# Challenges posed by non-standard neutrino interactions in the determination of at DUNE

###### Abstract

One of the primary objectives of Deep Underground Neutrino Experiment (DUNE) is to discover the leptonic CP violation and to identify it’s source. In this context, we study the impact of non-standard neutrino interactions (NSIs) on observing the CP violation signal at DUNE. We explore the impact of various parameter degeneracies introduced by non-zero NSI and identify which of these can influence the CP violation sensitivity and CP precision of DUNE, by considering NSI both in data and in theory. In particular, we study how the CP sensitivity of DUNE is affected because of the intrinsic hierarchy degeneracy which occurs when the diagonal NSI parameter and .

Introduction : Neutrino oscillation physics has entered an era filled with series of exceptional ongoing and forthcoming experiments using diverse sources and various detection techniques. These experiments are aimed at addressing some pressing questions like the determination of CP violation in the leptonic sector, ordering of neutrino masses i.e whether they obey normal hierarchy (NH, ) or inverted hierarchy (IH, ), octant of i.e whether it falls in the lower octant (LO, ) or in the higher octant (HO, ).

Numerous neutrino experiments have left a window open for several new physics scenarios among which non-standard neutrino interactions (NSIs) have received a lot of attention lately. A detailed review on NSIs can be found in Refs.Ohlsson (2013); Miranda and Nunokawa (2015); Farzan and Tortola (2017). NSIs were first proposed by Wolfenstein Wolfenstein (1978) as an alternative phenomenon to explain neutrino flavour transitions. However, experiments over past decades have established neutrino oscillations as the leading mechanism to flavor transitions leaving non-standard interactions of neutrinos with matter particles as potential next to leading order effects. Precision measurements from ongoing and future neutrino experiments could provide some insight into the existence of these new type of interactions. Among these, DUNE (Deep Underground Neutrino Experiment) is considered as one of the powerful experiments Acciarri et al. (2016). Therefore, it is important to phenomenologically understand the effect of these beyond SM interactions on the experimental sensitivities of DUNE.

DUNE is a long baseline accelerator neutrino experiment which will use very high intensity neutrinos produced by proposed Long Baseline Neutrino Facility’s (LBNF) beamline at Fermilab, USA Acciarri et al. (2015). Fermilab’s main injector accelerator provides a proton beam corresponding to an energy of 120 GeV which in turn generates a broad band neutrino beam of energy ranging from 100 MeV to 8 GeV with a peak at 2.5 GeV. This neutrino beam is then made to travel 1300 km towards Sanford Underground Research Facility (SURF) where it encounters a Liquid Argon Time-Projection Chamber (LArTPC). We consider a detector of 40 kt fiducial volume in our simulations.

CP violation in the leptonic sector holds a key to leptogenesis which in turn can shed some light on baryogenesis Fukugita and Yanagida (1986); Davidson et al. (2008). Several phases could contribute to leptonic CP violation. Among these - Dirac CP violation phase is one of the physical phases (apart from two Majorana phases) occurring in neutrino mixing matrices which can be determined via long baseline (LBL) neutrino oscillation experiments. Recent global analysis of neutrino oscillation measurements Capozzi et al. (2016); Esteban et al. (2017) have suggested to be and have shown a weak trend towards NH. However, these experiments suffer from various parameter degeneracies because of their shorter baselines as compared to DUNE Barger et al. (2002); Ghosh et al. (2016); Goswami and Nath (2017). Furthermore, given the potency of upcoming LBL neutrino oscillation experiments, like DUNE Acciarri et al. (2015) and Hyper-Kamiokande Abe et al. (2011), it is a feasible task to determine with high sensitivity. Additionally, there is a possibility that these experiments will be able to probe NSI and further identify any new sources of CP-violation. Recent studies considering the impact of NSI in the context of DUNE can be found in Masud et al. (2016); de Gouvea and Kelly (2016); Coloma (2016); Liao et al. (2016a); Soumya and Mohanta (2016); Blennow et al. (2016); Forero and Huber (2016); Huitu et al. (2016); Masud and Mehta (2016a); Coloma and Schwetz (2016); Masud and Mehta (2016b); Agarwalla et al. (2016); Liao et al. (2016b); Fukasawa et al. (2016); Blennow et al. (2017); Liao et al. (2017); Deepthi et al. (2017); Ghosh and Yasuda (2017a, b); Masud et al. (2017) and the references there in. Below is the brief account of the papers which dealt with the effect of NSI on the CP violation sensitivity of DUNE.

In Ref. Masud et al. (2016) the authors have studied the effect of non-zero complex NSI parameters on the CP violation sensitivity of DUNE and showed that these complex phases could mimic CP violation even for the CP conserving values. The authors of Ref. de Gouvea and Kelly (2016) have investigated how well DUNE can measure the new sources of CP violation assuming the presence of NSI. Ref. Liao et al. (2016a) discusses several parameter degeneracies between standard and non-standard interactions and the effect on the determination of mass hierarchy, octant of and the CPV sensitivity of DUNE. Authors of Ref. Forero and Huber (2016) have tested the robustness of the recent result of Dirac CP phase by assuming NSI in the simulated data (generated for many iterations) and fitting it with SI. While Ref. Masud and Mehta (2016a) deals with precise study of the impact of NSI on the CP violation sensitivity of the long-baseline experiments like T2K, NOA, DUNE and T2HK. In Ref. Huitu et al. (2016) the authors have constrained the NSI parameters while considering standard interactions in the data and non-standard interactions in theory. In Ref. de Gouvêa and Kelly (2016), authors have investigated false CPV signals from new physics scenarios like NSI and a four-neutrino scenario at DUNE. Ref. Liao et al. (2017) investigates the effect of matter NSI on the determination of Dirac CP phase at DUNE, T2HK and T2HKK while studying the role played by the generalized mass hierarchy degeneracy introduced by Bakhti and Farzan (2014); Coloma and Schwetz (2016).

In this work we investigate how matter induced non-standard neutrino interactions would affect the determination of CP violation by DUNE, during its proposed run time of (5 + 5 ) years.
We focus on the impact of various parameter degeneracies, introduced by the non-standard neutrino interactions. We consider the most general degeneracy of the form occurring for same as well as opposite hierarchies
and study how this can affect the CP sensitivity of DUNE.
In particular,
we study the effect of the generalized mass ordering degeneracy , stressing on the special case of and assuming NSI both in data and in theory. This particular case is interesting as nullifies the standard matter effect ^{1}^{1}1Note that though cancels the matter effect in the Earth it doesn’t cancel the matter effect in the Sun Deepthi et al. (2017). For the impact of large in the solar matter effect see Miranda et al. (2006). Additionally if there is maximal CP violation there exists an intrinsic hierarchy degeneracy that is independent of baseline and neutrino beam energy Deepthi et al. (2017). We study whether the intrinsic degeneracy can
alter the CP discovery potential and CP precision of DUNE.
In addition, we consider other true values of and obtain the fraction of values for which CP violation (CPV) can be discovered at . We also
investigate the impact of introducing non-zero value of the
off-diagonal NSI parameter along with .

We organize the paper as follows: We present a brief account of NSI formalism and their current bounds in section I. In section II, we present a discussion on the probability and CP asymmetry of DUNE. In section III we discuss the effect of diagonal and off-diagonal NSI on the CPV discovery capability of DUNE assuming the presence of NSI in both data and theory. Section IV, discusses the impact of NSI on the CP precision of DUNE. Concluding remarks are made in section V.

## I NSI formalism and current bounds

Neutral current NSIs that affect the propagation of neutrinos through earth matter are described using an effective four-fermion Lagrangian with dimension operators of the form

(1) |

where are NSI parameters, , , , and is the Fermi constant. The Schrödinger-like evolution equation of a neutrino , travelling a distance x, can be written as

(2) |

where the effective Hamiltonian characterizes both standard and non-standard interactions of neutrinos with matter fermions and can be expressed in the flavour basis as,

(3) |

The ‘1’ in the effective matter potential of eq. (3) corresponds to the standard charged-current matter interactions whereas, the NSI parameters describe the non-standard interactions of neutrinos with earth matter. Here, is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix Patrignani et al. (2016), , and , with being the number density of fermion .

## Ii Probability and CP asymmetry for DUNE :

The relevant oscillation probability for the super beam experiment DUNE is the appearance channel (). This can be expressed in terms of small parameters, , and except for normal hierarchy (NH) as Liao et al. (2016a)

(5) | |||||

where

(6) |

The expressions for the inverted hierarchy (IH) can be obtained by replacing (i.e. , (i.e. and ), ). Similar expressions for antineutrino probability () can be obtained by replacing ( ), . We note from the above that the only diagonal parameter to which appearance channel is sensitive to is . In the absence of off-diagonal NSI parameters, the probability expression is just the first line of eq.(5). The NSI contribution appears only in and terms via which is not treated as a small parameter in this formulation.

Fig. (1) shows vs of DUNE for E = 2 GeV with and assuming normal hierarchy. The plot in the left panel shows as a function of for different values of . The standard case without NSI corresponds to the pink solid curve with . It can be seen that there exists degeneracies of the form (, ) even for the same hierarchy. This can be understood by drawing a horizontal line intersecting any two curves. Moreover, if one plots vs for all non-integral values of there will be a continous degeneracy of this form which effects the determination of CP phase at DUNE. Note that this kind of degeneracies between opposite hierarchies have already been elaborated in Deepthi et al. (2017).

In the right panel we show the effect of non-zero on for . We consider a representative value of and to see the effect of complex diagonal NSI^{2}^{2}2In this paper we study the effect of only non-zero and not as the latter has tighter bounds..
It can be seen that the standard case (no NSI) shown by pink (solid) curve is degenerate with the non-zero NSI case represented by blue (dot-dashed)
and the violet (dotted) curves for four different values of .
Apart from that, one can also find many degeneracies when a horizontal line is drawn intersecting the lines. These degeneracies correspond to wrong solutions which play a role in CPV sensitivity of DUNE if NSI exists in nature (as shown in the next section) even when the hierarchy is known.

In neutrino oscillation experiments the CP-violating effects can be characterized by a quantity known as CP asymmetry which is defined as (). In the case of vacuum oscillations and clearly the Dirac CP phase can be easily determined. However for long baseline experiments like DUNE large matter effects induce fake CP phase () which cannot be distinguished from the intrinsic CP phase ().

Fig (2) shows the CP asymmetry of DUNE defined as for NH and . The experimentally measured asymmetry is . Here is the CP asymmetry quantifying the intrinsic Dirac CP phase and quantifying fake CP phase introduced by the asymmetry in the earth matter.

In the left panel of fig (2) the red (dotted) curve shows of DUNE when there is maximal CP violation with . The black (solid) curve represents which can be obtained in theory by taking . Now, the intrinsic CP phase can be quantified by plotting as shown by the blue (dashed) curve. Clearly the intrinsic CP phase is different from the experimentally measured phase that is quantified by .

The plot in the right panel of fig (2) shows all the three asymmetries defined earlier for . Since nullifies the matter effect one can see from the black (solid) curve that the asymmetry induced by the earth matter . Thus, which can be seen from the blue (dashed) and red (dotted) curves. That is in this special case, interestingly, the fake asymmetry introduced by the earth matter is nullified and the measured by the experiment is the same as . In this context it is worthwhile to study how the CPV sensitivity of DUNE is affected in this special case.

## Iii Results and discussions

In our numerical simulations we have used the General Long baseline Experiment Simulator (GLoBES) Huber et al. (2005) along with some additional packages Jaochim (2008). The experimental details are taken from Nath et al. (2016) except here, we have considered the detector volume to be 40 kt. The total in our analysis is defined by

(7) |

where , being the number of true events and corresponding to the number of test events. The effect of systematics is included through the method of pulls. We have added a prior on with an error of .

The true values that we have considered are, , , (unless mentioned otherwise), , and . These are consistent with the global analysis of neutrino oscillation data Capozzi et al. (2016); Esteban et al. (2017); de Salas et al. (2017). We neglect the production and detection NSIs as these are bounded by an order of magnitude stronger than the matter NSIs Biggio et al. (2009).

### iii.1 CPV sensitivity : Effect of non-zero NSI

Fig. (3) shows the significance with which CP violation, i.e. can be determined for different true values of . The CPV sensitivity is quantified by the test statistics which is defined as :

(8) |

where is calculated by varying true in the range .

We consider the true hierarchy to be NH and marginalized over the test hierarchy (hierarchy marginalization is done in all the plots unless otherwise mentioned), octant of and . The top panel of fig. (3) shows the CPV discovery potential of DUNE running for years assuming in the data. In the test we have marginalized over in addition to the other parameters stated above. Here, the horizontal solid brown lines represent and C.L. as labelled in the figure. In the left panel of the first row, the black (solid) curve corresponds to the standard case i.e. assuming no NSI in both data and theory while for other curves we assume that the data has non-zero NSI with . The green (dashed) curve corresponds to the case where is kept fixed in both true and test event spectra while the hierarchy is marginalized. We see that this gives a higher sensitivity than the standard case around . For other values of the CP discovery potential is not affected much. Thus, the intrinsic degeneracy by itself does not impair the CP discovery capability of DUNE. However, when one marginalizes over in it’s model-independent range of , the sensitivity is compromised for both in the upper () and the lower () half-planes which can be seen from the blue (dot-dashed) and magenta (dotted) curves. This is due to the continuous degeneracy of the form (, ). Note that the blue (dot-dashed) curve is for fixed hierarchy indicating that the continuous degeneracy can reduce the CPV discovery power of DUNE even for the same hierarchy. One can understand this from fig. (1) by drawing a horizontal line to intersect (blue dot-dashed) curve and (brown dashed) curve at which corresponds to (, ) degeneracy.

In the right plot of the upper panel of fig. (3), the green solid and dashed lines represent the case where both data and theory are consistent with the standard paradigm while true and respectively. While the red solid (dashed) line shows CP violation sensitivity of DUNE assuming true non-zero diagonal NSI and (). From the red curves it can be seen that the CPV sensitivity is compromised when compared to the standard case for all values of both in the lower and upper half-planes. The sensitivity for is only slightly greater than that of . Thus, one can conclude that there is very low correlation between and the octant of .

The lower panel of Fig. (3) shows CPV sensitivity of DUNE running for years assuming a non-zero diagonal and off-diagonal NSI in the data. In the previous work Deepthi et al. (2017) the authors have shown that in the presence of non-zero off-diagonal NSI parameter () the intrinsic hierarchy degeneracy that occurs at and gets transported to a different value of . Thus, to check the impact of this shift in the intrinsic hierarchy degeneracy on the CPV discovery potential, we have considered the same representative values of , , chosen in Deepthi et al. (2017) as our true values while we marginalize over these parameters in the test plane. Here, the horizontal solid brown lines represent and C.L. as labelled in the figure. The green solid and dashed lines represent the case where both data and theory are consistent with the standard paradigm while true and respectively. While the red solid (dashed) line shows CP violation sensitivity of DUNE while assuming true non-zero NSI with (). The wiggly pattern of the red lines indicate the presence of various degeneracies discussed in the Fig. (1). One can see that for these values the CPV sensitivity of DUNE gets enhanced as compared to the standard sensitivity even for true . Since the variation in the probability with is higher in the presence of (can be seen from fig. (1)), the CP discovery potential got improved.

In the above plots we have considered true . To show how the CPV sensitivity depends on we obtain the fraction of values for which CPV discovery potential is possible by varying true . CP fraction is one factor that can be used to quantify the effects of CP violation of a particular experiment. This fraction determines the values of (true) for which the CP violation sensitivity can be obtained above a particular significance level. In left panel of fig. (4), we obtained CP fraction of values for which significance is above for different values of in order to comprehend its complete effect. From the red (solid) and blue (dashed) curves in the left panel of the fig. (4) it can be seen that CP violation discovery with significance can be obtained for of the values when varies from for NH and for IH.

As the global oscillation data hints we plot vs in the right panel of fig. (4) for true , and assuming NH. It can be seen from the fig that the CPV discovery potential remains above for almost the full range of . For the is when we use NSI to fit the data. However, if we compare with the standard case presented in figure 3 the is . Thus the undergoes a large reduction in presence of NSI. And for no other values of the goes as high as . Thus the effect of is to reduce the CPV discovery potential of DUNE as compared to the standard case. This was also seen earlier in fig. 3 for .

## Iv CP precision of DUNE

Assuming the presence of diagonal NSI in nature, we check to what extent DUNE’s CP measurement will get affected. To address this question we assume non-zero diagonal NSI along with a maximal CP violation in true scenario and in the fit we vary in its full range . We also show the results of CP precision of DUNE obtained by considering only standard neutrino framework both in test and true, as a benchmark for comparison. Since, the correlation between and the octant of is very less henceforth we only plot the sensitivities corresponding to LO ().

Fig. (5) shows CP precision of DUNE in terms of vs test- for true . While considering the true hierarchy to be normal, we have marginalized over , , and . We have also added gaussian prior to .

The left plot in the upper panel of fig. (5) shows the CP precision of DUNE when true . The green (solid) line corresponds to the standard case and the blue (dot-dashed) line corresponds to the case assuming in true and marginalizing over its full model independent range in the test while keeping hierarchy fixed to NH both in true and test planes respectively. The red (dashed) line corresponds to the same case as the blue (dot-dashed) curve except here the hierarchy is marginalized in the test. For the SI case we see from the green curve that DUNE should be able to measure with a precision of at C.L. However when NSI exists, the blue (dot-dashed) and the red (dashed) curves reveal that the precision remain nearly the same at C.L. but become worse at . This is because of the additional hierarchy degeneracies of the form occurring between same as well as opposite hierarchy.

The right plot of fig. (5) corresponds to the CP precision of DUNE for true . For the standard case it can be seen from the green curve that DUNE has a CP precision of at C.L. The blue (dot-dashed) and the red (dashed) curves are plotted with the same assumptions as in the left plot. It can be seen that the precision at and got worse because of the degeneracies of the form occurring for the same hierarchy. However when the hierarchy is marginalized, there appears another minima at corresponding to ( – hierarchy) degeneracy which can be seen from the red (dashed) curve. This shows that the CP precision of DUNE gets more compromised for any value of other than (here we show for ) due to generalized hierarchy degeneracy which occurs for .

The bottom panel of fig. (5) shows the CP precision of DUNE for true and in the presence of non-zero off-diagonal NSI . It can be clearly seen that though the CP precision around the true value of gets better, there are other local minima occurring because of the additional degeneracies introduced by the presence of non-zero . This can give multiple solutions in test - test plane.

## V Conclusion:

In this paper we have studied the impact of the non-standard interactions on the CP sensitivity of DUNE. In particular, we have considered the impact of the diagonal NSI parameter . This is real and does not contain any extra complex phase. Nonetheless it can give rise to degeneracies of the form between same as well as opposite hierarchies. In particular, this parameter is responsible for the generalized hierarchy degeneracy which is known to adversely affect the hierarchy sensitvity of DUNE. This degeneracy can be resolved if the CP phase or the diagonal NSI parameter could be measured accurately. However for the special case of this with and the hierarchy would remain undetermined for all baselines and energies. This is known as the intrinsic hierarchy degeneracy. We have investigated how all these degeneracies can influence the CP discovery potential and precision measurement of at DUNE. Taking the true value of we obtain the CP discovery assuming presence of NSI in both data and theory. We find that the CP discovery potential gets negatively affected due to this parameter even when hierarchy is known. The degeneracy responsible for this is the continuous degeneracy. However, if we keep to be fixed as -1 in both data and theory i.e do not marginalize over this parameter then the CP discovery potential can be better than the standard case around . Thus we conclude that the intrinsic hierarchy degeneracy does not impair the CP discovery potential of DUNE. We have also shown that there is minuscule interplay between and the octant of . We have further studied the impact of introducing a non-zero off-diagonal parameter in the analysis. We find that for the representative value considered by us the CP discovery potential is enhanced even when the true value of the complex phase associated with this parameter is taken to be zero. Additionally, we study the fate of CP discovery potential of DUNE for other true values of by presenting the fraction of values for which CP sensitivity can be attained as a function of this parameter. We find that this fraction is 25% for approximately between 1 and -3. We also study the maximal CPV discovery reach as a function of for , which is the best-fit value from global analysis of current world neutrino data. In this case we find that although the CPV discovery stays above over almost the full allowed range of this is less than the standard case. We have also studied the CP precision at DUNE. We find that for as true value the precision at is comparable to the case with no NSI. However at the precision is worse. We have shown that this can again be attributed to degeneracy which occurs for same and opposite hierarchies. However if we take as different from the precision is compromised due to occurring in the generalized hierarchy degeneracy. For a representative non-zero the CP precision around the true value improves but there are other local minima which can give spurious solutions. Further detailed study may be required to understand whether these degeneracies can be resolved by combining various neutrino oscillation experiments.

Acknowledgement: The research work of NN was supported in part by the National Natural Science Foundation of China under grant No.11775231.

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