Neutrino versus antineutrino cross sections and CP violation

We discuss the nuclear interactions of neutrinos versus those of antineutrinos, a relevant comparison for CP violation experiments in the neutrino sector. We consider the MiniBooNE quasielastic-like double differential neutrinos and antineutrinos cross sections which are flux dependent and hence specific to the MiniBooNE set-up. We combine them introducing their sum and their difference. We show that the last combination can bring a general information, which can be exploited in other experiments, on the nuclear matrix elements of the axial vector interference term. Our theoretical model reproduces well the two cross sections combinations. This confirms the need for a sizeable multinucleon component in particular in the interference term.


I. INTRODUCTION
One of the challenging goals of neutrino experiments is the detection of CP violation in the neutrino sector. A convincing evidence would be the detection of an asymmetry between the oscillation rates of muon neutrinos and antineutrinos into electron neutrinos since, in the absence of CP violation, these rates are the same. For this test one needs a muonic neutrino beam and an antineutrino one. The electrons (positrons) produced in a far detector by the charged current interaction of the electron neutrinos (antineutrinos) are the signature for the ν µ → ν e (ν µ →ν e ) oscillation process, once the direct ν e (ν e ) background is eliminated.
Several obstacles can stand on the way of the detection of CP violation through the ν,ν asymmetry. One is that the interactions of neutrinos and antineutrinos with any nucleus are not identical but they differ by the sign of the axial vector interference term, creating an asymmetry unrelated to CP violation and which has to be fully mastered. This is not a trivial task due to the complexity of the nuclear dynamics. It reflects in the neutrino interactions and may obscure the message that one wants to extract on the oscillation mechanism.
One example concerns the role of the multinucleon emission process which in a Cherenkov detector is misidentified for a quasielastic one [1,2]. This misinterpretation produces an apparent increase of the neutrino quasielastic cross section, at the origin of the so called axial mass anomaly detected in the MiniBooNE experiments [3] as now widely accepted [2,[4][5][6][7][8][9][10][11][12][13][14][15]. The detection of CP violation which involves a comparison between neutrinos and antineutrinos events needs an even more detailed understanding of the multinucleon processes since it concerns the difference between neutrino and antineutrino cross sections. This is not a priori obvious and it is the object of the present article.

II. ANALYSIS AND RESULTS
In order to illustrate the respective roles of the multinucleon components in the ν andν cross sections, as we have introduced in a previous work [4], we start by giving below the following simplified expression (we remind however that for the actual evaluation we use a more complete one) for the double differential neutrino or antineutrino cross sections on a nuclear target such as 12 C where the plus (minus) sign applies to neutrinos (antineutrinos). In this Eq. (1) G F is the weak coupling constant, θ c the Cabibbo angle, G E , G M and G A are the nucleon electric, magnetic and axial form factors, E ν and E l the initial and final lepton energies, k l the modulus of the final lepton momentum, ω and q the energy and the momentum transferred to the nucleus and θ is the lepton scattering angle. The cross section on the nuclear target is expressed here in terms of the nuclear responses R to probes with various couplings to the nucleon, isovector (index τ ) or isovector with isospin and spin coupling (index στ ). For the last responses the isovector spin coupling can be spin transverse, στ (T ), or spin longitudinal, στ (L). The responses are q and ω dependent. The last term of Eq. (1) which changes sign between ν andν is the axial vector interference term, the basic asymmetry which follows from the weak interaction theory. It is expressed here in terms of isospin spin-transverse nuclear response. Its evaluation for complex nuclei is not trivial due to the presence of many-body effects. This is why it is important to obtain an experimental information on this term, one object of the present work. Now, for our description of the multinucleon component of the responses, we followed the experimental indications provided by the electron scattering data. In the transverse, or magnetic, data the dip between the quasielastic and the Delta part of the response is filled, which we interpreted as an indication for the presence of two nucleon emission [16]. In the charge response instead this component is absent. With these indications, for neutrinos we have introduced the multinucleon component only in the spin isospin response, the pure isovector one keeping its quasielastic character. This difference is at the origin of the statement in our work of Ref. [4] that the multinucleon effect should be relatively less pronounced for antineutrinos than for neutrinos, the reason being that the isovector response which is free from multinucleon effect plays a larger relative role for antineutrinos due to the negative sign of the interference term. This argument obviously holds only in the kinematical regions where the influence of the isovector component is significant.
We have previously tested our model on the MiniBooNE data for the differential cross sections [3,17], independently for neutrinos [8] and antineutrinos [13]. The good fit of the data provided by our model did not contradict our statement on the respective roles of np-nh. However in these works, the test was performed separately for the neutrinos and the antineutrinos and we did not specifically address the detailed comparison between the two.
As this comparison is essential for the CP violation detection we want to investigate in the present work this question in more details. For Cherenkov detectors, such as the MiniBooNE or the Super-Kamiokande ones, are we able to describe quantitatively the difference between neutrinos and antineutrinos nuclear interactions? The wealth of experimental data which has accumulated in the last years by the MiniBooNE experiment allows an experimental test for this comparison and we will explore the message which can be drawn from these data. The quantity best suited for this exploration is the measured double differential quasielastic-like (i.e. which incorporates also multinucleon states) cross section d 2 σ d cos θdEµ with respect to the muon energy E µ and the muon emission angle θ, which is not affected by the reconstruction problem of the neutrino energy [18], [11], [19][20][21]. However it is flux dependent, with where ω is the energy transferred to the nucleus and Φ(E ν ) is the neutrino (or antineutrino) normalized flux. Neutrino experiments measure d 2 σ d cos θdEµ while nuclear physics evaluations calculate the quantity d 2 σ d cos θdω , which is the basic ingredient for all analysis on neutrino data. The flux dependence in d 2 σ d cos θdEµ is a priori an obstacle to extract a universal comparison between the two cross sections for neutrinos and antineutrinos, applicable to CP violation experiments. For each set of flux profiles the measured differential cross sections are different.
In particular the ν andν asymmetry for d 2 σ d cos θdEµ has two sources, one arises from the basic weak interaction theory, as given in Eq. (1). The second one which arises from the flux asymmetry has no universal character and is specific for each experiment. Is there nevertheless something general and informative in the MiniBooNE data? It is the question that we will address in the following.
Let us consider the following combinations: the sum, sum, and the difference, dif , of the double differential cross sections for neutrinos and antineutrinos with respect to the lepton emission angle, θ, and to the energy ω transferred to the nucleus 1: (color online) Normalized MiniBooNE ν µ andν µ fluxes derived from Refs. [3] and [17] respectively. Their half sum (Φ + ) and half difference (Φ − ) are also shown.
while for the difference dif the sign plus is changed to a minus one. Notice from Eq. (1) that the difference, dif , contains only one term, the axial vector interference one. It is this quantity which governs the difference in cross sections of ν andν but it is not accessible directly from neutrino data since only derivatives with respect to the emitted lepton energy are measurable. We also introduce S and D, the corresponding flux integrated quantities, which are instead experimentally accessible in the MiniBooNE data, in which quantities such as d 2 σν d cos θdEµ have been defined previously in Eq.
(2). The MiniBooNE experimental distribution of D(cos θ, E µ ) has been given by Grange and Katori [22] in a tridimensional plot.
If the normalized neutrino and antineutrino flux profiles would be identical, Φ ν (E ν ) ≡ Φν(E ν ), their common value could be factorized in the integrals over the neutrino energies implicitly contained in the above Eqs. (4) and (5). In this case only the axial vector interference term would survive in the difference D, while the sum S would totally eliminate this term. This would allow a direct experimental test of the interference part. However this is not quite the case as appears in Fig. 1 which compares the two normalized MiniBooNE fluxes. In view of this difference we express the two cross section combinations in terms (6) and (7) We observe again that for identical fluxes, for which Φ − (E ν ) = 0, the quantity D probes only the quantity dif , i.e., the axial vector interference term, while the sum S is totally blind to  measurable differential cross sections between neutrinos and antineutrinos is dominated by the axial-vector interference term. This is the case for T2K [23] and also for the NuMI [24] beams, the ones used in the MINOS, MINERνA and NOνA experiments.
Having seen the message that the comparison with the mean flux curves carries i.e. that the axial vector interference term dominates the difference and has very little influence on the sum, in the following we calculate in our model the sum and the difference of the cross sections with the true neutrinos and antineutrinos fluxes, and not with the mean one, in order to avoid unnecessary errors. Our present results are then simply the sum and the difference of our previous theoretical results published in Ref. [8] for neutrinos and in Ref. states into nucleon-hole ones [26]. The question is then: if a good fit is obtained with the combined and opposite effects of RPA and of the multinucleon component, could it be that a similar good fit would be achieved by omitting both effects. Would the simplest quasielastic description also account for the data? In Figs. 4 and 5 the effect of RPA is suppressed in the quasielastic cross section, which indeed has some enhancement effect. In particular for the cross section difference this enhancement is moderate and not enough to account by itself, for the data. We can safely conclude that a large multinucleon component is needed to describe the data for the axial vector interference term which governs the cross sections difference. The same conclusion applies to the cross sections sum, i.e. to the remaining part of the interaction, it is also appreciably influenced by the multinucleon component. Our model for the neutrino nucleus interaction is able to describe both components. As an additional illustration of its success we report in Fig. 6 the single differential cross sections, with respect to the muon kinetic energy or to the muon emission angle. As previously we deal with the sum and the difference for neutrinos and antineutrinos calculated with the true and the averages fluxes. We can observe again that the sum shows practically no sensitivity to the flux difference, while the cross sections difference displays some a mild sensitivity, in particular in the forward direction or at large T µ values. Our predictions which include the multinucleon component reproduce well the data. In our model the relative importance of the multinucleon term in the cross sections combinations depends on the role of the isovector response which is shown in Fig. 6, the smaller this role the larger the multinucleon contribution. It is the largest in the cross section difference. Similarly the isovector response weight is larger in antineutrino cross sections than in neutrino ones, hence the smaller multinucleon contribution for antineutrinos. It is however not a large difference and the multinucleon influence remains important also for antineutrinos.
Finally for completeness we combine in Fig. 7 our previous evaluations of the neutrino and antineutrino Q 2 distributions, published in Refs. [8] and [13] respectively, to evaluate their sum and difference. We also display the result obtained with the averaged flux Φ + .
As previously pointed out the difference is more sensitive to the mean flux approximation which however gives the bulk of this difference. As a consequence the difference of the experimental MiniBooNE points dσ is directly related to the Q 2 distribution of the axial vector interference term. This conclusion would apply as well to the MINERνA neutrino [27] and antineutrino [28] Q 2 distributions, due to the closeness of the neutrino and antineutrino normalized fluxes. It will be the object of a future investigation.
Notice that the present study is done for the muonic neutrinos of the MiniBooNE experiment, for which data are available while CP violating experiments through the asymmetry of the oscillations rates of muonic neutrinos and antineutrinos into electron ones involve the detection of electrons or positrons produced in a detector by the charged current interactions of these electron neutrinos. We have already addressed the question of electron neutrino cross sections, versus the muon neutrino ones [21]. The effect of the small change in kinematics due to the smaller electron mass will not affect the present conclusions: the electron neutrino nuclear interactions produce by themselves an important ν eνe asymmetry.
The present study indicates that the multinucleon role is essential in this problem and to what precision this asymmetry can be mastered.

III. SUMMARY AND CONCLUSIONS
In summary we have investigated two different combinations of the neutrino and antineutrino MiniBooNE flux folded double differential cross sections on 12