresonant (f)||Only (b)||(b)||Nonresonant (b)|
Iv NC photon production
In this section, we study NC photon production from . The calculation is done in the same way as in pion production, except that the hadronic current in Eq. (5) is changed to the following:
The Feynman diagrams are the same as those in Fig. 8 with the final line substituted by the final line. See Ref. 1stpaper () for detailed discussion about them. Again we need to implement the change of the baryon spectrum when we apply the formula in 1stpaper (), as we do in previous calculations. Because of built in symmetries in our model, conservation of the vector current is automatically satisfied, which is important for photon production. The difference in the kinematic analysis, compared to that in pion production, is due to the zero mass of the photon. Moreover, we apply an energy cut on the photon energy in the laboratory frame, , motivated by the MiniBooNE’s detector efficiency. This also eliminates the infrared singularity and simplifies the calculation.
In Fig. 12, the total cross sections averaged over proton or neutron number are shown. Four different calculations are compared. The first “only ” is the same as before. “” calculations include all the diagrams. It turns out no contact diagrams contribute, and there are only two contact vertices contributing (See Ref. 1stpaper () for details):
As we have checked, the contributions of these two are small compared to those of the and existing nonresonant diagrams, which should be expected according to the power-counting. Here, we have assumed their strength are due to both the and meson anomalous interaction vertices ( and ) 1stpaper (); Hill10 (). Moreover for these calculations, changing and does not change the total cross section significantly, which is also observed in the differential cross section for pion electroproduction. In the three calculations for different channels, gives a bigger cross section than and is bigger than . This pattern has been explained in pion production. We also see that the NUANCE output is close to the “only ” calculation and smaller than the full calculations, which should be expected from the comparison in pion production.
In addition, in Tab. 3 we show how the significance changes from neutrino–nucleon scattering to neutrino–nuclus scattering (free nucleon scattering results are from 1stpaper () with a change on photon energy cut: GeV): the contribution is strongly reduced, and the nonresonant contribution is reduced less significantly. Since we have put a constraint on the minimum photon energy, the lower energy events are not included in the results and the Pauli blocking effect is not significant. That explains why the nonresonant contribution is not quite suppressed. And the reduction of the contribution is mainly due to the broadening of its width. We also expect the Pauli blocking effect to be less significant with higher energy neutrinos. Furthermore, the same pattern about the reduction of cross sections happens in antineutrino scattering. Based on Tab. 3, we need to include nonresonant contributions in photon production, as emphasized in pion production.
|only (f)||(f)||Nonresonant (f)||Only (b)||(b)||Nonresonant (b)|
Neutrinoproduction of photons and pions from nuclei provides an important background in neutrino-oscillation experiment and must be understood quantitatively. Especially, we are interested in the possible role of NC photon production in the excess events seen in the MiniBooNE experiment at low reconstructed neutrino energy. In Ref. 1stpaper (), we have calibrated our theory—QHD EFT with introduced—by calculating photon and pion production from free nucleons up to . In this work, the theory is applied to study the production from nuclei. Here we make use of the LFG approximation and Impulse Approximation, and include only one-body current contributions. In the mean-field approximation of the nuclear ground state, the change of the baryon spectrum is represented by introducing an effective mass for baryons, which leads to the change of one-body currents in this calculation. The calculation for electron quasi-elastic scattering and electroproduction of pion serves as a benchmark for our approximation schemes. We then proceed to calculate the neutrinoproduction of pion and photon from , and show the plots for total cross section in every channel. First, we present calculations for pion production up to next-to-leading-order with different and parameters as constrained by the phenomenological study. It turns out that total cross sections are not very sensitive to changes of these parameters. Then in NC production of photon, although we show the result up to order, there are no contributions from contact terms, and as we have checked already the contributions due to and , related to the so-called anomalous interactions, are tiny (the same has been shown for nucleon scattering in 1stpaper ()). Again, the total cross section of photon production is not sensitive to choice of different and . In all the plots, the contributions are singled out and compared with the full calculations. Moreover, we also compare our results with the output from NUANCE, and we find that the NUANCE output is close to our “only ” calculation with for both pion and photon production, which should be expected since the dominates in NUANCE.
In the calculation, the dynamics in nuclei is a key component. The dynamics has been investigated in a nonrelativistic framework and also initiated in the QHD model. Parallel to the modification of the nucleon’s spectrum, the -meson couplings (related to and ) introduced in our theory dictates the real part of the self-energy. The couplings are used to explain the S-L coupling of . Meanwhile the phenomenological result about S-L coupling based on nonrelativsic isobar-hole models puts an interesting constraint on the -meson coupling strengths, which is complementary to the constraints based on an EOS consideration. The width is treated in a simplified way, as we take advantage of the existing result that shows an increase of the width due to the opening of other decay channels. In pion electroproduction, the pion-production (without FSI) result gives a correct prediction for the location of the -peak. We argue that this deficit is due to the absence of other channels. By adding contributions from two-body currents (from other relativistic studies) to our quasi-elastic and pion production (and turning off broadening), we can explain the inclusive electron scattering strength. The investigation on dynamics and two-body currents, which plays an important role in nuclear response and other problems, certainly needs to be pursued further in QHD EFT.
Moreover, because of the broadening of the width, we expect that in both pion and photon productions, the contribution is much less in nuclear scattering than in nucleon scattering. But the reduction of nonresonant contributions would be less at higher energies (beyond 0.5 GeV), because the Pauli blocking effect should be less important. In Tabs. 1, 2, and 3, we have shown explicitly the cross sections at GeV due to and nonresonant contributions in both neutrino–nucleon and neutrino–nucleus scattering. Although we see the reduction of nonresonant contributions for pion production in Tabs. 1 and 2, we see a smaller reduction for photon production in Tab. 3. This is consistent with the picture that the nonresonant contribution is reduced because of Pauli blocking. The same situation occurs in antineutrino scattering. This conclusion is important for future investigations of higher energy neutrino scattering, which may be relevant to MiniBooNE’s excess event problem.
Since our calculation is based on a QHD EFT Lagrangian with all the relevant symmetries built in, conservation of vector current is manifest. This is crucial for photon production. Also partial conservation of the axial current is a necessary constraint in the problem. By using the mean-field approximation and the LFG model, these constraints are satisfied in a transparent way.
We are currently working on coherent pion and photon production from nuclei by applying this QHD EFT, which may also be relevant to the MiniBooNE low energy excess event problem.
Acknowledgements.XZ would like to thank T. William Donnelly, Gerald T. Garvey, Joe Grange, Charles J. Horowitz, Teppei Katori, J. Timothy Londergan, William C. Louis, Rex Tayloe, and Geralyn Zeller for their valuable information, useful discussions, and important comments on the manuscript. This work was supported in part by the Department of Energy under Contract No. DE–FG02–87ER40365.
Appendix A kinematics for quasi-elastic scattering
The analysis of the kinematics is for scattering from nuclear matter, and can be easily generalized in the LFG model. The kinematic variables are shown in Fig. 3, and discussed following Eq. (2). From the mean-field theory in QHD EFT, we know that the leading order Hamiltonian gives rise to the nucleon spectrum in nuclear matter as , . Then we can define . This can be generalized from the laboratory frame to an arbitrary frame. In the LFG model, we consider each neighborhood inside the nucleus as a homogeneous system; the field expectations, and , are space-time dependent (and in the laboratory frame, they only depend on the space coordinate). In the following, we always work in the nuclear laboratory frame. The covariance of our calculation is more transparent with the variables than with . For example energy momentum conservation is .
Next we derive the formula for the total cross section. Suppose is the covariant interaction amplitude between the probe and each individual nucleon with specific initial and final states. We have
Pauli blocking leads to constraints on the integration of and , i.e. and . Here is the Fermi momentum related with the local density. The two constraints can be expressed by using factors and . In the following, we will not include them explicitly. We know that
By using this, we have the total cross section as
Meanwhile to make our phase space analysis simple, we can integrate over and :
Now, we need to calculate the boundary of the phase space in Eq. (14). From the lepton kinematics, we can determine the boundary of ():
For a given , we have the following constraints based on the lepton kinematics:
However, there are further constraints on for a given due to the hadron kinematics. For a given set of , , has to be physical. This requires
Moreover, the constraint is not present in the former discussion, but is taken care of in the numerical calculation.
Appendix B kinematics for pion production
The kinematic variables are defined in Fig. 7 in the laboratory frame. Except for the momentum , all the others are defined in Appendix A. The variables defined in other frames will be mentioned explicitly. First we have
The constraints on and , i. e. and , are always implicit in the formula.
One way to think about the phase space as follows: Given specific values for and , the final pion and nucleon invariant mass are fixed, and then the degrees of freedom in the isobaric frame (final pion and nucleon’s center-of-mass frame) is the angle of , i.e. . So we have
In the above, we have made use of the following identities: