Degeneracy between octant and neutrino non-standard interactions at DUNE
We expound in detail the degeneracy between the octant of and flavor-changing neutral-current non-standard interactions (NSI’s) in neutrino propagation, considering the Deep Underground Neutrino Experiment (DUNE) as a case study. In the presence of such NSI parameters involving the () and () flavors, the and appearance probabilities in long-baseline experiments acquire an additional interference term, which depends on one new dynamical CP-phase . This term sums up with the well-known interference term related to the standard CP-phase creating a source of confusion in the determination of the octant of . We show that for values of the NSI coupling (taken one at-a-time) as small as (relative to the Fermi coupling constant ), and for unfavorable combinations of the two CP-phases and , the discovery potential of the octant of gets completely lost.
keywords:Neutrino, octant, CP-phase, Non-Standard Interactions, Long-baseline, DUNE
Although the interactions of neutrinos are well described by the Standard Model (SM) of particle physics, it is possible that these particles may participate to new non-standard interactions (NSI’s), whose effects are beyond the reach of the existing experiments. NSI’s may appear as a low-energy manifestation of high-energy physics involving new heavy states (for a review see Biggio et al. (2009); Ohlsson (2013); Miranda and Nunokawa (2015)) or, alternatively, they can be related to new light mediators Farzan (2015); Farzan and Shoemaker (2015). As first recognized in Wolfenstein (1978), NSI’s can profoundly modify the MSW dynamics Wolfenstein (1978); Mikheev and Smirnov (1985, 1986) of the neutrino flavor conversion in matter. As a consequence, they can be a source of confusion in the determination of the standard parameters regulating the 3-flavor oscillations if the estimate of these last ones is extracted from experiments sensitive to MSW effects. Recently, in the context of long-baseline (LBL) experiments, the potential confusion between the standard CP-violation (CPV) related to the 3-flavor CP-phase and the dynamical CP-phases implied by neutral-current flavor-changing NSI’s has received much attention Friedland and Shoemaker (2012); Rahman et al. (2015); Liao et al. (2016); Forero and Huber (2016); Huitu et al. (2016); Bakhti and Farzan (2016); Masud and Mehta (2016); Soumya and Mohanta (2016); de Gouvêa and Kelly (2016a)111Another notable degeneracy occurs between off-diagonal NSI’s and non-zero in long-baseline Huber et al. (2002a) and solar neutrino experiments Palazzo and Valle (2009); Palazzo (2011). Now, this degeneracy has been resolved with the help of data from reactor experiments (Daya Bay, Double Chooz, and RENO), which confirmed that is non-zero without having any dependency on matter effects..
In this paper, we explore in detail, a different kind of degeneracy affecting LBL experiments. It is still induced by the new CP-phases related to NSI’s, but concerns the octant of the atmospheric mixing angle . Such a degeneracy has been noted in the numerical simulations performed in Coloma (2016); Blennow et al. (2016); de Gouvêa and Kelly (2016b) and also briefly discussed at the analytical level in Liao et al. (2016) (see also Friedland and Shoemaker (2012); Soumya and Mohanta (2016)). But, to the best of our knowledge, it has not been addressed in a systematic way in the literature. We recall that present global neutrino data Capozzi et al. (2016); Gonzalez-Garcia et al. (2016); Forero et al. (2014) indicate that may be non-maximal with two degenerate solutions: one , dubbed as lower octant (LO), and the other , termed as higher octant (HO). Just a few days ago, at the Neutrino 2016 Conference, the NOA collaboration has reinforced the case of two degenerate solutions, excluding maximal mixing at the confidence level Vahle (2016). This makes the octant issue even more pressing than before. The identification of the octant is an important target in neutrino physics, due to the profound implications for the theory of neutrino masses and mixing (see Mohapatra and Smirnov (2006); Albright and Chen (2006); Altarelli and Feruglio (2010); King et al. (2014); King (2015) for reviews). In the presence of flavor-changing NSI’s involving the or sectors, the transition probability probed at LBL facilities acquires a new interference term that depends on one new dynamical CP-phase . This term sums up with the well-known interference term related to the standard CP-phase creating a potential source of confusion in the reconstruction of the octant. Taking the Deep Underground Neutrino Experiment (DUNE) Acciarri et al. (2016a); Acciarri et al. (2015); Strait et al. (2016); Acciarri et al. (2016b); Adams et al. (2013) as a case study,222Recent work on the impact of NSI’s at DUNE can be found in de Gouvêa and Kelly (2016b); Coloma (2016); Blennow et al. (2016); Masud and Mehta (2016); de Gouvêa and Kelly (2016a); Bakhti and Khan (2016). we show that for values of the NSI coupling as small as (relative to the Fermi constant ), for unfavorable combinations of the two CP-phases and , the discovery potential of the octant of gets completely lost.
2 Theoretical framework
A neutral-current NSI can be described by a four-fermion dimension-six operator Wolfenstein (1978)
where subscripts indicate the neutrino flavor, superscript labels the matter fermions, superscript denotes the chirality of the current, and are dimensionless quantities which parametrize the strengths of the NSI’s. The hermiticity of the interaction demands
For neutrino propagation through matter, the relevant combinations are
where denotes the number density of fermion . For the Earth, we can assume neutral and isoscalar matter, implying , in which case . Therefore,
The NSI’s modify the effective Hamiltonian for neutrino propagation in matter, which in the flavor basis reads
where is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, which, in the standard parameterization, depends on three mixing angles () and one CP-phase (). We have also introduced the solar and atmospheric wavenumbers and and the charged-current matter potential
where is the relative electron number density in the Earth crust. It is useful to introduce the dimensionless quantity , whose absolute value is given by
since it will appear in the analytical expressions of the transition probability. In Eq. (7), we have taken the energy of the DUNE first oscillation maximum as a benchmark.
In the present work, we limit our investigation to flavor non-diagonal NSI’s, that is, we only allow the ’s with to be non-zero. More specifically, we will focus our attention on the couplings and , which, as will we discuss in detail, introduce an observable dependency from their associated CP-phase in the appearance probability probed at the LBL facilities. For completeness, we will comment about the (different) role of the third coupling , which mostly affects the disappearance channel and has not a critical impact in the octant reconstruction. We recall that the current upper bounds (at 90% C.L.) on the two NSI’s under consideration are: , as reported in the review Biggio et al. (2009), and as derived from the most recent Super-Kamiokande atmospheric data analysis Mitsuka et al. (2011) under the assumption (see also Gonzalez-Garcia and Maltoni (2013)). As we will show in detail, the strengths of and that can give rise to a degeneracy problem with the octant of are one order of magnitude smaller than these upper bounds.
3 Analytical Expressions
Let us consider the transition probability relevant for the LBL experiment DUNE. Using the expansions available in the literature Kikuchi et al. (2009) one can see that in the presence of a NSI, the transition probability can be written approximately as the sum of three terms
where the first two terms return the standard 3-flavor probability and the third one is ascribed to the presence of NSI. Noting that the small mixing angle , the matter parameter and the modulus of the NSI can be considered approximately of the same order of magnitude , while is , one can expand the probability keeping only third order terms. Using a notation similar to that adopted in Liao et al. (2016), we obtain333Interestingly, a similar decomposition of the transition probability is valid in the presence of a light sterile neutrino Klop and Palazzo (2015). In that case, however, the origin of the new interference term is kinematical, and it is operative also in vacuum. In fact, the new term is related to the interference of the atmospheric oscillations with those induced by the new large mass-squared splitting implied by the sterile state.
where is the atmospheric oscillating frequency related to the baseline . For compactness, we have used the notation (, ), and following Barger et al. (2002), we have introduced the quantities
We observe that is positive definite being independent of the CP-phases, and gives the leading contribution to the probability. In one recognizes the standard 3-flavor interference term between the solar and the atmospheric frequencies. The third term brings the dependency on the (complex) NSI coupling and of course is non-zero only in the presence of matter (i.e. if ). In Eq. (11) we have assumed for the NSI coupling the general complex form
The expression of is slightly different for and and, in Eq. (11), one has to put
In the expressions given above for , and , one should bear in mind that the sign of , and is positive (negative) for NH (IH). In addition, we stress that the expressions above are valid for neutrinos, and that the corresponding ones for antineutrinos are obtained by inverting the sign of all the CP-phases, and of the dimensionless quantity .
Now let us come to the octant issue. As a first step it is useful to quantify the size of the perturbation from maximal mixing allowed by current data. We can express the atmospheric mixing angle as
where is a positive-definite angle. The positive (negative) sign corresponds to HO (LO). The current 3-flavor global analyses Capozzi et al. (2016); Gonzalez-Garcia et al. (2016); Forero et al. (2014) indicate that cannot deviate from by more than , i.e. must be in the range . Therefore, one has , and we can use the expansion
An experiment is sensitive to the octant if, in spite of the freedom provided by the unknown CP-phases, there is still a non-zero difference among the transition probability in the two octants, i.e.
In Eq. (18) one of the two octants should be thought as the true octant (whose value is used to simulate the data) and the other one as the test one (whose value is used to simulate the theoretical model). For example, if for definiteness we fix the HO as the true octant, then for a given combination of ) there is sensitivity to the octant if there exist some values of the test phases such that at a detectable level444In the numerical analysis the values of the test parameters are determined by minimization of the [see Eq. (29) in Section 4]..
According to Eq. (8), we can split in the sum of three terms
The first term is positive-definite, does not depend on the CP-phases and, at the first order in , it is given by
The second and third terms depend on the CP-phases and can have both positive or negative values. Their expressions are given by
where for compactness, we have introduced the amplitudes555In the expressions of , and we are neglecting terms proportional to powers of , which would give rise to negligible corrections.
The positive (negative) sign in front of the coefficient in Eq. (22) corresponds to (). In order to get a feeling of the size of the three terms of we provide a ballpark estimate adopting as a benchmark case (, NH), and fixing the energy at the value GeV corresponding to the first oscillation maximum (), in which case and . For the 3-flavor parameters, we have used the values provided at the beginning of the next section. For the first term we find
where we have left manifest the linear dependency on the deviation from maximal . The amplitude of the standard interference term is
while, for the two coefficients entering the NSI-induced term , one finds
where we have left evident the linear dependency on the NSI strength . From this last relation we see that for values of the NSI coupling , the difference induced by the new interference term () has approximately the same amplitude of that arising form the standard interference term (). Also, it is essential to notice that the third term in Eq. (22) depends not only on the standard CP-phase but also on the new dynamical CP-phase related to the NSI. Since the two CP-phases and are independent quantities, in the SM+NSI scheme there is much more freedom with respect to the SM case, where only one phase () is present. Therefore, for sufficiently large values of the NSI coupling, it is reasonable to expect a degradation of the reconstruction of the octant, which will depend on the amplitude of the deviation .
Figure 1 provides a useful geometrical representation of the situation. In such a plot, the ellipses refer to the SM case, while the colored blobs represent the SM+NSI scheme. In each panel we show the four cases corresponding to the different choice of the neutrino mass hierarchy (MH), which can be normal (NH) or inverted (IH), and to the different choice of the octant (LO and HO). We have taken as a benchmark value for the LO (HO) octant. In the left (right) panel we have switched on the () coupling taking for its modulus () and varying the associated CP-phase () in its allowed interval . The graphs neatly show that the octant separation existing in the SM case is lost in the presence of NSI’s since the two separate ellipses become overlapping blobs. We can understand how the blobs arise thinking them as a convolution of an infinite ensemble of ellipses (for more examples, see Agarwalla et al. (2016a, b)), each corresponding to a different value of the new phase ( or ). The orientation of the ellipses changes as a function of such new CP-phase covering a full area in the bi-event space. The shape of the colored blobs is slightly different between the two cases of and as a result of the different functional dependency of the transition probability. One can notice that in both panels there is also an overlap among the two hierarchies, which is more pronounced in the case (left panel) if compared with the case (right panel). This may indicate that the MH may be a source of degeneracy in the octant identification.
This is not the case, however, because in the DUNE experiment the energy spectrum brings additional information that breaks the MH degeneracy. In contrast, the energy spectrum is not able to offer much help in lifting the octant degeneracy. This behavior is elucidated by Fig. 2, which represents the reconstructed electron neutrino event spectra in DUNE for plotted for four representative cases. The left panel illustrates the comparison of two cases in which the total number of events is exactly the same for NH and IH. The right panel displays the comparison of two cases in which the total number of events is exactly the same for LO and HO. The two spectra on the left panel are calculated for the values of the CP-phases and indicated in the legend, which correspond to the same point in the bievent space located in the overlapping region of the two (red and blue) LO blobs (the cyan star in the left panel of Fig. 1). The two spectra on the right panel are calculated for the values of the CP-phases and indicated in the legend, which correspond to the same point in the bievent space located in the overlapping region of the two (red and green) NH blobs (the black square in the left panel of Fig. 1). Figure 2 clearly shows that, while the spectra are rather different for the two hierarchies, especially at low energies, they are almost identical for the two octants. We find a similar behavior in the electron antineutrino spectra (not shown) for the same choices of the CP-phases indicated in the legend of Fig. 2. This implies that the MH is not a source of confusion in the octant identification666In Liao et al. (2016); Coloma and Schwetz (2016), it has been pointed out that if is non-zero and , then DUNE alone cannot determine the correct MH. In such a scenario, the octant ambiguity that we are dealing with can be further exacerbated.. Nonetheless, for generality, in the numerical analysis presented in the next section, we will treat the MH as an unknown parameter.
4 Numerical results
For DUNE, we consider a 35 kiloton fiducial liquid argon far detector in our work, and follow the detector characteristics which are mentioned in Table 1 of Ref. Agarwalla et al. (2012). We assume a proton beam power of 708 kW in its initial phase with a proton energy of 120 GeV which can deliver protons on target in 230 days per calendar year. In our calculation, we have used the fluxes which were obtained assuming a decay pipe length of 200 m and 200 kA horn current Mary Bishai (2012). We take a total run time of ten years, which is equivalent to a total exposure of 248 kiloton MW year, equally shared between neutrino and antineutrino modes. In our work, we consider the reconstructed energy range of neutrino and antineutrino to be 0.5 GeV to 10 GeV. As far as the systematic uncertainties are concerned, we assume an uncorrelated 5% normalization error on signal, and 5% normalization error on background for both the appearance and disappearance channels. The same set of systematics are taken for both the neutrino and antineutrino channels which are also uncorrelated. In our simulations, we use the GLoBES software Huber et al. (2005, 2007). We incorporate the effect of the NSI parameters both in the appearance channel, and in the disappearance channel. The same is also applicable for the antineutrino run. The benchmark (central) values of the three-flavor oscillation parameters that we consider in this work are: = 0.304, = 0.085, = 0.42 (0.58) for LO (HO), = eV, (NH) = eV, (IH) = - eV, and the CP phase in the range [-, ]. These choices of the oscillation parameters are in close agreement with the recent best-fit values from Ref. Capozzi et al. (2016); Gonzalez-Garcia et al. (2016); Forero et al. (2014). For the cases, where the results are shown as a function of true value of , we consider the 3 allowed range of 0.38 to 0.63. For the DUNE baseline of 1300 km, we take the the line-averaged constant Earth matter density of estimated using the Preliminary Reference Earth Model (PREM) Dziewonski and Anderson (1981). To obtain the numerical results, we carry out a full spectral analysis using the binned events spectra for DUNE. In order to determine the sensitivity of DUNE for excluding the false octant, we define the Poissonian as
where () is generated for the true (test) values of (). To obtain the curves displayed in Fig. 3, for any given choice of the true parameters, we minimize the in Eq. (29) with respect to the test parameters varying in the false octant and () in the range . In addition, in Fig. 4 we marginalize also over in the range . Finally, in Fig. 5 with also marginalize over . We follow the method of pulls as described in Refs. Huber et al. (2002b); Fogli et al. (2002) to marginalize over the uncorrelated systematic uncertainties. To give our results at confidence levels for 1 d.o.f., we use the relation , which is valid in the frequentist method of hypothesis testing Blennow et al. (2014).
Figure 3 displays the sensitivity for excluding the wrong octant as a function of true . The two upper panels refer to while the two lower panels refer to . In each case we fix the modulus of the coupling ( or ) equal to 0.05. The two left (right) panels refer to the true choice LO-NH (HO-NH). In all panels, for the sake of comparison, we show the results for the 3-flavor SM case (represented by the black curve). Concerning the SM+NSI scheme, we draw the curves corresponding to four representative values of the (true) dynamical CP-phase ( or ). In the SM case we have marginalized over () (test). In the SM+NSI scheme, we have also marginalized over the test value of the new dynamical CP-phase ( or ). In all cases we have marginalized over the mass hierarchy. However, we have checked that the minimum of is never reached in the wrong hierarchy. This confirms that the neutrino mass hierarchy is not an issue in the determination of the octant in DUNE, as expected on the basis of the discussion about the energy spectral information made in the previous section.
In the analysis shown in Fig 3, we have fixed (0.58) as a benchmark value for the LO (HO), corresponding to a deviation . In general, one may want to know how things change for different choices of the true value of since it is unknown. Figure 4 gives a quantitative answer to this question. It displays the discovery potential of the true octant in the  (true) plane, assuming NH as true choice. The left panel corresponds to the SM case. The middle (right) panel represents the SM+NSI case, where we have “switched on” () with modulus 0.05. In the SM case we have marginalized away () (test). In the SM+NSI cases, in addition, we have marginalized over the true and test value of the new dynamical CP-phase ( in the middle panel, in the right panel). The solid blue, dashed magenta, and dotted black curves correspond, respectively, to the 2, 3, and 4 confidence levels (1 d.o.f.). From the comparison of the middle and right panels with the left one, we can see that the presence of NSI with strength compromises the octant sensitivity for all the phenomenologically interesting region allowed for by current data. This is of particular interest because such low strengths of the NSI’s are well below the current upper bounds both for and . Finally, it is interesting to ask how the deterioration of the octant discovery potential varies with the NSI strength. For this purpose one needs to treat the NSI strength as a free parameter, allowing the associated CP-phase to vary in the interval . The results of this general analysis are represented in Fig 5, which shows the discovery potential of the octant in the plane  (true), assuming NH as true choice. The left (right) panel corresponds to (). In both cases, the standard parameters () (test) and (true) have been marginalized away. In addition, in the left (right) panel the true and test values of the CP-phase () have been marginalized away. We observe that for NSI strengths below the 1% level, the sensitivity substantially coincides with that achieved in the SM case. In this case the NSI’s are harmless. For larger values, the sensitivity gradually deteriorates, until it basically goes below the level for all the interesting values of if .
Before concluding this section, a remark is in order concerning the off-diagonal coupling . First, one should note that this coupling is the most strongly constrained due to the high sensitivity of atmospheric neutrino data to the transitions. The most recent Super-Kamiokande analysis provides the upper bound at the 90% C.L. Mitsuka et al. (2011) (see also Gonzalez-Garcia et al. (2011)), whose results are corroborated by MINOS data Adamson et al. (2013). Second, in the context of long-baseline experiments, essentially affects only the disappearance probability, while its effects on the appearance probability are negligible. These two circumstances make the coupling less important for what concerns the discrimination of the octant of . This fact is corroborated by our numerical simulations. We have explicitly verified that even for , which is well above the present upper bound, the DUNE sensitivity to the octant of never goes below 4.4 (3.1) for the benchmark value for LO (HO). Also, we find only mild changes in the sensitivity when the associated CP-phase is allowed to vary in the interval .
We have investigated the impact of non-standard flavor-changing interactions (NSI) on the reconstruction of the octant of the atmospheric mixing angle in the next generation LBL experiments, taking the Deep Underground Neutrino Experiment (DUNE) as a case study. In the presence of such new interactions the transition probability acquires an additional interference term, which depends on one new dynamical CP-phase . This term sums up with the well-known interference term related to the standard CP-phase . For values of the NSI coupling as small as (relative to the Fermi constant ) the combination of the two interference terms can mimic a swap of the octant. As a consequence, for unfavorable values of the two CP-phases and , the discovery potential of the octant of gets completely lost. We point out that the degeneracy between the octant of and NSI’s discussed in this paper has now become more important in light of the new results from the NOA Collaboration presented a few days ago at the Neutrino 2016 conference, which suggest that maximal is disfavored at the confidence level Vahle (2016).
We close the paper with a general remark. In a previous work Agarwalla et al. (2016c), we found that a similar loss of sensitivity to the octant can occur due to the presence of a light eV-scale sterile neutrino. Also in that case a new interference term appears in the transition probability, which depends on one additional CP-phase. Therefore, albeit in the two cases the origin of the new CP-phase is completely different, having kinematical nature in the sterile neutrino case and dynamical nature in the NSI case, their phenomenological manifestation at the far detector of LBL experiments is very similar. On the basis of this observation, we can predict an analogous behavior also for other mechanisms which involve a new interference term in the transition probability, like for example the violation of unitarity of the PMNS matrix recently investigated in Miranda et al. (2016). Therefore, we can conclude that in general, whenever a new interference term due to any new physics crops up in the LBL appearance probability, the reconstruction of the octant may be in danger.
S.K.A. is supported by the DST/INSPIRE Research Grant [IFA-PH-12], Department of Science & Technology, India. A.P. is supported by the Grant “Future In Research” Beyond three neutrino families, contract no. YVI3ST4, of Regione Puglia, Italy.
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