Matter effect of light sterile neutrino: an exact analytical approach

The light sterile neutrino, if it exists, will give additional contribution to matter effect when active neutrinos propagate through terrestrial matter. In the simplest 3+1 scheme, three more rotation angles and two more CP-violating phases in lepton mixing matrix make the interaction complicated formally. In this work, the exact analytical expressions for active neutrino oscillation probabilities in terrestrial matter, including sterile neutrino contribution, are derived. It is pointed out that this set of formulas contain information both in matter and in vacuum, and can be easily tuned by choosing related parameters. Based on the generic exact formulas, we present oscillation probabilities of typic medium and long baseline experiments. Taking NOνA experiment as an example, we show that in particular parameter space sterile neutrino gives important contribution to terrestrial matter effect, and Dirac phases play a vital role.


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This paper is organized as follows. In section 2 we will give a brief introduction of neutrino oscillation with the consideration of matter effect. In section 3 we will derive the mass-square differences within matter effect, show related rotation matrix elements and propagation probabilities explicitly. The applications of such analytical solution will be presented in section 4. In section 5, we will summarize and give a conclusion. More details involved in section 3 are shown in appendix.

Theorectial framework
The picture of neutrino oscillation is well understood currently. The identity of neutrinos in flavor space and mass space is not identical, or they have a mixing. Due to such a mixing, described by rotation matrix U , the identity of neutrinos can be changed during its journey from source to destination, called neutrino oscillation. The oscillation probability, that is the probability for capturing neutrino as ν β from the initial beam ν α , is where ∆ ij ≡ ∆m 2 ij L/(2E) with ∆m 2 ij = m 2 i − m 2 j , while L is propagating distance, and E is the energy carried by neutrinos. Both appear mode and disappear mode are contained in eq. (2.1).
The oscillation probability is determined by universal parameters U αi , m i as well as experiment dependent parameters E and L. In Standard Model (SM) there are only three flavors of active neutrinos, thus the mixing matrix, named PMNS matrix, is parameterized by three rotation angles and one CP-violating phase. Within this theoretical framework the three angles (θ 12 , θ 23 , θ 13 ) are measured by solar neutrino, atmospheric neutrino and reactor neutrino experiments, respectively. The remaining undetermined parameter is CP phase δ, as well as the sign of ∆m 2 13 , could be reachable in the following ten years. On the other hand, the possibility to have one more light sterile neutrino still exists. The sterile neutrino (denoted as ν s ), unlike the active neutrinos (denoted as ν e , ν µ , ν τ in flavor state), are known for its absence from SM weak interactions. However, its effect appears indirectly by mixing with active neutrinos. Such a mixing is described by lepton mixing matrix U , given which characterizes the rotation between mass eigenstate and flavor eigenstate in vacuum.
With more degrees of freedom in U , the oscillation probability will contain richer information. There are 6 angles and 3 phases to parameterize mixing matrix U . Similar to -3 -

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the standard parameterization of PMNS matrix, putting the two extra phases in 1-4, 3-4 plane, we can write down the four dimensional mixing matrix as U = R(θ 34 , δ 34 )R(θ 24 )R(θ 14 , δ 14 )R(θ 23 )R(θ 13 , δ 13 )R(θ 12 ), (2.3) in which R (θ ij (, δ ij )) represents Euler rotation in i-j plane without (with) a CP phase. More details for four dimensional U are shown explicitly in appendix A. When passing through matter, active neutrinos interact with matter by weak interaction. More exactly ν e interacts via both charged current and neutral current while ν µ , ν τ only receive neutral current interaction by exchanging Z bosons. Though sterile neutrino itself does not take part in weak interaction, by removing the global neutral current which will not affect oscillation probability, ν s has an induced nonzero term in effective Hamiltonian while the corresponding ones for ν µ,τ vanish, giveñ where U is the lepton mixing matrix in vacuum and Without loss of generality, the Hamiltonian can always be written in a more compact form where the effective massm i and the new defined effective lepton mixing matrixŨ have incorporated information of matter effect. And hence the oscillation probability including matter effect has the same structure of the one in vacuum, that is Hereafter we will adopt P to stand forP for its clear meaning. Obviously if one works out explicitlyŨ αi andm 2 i , the probability will be well presented. However, such a calculation would be challenging. To avoid the difficulty, by working out the effective mass differences and some necessary combinations of entries ofŨ , we can obtain complete exact expressions for P as well. In the following section, we will derive the necessary parameters.

Effective parameters
Mass difference ∆m 2 ij and lepton mixing matrix U αi in vacuum are universal. The corresponding ones in matter will be corrected by matter parameters. We will give ∆m 2 ij firstly, based on whichŨ αi is also obtained.

Effective mass-square difference
In this part, we aim at solve ∆m 2 ij . It is known that a constant can be removed from diagonal entries simultaneously, as it contributes to a global phase which does not affect oscillation probability. Then by subtracting a global m 2 1 in eq. (2.4), we have The induced mass difference can also be written in the form of∆m 2 ij =∆m 2 i1 − ∆m 2 j1 . With the help of∆m 2 ij , the effective mass difference ∆m 2 ij which we seek for is constructed as The effective mass difference between two arbitrary effective masses can be resort to those differences fromm 2 1 . Thus the aim now is simplified to find out∆m 2 i1 , that is the diagonalization of the right-handed side of eq. (3.1).
In principle the key point of the diagonalization is to solve a quartic equation, which is fortunately solvable. The particular involved quartic equation as well as its solutions are shown in appendix B. With necessary new-defined parameters, we can obtain the exact analytical expressions for∆m 2 j1 (j = 1, 2, 3, 4), which depend on ∆m 2 i1 (i = 1, 2, 3, 4). Hence by the usage of eq. (3.2) the effective mass difference, including matter effect correction, now can be expressed explicitly in which b, p, q and S are intermediate parameters defined in appendix B. Note here we have assumed normal mass hierarchy (m 2 1 < m 2 2 < m 2 3 < m 2 4 ). Without loss of generality, other situations of mass ordering can be derived similarly.

Effective lepton mixing matrix
In addition to effective mass difference, the oscillation probabilities of neutrino propagation rely on lepton mixing matrix as well. Without relating to each entry of the matrix, only some particular combinations are concerned. We have shown how to solve these quantities in appendix C, in which the general expressions have been given in eqs. (C.7), (C.8), (C.9), (C.10). In this section, we restrict our interests typically in reactor neutrino and accelerator neutrino experiments. The relevant entries are listed below explicitly.
• The reactor neutrino experiments:ν e →ν e For the disappear mode of anti-electron neutrino, the only concerned entryŨ ei is in which the auxiliary quantities are For the first glance, |Ũ ei | 2 relies on ∆m 2 ij , ∆m 2 ij and U αi . Note ∆m 2 ij =∆m 2 i1 −∆m 2 j1 and∆m 2 i1 is the solution for quartic equation which further relies on ∆m 2 i1 and U αi as well as matter effect parameters A, A . So the free parameters for matter effect correction to |Ũ ei | 2 are ∆m 2 ij , U αi , A and A .
• The accelerator neutrino experiment: disappear mode ν µ → ν µ Both disappear mode and appear mode will be used in accelerator neutrino experiments. For the disappear mode, the required |Ũ µi | 2 is given as with the associated functions Except a difference in F ij α , all other terms are same as |Ũ ei | 2 up to a corresponding change e → µ. One may find a consistent result from [28].
• The accelerator neutrino experiment: appear mode ν µ → ν e In this case, a distinct difference from disappear mode is that the product of two entries,Ũ eiŨ * µi , are required. One can immediately have the following relation according to the general expression in appendix C, Note the corresponding result provided in [28] is not consistent with ours, while our calculation can be confirmed by numerical evaluation.

Exact oscillation probability
Armed with effective mass difference and effective mixing matrix entries, the oscillation probabilities are spontaneously presented as, where∆ ij ≡ ∆m 2 ij L/2E and L is the baseline of a particular neutrino experiment. The input parameters are (∆m 2 i1 , U αi , A, A ), where the description of U αi further relies on their parametrization, one of them can be found in appendix A.
Thorough out the whole derivation, no additional assumptions are adopted except the unitary condition of U andŨ . So the exact analytical expressions are applicable for both short baseline and long baseline experiments. Meanwhile we would like to point out that the formulas derived here are the most generic ones in 3+1 scheme, since all possible situations, including SM case, are all contained in. In particular, we may get the following extreme cases by tuning parameters in our formulas, i) active neutrino propagating in matter with sterile neutrino effect: ii) active neutrino propagating in vacuum with sterile neutrino effect: iii) active neutrino propagating in matter without sterile neutrino effect: U α4 = 0, U si = 0(, A = 0), A = 0, in which whether A vanishes doesn't give an effect.
iv) active neutrino propagating in vacuum without sterile neutrino effect: By setting θ 14 = θ 24 = θ 34 = 0, to close parameters U α4 and U si can be easily fulfilled.

Applications and discussion
The exact analytical solution keeps the original information of sterile neutrino without any approximation. Since sterile neutrino mass is still unknown, approximated formulas, though can speed up evaluation, still have a risk to lose some information. In this section, based on the exact solutions, we give a numerical analysis for typical neutrino experiments. For each experiment, two types of input parameters are relevant. One type is the universal parameters, including mixing matrix and mass difference, while the other nonuniversal one depends on experiment location, neutrino source and matter effect parameters. For illustration, we take input parameters as follows. There are 6 rotation angles and 3 Dirac phases in mixing matrix, while the oscillation irrelevant Majorana phases can be ignored here. We take sin 2 θ 13 = 0.0218, sin 2 θ 12 = 0.304, sin 2 θ 23 = 0.437 from the SM global fitting [30], the other 3 angles we choose sin 2 θ 14 = 0.019, sin 2 θ 24 = 0.015, sin 2 θ 34 = 0 [16]. Throughout the simulation, we fix one of the three Dirac phases as δ 34 = 0, and let the other two as free parameters for the purpose of illustration. As for the mass-square difference, two of the three are consistent with SM global fitting, ∆m 2 21 = 7.5 × 10 −5 eV 2 , ∆m 2 31 = 2.457 × 10 −3 eV 2 , the remaining one is fixed as ∆m 2 41 = 0.1 eV 2 . To describing matter effect, we adopt the relevant parameters from realistic oscillation experiment [31], which set matter density as ρ ≈ 2.6g/cm 3 and eletron fraction Y e ≈ 0.5.

Medium baseline experiment
Around a nuclear power plant (NPP), there are plenty of antielectron neutrinos produced via β decay in nuclear reactions. Detectors can be put in suitable places near to the nuclear plant to explore reactor neutrino events. Usually the baselines of such kind of experiments are in the range of short or medium baseline. For the exploring experiments, matter effect is not taken within the main considerations. But the situation could be changed in precise measurement as experiment sensitivity may be affected. The ongoing Jiangmen Underground Neitrino Observatory (JUNO) experiment [32], with its baseline L = 52.5 km, is one of such kind of experiments.
Regarding to matter effect, whether the oscillation probability will change with or without matter effect, both in purely 3 flavor active neutrinos case and in the framework of active plus sterile neutrino case, is what we are concerned. In figure 1 we take the relative difference for probability, from matter to vacuum, and plot it varying by energy. In order to discriminate the Dirac phases' effect, we choose typic values of the two phases δ 13 , δ 14 and make various combinations. In this analysis only normal hierarchy (NH) of neutrino mass situation is presented, while the inverted hierarchy (IH) case has similar behaviors,

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though not shown explicitly. 1) Around the most promising range E ∼ 3MeV, the relative difference can reach 4%, which could possibly be distinguished by JUNO detector.
2) The sterile neutrino contribution does not affect probability curve dramatically, that is to say for short/medium baseline experiment, sterile neutrino effect is quite limited.
3) The effect from Dirac phase seems bleak, no distinction can be reflected from different phase combinations.
Hence we may conclude that the short and medium baseline experiments are not sensitive to the matter effect of sterile neutrino, as well as the CP-violating Dirac phases.

Long baseline experiment
Neutrino beam produced from accelerators usually carries higher energy and can be detected in a long distance from source. In this part, we will take NOνA experiment [33], with its baseline L = 810km, as an example to illustrate the properties of long baseline experiments case.
Here we show the oscillation probability of appearance mode ν µ → ν e in long baseline accelerator neutrino experiment in figure 2, where the brown curves stands for oscillation in 3+1 scheme and blue curves correspond to SM case while solid (dashed) lines mean matter effect has (not) been contained. In the plot, we have chosen typical CP phase combination of (δ 13 , δ 14 ) in NH case, and consider its variation in energy range 1 ∼ 3GeV. One can address the following points: 1) The matter effect can not be negligible, on the contrary, it is important both in 3ν and 4ν case. At about 1 GeV range, the relative difference for probabilities can be as large as 50% in whichever scenario. This difference could be ∼ 20% around the maxima of oscillation probabilities.
2) No matter propagating in vacuum or in matter, sterile neutrino gives nonnegligible contribution to oscillation probability. In each graph, the dashed lines have obvious deviation from their solid correspondence.
3) The CP-violating Dirac phases also plays a non-ignorable role. By comparing figure 2a with figure 2c, one may see the oscillation probability has been affected. In the scenario of (δ 13 , δ 14 ) = (0, π 2 ), one can see the blue lines are almost in the middle of corresponding brown lines. But the blue curves has a distinct deviation from the average lines of brown ones in the scenario of (δ 13 , δ 14 ) = ( π 2 , π 2 ).
Therefore we may conclude that in the long baseline experiment, in the existence of sterile neutrino, the matter effect can not be ignored. The CP-violating Dirac phases in the mixing matrix may play an important role in sterile neutrino's matter effect. A more comprehensive analysis to display the entanglement of the phases is necessary, and we will show it in other places.

Conclusion
In this work, we have derived exact formulas of oscillation probabilities with matter in medium and long baseline experiment in presence of an additional light sterile neutrino. In particular, the key quantities contributing to oscillation probability, ∆m 2 ij andŨ αiŨ * βi , are shown explicitly. Based on exact formulas, we perform a detailed study of the matter effect correction in medium and long baseline experiments. We found that in medium baseline experiment, like JUNO, the matter effect contribution is negligible even in presence of light sterile neutrino. But in the long baseline experiment, taking NOνA as an example, the matter effect contribution plays a very important role especially when baseline grows.

A The parameterization of mixing matrix
In (3 + 1) scenario, the full neutrino mixing is characterized by a 4 × 4 matrix. To parameterize it, we need 6 rotation angles and 3 addtional Dirac phase angles. [34] The Majorana phase angles are closed here because it doesn't involve in the oscillation process. The mixing matrix can be constructed by 6 two-dimensional rotations R ij is a four dimensional rotation matrix, and in the (i, j) sublocks its elements reads -12 -

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Apparently if we close the angles related the forth neutrino, this 4 × 4 lepton mixing matrix will reduce to 3 × 3 PMNS matrix.
B Calculation of effective mass-square difference∆m 2 i1 In order to solve eq. (3.1), we resort to solving a quartic equation in below, where its root is denoted as λ i (i = 1, 2, 3, 4).
and the coefficients are defined as with Levi-Civita symbol ε ijk and ε ijkl . More auxiliary qualities are introduced to make the result more concise With the above notations, we find solutions of λ i where λ 1 < λ 2 < λ 3 < λ 4 . Notice that we don't assume the a normal or inverted mass hierachy. Thus eq. (B.4) and the inequality above are hierachy independent. If neutrinos -13 -
Since bothŨ and U are unitary, one can subtract a diagonal matrix from eq. (2.4) and eq. (2.4), leading tõ We can further write down with A αβ ≡ Aδ eα δ eβ + A δ sα δ sβ . Similarly by taking the square and cube of eq. (C.1), we can have equations corresponding to eq. (C.2) The square and cube on the right-handed side of (C.

1) α = β
The solution is with the auxiliary functions Solution in this case is obtained as in which we have especially introduced two important quantities, C αβ = A k,l ∆m 2 k1 ∆m 2 l1 U αk U * βl U sk U * sl + A k,l ∆m 2 k1 ∆m 2 l1 U * ek U el U αk U * βl (C.10) In particular the second equation eq. (C.8) is obtained by making use of the unitary property ofŨ .
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