Neutrino Mass Anarchy and the Universe

We study the consequence of the neutrino mass anarchy on cosmology, in particular the total mass of neutrinos and baryon asymmetry through leptogenesis. We require independence of measure in each mass matrix elements in addition to the basis independence, which uniquely picks the Gaussian measure. A simple approximate U(1) flavor symmetry makes leptogenesis highly successful. Correlations between the baryon asymmetry and the light-neutrino quantities are investigated. We also discuss possible implications of recently suggested large total mass of neutrinos by the SDSS/BOSS data.


I. INTRODUCTION
Neutrino physics is a unique area in particle physics that has many direct consequences on the evolution history and the current state of Universe. It was one of the first hypotheses for the non-baryonic dark matter. Excluding this possibility relied on rather surprising constraint that the density of neutrinos would exceed that allowed by Fermi degeneracy in the core of dwarf galaxies! [1] Because of the free streaming, massive neutrinos would also suppress the large-scale structure, which is still subject to active research. The explosion mechanism of supernova is tied to properties of neutrinos, and hence the chemical evolution of galaxies depend on neutrinos. The number of neutrinos is relevant to Big-Bang Nucleosynthesis. In addition, neutrinos may well have created the baryon asymmetry of the Universe [2] or create the Universe itself with scalar neutrino playing the role of the inflaton [3,4].
Many of the consequences of neutrino properties on the Universe rely on the mass of neutrinos. After many decades of searches, neutrino mass was discovered in 1998 in disappearance of atmospheric neutrinos by the Super-Kamiokande collaboration [5]. Subsequently the SNO experiment demonstrated transmutation of solar electron neutrinos to other active neutrino species [6] corroborated by reactor neutrino data from KamLAND [7]. Most recently, the last remaining mixing angle was discovered by the Daya Bay reactor neutrino experiment [8]. Other experiments confirmed this discovery [9,10].
Quite counterintuitively, however, it was pointed out that the pattern of neutrino masses and mixings can also be well accounted for by structureless mass matrices [17]. Mass matrices with independently random entries can naturally produce the small hierarchical mass spectrum and the large mixing angles. This provides us with an alternative point of view: instead of contrived models for the mass matrix, one can simply take the random mass matrix as a low energy effective theory [18]. This anarchy approach is actually more naturally expected, because after all, the three generations possess the exact same gauge quantum numbers. From the viewpoint of anarchy, however, the mass spectrum with large hierarchy and small mixings exhibited by quarks and charged leptons need an explanation. It turns out that introducing an approximate U (1) flavor symmetry [19,20] can solve this problem well [18]. This combination of random mass matrix and approximate U (1) flavor symmetry has formed a new anarchy-hierarchy approach to fermion masses and mixings [18].
To be consistent with the spirit of anarchy, the measure to generate the random mass matrices has to be basis independent [18]. This requires that the measure over the unitary matrices be Haar measure, which unambiguously determines the distributions of the mixing angles and CP phases. Consistency checks of the predicted probability distributions of neutrino mixing angles against the experimental data were also performed in great detail [21,22]. Although quite successful already, this anarchy-hierarchy approach obviously has one missing brick: a choice of measure to generate the eigenvalues of the random mass matrices. With the only restriction being basis independence, one can still choose any measure for the diagonal matrices at will. However, as we will show in this paper, if in addition to basis independence, one also wants the matrix elements to be distributed independent from each other, then the only choice is the Gaussian measure.
In this paper, we focus on the Gaussian measure to investigate the consequences. As pointed out in [18], the quantities most sensitive to this choice would be those closely related to neutrino masses. We study a few such quantities of general interest, including the effective mass of neutrinoless double beta decay m eff , the neutrino total mass m total , and the baryon asymmetry η B0 obtained through a standard leptogenesis [23,24]. We also present a correlation analysis between η B0 and light-neutrino parameters. Recently, the correlation between leptogenesis and light-neutrino quantities was also studied by taking linear measure in [25]. Their results are in broad qualitative agreement with ours in this paper.
The rest of this paper is organized as following. We first motivate our sampling model-Gaussian measure combined with approximate U(1) flavor symmetry-in Section II. Then the consequences of this sampling model is presented in Section III. We show our Monte Carlo predictions on light-neutrino mass hierarchy, effective mass of neutrinoless double beta decay, light-neutrino total mass, and baryon asymmetry through leptogenesis. In Section IV, we investigate the correlations between baryon asymmetry and light-neutrino quantities. A recent Baryon Oscillation Spectroscopic Survey (BOSS) analysis suggests that the total neutrino mass could be quite large [26]. Our Section V is devoted to discuss the consequence of this suggestion. We conclude in Section VI.

A. Gaussian Measure
To accommodate the neutrino masses, we consider the standard model with an addition of three generations of right-handed neutrinos ν R , which are singlets under SU (2) L × U (1) Y gauge transformations. Then there are two neutrino mass matrices, the Majorana mass matrix m R and the Dirac mass matrix m D , allowed by gauge invariance where With the spirit of anarchy, we should not discard either of them without any special reason. Both should be considered as random inputs. We parameterize them as where D R and D 0 are real diagonal matrices with nonnegative elements; U R , U 1 and U 2 are unitary matrices; M and D are overall scales. At this point, the requirement of neutrino basis independence turns out to be very powerful. It requires that the whole measure of the mass matrix factorizes into that of the unitary matrices and diagonal matrices [18]: Furthermore, it also demands the measure of U R , U 1 and U 2 to be the Haar measure of U (3) group [18]: where λ 3 = diag(1, −1, 0), λ 8 = diag(1, 1, −2)/ √ 3, and c 12 = cos θ 12 , etc.
Although basis independence is very constraining, it does not uniquely fix the measure choice of m R or m D , because arbitrary measure of the diagonal matrices D R and D 0 is still allowed. Now let us look at the entries of m R and m D . There are 9 complex free entries for m D and 6 complex free entries for the symmetric matrix m R = m T R . Generating each free entry independently is probably the most intuitive way of getting a random matrix. However, if one combines this free entry independence with the basis independence requirement, then it turns out there is only one option left-the Gaussian measure: Although this is a known result in random matrix theory [27,28], we also present our own proof in the appendix. On one hand, basis independence is necessary from the spirit of anarchy. On the other hand, free entry independence is also intuitively preferred. With these two conditions combined, we are led uniquely to the Gaussian measure. Now the only parameters left free to choose are the two units M and D. Following the spirit of anarchy, D should be chosen to make the neutrino Yukawa coupling of order unity, Throughout this paper, we choose D = 30 GeV, which yields y ν ∼ 0.6. Then we choose M to fix the next largest mass-square difference of light-neutrinos ∆m 2 l at 2.5 × 10 −3 eV 2 in accordance with the data.

B. Approximate U (1) Flavor Symmetry
Our model (Eq.(6)) is capable of generating a baryon asymmetry η B0 through thermal leptogenesis [2,29]. For the simplicity of analysis, we would like to focus on the scenario with two conditions: (1) There is a hierarchy among heavy-neutrino masses M 1 M 2 , M 3 , so that the dominant contribution to η B0 is given by the decay of the lightest heavy neutrino N 1 [23].
(2) If we use h ij to denote the Yukawa couplings of heavy-neutrino mass eigenstates then the condition h i1 1 for all i = 1, 2, 3 would justify the neglect of annihilation process N 1 N 1 → ll, and also help driving the decay of N 1 out of equilibrium [23,29]. This condition used to be taken as an assumption [29].
Both conditions above can be achieved by imposing a simple U (1) flavor charge on the right-handed neutrinos. For example, one can make charge assignments as shown in Table. I. Assuming a scalar field φ carries −1 of this U (1) flavor charge, one can construct neutral combinations ν φ as Now it only makes sense to place the random coupling matrices among these neutral combinations where m R0 and m D0 should be generated according to Gaussian measure as in Eq. (13)- (14). Upon U (1) flavor symmetry breaking φ = with 0.1, this gives and hence the mass matrices Let us parameterize the heavy-neutrino mass matrix as where one can identify Clearly a hierarchy among heavy neutrino masses is achieved. Furthermore, the heavy neutrino mass eigen- we obtain the Yukawa coupling h ij as We see that h i1 ∼ 2 1 is guaranteed for all i = 1, 2, 3. It is worth noting that the light-neutrino mass matrix m ν is not affected by our U(1) flavor change assignment on right-handed neutrinos (Table. I). The hierarchical matrix D cancels due to the seesaw mechanism [30][31][32][33][34][35]: Therefore all properties of light neutrinos do not depend on the particular U(1) flavor charge assignment nor the size of the breaking parameter . The leptogenesis aspect is the only discussion in the paper where this flavor structure is relevant.

III. CONSEQUENCES
In this section, we present our Monte Carlo results based on the sampling measure described in the last section.

A. Light-neutrino Mixings and Mass Splitting Ratio
We parameterize the light neutrino mass matrix as with U ν a unitary matrix and m a real diagonal matrix with non-negative elements. We also follow a conventional way to label the three masses of the light neutrinos: . We use ∆m 2 s and ∆m 2 l to denote the smaller and larger one respectively. If ∆m 2 s is the first one, we call this scenario "normal" and take the definitions m 1 ≡ m 11 , m 2 ≡ m 22 , m 3 ≡ m 33 . Otherwise, we call it "inverted" and take m 1 ≡ m 22 , m 2 ≡ m 33 , m 3 ≡ m 11 . The columns of the unitary matrix U ν should be arranged accordingly.
Predictions on light-neutrino mixings-the distributions of mixing angles θ 12 , θ 23 , θ 13 , CP phase δ CP , and other physical phases χ 1 , χ 2 -are certainly the same as in general study of basis independent measures [18], since U ν is totally governed by the Haar measure. A statistical Kolmogorov-Smirnov test shows that the predicted mixing angle distribution is completely consistent with the experimental data [21,22].
The mass-squared splitting ratio R ≡ ∆m 2 s /∆m 2 l is observed to be around [11] Fig. 1 shows our predicted distribution of this ratio. With a probability of R < R exp being 36.1%, the prediction is completely consistent with the data.
For the purpose of studying other quantities, we introduce the following Mixing-Split cuts as suggested by the experimental data [11] on light-neutrino mixings and mass-squared splitting ratio: is the light-neutrino total mass m total ≡ m 1 + m 2 + m 3 , due to its presence in cosmological processes. Our predictions on m eff and m total are shown in Fig. 3 and Fig. 4 respectively. For each quantity, we plot both its whole distribution under Gaussian measure and its distribution after applying the Mixing-Split cuts. Distributions of m eff and m total under Gaussian measure were also studied recently in [36]. Their results are in agreement with ours. The small difference is due to the difference in taking cuts on neutrino mixing angles.
We see from Fig. 3 that most of the time m eff 0.05 eV. It becomes even smaller after we apply the Mixing-Split Cuts, mainly below m eff 0.01 eV. This is very challenging to experimental sensitivity. For m total , Fig. 4 shows it being predicted to be very close to the current lower bound ∼ 0.06 eV. The kink near 0.1 eV is due to the superposition of the two mass hierarchy scenarios.
To understand each component better, we show the distributions of m eff and m total in Fig. 5 and Fig. 6 for both before/after applying the cuts and normal/inverted hierarchy scenario. As Fig. 6 shows, under either hierarchy scenario, m total is likely to reside very close to its corresponding lower bound, especially after applying the cuts. Very interestingly, however, recent BOSS analysis suggests m total could be quite large, with a center value ∼ 0.36 eV [26]. As seen clearly from Fig. 6, a large value of m total would favor inverted scenario. We will discuss some possible consequences of this suggestion in Section V.

C. Leptogenesis
As explained in Section II, with our approximate U (1) flavor symmetry, the baryon asymmetry η B0 ≡ n B nγ can be computed through the standard leptogenesis calculations [23,24]. For each event of m R and m D generated, we solve the following Boltzmann equations numerically.
where we have followed the notations in [23] and [24]. The argument z ≡ M 1 /T is evolved from z i = 0.001 to z f = 20.0. Evolving z further beyond 20.0 is not necessary, because the value of N B−L becomes frozen shortly after z > 10.0. The baryon asymmetry is then given by [23]. Due to randomly generated m R and m D , we have equal chances for ε to be positive or negative. It is the absolute value that is meaningful. We take the initial condition N B−L (z i ) = 0. We actually tried two typical initial conditions for N 1 , N 1 (z i ) = 0 and N 1 (z i ) = N eq 1 (z i ), and found no recognizable differences. The distributions of η B0 , both before and after applying the Mixing-Split cuts, are shown in Fig. 7. We see from figure that our prediction on η B0 is about the correct order of magnitude as η B0,exp ≈ 6 × 10 −10 [37].
Let us try to understand the results from some crude estimates. First, let us see why there is almost no difference between the two initial conditions N 1 (z i ) = 0 and N 1 (z i ) = N eq 1 (z i ). The decay function D(z) has the form [24]: where K 1 (z) and K 2 (z) are modified Bessel functions, and the constant α D is proportional to the effective neutrino massm 1 : So roughly speaking, after z 1, the modified Bessel functions become rather close and D(z) increases linearly with z. But in our prescription, the value of D(z) + S(z) becomes already quite large, typically around 50, at z = 1.0. So N 1 is forced to be very close to N eq 1 thereafter and the solution to the first differential equation (Eq. (33)) is approximately Of course, this initial-condition-independent solution only holds when D + S is large enough, typically for z > 1.0. The values of N 1 − N eq 1 at z < 1.0 certainly have a considerable dependence on N 1 (z i ). However, the solution to the second differential equation (Eq.(34)) is And due to the shape of W (z), yield of N 0 B−L at z < 1.0 is significantly suppressed by a factor e − ∞ z W (z )dz . Therefore η B0 turns out to be almost independent of N 1 (z i ).
Second, let us crudely estimate the order of magnitude of η B0 . In addition to D(z), the scattering functions S(z) and washout function W (z) are also proportional tõ m 1 . Som 1 is the key factor that significantly affects the evolution of Eq. (33) and (34) [24]. In our anarchy model, apart from the overall units, the mass matrix entries are O(1) numbers, so we expect This is just the light-neutrino mass scale. Because in our simulation, M is chosen such that ∆m 2 l = 2.5 × 10 3 eV 2 , we have Therefore most of the time, our model is in the "strong washout regime" [24], sincem 1 is much larger than the equilibrium neutrino mass m * : In this regime, the integral in Eq.(38), which is called efficiency factor κ f , should be around [24] Thus our baryon asymmetry is To estimate the CP asymmetry ε, we notice that (following the notation of [24]) and thus So the baryon asymmetry is expected to be around η B0 ∼ Our use of U (1) flavor symmetry breaking plays an essential role to produce the correct order of ε (ε ∼ 4 ) and thus η B0 . It is thus interesting to investigate what would happen if we had a different U (1) charge assignment. An arbitrary charge assignment would be, upon symmetry breaking, equivalent to an arbitrary choice of D parameterized as where 1 , 2 , 3 1, and we have defined r2 ≡ 2 / 3 and r1 ≡ 1 / 2 for convenience. To make the simplest scenario of leptogenesis work, we need the hierarchy among the heavy-neutrino masses. So we restrict ourselves to the case r1 , r2 1. The Majorana mass matrix now becomes The Dirac mass matrix becomes which gives the Yukawa coupling h and K ≡ h † h as So our ε is given by We see that under the condition r1 , r2 1, ε is only sensitive to the value of 1 .
On the other hand, the value ofm 1 is not affected by changing U (1) flavor charge assignments: Here a subscript "0" is used to denote the value when there is no U (1) flavor charge assignment, as we did in Eq. (18). So the strong washout condition (Eq. (41)) still holds, and we are again led to Eq. (43). Therefore, the baryon asymmetry η B0 can only be affected through ε, which in turn is only sensitive to 1 , under the condition r1 , r2 1.

IV. CORRELATIONS BETWEEN ηB0 AND LIGHT-NEUTRINO PARAMETERS
As we can see from Fig. 7, the baryon asymmetry is slightly enhanced after applying the Mixing-Split cuts Eq. (29)- (32). This indicates some correlation between η B0 and light-neutrino parameters. To understand this better, we would like to systematically investigate the correlations between η B0 and the light-neutrino mass matrix m ν = U ν mU T ν . * Although both of η B0 and m ν seem to depend on the random inputs m R and m D in a complicated way, it is not hard to see that there should be no correlation between η B0 and U ν (This was also pointed out in [25]). Recall that we parametrize m R and m D as in Eq. (7)- (8). And due to the decomposition Eq. (9)-(10), there are five independent random matrices: U 1 , U 2 , U R , D 0 and D R . The first thing to observe is that changing U 1 with the other four matrices fixed will not affect η B0 . This is because: 1. The baryon asymmetry η B0 we have been computing in this paper is the total baryon asymmetry, including all the three generations. So m D enters the calculation of leptogenesis only through the form of the matrix with m D = D · U 1 D 0 U † 2 . Obviously U 1 cancels in K.
2. Throughout the simulation, we are also applying a built-in cut ∆m 2 l = 2.5 × 10 −3 eV 2 by choosing the value of M to force it. Due to this cut, m D can potentially affect η B0 through the value of M. However, since the actual relation is we see that changing U 1 would only affect U ν , not m. So no further adjustment of M is needed when we change U 1 . * The correlation between leptogenesis and light-neutrino parameters has been recently studied in [25]. They have some overlap with our results.
The second point to observe is that any change in U ν can be achieved by a left translation This in turn, can be accounted for by just a left translation in U 1 : U 1a → U 1b = U L U 1a , with the other four random matrices unchanged (see Eq. (56)). This left translation in U 1 is thus a one-to-one mapping between the sub-sample generating U νa and the sub-sample generating U νb = U L U νa . Any two events connected through this one-to-one mapping generate the same value of η B0 , because changing U 1 does not change η B0 . In addition, the two events have the same chance to appear, because the measure over U 1 is the Haar measure, which is invariant under the left translation. Thus the sub-sample with U ν = U νa and U ν = U νb , for any arbitrary U νa and U νb , will give the same distribution of η B0 , namely that η B0 is independent of U ν . So immediately we conclude that η B0 cannot be correlated with the three mixing angles θ 12 , θ 23 , θ 13 , the CP phase δ CP , or the phases χ 1 , χ 2 . Three of the Mixing-Split cuts applied to η B0 are cuts on mixing angles which we just showed not correlated with η B0 . So clearly, the enhancement of η B0 is due to its non-zero correlation with R. To study more detail about the correlation between η B0 and the light-neutrino masses m, we apply a χ 2 test of independence numerically to the joint distribution between η B0 and quantities related to m, including lgR, m eff and m total . For each quantity with η B0 , we construct a discrete joint distribution by counting the number of occurrences O ij (i, j = 1, ..., 10) in an appropriate 10 × 10 partitioning grid. Then we obtain the expected number of occurrences E ij as where n is the total number of occurrences in all 10 × 10 partitions. If the two random variables in question were independent of each other, we would have the test statistic satisfying the χ 2 distribution with degrees of freedom (10 − 1) × (10 − 1) = 81. We then compute the probability P (χ 2 > X) for the hypothesis distribution χ 2 (81) to see if the independence hypothesis is likely. Our results from a n = 3, 000, 000 sample Monte Carlo are shown in Table. II. Unambiguously, η B0 has nonzero correlations with lgR, m eff , and m total . To see the tendency of the correlations, we draw scatter plots with 5, 000 occurrences (Fig. 8).
The plots show that all the three quantities are negatively correlated with η B0 . For example the left panel of Fig. 8 tells us that a smaller lgR would favor a larger η B0 . This  explains the slight enhancement of η B0 after applying Mixing-Split cuts. But as the scatter plots show, the correlations are rather weak.

V. POSSIBLE CONSEQUENCES OF A LARGE m total
As mentioned previously, a recent BOSS analysis suggests m total possibly quite large, m total = 0.36 ± 0.10 eV [26]. Currently their uncertainty is still large, and thus no conclusive argument can be made. If in future the uncertainty pins down near its current central value, anarchy prediction (Fig. 4) would be obviously inconsistent with it and becomes ruled out. On the other hand, if the central value also comes down significantly, it could still be well consistent with anarchy prediction.
Without the knowledge of future data, we would like to answer the following question: Assuming the future data be consistent with anarchy, could a relatively large m total dramatically change anarchy's predictions on other quantities? For this purpose, we introduce a heuristic m total cut: just to get a sense of how much our predictions could be changed if there turns out to be a large but still consistent m total . We collect 10 4 occurrences that pass both the Mixing-Split cuts and the m total cut. It turns out that the predictions change quite significantly. We see from Fig. 2 that the mass hierarchy prediction is overturned, with normal hierarchy only 40% and inverted scenario more likely. This can be expected from Fig. 6. The predictions of m eff and η B0 are shown in Fig. 9. We see that m eff exhibits a very interesting bipolar distribution. Its overall expectation value also becomes about an order of magnitude larger than before and thus much less challenging to the neutrinoless double beta decay experiments. The prediction on η B0 drops by about an order of magnitude, but the observed baryon asymmetry is still very likely to be achieved.

VI. CONCLUSIONS
We have shown that basis independence and free entry independence lead uniquely to Gaussian measure for m R and m D . We also showed that an approximate U (1) flavor symmetry can make leptogenesis feasible for neutrino anarchy. Combining the two, we find anarchy model successfully generate the observed amount of baryon asymmetry. Same sampling model is used to study other quantities related to neutrino masses. We found the chance of normal mass hierarchy is as high as 99.9%. The effective mass of neutrinoless double beta decay m eff would probably be well beyond the current experimental sensitivity. The neutrino total mass m total is a little more optimistic. Correlations between baryon asymmetry and light-neutrino quantities were also investigated. We found η B0 not correlated with light-neutrino mixings or phases, but weakly correlated with R, m eff , and m total , all with negative correlation. Possible implications of recent BOSS analysis result have been discussed.   13) and (14).