# Can nonstandard interactions jeopardize the hierarchy sensitivity of DUNE ?

###### Abstract

We study the effect of non-standard interactions (NSIs) on the propagation of neutrinos through the Earth matter and how it affects the hierarchy sensitivity of the DUNE experiment. We emphasize on the special case when the diagonal NSI parameter , nullifying the standard matter effect. We show that, if in addition, CP violation is maximal then this gives rise to an exact intrinsic hierarchy degeneracy in the appearance channel, irrespective of the baseline and energy. Introduction of off-diagonal NSI parameter, , shifts the position of this degeneracy to a different . Moreover the unknown magnitude and phases of the off-diagonal NSI parameters can give rise to additional degeneracies. Overall, given the current model independent limits on NSI parameters, the hierarchy sensitivity of DUNE can get seriously impacted. However, a more precise knowledge on the NSI parameters, specially , can give rise to an improved sensitivity. Alternatively, if NSI exists in nature, and still DUNE shows hierarchy sensitivity, certain ranges of the NSI parameters can be excluded. Additionally, we briefly discussed the implications of (in the Earth) on MSW effect in the Sun.

###### pacs:

13.15.+g, 14.60.Pq, 14.60.StIntroduction: Phenomenal experiments over the past decades have established neutrino oscillations and led us into an era of precision measurements in the leptonic sector. Current data determines the two mass squared differences (, , being the mass states) and three leptonic mixing angles (, , ) with considerable precision Capozzi:2016rtj (). This leaves determination of neutrino mass hierarchy i.e. whether (normal hierarchy (NH)) or (inverted hierarchy (IH)), octant of i.e whether and lies in lower octant (LO) or it is and is in the higher octant (HO) and measurement of as the major objectives of ongoing and future experiments. Recently, the on-going T2K experiment Abe:2013hdq () and global analysis of data Capozzi:2016rtj () have hinted that the Dirac CP phase is maximal i.e. although at the full range () remains allowed. Whereas, recent NOA result Adamson:2017gxd () suggested that there are two best fit points if neutrinos obey normal hierarchy, (i) , and (ii) , . Also, inverted mass hierarchy with is disfavoured at 93 C.L. for all values of .

Although neutrino oscillation has been identified as the dominant phenomenon to explain the results of various experiments, the possibility of sub-leading effects originating from new physics beyond the Standard Model (SM) cannot be ignored. Among these, non-standard interactions have received a lot of attention lately specially with the emergence of proposed next generation experiments like DUNE,T2HK, T2HKK etc. Fukasawa:2016gvm (); Coloma:2015kiu (); Miranda:2004nb (); Agarwalla:2016fkh (); Blennow:2016etl (); Liao:2016hsa (); Coloma:2016gei (); Masud:2016bvp (); Masud:2016gcl (); Fukasawa:2016lew (); deGouvea:2015ndi ().

In this work, we emphasize on an interesting case when
standard matter effects during neutrino propagation through the Earth matter gets nullified
due to NSI effects ^{1}^{1}1We neglect the production and detection NSI,
bounds on which are stronger by an order of magnitude than matter NSI Biggio:2009nt ()., creating degeneracies which affect the
determination of hierarchy in any long-baseline (LBL) experiment.
Though additional hierarchy degeneracy in LBL experiments due
to NSI have been discussed earlier
GonzalezGarcia:2011my (); Coloma:2016gei ()
the new points that we make are :
(i) if the NSI parameter
characterizing new interactions between
electrons neutrinos and
electrons has the value
, the NSI effect cancels the standard matter effect;
(ii) if in addition
,
there exists an exact intrinsic hierarchy degeneracy
in the appearance channel, which is independent of baseline and the
neutrino beam energy making it unsolvable in LBL experiments, in particular
DUNE.
Note that this degeneracy cannot be lifted even if
and are precisely measured around these values.
This result assumes more importance in the light of current data from the
T2K experiment
hinting at .
We also consider the simultaneous presence of
and non-zero values of the NSI parameter
and show that the intrinsic degeneracy still
exists, albeit at a different .
Moreover, given the current model independent bounds on the NSI parameters,
hierarchy sensitivity in DUNE does not improve. Rather the phases
associated with can give rise to additional
degeneracies which seriously impact the sensitivity.

We focus on the matter NSI effects on the neutrino propagation. This is described by dimension-six four-fermion operators of the form Wolfenstein:1977ue ()

(1) |

where are NSI parameters, , denotes the chirality, , and is the Fermi constant. The Hamiltonian in the flavor basis can be written as,

(2) |

where is the PMNS mixing matrix having three mixing angles () and a CP phase . is the matter potential due to the the interactions of neutrinos,

(3) |

where, . The unit contribution to the (1,1) element of the matrix is the usual matter term arising due to the standard charged-current interactions. Here, diagonal elements of the matrix are real due to the Hermiticity of the Hamiltonian in eq. (2). , with being the number density of fermion along the trajectory ‘’ in the matter. can be written in terms of for neutral matter as

(4) |

Eq. (4) is a general equation valid for both Earth and the Sun.
Usually, the symbols , are
used to represent for the matter composition
of the Earth and the Sun respectively.
However, for convenience we have used the
notations in the Earth and in the Sun, throughout the paper.

For the case with only diagonal NSI parameter,
we observe from eq. (3) that for
, the NSI effect cancels the standard matter effect.
The relevant appearance channel probability ()
for normal hierarchy (NH), can be expressed
in terms of diagonal NSI parameter and
small parameters, , and the off-diagonal NSI parameter as, Liao:2016hsa (),

(5) | |||||

(6) |

In the above expressions . The expressions for IH can be obtained by replacing (implying , (i.e. and ), ). Similar expressions for anti neutrino probability () can be obtained by replacing (i.e. ), , . From eq. (6) the only diagonal parameter to which appearance channel is sensitive is . Also note that since contains terms of the form the transformation is not very important in our discussion and we concentrate on the region Miranda:2004nb ().

While assuming the presence of only the diagonal NSI parameters if
, we observe from eq. (3) that the NSI effect cancels the standard matter effect and
, for NH,
can be expressed as,
^{2}^{2}2At , and for NH (IH).

(7) |

This has a and
degeneracy. If is measured accurately such that is
not allowed then this degeneracy can be alleviated. However for
true value of since both and are same
there is an intrinsic degeneracy which cannot be
removed even if is measured precisely around these values.
Note that for ,
the third term
becomes and there is no hierarchy sensitivity.
Note that this is a special case of the and
degeneracy discussed in
Coloma:2016gei ().
We emphasize on the intrinsic nature of this degeneracy for
^{3}^{3}3In Yasuda:2015lwa (), a fit to the SuperKamiokande data assuming NSI gives the best-fit as , . However, the author mentions that this
could be because of the discrepancies between the Monte Carlo simulation of the Superkamiokande group and theirs. and .

The behaviour of as a function of is shown in fig. (1) for a fixed energy E = 2 GeV and for DUNE experiment. The 4 bands correspond to different combinations of hierarchy and octant as labelled in the figure. The width of the bands are due to variation over . We observe that the probability is a rising (falling) function of for NH (IH) for both the octants. The value , represents the standard oscillation case for which a huge separation between NH and IH bands can be seen for neutrinos, as the DUNE baseline has large matter effect.

But in presence of NSI, DUNE loses hierarchy sensitivity due to additional degeneracy creeping in through the NSI parameters. The figure exhibits the degeneracy discussed in Coloma:2016gei (). One further observes that, for , the bands for NH-LO(NH-HO) overlap with those of IH-LO(IH-HO) demonstrating the intrinsic nature of the hierarchy degeneracy.

It’s also noteworthy that this degeneracy will be there for all baselines and hence no LBL experiment will be able to resolve this degeneracy. We have checked that similar plots for antineutrinos show that the nature of the octant degeneracy as a function of is opposite and hence, running in () mode will take care of the wrong octant solutions. For known octant, the probability figures show the absence of hierarchy degeneracy for .

So far we have discussed hierarchy degeneracy for .
In fig. (2), we have plotted the probability vs for
various values
of to understand the degeneracies due to
in presence of NSI.
The various degenerate solutions observed are,
(i) The WH-RO-R^{4}^{4}4Note that WH = Wrong Hierarchy, RO = Right Octant, R = Right , W = Wrong . solution, discussed above,
occurs at .
This is seen by comparing the blue and magenta bands
or the brown and the grey bands.
Mathematically the above implies .
(ii) The WH-RO-W solutions which can be observed by comparing
the blue band with magenta band or brown band with grey band by drawing a horizontal line
corresponding to a given probability.
This can be defined as .
(iii) Apart from the degenerate solutions corresponding to one can have a more general form of the degeneracy .
This can be seen by comparing the yellow band with the dark-red band.
The conclusions made above are also true for antineutrinos.

Off-diagonal NSI :
We also study the cases of whether inclusion of off-diagonal NSI parameter
can resolve the intrinsic hierarchy degeneracy
occurring at and .
^{5}^{5}5Since, the bounds on are more tightly constrained than we consider the effect of latter.
From eq. (5)
the difference in hierarchy for
and and same
in both NH and IH can be expressed as,

(8) |

Thus, we can see that there is a finite difference between the NH and IH probabilities which vanishes if is zero or if . However, the intrinsic degeneracy may shift by an amount which depends on the values of the off-diagonal NSI parameters. For instance, assuming a small shift ‘q’ in presence of one can write . Then assuming and , can be calculated at the oscillation maxima () as,

(9) |

For instance, for and , eq. (9) gives . This shift can be observed from the green(blue) dashed and solid lines of fig. (3). These two cases demonstrate degeneracy of the form with and .

Additionally, we can also observe more general degeneracy of the form ) by drawing horizontal lines intersecting the green(blue)-dashed and blue(green)-solid lines. Fig (3) is drawn for fixed values of and . Allowing these parameters to vary can generate additional degeneracies.

Note that eq (9) and fig. (3) correspond to the energy at which the oscillation maxima occurs. We have verified that in general the amount of this shift depends on the energy and for a different energy the intrinsic degeneracy occurs at a different value of .

Thus the intrinsic degeneracy becomes energy dependent in presence of and hence spectrum information can be useful for removal of these degeneracies.

In fig. (4), we plot hierarchy Vs (Test) to understand how the diagonal and off-diagonal NSI affect the mass hierarchy sensitivity of DUNE while assuming NSI in nature.

We have used General Long baseline Experiment Simulator (GLoBES) globes1 () with additional tools from globes-nsi1 () in our numerical calculations. The experimental specifications and other numerical details are taken from Nath:2015kjg () except that this analysis is done for 40 kt detector mass. The true values that we have considered are, , , , , and Capozzi:2016rtj (). Different true values of NSI parameters are as mentioned in the fig. (4). We marginalize over , , and . The remaining NSI parameters are taken to be zero. The bounds on NSI parameters Biggio:2009nt (); Ohlsson:2012kf () that occur in , eq. (5), at 90 C.L. are considered to be , and and the corresponding off-diagonal phase .

The black solid curve shows that there is no hierarchy sensitivity for
in the absence of other NSI parameters because of
the intrinsic degeneracy in the appearance channel.
Here, the small non-zero at
arises from the disappearance channel . Because of the intrinsic nature of this degeneracy,
if hierarchy sensitivity is observed in DUNE and NSI exists in nature then
certain ranges of NSI parameters will be ruled out.
For instance assuming existence of only diagonal NSI,
the range will be ruled out if
DUNE observes hierarchy sensitivity. For other true values
of the corresponding degenerate parameter space will be
excluded.
The magenta curve shows the hierarchy sensitivity for a non-zero true value
of and .
We find that for this the global minima comes at
and there is no hierarchy sensitivity.

Note that there is also a local minima for , where we observed an intrinsic degeneracy in fig. (3). The reasons for which we do not get the global minima at this value are (i) marginalization over the phase , which had been kept fixed in fig. (3), (ii) the broadband nature of the beam at DUNE. The green dotted curve depicts the hierarchy sensitivity for a different true value of . The global minima comes at as well as a very close minima near and several other local minima. Here the global minima exhibits the most general form of the degeneracy: . However in this case sensitivity can be achieved by DUNE if the region is excluded.

Our study shows that in presence of true non-zero NSI
parameters DUNE will not have any hierarchy sensitivity if marginalized over
the model independent ranges of NSI parameters.
If however, we use the more restrictive model dependent bounds then
the hierarchy sensitivity of DUNE can improve considerably.
For instance assuming only diagonal NSI and the model dependent bound
Biggio:2009nt (), hierarchy sensitivity can be achieved for and it can be very large for
higher values of Masud:2016gcl ().

Implications of in the Sun :
Neutrinos, while travelling through the matter of varying density possibly undergo resonant flavor transitions through Mikheyev–Smirnov–Wolfenstein (MSW) effect. This resonance phenomenon accounts for the flavor transitions of solar neutrinos while they propagate from the core to the surface of the Sun.

The solar neutrino evolution equation can be written in an effective two flavor model under one-mass scale dominance (OMSD) approximation as :

(10) |

where is the coordinate along the neutrino trajectory and is the effective hamiltonian given by sum of vacuum, standard matter and NSI parts as

(11) |

where , and . Here, the usual MSW effect induced by standard matter potential is in the element of the second term in . The diagonal (real) and off-diagonal (complex) NSI parameters are related to by Kuo:1986sk (); Gonzalez-Garcia:2013usa (),

(12) | |||||

The survival probability of solar neutrinos under OMSD approximation is given by :

(13) |

where can be obtained by solving eq. (10) in the effective 2-flavor system. By diagonalizing the effective hamiltonian in eq. (11) one can obtain the effective mixing angle in the matter as :

(14) |

It can be noted that the eq. (14) reduces to the effective mixing angle of a neutrino encountering only the standard matter potential when the NSIs cease to exist i.e. .

In the previous section, while considering the presence of only diagonal NSI parameter we have taken a special case of for the Earth. Below, we have explored how this choice of NSI parameters would affect the MSW resonance in the Sun.

For the matter composition of the Earth the average of is given by the PREM model as .
Thus from eq. (4) we have

(15) |

Whereas, in the Sun, varies from in the core to at the surface. Relevant expression for can be obtained by substituting these values in eq. (4).

For instance, let us consider and which are within 1 allowed region of CHARM results CHARM (). Substituting and (at the core), in eq. (4) gives

(16) |

Thus, eq. (Can nonstandard interactions jeopardize the hierarchy sensitivity of DUNE ?) reduces to

(17) |

Now, by substituting the above equations in eq. (14) one can obtain a simplified form of

(18) | |||||

(19) |

Thus, MSW resonance in Sun occurs when the condition

(20) |

is satisfied. The electron density at resonance is,

(21) |

We have observed that for neutrino energy MeV MSW resonance occurs in the Sun, i.e. the matter mixing angle is and the corresponding survival probability is for and Large Mixing Angle solution with . This analysis can be extrapolated to the case of non-zero off-diagonal NSI parameters (say ). However, this would require a thorough study of non-standard neutrino interactions in the Sun which is beyond the scope of this paper.

Conclusions : In this paper, we make the striking observation that if the parameter , the standard matter effect gets cancelled by the NSI effects and the probability is just the vacuum oscillation probability in absence of off-diagonal NSI parameters. If in addition, the CP phase one gets an intrinsic hierarchy degeneracy which cannot be removed even if both and are known precisely. This result acquires more relevance in the light of the preference of T2K data and global oscillation analysis results implying . Although this is a special case of the generalized hierarchy degeneracy , , considered in Coloma:2016gei (), for any other value of and , a precise measurement of these parameters will alleviate the degeneracies. However for together with that is not true. To the best of our knowledge this particular point has not been highlighted before. This conclusion, being independent of baseline and energy, can seriously impact the hierarchy sensitivity of the DUNE experiment.

The matter effects can reappear, if for instance, the off-diagonal NSI parameter is included. But we show that for this case the intrinsic hierarchy degeneracy is transported from to a different value depending on the off-diagonal NSI parameters as well as energy. Moreover the uncertainty in the magnitude and phase of the off-diagonal NSI parameters can give rise to additional degeneracies affecting the hierarchy sensitivity. A more precise knowledge of the parameter , can however, give rise to an enhanced sensitivity provided . This underscores the importance of independent measurement of the NSI parameters from non-oscillation experiments Akimov:2015nza (). Furthermore, we also discussed the implications of (in the earth) on the matter effect in the Sun.

Acknowledgement: Authors thank Monojit Ghosh for his help with the code and many useful discussions.

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