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March, 1995 hep-ph/9503436
SNUTP 95-024 Brown-HET-990-Rev
A simple modification of the maximal mixing
arXiv:hep-ph/9503436v4 16 Aug 1996
scenario for three light neutrinos Kyungsik Kang Department of Physics, Brown University Providence, Rhode Island 02912, USA
Jihn E. Kim Center for Theoretical Physics and Department of Physics Seoul National University, Seoul 151-742, Korea
Pyungwon Ko Department of Physics, Hong-Ik University Seoul 121-791, Korea
Abstract We suggest a simple modification of the maximal mixing scenario (with S3 permutation symmetry) for three light neutrinos. Our neutrino mass matrix has smaller permutation symmetry S2 (νµ ↔ νe ), and is consistent with all neutrino experiments except the
experiment. The resulting mass
eigenvalues for three neutrinos are m1 ≈ (2.55 − 1.27) × 10−3 eV, m2,3 ≈ (0.71 − 1.43) eV for ∆m2LSN D = 0.5 − 2.0 eV2 . Then these light neutrinos can account for ∼ (2.4 − 4.8)% (6.2 − 12.4%) of the dark matter for h = 0.8 (0.5). Our model predicts the νµ → ντ oscillation probability in the range sensitive to the future experiments such as CHORUS and NOMAD.
The minimal standard model (MSM) has been highly successful in describing interactions among elementary particles from low energy up to ∼ 100 GeV. The only possible exception may be various types of neutrino oscillation experiments. There have been positive indications from large scale experiments for solar and atmospheric neutrinos that a certain amount of mixing between neutrino species may be present . The recent report from the LSND experiment at the laboratory scale provides us with another hint of such a possible neutrino mixing . Since neutrinos in the MSM are exactly massless, there can be no mixing among them, and it is impossible to accommodate such neutrino mixing data in the framework of the MSM. This situation is rather encouraging, since it is at present the only place where we can grasp a hint of new physics beyond the MSM. In view of this, it is quite interesting to speculate what type of neutrino mass matrix can fit all the data from the various types of neutrino oscillation experiments. It is our purpose to present one such mass matrix in this work. Most analyses on the neutrino oscillation assume two neutrinos oscillating with one mass difference parameter, ∆m2 . However, the LSND experiment and the atmospheric and solar neutrino data hint at least two mass difference parameters, requiring oscillations among at least three neutrinos. For oscillations with three neutrinos, we have two mass differences, three real angles and one phase. In order to simplify the analysis, a certain ansatz for the mass matrix is required. In this vein, we first briefly discuss the maximal mixing scenario for the neutrino sector. We then present our ansatz for the neutrino mass matrix as well as the numerical analyses to fit the atmospheric, LSND, and solar neutrino data from GALLEX and SAGE. In this work, we consider oscillations among three neutrinos only, να → νβ . One of the popular ansatz for the neutrino mass matrix is the maximal mixing one (equivalent to a cyclic permutation symmetry among three generations)  : Mmaximal
a b b∗ = b∗ a b , b b∗ a
with the mixing matrix U given by Umaximal
ω ω ω 1 1 1 1 = √ ω1 ω2 ω3 . 3 ω1 ω3 ω2
Here, ω1,2,3 are three complex roots of ω 3 = 1 with ω1 = 1. This ansatz was originally proposed in the neutrino sector , and extended to the quark sector with a partial success in explaining the quark masses and the Kobayashi-Maskawa matrix elements . The maximal mixing scenario has many interesting features . For example, the survival probability for a neutrino is independent of its flavor. The νe survival probability has two plateaus, 5/9 in the intermediate step, and 1/3 for EL ≫ (∆m21 )min through vacuum ij
oscillations. With vacuum oscillation, one cannot explain solar neutrino data from 37 Cl and the Ga data simultaneously. Thus one must make a choice between the 37 Cl and Ga experiments. Here, we choose to interpret the Ga data (GALLEX and SAGE experiments)
through vacuum oscillation and disregard the 37 Cl data 1 , following Ref. . The Ga data requires νe survival probability of ∼ 5/9, which implies EL for the solar neutrino is smaller than (∆m21 )min . The atmospheric neutrino data and km range laboratory experiments require ij
another ∆m2ij . Hence, two mass difference scales have been all used, and there is none left for a new scale suggested by the LSND data around ∆m2LSN D ∼ O(1) eV2 with a mixing angle ∼ (a few) × 10−3 . The only possibility to explain both the mass shifts at LSND point and at atmospheric data points in the maximal mixing scenario is that there are two thresholds corresponding to a larger ∆m2 at ∼ O(1) eV2 and a smaller ∆m2 at around 10−2 eV2 . In this case, the νe survival probability for the solar neutrino problem is 1/3, and is too small to accommodate the Ga data. Therefore, although the qualitative features of the maximal mixing scenario is encouraging, it is not viable if the LSND data is confirmed in the future. Another way to see this is as follows : the maximal mixing scenario predicts the transition probability for νµ → νe to be 4/9 in the range of the LSND experiment, which clearly contradicts the reported transition probability, (a few) ×10−3 . Furthermore, the best χ2 fit to the atmospheric and the solar neutrino data indicates that the masses of three light neutrinos are m3 ≃ (85 ± 10) meV, and m1,2 < 3µeV , which are too light to be cosmologically interesting as a hot dark matter component of the missing mass of the universe. Therefore, we make an ansatz for the neutrino mass matrix which is a simple modification of the maximal mixing one, (1), and study its consequences in this work. We assume that neutrinos are Dirac particles so that the lepton number is to be conserved in our model. Then, each left-handed neutrino (νLi ) is accompanied by the right-handed partner (νRi ) which is sterile under electroweak interactions. Note that any 3 × 3 matrix Mij can be decomposed as M = X − iY with both X and Y hermitian. Also any hermitian matrix X can be written as X = S + iA, where S (A) is a real (anti)symmetric matrix. Finally, the symmetric matrix S can be decomposed as the trace part proportional to δij and the traceless symmetric matrix. One can combine the trace part of the symmetric mass matrix S and the real antisymmetric part A in order to get the neutrino mass matrix,
1 ic id M = µ −ic 1 ib , −id −ib 1
where µ is the mass scale, b, c and d are all real. We have chosen a basis in which the charged lepton mass matrix is diagonal. Note that the diagonal terms are still universal, and only the off-diagonal elements are modified from the maximal mixing one, (1). Note that this ansatz becomes the maximal mixing one if b = c = −d (i.e., if there is a permutation symmetry among three generations). This form for the mass matrix is sufficiently simple but rich
matter oscillation effect, i.e., the MSW mechanism, has also been suggested to interprete this Homestake experiment in Ref. . In this case, the relevant ∆m2 is around ∼ 10−4 eV2 . Since we have only two ∆m2 around ∼ O(1) eV2 and ∼ 10−2 eV2 , the MSW mechanism is irrelvant to our study in this work.
enough to give nontrivial analytic formulae for the survival and transition probabilities for three neutrinos. If there is a solution when two of the off-diagonal elements are the same (say, b = d for example), then the permutation symmetry among three generations (S3 ) in the original maximal mixing case (1) breaks down to S2 . This would imply that our mass matrix ansatz depends on three real parameters and, thus it is one of the simplest modifications to the maximal mixing ansatz, which can accommodate LSND, atmospheric and solar neutrino data. In fact, this is the case (with b2 = d2 ) as discussed in the following. It breaks the original S3 possessed by (1) into S2 in a particular way. There may be several other ways to break this permutation symmetry, which will be considered elsewhere. There may be some underlying dynamical reasons for the above form of the mass matrix, but we take it as a simple phenomenological ansatz for the moment. Three eigenvalues of the mass matrix (3) are m1 = µ,
m2,3 = µ(1 ± N),
where N = (b2 + c2 + d2 )1/2 . The corresponding eigenvectors form the mixing matrix U which relates the weak eigenstate να to the mass eigenstates νi as να = Σi Uαi νi . The indices α = e, ν, τ label the flavor eigenstates, and i = 1, 2, 3 label the mass eigenstates of three neutrinos. Then one can easily verify that 1 1 P (νe → νe ) = 1 − 4 (∆21 + ∆31 ) b2 (N 2 − b2 ) + ∆32 (N 2 − b2 )2 , N 2 1 1 P (νµ → νe ) = 4 (∆21 + ∆31 )b2 d2 + ∆32 (c2 N 2 − b2 d2 ) , N 2 1 h P (νµ → νµ ) = 1 − 4 (∆21 + ∆31 )d2 (N 2 − d2 ) N( ! )# c2 N 2 − b2 d2 (c2 N 2 + b2 d2 )(b2 c2 + N 2 d2 ) 2 2 +∆32 + , −c d 2 2(c2 + d2 )2 !# " 1 (c2 N 2 + b2 d2 )(b2 c2 + d2 N 2 ) 2 2 2 2 P (νµ → ντ ) = 4 (∆21 + ∆31 )c d + ∆32 , −c d N 2(c2 + d2 )2
(5) (6) (7)
where ∆ij = 2 sin
1.27 L ∆m2ij E
with ∆m2ij ≡ m2i − m2j in eV2 and L/E in km/GeV. Since ∆m2ij = 0, there exist only two independent mass difference parameters. Note that the heights of the plateaus for the νe survival probability are functions of 2 b /N 2 only, even in the presence of nonvanishing c and d. Since P (νµ → νe ) depends on two mass differences, one can identify the mass difference ∆m2LSN D ∼ O(1) eV2 either as ∆m231 or as ∆m232 . The other mass difference is taken to be 0.72 × 10−2 eV2 in order to solve the atmospheric neutrino problem. Thus, there is a reasonable hierarchy between two mass differences. In the following, we discuss two possibilities separately. (I) : ∆m231 = ∆m2LSN D ∼ O(1) eV2 and ∆m232 = 0.72 × 10−2 eV2 : P
In this case, the heights of the intermediate plateaus for the νe and νµ survival probabilities are given by b2 P (νe → νe ) = 1 − 2 2 N d2 P (νµ → νµ ) = 1 − 2 2 N
b2 1− 2 , N ! d2 1− 2 , N !
and the transition probabilities can be approximated as 1.27L∆m231 b2 d2 , P (νµ → νe ) = 4 4 sin2 N E ! 2 c2 d 2 2 1.27L∆m31 P (νµ → ντ ) = 4 4 sin . N E !
The transition probability is often described in terms of two parameters, ∆m2 and θαβ for which 2
P (να → νβ ) = sin 2θαβ sin
1.27L∆m231 . E !
Therefore, we can make the following identifications : 4b2 d2 , N4 4c2 d2 = , N4
sin2 2θeµ =
in our model (for ∆m231 >> ∆m232 ). Experimental results for the neutrino oscillations are shown in the (sin2 2θ, ∆m2 ) plane. In the plot presented by the LSND group, there are small regions in this plane which indicates a possible transition of ν¯µ → ν¯e . This region is not ruled out by other laboratory searches such as BNL E776 , KARMEN , BUGEY  and others - . For each possible ∆m2LSN D , we show the possible value(s) of sin2 2θeµ in Table 1. For the same ∆m2LSN D , there is an upper bound on sin2 θµτ from FNAL E531 , CHARM II  and CDHSW , and we also list these numbers in Table 1. For each ∆m231 = ∆m2LSN D given in Table 1, one can solve Eq. (4) to get the neutrino masses. For example, ∆m231 = 6 eV2 leads to m1 = 7.35 × 10−4 eV, m2 ≈ −m3 ≈ 2.45 eV,
with Σi |mνi | = 4.9 eV. (The negative m3 can be remedied by a chiral transformation of ν3 field.) For other values of ∆m231 , we show the resulting neutrino masses from our mass matrix ansatz (3) in the fourth column of Table 1. These light neutrinos can contribute to the missing mass of the universe (the hot dark matter) in amount of  5
Ωh2 = 7.83 × 10−2
gef f g∗s (TD )
mν , eV
where g∗s (TD ) = 10.75 and gef f = (3g)/4 = 3/2 are the effective degrees of freedom contributing to the entropy density s and to the ratio Y = n/s, n being the number density, respectively. The parameter h is related to the Hubble constant H0 as H0 = 100h km sec−1 Mpc−1 . So, for the solution (17), three light neutrinos can constitute 8.3 % (21.4 %) of the missing mass of the universe for h = 0.8 (0.5), which is again cosmologically interesting. (Here, we have assumed that three sterile right-handed neutrinos decouple much earlier than the left-handed neutrinos, and that they don’t affect the results of the standard cosmology.) The results for other values of ∆m231 are listed in the last column of Table 1. When we determine b2 and d2 , it is important to satisfy all the constraints shown in Table 1. One might try to perform the χ2 fit to the available data on the neutrino oscillations. Instead, we choose to scan d2 /N 2 for each ∆m2 in the first column of Table 1. For each d2 /N 2 , the parameter b2 is determined by the mixing angle given by the LSND experiments, and c2 /N 2 = (1 − b2 /N 2 − d2 /N 2 ). Then, we require that the resulting sin2 2θµτ satisfy the upper limit given in the third column of Table 1. We also calculate the survival probabilities for νe and νµ at the intermediate level (the laboratory and the km range scale) and require them to be larger than 0.95 in order to satisfy the null results in various types of disappearance experiments for the νµ and ν¯e beams. For ∆m2 = 6 eV2 or larger (the first and the second rows), the resulting d2 /N 2 ≈ 0.995−1.00, which corresponds to almost no disappearance of νµ for all ranges of L and E. Thus, we reject d2 /N 2 around 1. For ∆m231 ≤ 2 eV2 , the allowed ranges for d2 /N 2 are typically around 0.010–0.020. The corresponding b2 /N 2 ’s are also in the same range as d2 /N 2 . Thus, as discussed in the following, we can accommodate the laboratory scale and the large scale neutrino experiments, by choosing small (but nonvanishing) b2 and d2 . In particular, there is a small region in which b2 = d2 gives acceptable fits to all available data on neutrino oscillation experiments except for the 37 Cl data. This is quite interesting, since it corresponds to residual permutation symmetry (S2 ) between e and µ in (3) with three real parameters. In other words, we have |Meµ | = |Mµτ |. Our matrix with b2 = d2 breaks the original symmetry of the maximal mixing one (S3 ) into S2 , and thus may be regarded as one of the simplest modifications to the maximal mixing one, (1). In the following, we demonstrate that the ansatz (3) can reasonably fit all the data with b2 /N 2 = d2 /N 2 = 0.015 for ∆m231 = 2 eV2 , except for the HOMESTAKE data, with a reasonable accuracy at present. In Figure 1, we show the resulting survival probability for νe in the solid curve 2 along with various types of neutrino oscillation data, the νe disappearance experiments at reactors -  and the solar neutrino experiments - . Using three neutrino mixing with four parameters, we get two step survival probability for νe → νe . The plateau for the large L is about 0.49, a little bit lower than 5/9 used before in the maximal mixing case. Thus, the solar neutrino deficit is solved in terms of vacuum oscillations. The intermediate plateau of
In order to average the oscillation probabilities, we adopt the prescription by Harrison et al. , which amounts to replacing cos(x/2) by sin x/x.
∼ 0.97 for νe is the prediction of our specific form of the mass matrix, and is consistent with all the existing data. The KRASNOYARSK data has a relatively large error bar, and our curve is within two σ of the data. In Figure 2, the survival probability for νµ using our ansatz is shown in the solid curve, along with the νµ disappearance experiment data  . The survival probability for νµ at large L is 0.48, and the intermediate plateau has a height of 0.97. The curve agree with the available data quite well. For the atmospheric neutrino data, we show the data point for the so-called R defined by R≡
(Nµ /Ne )Data , (Nµ /Ne )M C
Pµµ + Peµ /r Pee + rPµe
along with our prediction
in Table 2, where r is the incident (µ/e) ratio. From Table 2, we observe that most of the predicted R values are consistent with all the atmospheric neutrino data from KAMIOKANDE, IMB, and others - , considering some data points have large errors. For these numbers with ∆m231 = 2 eV2 , we predict sin2 2θµτ ≈ 6 × 10−2 , which is just below the current upper limit, 8 × 10−2 . This range of sin2 2θµτ may be probed at CHORUS, NOMAD, FNAL P803, CERN/ICARUS and FNAL/SOUDAN2 . It would be interesting to test our predictions for the νµ → ντ oscillation in the future. Similar results can be drawn for other values of ∆m231 . Thus, our mass matrix ansatz (3) not only describes all the available data on neutrino oscillations, but also predicts the mixing angle for νµ → ντ in an interesting range which lies within sensitivity of the near-future experiments. Let us briefly discuss the second case : (II) ∆m231 = 0.72 × 10−2 eV2 , and ∆m232 = ∆m2LSN D . In this case, it is easy to verify that there is no solution for b2 /N 2 and d2 /N 2 which satisfy the constraints from the laboratory scale experiments from BUGEY, BNL E766, and those in the second and the third columns of Table 1. So, our mass matrix ansatz prefer the solution (I) for which m1 is smallest around 10−3 − 10−4 eV, and the other two are nearly degenerate with m2 ≈ m3 ≈ O(1) eV. In summary, the neutrino mass matrix ansatz (3) with four real parameters is one of the simple modifications to the maximal mixing ansatz (that fails to fit the new LSND data) which can fit various types of neutrino experiments except for the 37 Cl solar neutrino data . In particular, there are solutions with b2 = d2 with residual permutation symmetry among two generations (νµ ↔ νe ). In this sense, our mass matrix ansatz could be regarded as one of the simplest modifications to the maximal mixing ansatz. The resulting light neutrinos have masses (17), and thus they can constitute about 2.4–4.8 % (6.2–12.4 %) of the missing mass of the universe for h = 0.8(0.5), and thus cosmologically interesting unlike the maximal mixing scenario. In our model, the transition probability for νµ → ντ is close to the current upper limit, depending on ∆m2LSN D as shown in the third column of Table 1. Since it lies within the reach of various future experiments such as CHORUS, NOMAD, etc., the observation of νµ − ντ oscillation would constitute a definite test of our mass matrix ansatz along with the confirmation of the LSND data. 7
We thank Bob Lanou for helpful discussions on the LSND and other neutrino experiments, and Dr. Hang Bae Kim for his assistance to draw the figures. One of us (JEK) thanks the Brown High Energy Theory Group for the hospitality extended to him during the visit. This work is supported in part by the Korea Science and Engineering Foundation through Center for Theoretical Physics at Seoul National University (JEK, PK), SNU-Brown Exchange Program (KK, JEK), Korea-Japan Exchange Program (JEK), the Ministry of Education through the Basic Science Research Institute, Contract No. BSRI-94-2418 (JEK) and through Contract No. BSRI-94-2425 (PK), SNU Daewoo Research Fund (JEK), and also the US DOE Contract DE-FG-02- 91ER40688 - Task A (KK).
Note Added in Proof In Eqs. (5)-(8), we did not show terms involving sij ≡ sin
, since these terms
vanish under taking averages, or they are irrelvant to our study. See Ref.  for more details.
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FIGURES FIG. 1. The survival probabilities P (νe → νe ) using our ansatz (3) for the case (I), along with the reactor experiment data from KARMEN, ILL/GOSGEN, BUGEY, KRASNOYARSK, and the solar neutrino data from KAMIOKA, HOMESTAKE, SAGE and GALLEX. FIG. 2. The survival probabilities of νµ using our ansatz (3) for the case (I), along with the accelerator experiments from CDHS-SPS and CHARM-PS.
TABLES TABLE I. The allowed regions for ∆m2 and sin2 2θeµ consistent with the LSND as well as BNL E766, and the corresponding upper limit for sin2 2θµτ from FNAL E531 and CDHSW. The fourth column is the predicted neutrino masses by our mass matrix ansatz (3). The last column shows contributions of three light neutrinos to the missing mass of the universe for h = 0.8 (0.5). See the text for details. ∆m2 (eV 2 ) 20 6 2 1 0.5
TABLE II. The atmospheric neutrino data R for various L/E along with our predictions for ∆m231 = 2 eV2 , ∆m232 = 0.72×10−2 eV2 and b2 /N 2 = d2 /N 2 = 0.015. We show the r = (µ/e)incident values for each data point also. Experiments KAMIOKA  (Multi-GeV)