Checking T and CPT violation with sterile neutrino

Post LSND results, sterile neutrinos have drawn attention and motivated the high energy physics, astronomy and cosmology to probe physics beyond the standard model considering minimal 3+1 (3 active and 1 sterile) to 3+N neutrino schemes. The analytical equations for neutrino conversion probabilities are developed in this work for 3+1 neutrino scheme. Here, we have tried to explore the possible signals of T and CPT violations with four flavor neutrino scheme at neutrino factory. Values of sterile parameters considered in this analysis are taken from two different types of neutrino experiments viz. long baseline experiments and reactor+atmospheric experiments. In this work golden and discovery channels are selected for the investigation of T violation. While observing T violation we stipulate that neutrino factory working at 50 GeV energy have the potential to observe the T violation signatures for the considered range of baselines(3000 km-7500 km). The ability of neutrino factory for constraining CPT violation is enhanced with increase in energy for normal neutrino mass hierarchy(NH). Neutrino factory with the exposure time of 500 kt-yr will be able to capture CPT violation with $ \delta c_{31}\geq 3.6\times10^{-23} $ GeV at 3$ \sigma $ level for NH and for IH with $ \delta c_{31}\geq 4\times10^{-23} $ GeV at 3$ \sigma $ level.


Introduction
The standard model of particle physics considers neutrinos to be massless. Sudbury Neutrino Observatory [1] [2] gave evidence of neutrino oscillations which was further confirmed by KamLAND experiment [3]. This landmark research assigned mass to the neutrinos and gave a clear indication of new physics beyond the standard model. A simple stretch in the standard model was able to stand up with the mass of neutrino. In neutrino physics the standard three flavour neutrino oscillations can be explained with the help of six parameters namely θ 12 , θ 13 , θ 23 , ∆m 2 12 , ∆m 2 31 and δ CP . Amongst these six parameters, solar parameters(θ 12 , ∆m 2 12 ) and atmospheric parameters (θ 23 , ∆m 2 31 ) have been measured with high precision. Furthermore, Daya Bay and RENO reactor experiments have strongly constrained the value of mixing angle θ 13 . Now we are in need of such neutrino experiments which can impose tight constraints on the value of δ CP and mass hierarchy. Some anomalies popped up while observing appearance channel and disappearance channel of ν e at LSND experiment. While observingν µ →ν e appearance channel, LSND [4] [5] [6] [7] [8] [9] was the first experiment to publish evidence of a signal at ∆m 2 ∼ 1eV 2 . Later in 2002, MiniBooNE [10] [11] checked the LSND result for ν e → ν µ (ν e →ν µ ) appearance channel. In MiniBooNE experiment, while observing the CCQE events rate through ν e n → e − p(ν e p → e + n) above 475 MeV energy, no excess events were found but for energies < 475 MeV ν e (ν e ) excess events were observed. In this way, MiniBooNE supported the LSND result. The LEP data [12] [13] advocates the number of weakly interacting light neutrinos, that couple with the Z bosons through electroweak interactions, to be 2.984 ± 0.008; thus closing the door for more than three active neutrinos. Hence, the heavy neutrino announced by LSND group should be different from these three active neutrinos. This higher mass splitting in the standard three active neutrino model was accommodated by introducing sterile neutrinos. Sterile neutrinos carry a new flavor which can mix up with the other three flavors of standard model but they do not couple with W and Z bosons. The number of sterile neutrinos can vary from minimum one to any integer N.
The four flavors of neutrino can be studied in either of the two different neutrino mass schemes, 3+1 or 2+2 schemes [36]. For our work we have selected (3+1) four flavor neutrino mass scheme. In this framework, Maki-Nakagawa-Sakata (MNS) mixing matrix (4 × 4), includes six mixing angles θ ij , three dirac phases and three majorana phases. In our analysis majorana phases are not taken into consideration.
Neutrino factory [37] [38] provides excellent sensitivity to the standard neutrino oscillation parameters and therefore seems to be one of the promising option to explore and reanalyze the global fits for sterile neutrino parameters too. To mention, it provides a platform to constrain one of the most searched CP violation in leptonic sector [39] [40].
Hence neutrino factory seems to provide a promising environment for the study of T and CPT violation. The neutrino factory set up considered here is based on the International Design Study of Neutrino Factory [IDS-NF] [41] [42]. From the measure of ∆P CP we can not directly constrain CP phase because the value of ∆P CP in the presence of matter will contain in itself some CP odd effects even in the absence of CP phase. Therefore, instead of checking ∆P CP , variation in ∆P T can be studied to probe extent of true CP violation.
Our work is organized as follows. In section 2, we illustrate 3+1 neutrino matrix parametrization. In next section, T violating effects are checked for different channels. In section 4, bounds on CPT violating terms are checked in presence of sterile neutrino. In the last section, we have summarized our study and discussed the results observed.
2 Standard Parametrization in 3 + 1 neutrino scheme The 3 (active) + 1 (sterile) neutrino scheme can be looked upon as 3+1 or 2+2 scheme depending on the selection of mass ordering of the neutrinos. To check T and CPT violation we have selected 3+1 scheme for our analysis. In this scheme the flavor eigenstates ν α (α = e, µ, τ, s) and mass eigenstates ν j (j = 1, 2, 3, 4) are related by the given unitary transformation equation Here unitary matrix (U) can be parametrized in terms of six mixing angles(θ 12 , θ 13 , θ 23 , θ 14 , θ 24 , θ 34 ), three Dirac phases δ l (δ 1 , δ 2 , δ 3 ) and three majorana phases. Majorana phases are neglected in our study as they do not affect the neutrino oscillations in any realistically observable way. In principle, there are different parametrization schemes for the neutrino mixing matrix as their order of sub-rotation is arbitrary. Our selection for parametrization of neutrino mixing matrix is where U ij (θ ij , δ l ) are the complex rotation matrices in the ij plane, defined as The order of rotation between 14 and 23 is arbitrary since these matrices commute. When neutrinos pass through the earth matter, the charge current interactions (CC) of ν e and neutral current interactions (NC) of ν e , ν µ , ν τ with the matter give rise to a CC and NC potentials V e and V n respectively. While studying the sterile neutrinos, potential V n can not be neglected. The effective CPT violating hamiltonian (H f ) of neutrinos can be expressed as Here where H 0 , H 1 and H 2 are the hamiltonians corresponding to zeroth, first and second order in η respectively. The evolution equation for neutrino oscillation probability is defined as where S(t, t 0 ) is the evolution matrix of neutrino which is also called oscillation probability amplitude The evolution matrix of neutrinos in terms of eigenvalues of H D can be written as where L ≡ t − t 0 From equation (7) the neutrino oscillation probability P αβ from flavor α to flavor β can be written as This is the general form of equation for neutrino oscillation probability.

T violation in (3+1) framework
In neutrino oscillations the flavor conversion probabilities from flavor α to flavor β can be written as Redefining the above probability equation as sum of P CP −even and P CP −odd terms CP even terms are CP conserving and can be written as CP odd term are CP violating and can be written as Assuming CPT to be conserved, the magnitude of CP violation (∆P CP ) will be equal to the magnitude of T violation (∆P T ), i.e.
therefore we can write When neutrinos passes through the earth matter, the interaction of neutrinos with matter gives rise to an extra potential. This potential is positive for neutrinos and negative for antineutrinos leading to different eigenvalues of hamiltonian for them. Further, this difference in hamiltonian for ν's andν's give rise to fake(extrinsic) CP violation. Hence, check on T violation appears to be a better choice in the presence of matter. From equation (16) the T violation can be looked upon as If we considerŨ αjŨ * βj =Ṽ αβ j and ∆ ij = 2∆Ẽ jk /L = 2(Ẽ j −Ẽ k )/L the above equation The term Im(Ṽ j βαṼ k βα * ) is known as Jarlskog factor andẼ ′ j s are energy eigenvalues of hamiltonian in matter.
The energy eigenvalues in matter can be connected to the energy eigenvalues in vacuum using the following expressions [44].
A α = A e δ αe −A n δ αs is the diagonal element of the matter potential matrix in four neutrino scheme and A is the diagonal element of matter potential matrix in three neutrino scheme.
Since T violating effects can only be studied in appearance channels so α = β. In an effort to put constraints on △P T we have studied two appearance channels. These are ν e → ν µ (golden channel) and ν µ → ν τ (discovery channel).
The T violation probability difference expression for the golden channel can be expressed Since large value of ∆m 2 41 gives rise to rapid oscillations, hence ∆m 2 41 terms can be averaged out. Solving the above expression up to the power s ∆ e = A e L/4E is matter dependent term. The change in ∆ e will change the value of (∆P T ) µe .
Further we have developed equation of ∆P T for discovery channel. The discovery channel is not very useful in the standard three neutrino flavor framework, nevertheless while studying physics beyond three active neutrino flavor framework, it becomes very important. For discovery channel(ν µ → ν τ ) the probability difference is given as Solving the above expression up to the power s 4 ij we get, The ∆P T for three neutrino framework [45] is given by Keeping the best fit values of neutrino oscillation parameters and assigning maximum value to dirac phases i.e keeping mod of sin of dirac phases to be unity will lead us to maximum value of ∆P T . This assumption will render the maximum limit on the bounds which can be imposed on T violation arising due to the presence of dirac phases if all other oscillation parameters are known with utmost accuracy. From equation (32) gives the value of (∆P T ) max = 0.137 for three neutrino flavor framework [46]. This value is independent of the selection of probing channel and presence of matter effects. Whereas in 4 flavor framework it will depend on the selection of channel through which we want to probe CP or T violation and it will vary with matter effects too. Within 4 flavor neutrino framework the magnitude of ∆P T will depend on active flavor neutrino mixing angles (known with accuracy), sterile neutrino mixing angles (still needs better bounds), matter effects, baseline, energy and dirac phases (not known) . Imposition of constraints on dirac phase (three neutrino flavor) or phases (four neutrino flavor) is still in research phase. Lorentz invariance must violate but if Lorentz invariance is violated it is not necessary that CPT invariance must violate. In our work ν µ → ν µ disappearance channel is probed to check CPT violation. The CPT violating probability difference can be written as The intrinsic CPT violation arises due to the violation of CPT invariance theorem. A hamiltonian H f containing CPT violating terms is defined by equation (4)  is large, so we average out the effects produced due to ∆m 2 41 in the probability equations. Neutrino oscillation probabilities for different oscillation channels containing CPT violating parameters can be developed as  For small angles (θ ij ≃ sin θ ij ≃ s ij ) these oscillation probabilities can be written as In order to analyse CPT violation at probability level in 4 flavor neutrino framework the value of ∆P CP T αβ considered in our analysis is given by  useful muon decays per polarity, with parent muon energy E µ = 50GeV . We have done our analysis for 10 years running of neutrino factory. In particle physics meaningful observations always demands a detector with very good energy and angular resolutions.
This view point lead us to select Liquid Argon detector for particle detection. The energy resolution of the detector for muon is σ(GeV ) = 0.20/ E ν (GeV ).
A near detector is placed at a distance 20 m from the end of the decay straight of the muon storage ring. Effective baseline (L ef f ) is used in place of baseline (L), which is calculated using L ef f = d(d + s) [55]. Fiducial mass of near detector is 200 tons. Presence of near detector will minimize the systematic uncertainties in our observations.
A 50 Kt far detector is placed at a distance of 7500 Km. The systematic uncertainties considered for this analysis are given in Table 1  and energy is shown in Figure 2 and Figure 3 oscillographs respectively. The total CPT violation captured by any experiment will be the sum of extrinsic CPT violation (CPT violation arising due to matter effects) and intrinsic or genuine CPT violation(which we are probing in present work). In an endeavour to constraint intrinsic CPT violating parameters we must look for places where extrinsic CPT violation is negligible or very less. With three active neutrinos the extrinsic CPT violation is checked in reference [67] whereas with 3 (active) + 1 (sterile) neutrinos it is checked in reference [68]. These  (34) to (44) we found that δc 21 appears with ∆m 2 21 term and δc 31 term appears with ∆m 2 31 term. The solar and atmospheric mass square difference (∆m 2 21 and ∆m 2 31 ) are of the order of 10 −23 and 10 −21 respectively. Hence, any change in mass terms due to the presence of CPT violating parameter will be better observed in δc 31 term.
To hook CPT violating impression with neutrino factory we have investigated some observable parameters like R, △R and asymmetry factor. These terms are defined by equations (45) and (46). The ratio R and ratio difference ∆R are examined as where N(ν µ → ν µ ) denotes number of muon neutrinos reaching at detector as muon neutrinos and producing a µ − lepton and N(ν µ →ν µ ) denotes number of anti muon neutrinos reaching at detector as anti muon neutrinos and producing a µ + lepton. In △R, (R 4ν ) δc ij =0 denotes the ratio R in presence of CPT violating terms and (R 4ν ) δc ij =0 denotes the ratio R in absence of CPT violating terms.  In presence of matter the observable R will not be equal to one, even if pure CPT violation is absent. It will be equal to a numerical value representing the ratio of neutrino and antineutrino interaction cross-sections. If we want to analyse the extent of deviation produced by pure CPT violation, we have to hide or filter out the deviation produced by any other phenomenon. In an attempt to filter out pure CPT violating contribution from the total observed deviation we take into record a new observable ∆R. This parameter is defined in the equation (45). The next observable asymmetry factor A µ is defined as       This observation makes 15 GeV energy important for studying CPT violating effects.
A proposal of neutrino factory producing neutrino beam of 25 GeV muons is described in reference [69] whereas in a different proposal we have a 50 GeV muon beam for the production of neutrinos in neutrino factory [70]. Therefore, by checking at the extent of bounds imposed on CPT violating parameters with energies 25 GeV and 50 GeV we want to check that by what order the results will improve if we move towards higher energies. By looking at different energy contours we conclude that amongst the three selected energies, 50 GeV energy is the best suited energy to constrain CPT violating parameter δc 31 , if nature allows NH to be true hierarchy. At the same time we observe that for long baseline experiment IH will be favourable hierarchy for determination of bounds on CPT violating parameters for energies less than 15 GeV.
As discussed earlier that, out of two parameters δc 31 and δc 21 considered in our analysis, the variation in δc 31 will produce larger variation in the detectable observables which are used in our work for checking CPT violation. Figure 12 shows value of χ 2 as a function of CPT violating parameter δc 31 . It is plotted by marginalizing over oscillation parameters ∆m 2 31 in 3σ range of their best fit values and δ CP from 0 to 2π. Looking at figure we observe that the presence of CPT violation can be detected for δc 31 3.6 × 10 −23 GeV with neutrino factory for NH within 3σ limit.

Conclusions
Neutrino factory will provide us a potential setup for observing T violation and setting significant bounds on CPT violation in neutrino sector. In four(3+1) neutrino flavor framework the angular mixing parameters of three active neutrinos are well constrained while the sterile parameters still needs better bounds on them. With the change in the value of sterile parameters a notable variation in bounds on CPT violating parameter and on the extent of T violation is captured by neutrino factory. Hence, well constrained values of sterile parameters will allow any neutrino experiment to impose better constraints on T violation and CPT violating parameters. Amongst two selected sets of values of sterile parameters i.e. from long baseline experiments and reactor+atmospheric experiments we observed that neutrino factory potential for investigating T and CPT violation enhances when the sterile parameters values are equal to those which are constrained by reactor and atmospheric experiments. Neutrino factory with 50 GeV energy is sensitive to probe T violation when true values of sterile parameters will be equal to those predicted by reactor+atmospheric experiments. We stipulate that a pure CPT violating effects can be observed along short baseline i.e 1300 km-2000 km with energies 4 GeV to 6 GeV where extrinsic CPT violation is negligible. On the other hand at long baselines we can observe these effects with energies in the range 20 GeV-40 GeV along baselines 4000 km-7500 km. CPT violating parameters δc 31 3.6 × 10 −23 GeV for NH and δc 31 4 × 10 −23 GeV for IH will make neutrino factory capable to capture signatures of CPT violation at 3 σ level.

A Eigenvalues and Eigenvectors of Hamiltonian to second order
Using the time independent perturbation theory we calculate the eigenvalues and eigenvectors of hamiltonian H f up to the order of η 2 correctly.
Eigenvalues of H 0 are given by E (0) 1 = a e + a n , E 0 2 = a n , E respectively.
The total eigenvalues and eigenvectors of H f up to second order is given by Using the set of four normalized eigenvectors we form the unitary matrixŨ as.
where V jm is normalized vector.
Now hamiltonian H f can be diagonalised by using the above derived unitary matrixŨ and the diagonalized hamiltonian H D can be expressed as