Probing CP violation with non-unitary mixing in long-baseline neutrino oscillation experiments: DUNE as a case study

Here we show how, from general considerations, the constraints from weak no-universality translate into restrictions on non-diagonal ${\alpha }_{{ij}}$ parameters. In order to see this, we consider the parametrization of the non-unitary lepton mixing matrix in equation (1). The diagonal entries of the pre-factor matrix are given as a product of cosines [30]:

$$\begin{eqnarray}&&{\alpha }_{11}={c}_{1n}\ {c}_{1n-1}{c}_{1n-2}\ldots {c}_{14},\end{eqnarray} \tag{ 2 }$$

while the non-diagonal parameters are expressed as [30]:

$$\begin{eqnarray}{\alpha }_{21} & = & {c}_{2n}\ {c}_{2n-1}\ldots {c}_{25}\ {\eta }_{24}{\bar{\eta }}_{14}\ +\ {c}_{2n}\ \ldots {c}_{26}\ {\eta }_{25}{\bar{\eta }}_{15}\ {c}_{14}\\ & & +\ \ldots +\,{\eta }_{2n}{\bar{\eta }}_{1n}\ {c}_{1n-1}\ {c}_{1n-2}\ \ldots {c}_{14}\end{eqnarray} \tag{ 3 }$$

with the phase factors ${\eta }_{{ij}}={{\rm{e}}}^{-{\rm{i}}{\phi }_{{ij}}}\,\sin {\theta }_{{ij}}$ and ${\bar{\eta }}_{{ij}}=-{{\rm{e}}}^{{\rm{i}}{\phi }_{{ij}}}\,\sin {\theta }_{{ij}}$ [37]. Since the 'heavy' iso-singlet admixture is assumed to be small, within the framework of seesaw schemes, as well as from experimental evidence [44], we now treat unitarity violation as a perturbation, making use of an small-angle expansion in ${\theta }_{i\beta }$, with $\beta \gt 3$, so that

$$\begin{eqnarray}&&{\alpha }_{11}^{2}\ \simeq \ 1-\displaystyle \sum _{i=4}^{N}{\theta }_{1i}^{2}.\end{eqnarray} \tag{ 4 }$$

On the other hand one can show that

$$\begin{eqnarray}&&{\alpha }_{21}\simeq -{\theta }_{24}{\theta }_{14}{{\rm{e}}}^{-{\rm{i}}({\phi }_{24}-{\phi }_{14})}-{\theta }_{25}{\theta }_{15}{{\rm{e}}}^{-{\rm{i}}({\phi }_{25}-{\phi }_{15})}\ldots -\,{\theta }_{2n}{\theta }_{1n}{{\rm{e}}}^{-{\rm{i}}({\phi }_{2n}-{\phi }_{1n})}\end{eqnarray} \tag{ 5 }$$

so that

$$\begin{eqnarray*}| {\alpha }_{21}{| }^{2} & \simeq & | {\theta }_{24}{\theta }_{14}{{\rm{e}}}^{-{\rm{i}}({\phi }_{24}-{\phi }_{14})}+{\theta }_{25}{\theta }_{15}{{\rm{e}}}^{-{\rm{i}}({\phi }_{25}-{\phi }_{15})}\ldots \,+\,{\theta }_{2n}{\theta }_{1n}{{\rm{e}}}^{-{\rm{i}}({\phi }_{2n}-{\phi }_{1n})}{| }^{2}\\ & \leqslant & \displaystyle \sum _{i=4}^{N}| {\theta }_{2i}{\theta }_{1i}{{\rm{e}}}^{-{\rm{i}}({\phi }_{2i}-{\phi }_{1i})}{| }^{2}=\displaystyle \sum _{i=4}^{N}{\theta }_{2i}^{2}{\theta }_{1i}^{2}\,.\end{eqnarray*}$$

Now from the triangle inequality relation one can write

$$\begin{eqnarray}&&\displaystyle \sum _{i=4}^{N}{\theta }_{2i}^{2}{\theta }_{1i}^{2}\leqslant \left(\displaystyle \sum _{i=4}^{N}{\theta }_{2i}^{2}\right)\left(\displaystyle \sum _{i=4}^{N}{\theta }_{1i}^{2}\right)\end{eqnarray} \tag{ 6 }$$

which implies the relation

$$\begin{eqnarray}&&| {\alpha }_{21}| \leqslant \sqrt{(1-{\alpha }_{11}^{2})(1-{\alpha }_{22}^{2})}\end{eqnarray} \tag{ 7 }$$

and similar relations will hold for ${\alpha }_{31}$ and ${\alpha }_{32}$, namely,

$$\begin{eqnarray}| {\alpha }_{31}| & \leqslant & \sqrt{(1-{\alpha }_{11}^{2})(1-{\alpha }_{33}^{2})},\\ | {\alpha }_{32}| & \leqslant & \sqrt{(1-{\alpha }_{22}^{2})(1-{\alpha }_{33}^{2})}.\end{eqnarray} \tag{ 8 }$$

One sees that in the limit of small heavy singlet messenger admixture one has,

$$\begin{eqnarray*}&&{\alpha }_{{ii}}\sim 1\,\,\,\mathrm{and}\,\,\,{\alpha }_{{ij}}\ll 1.\end{eqnarray*}$$

These relations imply additional restrictions on non-diagonal entries coming from constraints on the diagonal ones. In the next section we will see that bounds on diagonal entries are relatively strong reinforcing the bounds on non-diagonal ones. This also implies that lepton flavor violation and CP violation rates in the charged sector are constrained mainly by universality restrictions, not by the smallness of neutrino masses themselves. This important observation has previously been made in a number of papers and reviews [45–47]. In the next subsection we compile bounds from universality as well as from the relevant neutrino oscillation experiments.

The non-unitarity of the light neutrino mixing matrix can be constrained by several observables related to weak universality.

• $CKM\,unitarity$ As has been widely discussed in the literature [48–56], the comparison of measurements of muon and beta decay rates can constrain the non-unitarity of the neutrino mixing matrix. For example, the Fermi constant value for muon and beta decay will be proportional to different non-unitary parameter combinations:
  $$\begin{eqnarray}&&{G}_{\mu }={G}_{F}\,\sqrt{{({{NN}}^{\dagger })}_{11}{({{NN}}^{\dagger })}_{22}}={G}_{F}\,\sqrt{{\alpha }_{11}^{2}({\alpha }_{22}^{2}+| {\alpha }_{21}{| }^{2})},\end{eqnarray} \tag{ 9 }$$
  and
  $$\begin{eqnarray}&&{G}_{\beta }={G}_{F}\,\sqrt{{({{NN}}^{\dagger })}_{11}}={G}_{F}\,\sqrt{{\alpha }_{11}^{2}}.\end{eqnarray} \tag{ 10 }$$
  This will imply that the CKM elements V_ud and V_us, proportional to the Fermi constant ${G}_{\mu }$, should be corrected by the corresponding factor and expressed as [49–51]:
  $$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{3}| {V}_{{ui}}{| }^{2}={\left(\displaystyle \frac{{G}_{\beta }}{{G}_{\mu }}\right)}^{2}={\left(\displaystyle \frac{{G}_{F}\sqrt{{({{NN}}^{\dagger })}_{11}}}{{G}_{F}\sqrt{{({{NN}}^{\dagger })}_{11}{({{NN}}^{\dagger })}_{22}}}\right)}^{2}=\displaystyle \frac{1}{{({{NN}}^{\dagger })}_{22}},\end{eqnarray} \tag{ 11 }$$
  The experimental value of this expression is given by [57]:
  $$\begin{eqnarray}&&\displaystyle \sum _{i=1}^{3}| {V}_{{ui}}{| }^{2}=\displaystyle \frac{1}{{\alpha }_{22}^{2}+| {\alpha }_{21}{| }^{2}}=0.9999\pm 0.0006\,.\end{eqnarray} \tag{ 12 }$$
• $W\,mass\,measurements$ The mass of the W boson, M_W, is related with the values of the weak mixing angle, s_W, and the Fermi constant. Including radiative corrections, in the on-shell renormalization scheme, this relation can be written as [57]:
  $$\begin{eqnarray}&&{M}_{W}=\displaystyle \frac{{A}_{0}}{{s}_{W}{(1-{\rm{\Delta }}r)}^{1/2}},\end{eqnarray} \tag{ 13 }$$
  with
  $$\begin{eqnarray}&&{A}_{0}={\left(\displaystyle \frac{\pi \alpha }{\sqrt{2}{G}_{F}}\right)}^{1/2},\end{eqnarray} \tag{ 14 }$$
  $$\begin{eqnarray}&&{s}_{W}^{2}=0.223\,36\pm 0.000\,10,\end{eqnarray} \tag{ 15 }$$
  $$\begin{eqnarray}&&{\rm{\Delta }}r=0.036\,48\pm 0.000\,31,\end{eqnarray} \tag{ 16 }$$
  where ${\rm{\Delta }}r$ includes the radiative corrections relating α, $\alpha ({M}_{Z})$, G_F, M_W and M_Z. In the non-unitary case, the Fermi constant should take into account the corresponding corrections and the prescription for A₀ will be:
  $$\begin{eqnarray}&&{A}_{0}={\left(\displaystyle \frac{\pi \alpha \sqrt{{\alpha }_{11}^{2}({\alpha }_{22}^{2}+| {\alpha }_{21}^{2}| )}}{\sqrt{2}{G}_{\mu }}\right)}^{1/2}.\end{eqnarray} \tag{ 17 }$$
• $Semileptonic\,weak\,decays$ The couplings between leptons and gauge bosons are dictated by gauge symmetry. For the standard case of lepton unitarity these are flavor independent. This feature is no longer true in the presence of non-unitarity. As a result, the ratio between two different semileptonic decay rates would constrain non-unitarity parameters. For example, for the case of pion decay we have [52]:
  $$\begin{eqnarray}&&{R}_{\pi }=\displaystyle \frac{{\rm{\Gamma }}({\pi }^{+}\to {e}^{+}\nu )}{{\rm{\Gamma }}({\pi }^{+}\to {\mu }^{+}\nu )}=\displaystyle \frac{{({{NN}}^{\dagger })}_{11}}{{({{NN}}^{\dagger })}_{22}}=\displaystyle \frac{{\alpha }_{11}^{2}}{{\alpha }_{22}^{2}+| {\alpha }_{21}{| }^{2}}\,.\end{eqnarray} \tag{ 18 }$$
  Here we include the updated measurement from [43] and theoretical prediction in [58, 59]:
  $$\begin{eqnarray}&&{r}_{\pi }=\displaystyle \frac{{R}_{\pi }}{{R}_{\pi }^{{\rm{SM}}}}=\displaystyle \frac{(1.2344\pm 0.0029)\times {10}^{-4}}{(1.2352\pm 0.0002)\times {10}^{-4}}=0.9994\pm 0.0030\,.\end{eqnarray} \tag{ 19 }$$
  Notice that this constraint is more restrictive than the previously reported value, ${r}_{\pi }=0.9956\pm 0.0040$ [60]. One also has the corresponding bound from Kaon-decay [52]
  $$\begin{eqnarray}&&{r}_{K}=\displaystyle \frac{{R}_{K}}{{R}_{K}^{{\rm{SM}}}}=\displaystyle \frac{{({{NN}}^{\dagger })}_{11}}{{({{NN}}^{\dagger })}_{22}}=\displaystyle \frac{{\alpha }_{11}^{2}}{{\alpha }_{22}^{2}+| {\alpha }_{21}{| }^{2}}=\displaystyle \frac{(2.488\pm 0.010)\times {10}^{-5}}{(2.477\pm 0.001)\times {10}^{-5}}=1.004\pm 0.010\,.\end{eqnarray} \tag{ 20 }$$
  However, this limit does not play a significant role, since the pion decay measurements are more restrictive.
• $\mu \mbox{--}\tau \,universality$ Likewise, for the case of $\mu \mbox{--}\tau$ universality there are restrictions that follow from the ratio of the decay of the meson (${\pi }^{-}$ or ${K}^{-}$) to a muon plus a muon neutrino, or from the tau decay to a meson and a tau neutrino [30, 51, 61]:
  $$\begin{eqnarray}&&{R}_{\tau /P}=\displaystyle \frac{{\rm{\Gamma }}({\tau }^{-}\to {P}^{-}{\nu }_{\tau })}{{\rm{\Gamma }}({P}^{-}\to {\mu }^{+}{\nu }_{\mu })}\propto {\left|\displaystyle \frac{{g}_{\tau }}{{g}_{\mu }}\right|}^{2}=\displaystyle \frac{{\alpha }_{33}^{2}+| {\alpha }_{32}{| }^{2}+| {\alpha }_{31}{| }^{2}}{{\alpha }_{22}^{2}+| {\alpha }_{21}{| }^{2}}\,,\end{eqnarray} \tag{ 21 }$$
  where ${P}^{-}$ stands for either ${\pi }^{-}$ or ${K}^{-}$ mesons. Several ratios can be considered and included in the analysis. In particular we have considered the results reported in [61].
• $e\mbox{--}\tau \,universality$ On the other hand, for the e–τ sector, we have considered only pure leptonic decays as well as direct leptonic decays of W boson, which lead to
  $$\begin{eqnarray}&&{\left|\displaystyle \frac{{g}_{e}}{{g}_{\tau }}\right|}^{2}=\displaystyle \frac{{\alpha }_{11}^{2}}{{\alpha }_{33}^{2}+| {\alpha }_{32}{| }^{2}+| {\alpha }_{31}{| }^{2}}\,.\end{eqnarray} \tag{ 22 }$$
  The value of $| {g}_{e}/{g}_{\tau }|$ ratio for each process was presented in [61].
• $Invisible\text{}Z\,decay\,width$ Non-unitarity can affect the neutral current couplings. As noted in [37], these are no longer 'trivial' as in the standard model since the couplings of light neutrinos to the Z-boson can be non-diagonal in the mass basis. Moreover the diagonal coupling strengths are smaller than in the standard model thereby decreasing the invisible Z width, well measured at LEP and reported to be slightly smaller than three (2.9840 ± 0.0082) [62]. However, neutral currents have a more complex structure that will depend both on the values of the α parameters as well as on the values of the three by three matrix ${U}^{3\times 3}$. Given this complexity and the quadratic dependence on the α, it is safe not to include this observable into the analysis.

Concerning searches for lepton flavor and CP violating processes we notice that these do not give us any independent robust constraint on unitarity violation. Indeed, such processes may proceed in the absence of neutrino mass and are only restricted by weak universality tests [45, 47, 63].

Direct constraints on the non-diagonal elements of the N matrix come from the so-called zero distance (0d) effect in the conversion probability [32]. For example, the conversion probability from muon to electron neutrinos can be written as [30],

$$\begin{eqnarray}&&{P}_{\mu e}\simeq {({\alpha }_{11}{\alpha }_{22})}^{2}{P}_{\mu e}^{3\times 3}+{\alpha }_{11}^{2}{\alpha }_{22}| {\alpha }_{21}| {P}_{\mu e}^{I}+{\alpha }_{11}^{2}| {\alpha }_{21}{| }^{2},\end{eqnarray} \tag{ 23 }$$

after neglecting cubic products of the small parameters ${\alpha }_{21}$, $\sin {\theta }_{13}$ and ${\rm{\Delta }}{m}_{21}^{2}$. Here, ${P}_{\mu e}^{3\times 3}$ stands for the standard conversion probability in the unitary case, while the interference probability term ${P}_{\mu e}^{I}$ depends on the non-unitarity parameters, including an additional CP phase. Finally, the last term in this expression is a constant factor, independent of the distance traveled by the neutrino and its energy. Therefore, any neutrino appearance experiment in the ${\nu }_{\mu }\to {\nu }_{e}$ channel would be sensitive to this 0d contribution:

$$\begin{eqnarray}&&{P}_{\mu e}^{0{\rm{d}}}={\alpha }_{11}^{2}| {\alpha }_{21}{| }^{2}.\end{eqnarray} \tag{ 24 }$$

There is a similar expression for the conversion probability in the ${\nu }_{e}\to {\nu }_{\tau }$ and ${\nu }_{\mu }\to {\nu }_{\tau }$ channels. In the latter case, the oscillation probability formula is slightly more complicated, but at leading order in the non-unitary parameters, one can approximate both 0d appearance probabilities by

$$\begin{eqnarray}&&{P}_{e\tau }^{0{\rm{d}}}={\alpha }_{11}^{2}| {\alpha }_{31}{| }^{2},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{P}_{\mu \tau }^{0{\rm{d}}}\simeq {\alpha }_{22}^{2}| {\alpha }_{32}{| }^{2}.\end{eqnarray} \tag{ 26 }$$

We have used these expressions to obtain direct constraints on the parameters $| {\alpha }_{21}|$, $| {\alpha }_{31}|$ and $| {\alpha }_{32}|$ using the negative searches from the NOMAD and CHORUS short-baseline experiments. NOMAD [44, 65] reported limits on the search for ${\nu }_{\mu }\to {\nu }_{e}$ as well as ${\nu }_{\mu }\to {\nu }_{\tau }$ oscillations in a predominantly ${\nu }_{\mu }$ neutrino beam produced by the SPS at CERN, while CHORUS [66, 67] used the same beam to search for ${\nu }_{\mu }\to {\nu }_{\tau }$ oscillations. Additionally, from the contamination of electron neutrinos in the beam, they were also able to constrain the oscillation channel ${\nu }_{e}\to {\nu }_{\tau }$. The stronger bounds at 90% C.L. from these experiments, obtained by NOMAD, can be summarized as:

$$\begin{eqnarray}{P}_{\mu e}^{0{\rm{d}}} & \lt & 7.0\times {10}^{-4},\\ {P}_{\mu \tau }^{0{\rm{d}}} & \lt & 1.6\times {10}^{-4},\\ {P}_{e\tau }^{0{\rm{d}}} & \lt & 0.74\times {10}^{-2}.\end{eqnarray} \tag{ 27 }$$

Similar constraints can also be obtained from the NuTeV data [68]. Note that, in addition to the short-baseline experiments discussed above, there are also nontrivial constraints arising from medium and long-baseline experiments in combination with atmospheric and solar neutrino data [64, 69]. For maximal values of the diagonal parameters ${\alpha }_{{ii}}$, one can summarize the bounds obtained in [64] in terms of 3σ limits on the non-diagonal parameters:

$$\begin{eqnarray}| {\alpha }_{21}| & \lt & 0.03,\\ | {\alpha }_{31}| & \lt & 0.11,\\ | {\alpha }_{32}| & \lt & 0.12\,.\end{eqnarray} \tag{ 28 }$$

We stress that the above constraints coming from neutrino oscillations are independent of the mass scale of the heavy neutrinos. Therefore, they can be used to constrain the non-unitarity of the lepton-mixing matrix independent on the heavy mass scale. These are the only fully model-independent constraints. Therefore, such neutrino-data-only bounds play a special role and for this reason have been separated as the lower part in table 1. This table summarizes all the available non-unitarity bounds discussed in this section. Clearly, as seen from the upper part of table 1, one can see that universality tests provide strong constraints on the diagonal parameters, ${\alpha }_{{ii}}$, that are very close to unity, independently of the number of degrees of freedom considered. In addition, one can also combine with the relations in equations (7) and (8) in order to obtain stronger constraints on the non-diagonal α parameters. Indeed, by combining universality bounds with these relations one finds that the constraints on the non-diagonal parameters are of order 10⁻³.

Table 1.  Bounds on the magnitudes of the non-unitarity parameters at 90% C.L. and $3\sigma$ (for 1 and 6 d.o.f.). Upper table: constraints coming from neutrinos and charged leptons. Lower part: constraints derived from direct neutrino oscillation searches [44, 64, 65].

One parameter (1 d.o.f.)			All parameters (6 d.o.f.)
	90% C.L.	$3\sigma$	90% C.L.	$3\sigma$
Neutrinos + charged leptons
${\alpha }_{11}\gt$	0.9974	0.9963	0.9961	0.9952
${\alpha }_{22}\gt$	0.9994	0.9991	0.9990	0.9987
${\alpha }_{33}\gt$	0.9988	0.9976	0.9973	0.9961
$| {\alpha }_{21}| \lt$	$\ 1.7\times {10}^{-3}\$	$\ 2.5\times {10}^{-3}\$	$\ 2.6\times {10}^{-3}\$	$\ 4.0\times {10}^{-3}\$
$| {\alpha }_{31}| \lt$	$2.0\times {10}^{-3}$	$4.4\times {10}^{-3}$	$5.0\times {10}^{-3}$	$7.0\times {10}^{-3}$
$| {\alpha }_{32}| \lt$	$1.1\times {10}^{-3}$	$2.0\times {10}^{-3}$	$2.4\times {10}^{-3}$	$3.4\times {10}^{-3}$
Neutrinos only
${\alpha }_{11}\gt$	0.98	0.95	0.96	0.93
${\alpha }_{22}\gt$	0.99	0.96	0.97	0.95
${\alpha }_{33}\gt$	0.93	0.76	0.79	0.61
$| {\alpha }_{21}| \lt$	$1.0\times {10}^{-2}$	$2.6\times {10}^{-2}$	$2.4\times {10}^{-2}$	$3.6\times {10}^{-2}$
$| {\alpha }_{31}| \lt$	$4.2\times {10}^{-2}$	$9.8\times {10}^{-2}$	$9.0\times {10}^{-2}$	$1.3\times {10}^{-1}$
$| {\alpha }_{32}| \lt$	$9.8\times {10}^{-3}$	$1.7\times {10}^{-2}$	$1.6\times {10}^{-2}$	$2.1\times {10}^{-2}$

However we note that these limits are all derived from charged current induced processes under the restrictive assumption that there is no new physics other than that of non-unitary mixing. As an example, we note that the presence of neutrino-scalar Yukawa interactions, absent in the standard model but present in models with extra Higgs bosons, such as multi-Higgs schemes (e.g. incorporating flavor symmetries), would potentially avoid these bounds. Likewise, the presence of right-handed charged current contributions expected within a left–right symmetric seesaw scheme would have the same effect. This happens if the extra scalar or vector-mediated contributions compensate the unitarity violation effect⁵ . Of course one may go beyond the above well-motivated assumptions and consider, for the sake of generality, the most general Lorentz structure for the charged weak interactions⁶ . In such case these limits would be invalidated, leading us to regard them as fragile. In contrast, the constraints from neutrino experiments provide a direct restriction on the non-unitarity α parameters. These bounds are significantly less stringent, of the order of 10⁻² for the non-diagonal ${\alpha }_{{ij}}$, and correspond to the lower entries in table 1.

Finally, there are also direct bounds from searches for neutral heavy leptons. These depend on the mass of the heavy neutrinos, and do not apply beyond the kinematical reach of the high energy experiments, such as LEP [72–74] and LHC [75]. All mass-dependent limits on light and heavy singlet neutrinos have been compiled in [30, 33, 76–78].

In short, at this stage one may adopt two approaches:

• to use as reference the more restrictive bounds coming from charged current weak processes (upper part of table 1)
• to use as benchmarks bounds taken strictly from the neutrino sector (lower part of table 1).

While the top limits on the α's are stronger, they are not robust enough for our purposes, so we would recommend to focus on the most direct constraints coming from the bottom part of table 1. In any case in our simulations for the DUNE experiment in order to evaluate its potential in probing leptonic CP violation in the presence of unitarity violation we include as benchmark values not only the conservative, but also the model-dependent bounds, for comparison. The bottom-line is that the DUNE experiment will have the potential of providing independent and robust probes of neutrino properties beyond standard oscillations, properties which can not be probed otherwise in a model-independent way.

The standard derivation of the effective potential that neutrinos feel when traversing a material medium assumes unitary mixing between the light neutrino species⁸ . In order to derive the neutrino potential in matter for a model with neutral heavy leptons, we note that the complete expression for the neutrino in a flavor state will be given by

$$\begin{eqnarray}&&{\nu }_{\alpha }=\displaystyle \sum _{i}^{n}{K}_{\alpha i}{\nu }_{i},\end{eqnarray} \tag{ 29 }$$

with the α subscript indicating flavor and i mass eigenstates. Charged current matter effects in neutrino propagation are illustrated in the Feynman-like diagram in figure 1 and will be proportional to

$$\begin{eqnarray}&&{K}_{\alpha i}{K}_{{ei}}^{* }{K}_{{ej}}{K}_{\beta j}^{* }={({{KK}}^{\dagger })}_{\alpha e}\,{({{KK}}^{\dagger })}_{e\beta }\,.\end{eqnarray} \tag{ 30 }$$

Therefore, the charged current potential will be given by

$$\begin{eqnarray}&&{V}_{{\rm{CC}}}^{\alpha \beta }=\sqrt{2}\,{G}_{F}{N}_{e}{({{KK}}^{\dagger })}_{\alpha e}{({{KK}}^{\dagger })}_{e\beta }\,.\end{eqnarray} \tag{ 31 }$$

where N_e is the number density of electrons in the medium and G_F is the Fermi constant. However, the heavy states will not take part in a long baseline neutrino oscillation set up. As a result the sum in equation (29) must be performed only up to the third mass eigenstate. Therefore, effectively, one has:

$$\begin{eqnarray}&&{\nu }_{\alpha }=\displaystyle \sum _{i}^{3}{K}_{\alpha i}{\nu }_{i}=\displaystyle \sum _{i}^{3}{N}_{\alpha i}{\nu }_{i}\,,\end{eqnarray} \tag{ 32 }$$

and the effective CC potential in the presence of non-unitarity will be given by:

$$\begin{eqnarray}&&{V}_{{\rm{CC}}}^{\alpha \beta }=\sqrt{2}\,{G}_{F}{N}_{e}{({{NN}}^{\dagger })}_{\alpha e}{({{NN}}^{\dagger })}_{e\beta }\,.\end{eqnarray} \tag{ 33 }$$

which is expressed in terms of the α parameters as:

$$\begin{eqnarray}{({{NN}}^{\dagger })}_{\alpha e}{({{NN}}^{\dagger })}_{e\beta }={\alpha }_{11}^{2}\left(\begin{array}{ccc}{\alpha }_{11}^{2} & {\alpha }_{11}{\alpha }_{21}^{* } & {\alpha }_{11}{\alpha }_{31}^{* }\\ {\alpha }_{11}{\alpha }_{21} & | {\alpha }_{21}{| }^{2} & {\alpha }_{21}{\alpha }_{31}^{* }\\ {\alpha }_{11}{\alpha }_{31} & {\alpha }_{21}^{* }{\alpha }_{31} & | {\alpha }_{31}{| }^{2}\end{array}\right).\end{eqnarray} \tag{ 34 }$$

Clearly in the unitary limit (${\alpha }_{{ii}}=1$ and ${\alpha }_{{ij}}=0$), one recovers the well-known Wolfenstein form for the effective CC potential:

$$\begin{eqnarray}&&{V}_{{\rm{CC}}}^{\alpha \beta }=\sqrt{2}{G}_{F}{N}_{e}{\delta }_{\alpha e}{\delta }_{\beta e}\,.\end{eqnarray} \tag{ 35 }$$

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For the neutral current case we proceed in a similar way. Again we consider the Feynman-like diagram of the NC process as described in figure 2 and the neutral current potential is given by

$$\begin{eqnarray}&&{V}_{{\rm{NC}}}^{\alpha \beta }=-\displaystyle \sum _{\rho }\displaystyle \frac{1}{\sqrt{2}}{G}_{F}{N}_{n}{({{KK}}^{\dagger })}_{\alpha \rho }{({{KK}}^{\dagger })}_{\rho \beta }\,.\end{eqnarray} \tag{ 36 }$$

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After truncating the rectangular K matrix into the square matrix N, we obtain that the NC contribution to the matter potential is given by:

$$\begin{eqnarray}&&{V}_{{\rm{NC}}}^{\alpha \beta }=-\sqrt{2}{G}_{F}\displaystyle \frac{{N}_{n}}{2}\displaystyle \sum _{\rho }{({{NN}}^{\dagger })}_{\alpha \rho }{({{NN}}^{\dagger })}_{\rho \beta }=-\sqrt{2}{G}_{F}\displaystyle \frac{{N}_{n}}{2}{[{({{NN}}^{\dagger })}^{2}]}_{\alpha \beta },\end{eqnarray} \tag{ 37 }$$

where the matrix product ${({{NN}}^{\dagger })}^{2}$ at leading order in the non-diagonal α's is given by:

$$\begin{eqnarray}\left(\begin{array}{ccc}{\alpha }_{11}^{4} & {\alpha }_{11}{\alpha }_{21}^{* }({\alpha }_{11}^{2}+{\alpha }_{22}^{2}) & {\alpha }_{11}{\alpha }_{31}^{* }({\alpha }_{11}^{2}+{\alpha }_{33}^{2})\\ {\alpha }_{11}{\alpha }_{21}({\alpha }_{11}^{2}+{\alpha }_{22}^{2}) & {\alpha }_{22}^{4} & {\alpha }_{22}{\alpha }_{32}^{* }({\alpha }_{22}^{2}+{\alpha }_{33}^{2})\\ {\alpha }_{11}{\alpha }_{31}({\alpha }_{11}^{2}+{\alpha }_{33}^{2}) & {\alpha }_{22}{\alpha }_{32}({\alpha }_{22}^{2}+{\alpha }_{33}^{2}) & {\alpha }_{33}^{4}\end{array}\right)\,.\end{eqnarray} \tag{ 38 }$$

So, one sees how, starting with diagonal CC and NC potentials, due to the non-unitarity one ends up in general with non-diagonal forms for the effective matter potentials.

Notice that the non-unitarity parameters ${\alpha }_{31}$, ${\alpha }_{32}$ and ${\alpha }_{33}$, which do not enter in the expression of ${P}_{\mu e}$ in vacuum [30, 33], do appear in the calculation of ${P}_{\mu e}$ in matter due to the form of the effective matter potential. The effect of the non-diagonal parameters ${\alpha }_{31}$ and ${\alpha }_{32}$ is not as important as the role of ${\alpha }_{21}$. The ${\alpha }_{31}$ parameter enters linearly in the CC and NC potential in the 13 entry. Its effect will be analogous to that of the parameter ${\epsilon }_{e\tau }$ in the case of non-standard interactions, so that the resulting degeneracy with the reactor angle ${\theta }_{13}$ [82, 83] will imply a deterioration of the sensitivity to CP violation [28]. In contrast, ${\alpha }_{32}$ will enter only in the neutral current potential in the 23 entry and, therefore, is expected to have a negligible impact.

Adding the two contributions to the effective potential in matter we will have⁹ :

$$\begin{eqnarray}&&{V}^{\alpha \beta }={V}_{{\rm{CC}}}^{\alpha \beta }+{V}_{{\rm{NC}}}^{\alpha \beta }=\sqrt{2}\,{G}_{F}{N}_{e}{({{NN}}^{\dagger })}_{\alpha e}{({{NN}}^{\dagger })}_{e\beta }-\sqrt{2}{G}_{F}\displaystyle \frac{{N}_{n}}{2}\displaystyle \sum _{\rho }{({{NN}}^{\dagger })}_{\alpha \rho }{({{NN}}^{\dagger })}_{\rho \beta },\end{eqnarray} \tag{ 39 }$$

where α and β stands for the initial and final neutrino flavor, respectively, and ρ implies a sum over the three active flavors. N_e is the electron density in the medium while N_n is the neutron density. In matrix form one has the following expression for the matter potential in the presence of non-unitarity:

$$\begin{eqnarray}{V}_{{\rm{NU}}}=({{NN}}^{\dagger })\left[\sqrt{2}\,{G}_{F}{N}_{e}\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{array}\right)-\displaystyle \frac{\sqrt{2}}{2}\,{G}_{F}{N}_{n}\left(\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{array}\right)\right]({{NN}}^{\dagger })\end{eqnarray} \tag{ 40 }$$

leading to a very simple compact form

$$\begin{eqnarray}&&{V}_{{\rm{NU}}}=({{NN}}^{\dagger }){V}_{\mathrm{unitary}}({{NN}}^{\dagger })\,.\end{eqnarray} \tag{ 41 }$$

Notice that, in contrast to the standard procedure used in the three-neutrino unitary case, the contribution of the neutral current potential can no longer be neglected when treating the non-unitary case.

One can also see how to get this result from the truncation of the N × N mixing matrix, U. Therefore, the Hamiltonian in matter in the flavor basis will be given by:

$$\begin{eqnarray}{H}_{{\rm{N}}{\rm{U}}}=N\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & \displaystyle {\textstyle \tfrac{{\rm{\Delta }}{m}_{21}^{2}}{2E}} & 0\\ 0 & 0 & \displaystyle {\textstyle \tfrac{{\rm{\Delta }}{m}_{31}^{2}}{2E}}\end{array}\right){N}^{\dagger }+({NN}^{\dagger })\left(\begin{array}{ccc}{V}_{{\rm{c}}{\rm{c}}}+{V}_{{\rm{n}}{\rm{c}}} & 0 & 0\\ 0 & {V}_{{\rm{n}}{\rm{c}}} & 0\\ 0 & 0 & {V}_{{\rm{n}}{\rm{c}}}\end{array}\right)({NN}^{\dagger })\end{eqnarray} \tag{ 42 }$$

with ${V}_{{\rm{cc}}}=\sqrt{2}{G}_{F}{N}_{e}$ and ${V}_{{\rm{nc}}}=-\tfrac{\sqrt{2}}{2}{G}_{F}{N}_{n}$.

DUNE is a long-baseline neutrino oscillation experiment that will measure neutrino oscillations over a broad energy range, from hundreds of MeV to few tenths of GeV. This experiment will detect neutrinos and anti-neutrinos produced in the NuMI beam line at Fermilab 1300 km away from the source, with relevant matter effects in the neutrino propagation. The effect of non-unitary neutrino mixing in the DUNE simulation will modify the standard calculation of neutrino oscillation probabilities. Besides the non-unitary neutrino mixing matrix, to calculate the neutrino conversion probability in DUNE one must take into account the modified matter potential affecting neutrino propagation through the Earth, as discussed in the previous section. Using the neutrino Hamiltonian with matter effects as given by equation (42), we have solved numerically the evolution equation, obtaining the corresponding conversion probability from muon to electron neutrino in the case of DUNE. We illustrate the behavior of the modified neutrino appearance probability ${P}_{\mu e}$ for the DUNE experiment in figure 3. The left panel corresponds to neutrino probability and the right panel to antineutrino probability. Each band corresponds to a different value of the standard CP phase, ${\delta }_{\mathrm{CP}}$, while the width of the band is due to the variation over the non-unitary phase ${\phi }_{21}={\rm{Arg}}({\alpha }_{21})$. The only deviation from unitarity in this calculation comes from the ${\alpha }_{21}$ parameter, set to 0.02. The overlap of the different bands indicates the presence of degeneracies in the neutrino oscillation probability in DUNE. This ambiguity, present at the probability level has already been noticed in [31, 33].

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The DUNE experimental setup assumed for this analysis corresponds to a 40 ton liquid argon far detector with optimized neutrino fluxes, cross sections, detector efficiency and energy resolution effects as provided in the form of GLoBES [86, 87] files in [88]. The calculation of the neutrino oscillation probabilities in the presence of non-unitarity has been implemented in the GLoBES package with an adequate modification of its probability engine. In our analysis, we have used the spectral event information from the four neutrino oscillation channels: electron (anti)neutrino appearance and muon (anti)neutrino disappearance. To statistically quantify the effect of the non-unitary lepton mixing parameters we have used the usual ${\chi }^{2}$ definition adding penalties on the 'unitary' oscillation parameters ${\theta }_{{ij}}$ and ${\rm{\Delta }}{m}_{k1}^{2}$ [89]. The relative error on these parameters and the systematic uncertainties in the normalization of signal and background for each oscillation channel, ranging from $0.2 \%$ to 20% depending on the channel, were set to the values given in [88].

For the rest of this section we denote the three mixing angles collectively as a vector $\vec{\lambda }=\{{\theta }_{{ij}},{\rm{\Delta }}{m}_{k1}^{2}\}$. We have included penalties to the ${\chi }^{2}$ accounting for the allowed values of the $\vec{\lambda }$ parameters. Likewise, we denote the non-unitarity parameters in a compact form as $\vec{\alpha }=\{{\alpha }_{{ii}},{\alpha }_{{ij}}\}$, including both their diagonal and non-diagonal components. Note that we treat the CP phase ${\delta }_{{\rm{CP}}}$ separately.

In this section we analyze how DUNE sensitivity to the standard CP violation is affected by the presence of non-unitarity. To the oscillation parameters present in the standard unitary scenario, this analysis implies the addition of nine real parameters describing the non-unitary mixing: the three real ${\alpha }_{{ii}}$ plus the three complex non-diagonal ${\alpha }_{{ij}}$.

In order to simplify the analysis, we consider however only five non-zero non-unitary parameters at a time: the three diagonal ones, plus one of the non-diagonal ones, with its complex phase at a time. The resulting CP sensitivity in the presence of non-unitarity is shown in figure 4. As in the standard ${\delta }_{\mathrm{CP}}$-sensitivity plot, the CP-violation hypothesis is tested with respect to a CP-conserving scenario [11]:

$$\begin{eqnarray}&&{\rm{\Delta }}{\chi }^{2}({\delta }_{{\rm{CP}}}^{{\rm{true}}})=\mathrm{Min}[{\rm{\Delta }}{\chi }_{{\rm{CP}}}^{2}({\delta }_{{\rm{CP}}}^{{\rm{true}}},{\delta }_{{\rm{CP}}}^{{\rm{test}}}=0),{\rm{\Delta }}{\chi }_{{\rm{CP}}}^{2}({\delta }_{{\rm{CP}}}^{{\rm{true}}},{\delta }_{{\rm{CP}}}^{{\rm{test}}}=\pi )]\,.\end{eqnarray} \tag{ 43 }$$

The remaining standard $\vec{\lambda }$ as well as the non-unitarity parameters $\vec{\alpha }\equiv \{{\alpha }_{{ii}},{\phi }_{{ij}}\}$, which are included in both the simulated and reconstructed event rates in DUNE, $n(\vec{\lambda },{\delta }_{\mathrm{CP}};\vec{\alpha })$ , have been marginalized over. The left panel has been obtained for different values of $| {\alpha }_{21}|$, while the right panel corresponds to the results of the corresponding analysis performed for the non-diagonal non-unitarity parameter ${\alpha }_{31}$. One sees from the left panel that the sensitivity to the Dirac CP phase decreases in the presence of non-unitarity with respect to the standard 'unitary' case, shown in the black-solid line. The remaining lines correspond to the non-unitary case with different values for the ${\alpha }_{21}$ parameter, as indicated. We have selected three different benchmark values, the smaller one, 0.003, consistent with the upper part of table 1 and 0.010 and 0.025 consistent with sensitivities displayed in the lower part of the table, obtained from neutrino data only. Even taking at face value the 'aggressive' sensitivity $| {\alpha }_{21}| =0.003$, the significance of a CP-violation measurement decreases by $0.85\sigma$, compromising the possibility of testing any range of values of the CP phase at $5\sigma$. For more conservative and reasonable choices $| {\alpha }_{21}|$ at the 1% level one sees that the presence of non-unitarity precludes our ability to probe CP violation at $3\sigma$ for nearly all of the ${\delta }_{{\rm{CP}}}$ range. One sees that probing maximal CP violating values $\pm \pi /2$ with high significance in the presence of non-unitarity for 'large' $| {\alpha }_{21}|$ constitutes a big challenge for DUNE. In the right panel, we show the results of the same analysis for the non-diagonal parameter ${\alpha }_{31}$. As we discussed before, the impact of this parameter on the neutrino oscillation probabilities in DUNE is significantly less relevant in comparison with ${\alpha }_{21}$. As a result, one can see from the figure that even if the fraction of CP at $5\sigma$ is largely reduced respect to the unitary case, the reduction in the significance of CP tests is much smaller than for $| {\alpha }_{21}|$. This result holds for relatively large values of the parameter compatible with the bounds in the lower part of table 1, namely $| {\alpha }_{31}| =0.05$. The effect of the third non-diagonal parameter, ${\alpha }_{32}$, is not displayed in the figure. We have checked that it plays nearly no role in the analysis, which confirms our discussion in section 3.

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Here we note that [90] has also discussed the possibility of probing CP violation with T2K, NOvA and DUNE in the presence of non-unitarity. Although we have a qualitative agreement in the loss of CP-sensitivity due to the presence of non-unitarity, our results show some quantitative differences. We ascribe these discrepancies to the treatment of the DUNE simulation. Here we are using the official description released by the DUNE Collaboration, and we have validated our method against the official DUNE CP-sensitivity result for the standard (unitary) oscillation analysis.

In this section we analyze the potential of DUNE in constraining the non-unitarity of the neutrino mixing matrix. As we have discussed in section 2, the most robust and direct of these constraints come from neutrino oscillation experiments and are not very strong. Therefore, we wish to explore the capability of DUNE in further constraining non-unitarity. For this purpose we will focus in the analysis of the neutrino signal at the DUNE far detector. The capability of the near detector will be analyzed in the future. As we have discussed in the previous subsection, the parameter with the most impact on the DUNE sensitivity to CP violation is ${\alpha }_{21}$. As a result we will focus on ${\alpha }_{21}$ as the key parameter to be constrained in order to characterize the loss of sensitivity in CP searches at DUNE. Following the usual procedure in analyzing the sensitivity of a given experiment to an unknown parameter (the non-unitary parameter ${\alpha }_{21}$ in this case), we have simulated DUNE events under the hypothesis of unitary mixing ${n}^{\mathrm{true}}(\vec{\lambda },{\delta }_{\mathrm{CP}}^{\mathrm{true}})$. Afterwards, we have tried to reconstruct DUNE data in terms of the non-unitary neutrino mixing ansatz, ${n}^{\mathrm{test}}(\vec{\lambda },{\delta }_{\mathrm{CP}}^{\mathrm{test}};\vec{\alpha })$. It is worth noticing that the treatment of the non-unitarity here is different from the analysis performed in the previous subsection and therefore a direct comparison between the results presented in figures 4 and 5 is not straightforward. For this analysis, the true value of the Dirac CP phase has been fixed to its current preferred value, ${\delta }_{\mathrm{CP}}^{\mathrm{True}}=-\pi /2$. After marginalizing over the diagonal non-unitary parameters and all the oscillation parameters but ${\delta }_{\mathrm{CP}}$, we obtained the allowed parameter regions (at $1\mbox{--}4\sigma$ for 2 d.o.f) shown in figure 5. In the left panel, the allowed regions in the ${\delta }_{\mathrm{CP}}-| {\alpha }_{21}|$ plane show that DUNE is sensitive to values of $| {\alpha }_{21}|$ at the percent level at $1\sigma$. As expected, the best fit point for the Dirac CP phase is equal to the assumed 'true' value. However, for large enough values of $| {\alpha }_{21}|$, degenerate solutions around ${\delta }_{\mathrm{CP}}=\pm \pi$ appear at higher C.L.

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Finally, we present an estimate of the absolute sensitivity of DUNE to the non-unitary parameter ${\alpha }_{21}$. In order to do this, we extended our previous analysis, considering all the possible values of ${\delta }_{\mathrm{CP}}^{\mathrm{True}}$ and marginalizing over ${\delta }_{\mathrm{CP}}$ and ${\phi }_{21}$. Figure 6 shows the ${\chi }^{2}$ profile obtained as a function of $| {\alpha }_{21}|$ after marginalizing over all the remaining parameters, including ${\delta }_{\mathrm{CP}}^{\mathrm{True}}$. The best fit point, denoted by a black point in the figure, is obtained for $| {\alpha }_{21}| =3\times {10}^{-4}$. Nevertheless, the preference over the unitary hypothesis is not significant at all, as can be seen from the figure. The shaded band in figure 6 indicates the three benchmark values of ${\alpha }_{21}$ used in the analysis of CP sensitivity in DUNE (see the left panel of figure 4), while the horizontal dotted black line defines the parameter region allowed by DUNE at 90% of C.L., corresponding to the limit $| {\alpha }_{21}| \lt 0.046$. This bound is somewhat weaker than the constraints derived from neutrino oscillation searches, indicating that the analysis of long-baseline neutrino oscillations in DUNE is not expected to improve our current knowledge on the non-unitarity of the neutrino mixing matrix. However, it is worth mentioning that this constraint can also be regarded as independent and complementary to the bounds in table 1.

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