Probing CPT breaking induced by quantum decoherence at DUNE

We show that, the decoherence phenomena applied to the neutrino system could lead us to have an observable breaking of the fundamental CPT symmetry. We require a specific textures of non-diagonal decoherence matrices, with non-zero $\delta_{CP}$, for having such observations. Using the information from the CPT conjugate channels: $\nu_{\mu} \rightarrow \nu_{\mu}$ and $\bar{\nu}_{\mu} \rightarrow \bar{\nu}_{\mu}$ and its corresponding backgrounds, we have estimated the sensitivity of DUNE experiment for testing CPT under the previous conditions. Four scenarios for energy dependent decoherence parameters $\Gamma_{E_\nu}=\Gamma \times (E_\nu/\mathrm{GeV})^n$, $n=-1,0,1,$ and $2$ are taken into account, for most of them, DUNE is able to achieve a 5$\sigma$ discovery potential having $\Gamma$ in $\mathcal{O} (10^{-23}$ GeV) for $\delta_{CP}=3\pi/2$. Meanwhile, for $\delta_{CP}=\pi/2$ we reach 3$\sigma$ for $\Gamma$ in $\mathcal{O} (10^{-24}$ GeV).


I. INTRODUCTION
The neutrino oscillation is provoked by the existence of non-zero neutrino masses allied with the mismatch between its corresponding eigenstates and the neutrino flavor eigenstates. This phenomenon is supported by an overwhelming experimental evidence which spans more than two decades ago [1][2][3][4][5][6][7][8][9][10]. Notwithstanding, the neutrino oscillation is well-established, the coexistance of new physics as sub-leading effect of it has not been yet rule out. In some occasions, this new physics bring about the option of breaking fundamental laws of nature, for instance, the violation of the equivalence principle [11][12][13] or the violation of lorentz invariance [14][15][16]. The latter is described within the Lagrangian of the Standard Model Extension (SME) [17] where you can find terms that explicitly violate the combined action of the conjugation (C), the parity inversion (P) and the time inversion(T), symmetries, known in short as the CPT symmetry. This combined symmetry holds for a local, lorentz invariant and unitary quantum field theory. There has been a lot of work on testing CPT violation (CPTV) on the side of the SME [18][19][20][21]. We must remark that, there is CPTV in the neutrino oscillation in matter, originated by the unequal number of particles and antiparticles in ordinary matter [22].
On the other hand, there are a set of theoretical hypotheses such as string and branes [23,24], and quantum gravity [25] which effects can be encoded behind an omnipresent environment that could be weakly interacting with neutrinos [26,27]. This type of interactions is written according to the open system formalism and have as a typical trait (in its simplistic version) the appearance of decoherence (damping) factors exp −Γt within the neutrino oscillation probabilities [28][29][30][31][32][33][34][35]. Here, arises other type of CPTV rooted in the impossibility of defining a CPT operator by virtue of the evolution from pure states to mixed states, caused by decoherence [36,37]. In the open system approach the effects of the environment are enclosed, in a model independent way, in the so-called dissipative/decoherence matrix (after tracing out the environment degrees of freedom). Thereby, it is uncertain to claim that this type of CPTV is genuine, i.e. there is a fundamental arrow of time, or it is only an apparent CPTV, in view of our lack of knowledge of the complete system. One way or another, an eventual observation of CPTV will be shaking our current understanding of fundamental physics.
In this paper, we will focus on study different nondiagonals textures of the dissipative matrix paying special attention to those which can produce an observable non-zero CPTV. An equal response of the environment for neutrinos and antineutrinos will be one of our working hypothesis. In constrast with the hypothesis use, for instance, in [31]. We will also consider the possibility that the parameters of the dissipative matrix can be energy dependent [38,39]. We will use the DUNE experiment [40,41] as the scenario for assesing how significant would be a CPTV signature caused by quantum decoherence.

A. Neutrino as open quantum system
Our aim is to treat the neutrino as a subsystem interacting, weakly, with a large (unknown) environment. In situations of this kind, the linear evolution of the reduced density matrix of the subsystem is represented by means of the Lindblad Master equation [26,27]: where ρ(t) is the neutrino density matrix, H is the hamiltonian of the neutrino subsystem and D[ρ(t)] is the dissipative term where the decoherence phenomena is encoded. This dissipative factor is written as follows: Considering a three-level system we can expand the operators in Eq. (1) in the basis of the Gell-Mann matrices from SU (3) group plus the identity matrix: where µ is running from 0 to 8, being t 0 the identity matrix and t k the Gell-Mann matrices (k = 1, ..., 8), which satisfy [t a , t b ] = i c f abc t c , where f abc are the structure constants of SU (3). Imposing the increasing with time of the Von Neumman entropy the hermiticity of theÂ j is assured, having, as a consequence, that the dissipative matrix can be expressed as [30]: being the matrix D ≡D kj symmetric, with components D µ0 = D 0µ = 0, and a r = {a 1 r , a 2 r , ..., a 8 r }. The complete positivity condition requires that the eigenvalues of the mixing matrix ρ(t) should be positive at any time, this is achieved demanding that the matrix A ≡a nl is positive [26,27]. The scalar product structure present in the elements D kj makes them to respect the Cauchy-Schwartz inequalities. Gathering the conservation of the probability to all that we have said, we have that the evolution equation of ρ(t) is given by: where The solution of the Eq. (5) written in matricial form is: where is an eight column vector compose by the ρ k and M≡ M kj . Therefore, we can obtain a general expression for the neutrino oscillation probability ν α → ν β : Since in our analytical approach we will use the vacuum case, then, the ρ α i are already defined and they are given by: where the U αj refers to an element of the Pontecorvo-Maki-Nakagawa-Sakata(PMNS) [42,43]. If we want to solve the Eq. (7) for the antineutrino case is enough to make U αj → U * αj .

B. CPT violation and quantum decoherence
We will test the CPT symmetry in the context of DUNE using the simulated total rates associated to the ν µ and theν µ survival channels, where the matter effects are unimportant. The latter fact implies that the vaccum probabilities formulae for oscillation (plus decoherence) are going to be well enough for understanding the corresponding features of CPTV effects. Thus, all the formulae in this section will be developed under the vacuum framework. Before start, it is the utmost importance to remark that the decoherence phenomena entails the transition from pure to mixed states, which implies that the time reversal operation is, as itself, meaningless for this situation [36]. The tool for revealing these, implicit, CPTV effects is the difference between the ν µ andν µ survival probabilities channels, which written for a generic flavor ν α is: With the aim of simplyfying of the analytical form of the latter expression we work under three assumptions: the diagonal elements (damping parameters) of the dissipative matrix D are all equal to a single parameter Γ, the dissipative matrix for neutrinos is equal to the corresponding for antineutrinos, D =D, and the last is that the D matrix is containing no more than one nondiagonal elements at a time we study the ∆P CPT . As a general feature, we have that a non-zero ∆P CPT is obtained when in the survival neutrino oscillation probability there is a term with β ij (non-diagonal term) coupled to ρ α i ρ α j that contains sin δ CP , therefore, when its corresponding antineutrino term is substracted for getting ∆P CPT they do not cancel each other because of the flipping of the sign of sin δ CP . We find that the aforementioned situation (i.e. non null ∆P CPT ) is fulfilled by fifteen β ij where one coefficient in the product ρ α i ρ α j is: ρ α 2 , ρ α 5 or ρ α 7 and the other one : ρ α 1 , ρ α 3 , ρ α 4 , ρ α 6 or ρ α 8 sumarizing in total fifteen cases. The remaining β ij does not produce non-null ∆P CPT given that they are not connected with ρ α i ρ α j terms that contains sin δ CP , similar to what happen for the survival probabilities, in the pure oscillation case, where there are no terms involving sin δ CP then these do not flip sign when we switch neutrinos to antineutrinos conserving CPT.
Based on the similarities of the structure of the form for ∆P CPT we can divide these fifteen cases in two groups, each group related to different set of β ij , that we present at follows.

∆P
CPT for group one

The ∆P
CPT expression for the first group is given by: where Ω βij = β 2 ij − ∆ βij 2 , with ∆ βij = ∆m 2 βij /2E, where E is energy and ∆m 2 βij , corresponds to standard square mass differences of neutrino masses, according to its indexes ij (see Table I). This formula applies for nine β ij , the details are given in Table I. On the other hand, in Appendix A, as an example, we display in Eq. (A2) the exact probability from where we can extrapolate the ∆P CPT for β 12 .
TABLE I: Here it is displayed each group of indexes (i, j), which corresponds to a one of the nine βij. The (i, j) in the same row are associated to the ∆ β ij in the same line

CPT for group two
The ∆P CPT for the remaining six β ij : β 15 , β 24 , β 17 , β 26 , β 47 and β 56 , is also proportional to β ij , but it is rather a cumbersome expression in comparison to the one in Eq. 10. In fact, it is the addition of two terms, one of them is proportional to ρ α i ρ α j while the other one, is proportional to ρ α k ρ α l . For a given ij indexes, there is a specific kl, with each one of these indexes associated to an specific mass squared diference value, for the complete details see Table II. The six expressions for the CPTV formula are obtained per each pair ij, kl plus exchanging ij ↔ kl, with all its correspondent terms associated with them. The explicit formula is given by: where As in the case of group one, it is shown in Appendix A the probability for β 24 in Eq. (A3). From there, the corresponding ∆P CPT can be extracted.

CPT analytical results
It is important to point out that, from now on, all the results that we will present the ∆P CPT wil be calculated Here it is shown how is the relation between the six indexes (i, j) and (k, l), each of them associated to its corresponding β and its neutrino mass square differences.
The remaining parameters are given in Table III  for α = µ. In Fig. 1, we present ∆P CPT for a set of β ij per each group, which are: β 28 , β 12 and β 47 , β 56 for the group one and group two, respectively, and for neutrino energies from 0.1 to 20 GeV, which encloses the DUNE energy range. The selected β's are those who produce a bigger amplitudes in the ∆P CPT . We have evaluated this effect in an isolated manner per each β (i.e. considering all the rest of β's as zero), considering its maximum value which is obtained from the inequalities and positivity conditions given in Appendix B, having as result the following values: |β 28 | = Γ/ √ 3, |β 12 | = Γ/3 and |β 47 | = |β 56 | = Γ/ √ 3, for these plots we have taken their positive values. For all these plots it is also fixed δ CP = 3π/2 and Γ = 10 −23 GeV, being that the remaining parameters are displayed in Table III. The parameters given in Table III will be used througout this paper. We note, for group one, that the β 28 is producing the highest amplitude for ∆P CPT , in all the energy range where a slightly minor effect for β 12 is observed. In the case of group two, β 47 and β 56 give the maximum values of amplitudes of ∆P CPT up to neutrino energies a bit less than 5 GeV. In Fig. 2, we have two plots which show iso-contour curves of ∆P CPT at the plane Γ versus δ CP . For both plots the neutrino energy is fixed at 2.4 GeV keeping the remaining parameters at the same values than those used for Fig 1. One plot is for β 28 (group one) and the other for β 56 (group two), both taken equal to Γ/ √ 3. As we said before these particular β's are the ones who generate the biggest amplitudes for ∆P CPT per each group. Among the general features, we have that other than the maximum (and minimum) value of the ∆P CPT the behaviour of both plots is rather equal. Other common detail is that the ∆P CPT grows with Γ until reaching a region where the maximum amplitude is located, then starts to decrease. Outside the regions around the peaks, i.e. for lower and higher values than the Γ at the peak, the ∆P CPT is zero. For getting a full understanding of why happens this behaviour it is enough to look the formula given for group one, Eq. (10), since there is no a qualitative difference between the plots for β 28 (group one) and β 56 (group two). Hence, from Eq. (10), we see that ∆P CPT is suppressed for low values of Γ, which implies low values of β 28 (= Γ/ √ 3) that are directly proportional to the value of ∆P CPT . On the other hand, ∆P CPT is reduced for higher values of Γ, given that the latter diminishes the factor exp −Γt . From the maximization of the Eq. (10) the value of the Γ at the peak can be extracted, for β 28 the peak is at Γ ∼ 1.7 × 10 −22 GeV, similarly, if we maximize the Eq. (11) we obtain the peak for β 56 at Γ ∼ 1.6 × 10 −22 GeV. In general, a very reasonable estimation for the value of Γ at the peak is obtained from ΓL ∼ 1 then Γ ∼ 1/L, which for L = 1300 km is ∼ 1.5 × 10 −22 GeV. The corresponding values of δ CP = π/2 and 3π/2, for β 28 , can be directly inferred from the unique presence of sin δ CP in the factor ρ µ 2 ρ µ 8 , the values of δ CP for β 56 are very close to π/2 and 3π/2 for similar reasons, but they are slightly distorted due to ρ µ 6 is composed by two terms, being one of them is proportional to cos δ CP . It is important to add that, in spite of they have been not showed here, we have checked that the equivalent plots of the Fig. 2, when matter effects are included, do not reveal significant differences in comparison with the vacuum case presented here. Besides, of course, that given the presence of matter effects, a ∆P CPT = 0 is expected even in the absence of decoherence. Actually, in the experimental (simulated) searches of CPTV that we will present in the following sections, the CPTV, due to matter effects, will play the role of normalization factor.

Decoherence parameters with energy dependency
From a more general view the entries of the decoherence matrix could be energy dependent, particularly, in this paper we will adopt this dependence as follows: where n can be −1, 0, 1 and 2. The n = −1 is taken because it imitates the oscillation energy dependence being that the motivation for n = 1 and n = 2 can be found in [38] and [39], respectively. In Fig. 3 we study the ∆P CPT for the aforementioned energy dependence and setting β 28 = Γ/ √ 3, the neutrino energy in 2.4 GeV (the DUNE energy peak) and δ CP = 3π/2. In this figure we note that the energy dependency on Γ only change its value at the peak but do not affect the amplitude of ∆P CPT . As we have discussed in the section II B 3, at the peak is satisfied approximately the next relation: Γ Eν L ∼ 1 then Γ ∼ 1/(LE n ) which turns out to be in Γ ∼ {4.0, 1.5, 0.6, 0.3}×10 −22 GeV for n = −1, 0, 1 and 2 respectively.

CPT
For maximizing the ∆P CPT we simultaneously turn on β 28 , β 12 , β 56 and β 47 in the following values: β 28 = Γ/ √ 3, β 12 = ( 2/3)Γ/3 and β 56 = −β 47 = Γ/3. These values has been set according to the following steps: First, we fix β 28 = Γ/ √ 3, given that this β produces the major effect on ∆P CPT . Second, once β 28 have been defined we obtain the maximum allowed value for β 12 , which is the second in importance regarding to its impact on ∆P CPT . By last, with β 28 and β 12 already set up, we get the maximum values of β 56 and β 47 , where we have taking β 56 = −β 47 in order to obtain a constructive effect between them. The restrictions imposed by the Schwarz inequalities and positivity conditions, fully described in the Appendix B, have been considered for getting the aforementioned values of β's.

CPT violation in matter
As we have already mentioned, when the neutrinos are travelling through matter, we have a non-zero CPTV value for pure standard oscillation, even for zero CP phase. From now on, when we refer to the term standard oscillation (SO), it means that the matter effects are included. If we add the decoherence to SO, the non-zero value of CPTV is still preserved, but, it has a different magnitude with respect to its corresponding in the pure SO, because, as expected, it is distorted by the presence of the quantum decoherence parameters. In particular, it is interesting to note that this happens even when a single parameter diagonal decoherence matrix (DDM) (proportional to the identity) is considered. In constrast with the DDM case in vacuum, where a non-zero CPTV is not brought to light. The matter neutrino oscillation probabilities for a single parameter DDM can be derived only replacing the vacuum mixing angles and mass squared for their corresponding ones in matter, in, for instance, the three generation formula displayed in [30]. Of course, it also includes the replacement of a singular decoherence parameter. The application of the latter procedure is fully justified and it has been very well explained in [35]. Therefore, we have that the structure of the formula is given by: where α, ρ, are neutrino flavours, and SO (SO DDM) stands for standard oscillation (standard oscillation plus diagonal decoherence). It is clear that: ∆P SO DDM CPT = e −Γt ∆P SO CPT , which goes to zero for high values of Γ. Nonetheless, when we deal with a real situation, the latter does not occurs, since, we have to convolute the neutrino (antineutrino) oscillation probabilities with the neutrino (antineutrino) fluxes, cross sections, efficiencies and resolution, being that, for this context, the 1/3 from the first term at the RHS in Eq. (13) is the only that survives for high values of Γ, leading us to find a non-zero constant value. We will see this type of behaviour further ahead in our section of results.
In this paper we are not going derive analytical formula for the neutrino matter oscillation probability, for the non-diagonal decoherence matrices (NDM) cases that we have presented before. This is because it is a rather complicated task and, besides, as we have already argued, the vacuum oscillation probabilities formulas are going to be enough for having a qualitative understanding of our results.

III. EXPERIMENT, SIMULATION AND RESULTS
The DUNE experiment will be able to unravel several non-standard neutrino physics scenarios through oscillation measurements [45][46][47]. It will consist in a muon neutrino(antineutrino) beam traversing the Earth from Fermilab to Sanford Underground Research Facility (SURF) which comprises a distance of 1300 km and average matter density of ρ DUNE = 2.96 g/cm 3 . At SURF the neutrino beam will hit a massive liquid argon time-projection chamber (LArTPC) of 40 Ktons [40].
For this work, it is assumed the configuration of 80 GeV energy with 1.07 MW power in the primary proton beam from the Main Injector runing over 5 years for exposure for each mode (FHC and RHC). In our simulation of DUNE, the GLoBES package [48,49] is used and feeding with the information of the cross section, neutrino fluxes, resolution function and efficiency extracted from [41]. While, the matter neutrino oscillation probabilities plus decoherence was calculated with nuSQuIDS [50].
For testing the CPTV effects the following experimental observable is defined: where ∆N SO (SO DEC) = N νµ − Nν µ is the difference between the total events rates for neutrino and antineutrino, respectively, and DEC stands for any case of decoherence. The total event rates has been calculated using the prescription given in [40]. Our observable is normalized with the SO difference of events ∆N SO , which is nonzero due to matter effects plus the intrinsic differences between the cross sections, fluxes, etc for neutrinos and antineutrinos. Given our definition, when decoherence is absent R = 1.
For giving an idea of the impact of decoherence into SO physics, we display in Table IV, the total rates for four energy dependent decoherence scenarios. In Fig. 4 we are showing iso-contour curves for the observable R in the plane Γ versus δ for four plots which corresponds to n = −1, 0, 1, and 2. In these plots, the maximum amplitudes are located at similar δ CP , δ CP π/2 and 3π/2, to those presented in Fig. 2. In relation to the Fig. 2, there is a dislocation between the values of Γ at the maximum amplitudes for δ CP π/2 and 3π/2. This is mainly because of the differences in the inputs used when we convolute the probabilities for the neutrino and antineutrino mode. In addition, the Γ for δ CP π/2 and 3π/2 is shifted to its lower values whenever n increases, gaining more sensitivity to lower values of Γ. The latter kind of behaviour is expected and resembles the one we have seen for ∆P CPT in Fig. 3 (but here is a one dimensional view). Moreover, we also see the existance of degeneracies in (Γ, δ) likewise we have in [35].
In Fig. 5, we present the observable R, with its corresponding error bands for 1σ, 3σ and 5σ, versus Γ, for n = −1, 0, 1, and 2. We take δ CP = 3π/2 given that we learn from Fig. 4 that one of the maximum amplitude of ∆P CPT is obtained at this δ CP . The behaviour displayed in this plot for small and medium values of Γ, at the given scale, is rather similar than that observed in Fig. 3. However, for large values of Γ the observable R ∼ 1.17, and not ∼ 1.0, which it could be expected taking into account only the signal. This discrepancy is due to the inclusion of the backgrounds in our calculations, as described at Table IV. In order to make a comparison, we introduce in this plot the R corresponding to the single parameter DDM. We see that at small and large values of Γ, R tends to be ∼ 1. and ∼ 1.17, respectively, for the DDM and NDM, irregardless the dependency on n, as well. As we have anticipated in section II B 6, the diagonal case also produces non-zero ∆P CPT but in a lower magnitude than the NDM case. In fact, we have that for the NDM case a 5σ discrepancy, respect to the expectation value for SO ( R = 1), is reached at the following Γ = {13.1, 4.6, 2.1, 0.8} × 10 −23 GeV for n = −1, 0, 1 and 2, respectively. It is interesting to note that at these values of Γ the DDM is compatible with the SO prediction. Thus, here, we would be able to distinguish the NDM from the DDM.   An analogous result is shown n Fig. 6, but taking δ CP = π/2. In this case, the following values of Γ achieve the 3σ significance: {21.6, 6, 0.8, 0.09} × 10 −23 GeV for n = −1, 0, 1 and 2, respectively. All of them have R < 1 For the cases n = −1, 0, we can discriminate between the NDM and DDM, since we have: R < 1 and R > 1, respectively. For n = 1, the DDM case is congruent with the SO, meanwhile, for n = 2, the DDM and NDM can be confused.

IV. SUMMARY AND CONCLUSIONS
We have shown that an apparent breakdown of the fundamental CPT symmetry can take place when the neutrino system is affected by the environment. This CPTV is produces by the combination of having δ CP in the neutrino sector with a certain set of some nonnull coherences terms in the dissipative matrix. Furthermore, we have quantified a possible measurement of this CPTV using the dissapearance channels ν µ → ν µ and ν µ →ν µ , with their corresponding backgrounds, and an observable R. All in the context of the DUNE experiment. The simulated measurements of R have been performed considering four hypothesis of energy depen-  dence on the decoherence parameters: n = −1, 0, 1, and 2, where Γ Eν = Γ(E ν /GeV) n . For δ CP = 3π/2, which is rather close to the current value of δ CP given by the global fit [44], and a NDM, we achieve a 5σ for R with respect to its expectation value at the SO case, R = 1, for the following Γ: {13.1, 4.6, 2.1, 0.8} × 10 −23 GeV, for n = −1, 0, 1 and 2, respectively. At all these points, the DDM is compatible with the SO case. For δ CP = π/2, we reach discrepancies of the order of 3σ. In our best case for n = 2 we have Γ 10 −24 GeV, but with the inability of discriminating from the DDM case. We have to keep on mind that the aforementioned observations of CPTV appear when the neutrino system is treated as an open system. The latter means that it is likely that if we had access to the information of the environment, i.e. to the whole system, the overall CPT symmetry is conserved. For this reason, it deserves a more profound discussion to ascertain if this CPTV is a breaking at the fundamental level or it is only an apparent one, because of our lack of information from the environment. In some way, this CPTV represents a loss of information that in order to show that this information is not destroyed we need to know how this CPTV is compensated with the environment, probing the conservation of the information.
However, we will focus on the survival probabilities for only two cases: β 12 and β 24 , below we display these probabilities.