Minimally Allowed \beta\beta 0\nu Rates From Approximate Flavor Symmetries

Minimally Allowed Rates From Approximate Flavor Symmetries

James Jenkins Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Northwestern University, Department of Physics & Astronomy, Evanston, IL 60208, USA

Neutrinoless double beta decay () is among the only realistic probes of Majorana neutrinos. In the standard scenario, dominated by light neutrino exchange, the process amplitude is proportional to , the element of the Majorana mass matrix. Naively, current data allows for vanishing , but this should be protected by an appropriate flavor symmetry. All such symmetries lead to mass matrices inconsistent with oscillation phenomenology. I perform a spurion analysis to break all possible Abelian symmetries that guarantee vanishing rates and search for minimally allowed values. I survey 230 broken structures to yield values and current phenomenological constraints under a variety of scenarios. This analysis also extracts predictions for both neutrino oscillation parameters and kinematic quantities. Assuming reasonable tuning levels, I find that at confidence. Bounds below this value might indicate the Dirac neutrino nature or the existence of new light (eV-MeV scale) degrees of freedom that can potentially be probed elsewhere.

preprint: LA-UR-08-06358preprint: NUHEP-TH/08-07

I Introduction

Neutrino oscillation experiments have given conclusive evidence that neutrinos have mass and mix. This constitutes the first terrestrial evidence of physics beyond the Standard Model (SM) and leads to many important questions. For a recent review of neutrino physics see Mohapatra et al. (2007); de Gouvea (2004a, b). Broadly, it is puzzling why the neutral lepton sector is so different from the other SM fermions in both mass scale and mixing pattern. The resolution to this mystery has implications for both particle and astrophysics and provides deep theoretical insight into the nature of other high scale phenomena. The neutrality of the neutrino under the only unbroken gauge symmetry is the likely key to this problem, as it offers the possibility that the neutrino is its own antiparticle via direct coupling within a Majorana mass term. Such a mass, as opposed to the more common Dirac mass, is composed of only one field and violates all non-zero quantum numbers by two units. The charge conjugation properties of the neutrinos, their Dirac vs Majorana nature, are currently unknown and their determination is arguably the most important task facing the neutrino community.

The favored means of probing Majorana neutrinos is via the process of neutrinoless double beta decay () where, within a nucleus, two neutrons decay into two protons with no neutrinos Elliott and Vogel (2002). This process violates lepton number by two units and may proceed via the virtual exchange of Majorana neutrinos. In this case, the decay amplitude is directly proportional to the mass of the exchanged electron-type neutrino, or more precisely the element of the Majorana neutrino mass matrix () taken in the flavor basis where the charged lepton masses are diagonal. Current experimental limits on the Ge isotope constrain the half-life below years, corresponding to eV at 90% confidence***The translation between measured half-life and is not straightforward, as it depends critically on isotope dependent nuclear matrix element calculations, where uncertainties currently range within a factor of three (Rodin et al., 2005, 2006; Menendez et al., 2008). This is likely to improve within the next several years. (Aalseth et al., 2002; Klapdor-Kleingrothaus et al., 2001; Bilenky et al., 2003; Strumia and Vissani, 2005). Next generation experiments are poised to extend this reach by roughly an order of magnitude to eV (Avignone, 2005; Bilenky et al., 2003; Zuber, 2006).

Of course, other exotic interactions can mediate , but it was argued in the Blackbox theorem (Schechter and Valle, 1982; Hirsch et al., 1997) that any such Lepton Number Violating (LNV) process will necessarily yield a Majorana neutrino mass at some order in perturbation theory, just as a Majorana neutrino mass term will lead to LNV processes. This notion was extended in (Hirsch et al., 2006) to the realistic three neutrino system. The authors showed, among other things, that there exists a one-to-one relationship between LNV rates and elements of the Majorana neutrino mass matrix such that, in particular, and . Additionally, using general symmetry arguments, they demonstrated that there exists a non-trivial relationship between various mass matrix elements, implying a finite set of textures with vanishing . None of these are consistent with the observed oscillation data. This leads naturally to the conclusion that if neutrinos are Majorana particles, must occur at a nonzero rate.

Here, using similar logic, I explore exactly how small can be under a variety of circumstances. If light Majorana neutrino exchange is the only mode of , this can be applied directly to the interpretation of experimental results. The situation is not as straightforward in the face of other contributions, as these will generally effect the mass and LNV rate differently (de Gouvea and Jenkins, 2008a). Still, the one-to-one correspondence between and adds confidence to the conclusion that is a good measure of the rate for small , even in the presence of arbitrary new physics. This statement becomes exact in the limit of vanishingly small .

This analysis is conducted within the framework of Abelian flavor symmetries acting within a three light Majorana neutrino system. It will become clear that the results can be extended beyond this paradigm to include non-Abelian groups. Additionally, by the above argument, there is reason to believe that the qualitative existence of a lower bound, as well as its connection to , should remain valid in the face of arbitrary new physics at scales down to approximately . Beyond this point, new light degrees of freedom can contribute directly to the system and restrict the result validity. Thus, limits that fall below extracted minimum values are evidence for new light physics or the Dirac neutrino nature. A measurement near the derived lower bound would indicate a slightly broken symmetry mechanism at work.

It is natural to wonder how small can be without the introduction of these flavor symmetry suppressions. It is well known that, given current neutrino data, there is a well-defined range of allowed values. This depends on the neutrino mass hierarchy and oscillation parameters within the light Majorana neutrino exchange hypothesis. Bahcall et al. (2004); Petcov (2005); Choubey and Rodejohann (2005); de Gouvea and Jenkins (2005); Strumia and Vissani (2005). Neutrinos with normal mass spectra can yield vanishing provided appropriate phase and parameter choices. Next generation experiments will probe the quasi-degenerate and inverted hierarchy region of the allowed range Elliott and Vogel (2002). Clearly, a positive measurement at this relatively large level would not indicate a flavor suppression of any kind. The question becomes more involved as smaller values within the normal mass hierarchy are explored. When is small no longer an accident? To answer this question, it is instructive to take the structure free limit and consider the neutrino mass matrix anarchy hypothesis Haba and Murayama (2001); Hall et al. (2000). Here, the underlying neutrino model is sufficiently complicated such that the low energy mass matrix appears random and must be treated statistically. In other words, the flavor basis is some random rotation from the mass eigenbasis. An analysis of the allowed mixing angle distribution is straightforward and described in de Gouvea and Murayama (2003). However, there is an added level of ambiguity introduced whenever mass values are discussed, due to freedom in assigning an integration measure to the probability distributions. These issues were studied in Haba and Murayama (2001); Jenkins (2008). The distribution of values within the anarchy scenario was surveyed in Jenkins (2008) under a variety of conditions to conclude that implies the existence of a flavor symmetry mechanism, new light degrees of freedom, or the Dirac neutrino nature. A measurement above this limit could be attributed to either a flavor symmetry or random fluctuations of the neutrino mass matrix. Below the anarchy bound, the present analysis sets a limit on minimum values and identifies what symmetries are responsible for the suppression. Above the anarchy bound, it selects those broken symmetries that are allowed by the data and makes predictions for other observables that can further constrain the system.

This paper is organized as follows. In Section II, I review Majorana neutrino masses and the status of current neutrino data used as constraints in the remainder of the analysis. I then introduce flavor symmetries in the context of the neutrino mass matrix and motivate the utility of Abelian groups as an ideal laboratory for a comprehensive search for minimal rates. In Subsection II.1, I exhaustively enumerate the flavor symmetry structures that lead to exact and explore their consequences for neutrino oscillation phenomenology. These results are summarized in Table 2, which illustrates that of the eleven possible symmetry classes, none are consistent with current data. I break these flavor symmetries in Subsection II.2 with the introduction of a single spurious charged scalar field that acquires a real vacuum expectation value (vev). Subsubsection II.2.1 forms the bulk of the analysis, where I numerically survey a comprehensive set of broken symmetry structures. For each case (referred to loosely as models), I determine current constraint from data, extract the minimally allowed values, and make predictions for future neutrino experiments. Variations of these results, subject to improvements in future oscillation parameter measurements, are also studied. I conclude in Section III with a summary of the results and a discussion of the limitations of the analysis in the face of new physics.

Ii Abelian Flavor Symmetries

If lepton number is violated by physics at some high scale , the effective low energy Majorana neutrino mass Lagrangian term may be written as


In the weak interaction basis where the charged leptons are diagonal, the Greek subscrips are flavor indices that run over the three generations , and . The symmetric mass matrix may be diagonalized to yield positive real mass eigenvalues , and by the neutrino mixing matrix in the PDG parametrization Yao et al. (2006)


that describes the rotation from the flavor basis to the mass basis. I use the shorthand and for notational convenience. Due to symmetries of the mixing matrix, the mixing angles may be constrained within de Gouvea et al. (2000); de Gouvea and Jenkins (2008b)with the Majorana and Dirac phases and without loss of generality de Gouvea and Jenkins (2008b); Jenkins and Manohar (2008). By conventionSee for example de Gouvea and Jenkins (2008b) for a summary of mass naming conventions and their relationship with other mixing parameters., the neutrino eigenstates are ordered in mass squared such that , and have the smallest separation, the so-called solar mass squared splitting , while is the most distant state. The identity of the lightest eigenstate depends on the mass ordering, with and existing as the lightest state for the normal and inverted hierarchies, respectively. Neutrino oscillation data constrains the mass eigenvalue squared differences and mixing angles as can be seen from the first five entries of Table 1 which lists both the current best fit values and uncertainties, as adapted from Schwetz et al. (2008). See also Maltoni et al. (2004); Strumia and Vissani (2005); Fogli et al. (2006). Additionally, kinematic probes of the endpoint of the tritium beta decay spectrum, cosmological observations and even constrain the absolute neutrino mass values Bilenky et al. (2003) as shown in entries six, seven and eight. To date, these have yet to observe positive signals but have been successful in bounding neutrino masses below the eV level Strumia and Vissani (2005). Next generation experiments will extend the reach of tritium decay and cosmological measurements to Osipowicz et al. (2001) and Wang et al. (2005); Abazajian and Dodelson (2003); Lesgourgues et al. (2004), at confidence, respectively. The single Dirac phase and two Majorana phases , and are currently unconstrained by experiment. The first column of Table 1 lists the parameter name conventions incorporated in this analysis. The subscripts , and attached to the oscillation parameters refer respectively to “solar”, “atmospheric” and “reactor,” after the primary/historical neutrino sources used in their measurement. These parameter constraints must be satisfied by all viable neutrino mass models.

Name Parameter Combination Value Uncertainty
Table 1: Summary table of current neutrino results together with naming conventions and parameter definitions. Columns three and four list best fit central parameter values and uncertainties. The first five entries were adapted from the global oscillation analysis of Schwetz et al. (2008). The upper bounds for the last three entries come from endpoint measurements of Tritium -decay (), cosmological observations () and , respectively Strumia and Vissani (2005). These are used as model constraints through the remainder of this analysis.

The entries of the symmetric matrix are constrained to be small by a combination of the kinematic and oscillation neutrino data. The suppression of this term with respect to the other charged fermions is likely due to the high scale of new physics. The mass term of Eq. (II.1) is the relic of an effective operator after electroweak symmetry breaking and as such, where is the Higgs vev and is a matrix of complex constants. This may arise from a simple dimension five operator as in the seesaw mechanisms Ma (1998); Minkowiski (1977); Yanagida (1979); Glashow (1980); Mohapatra and Senjanovic (1980); Schechter and J.W.F. Valle (1980) or from more exotic high dimensional interactions Chang and Zee (2000); de Gouvea and Jenkins (2008a); Babu and Leung (2001). The specific UV completion is irrelevant for the purposes of this paper. All that is required is a possibly broken flavor symmetry principle that is manifest in the low energy effective system. Then the neutrino fields of Eq. (II.1) will transform under a representation of the symmetry group and only those mass matrices that render invariant will be allowed. Some of these symmetries will require particular texture zeros for full invariance that when broken will induce small deviations from zero that may be probed experimentally.

Assuming the Majorana nature of neutrinos, I extract the minimal value of for a variety of model classes. Abelian flavor symmetries are useful in this endeavor due to the freedom of charge assignments to the neutrino fields. Under non-Abelian symmetries, on the other hand, the fields will transform under some representation of the group and the Lagrangian terms will be composed of invariant field combinations. Members of the field multiplets are assigned quantum numbers, as in the Abelian case, except that these are imposed by the group representation. These charges are an important factor in mass matrix construction and may be mimicked by a properly constructed Abelian symmetry. There is more freedom in mass matrix construction associated with Abelian symmetries, which proves to be useful when scanning for small values. The restrictions imposed by non-Abelian symmetries can only push the extracted values up. Thus, in what follows, the discussion is restricted to Abelian flavor symmetries.

ii.1 Unbroken Abelian Flavor Symmetries

Here I assume that a zero element is protected by an Abelian flavor symmetry. Given the charges , and supplied respectively to the , and neutrinosThe neutrinos are components of the SM left-handed doublet fields , thus left-handed charged leptons will also be charged under . The symmetry dictates the identity of the flavor basis by well defined charge assignments. For my purposes, freedom in the flavor charge structure of the right-handed leptons can be used to propertly construct the diagonal charged lepton mass matrix. , it is simple to derive the form of the resulting symmetric Majorana neutrino mass matrix invariant under


is some, presumably , complex constant and is the neutrino mass scale. Currently, is bounded at eV from below by the atmospheric mass squared difference (Strumia and Vissani, 2005; Maltoni et al., 2004) and from above at roughly eV by cosmological data (Strumia and Vissani, 2005). Notice that the electron neutrino mass term has charge . To preserve the imposed flavor symmetry, either or . Thus, a non-trivial transformation of under guarantees and consequently a vanishing rate. If one assumes that all allowed entries are nonzero, there are eleven possible mass matrix classes that can be obtained from Eq. (II.3) by scanning charge assignments§§§The trivially zero mass matrix, obtained when for all flavors and , is not included in this listing.. These are listed in Table 2, using the charge assignment notation , along with their associated mass matrix and neutrino mixing predictions. There are only a small number, between one and three, of free parameters in each matrix entry. These must be used to construct three mixing angles and two mass squared difference predictions upon diagonalization. Thus, one obtains large correlations among the derived oscillation parameters. If the coupling coefficients are allowed to take on any value, the class assignments are superfluous in that some entries are just special limiting cases of other classes. For example, is just a special case of with . These distinctions are made here due to the expectation that all matrix elements allowed by should be of the same order, in which case each class yields different predictions. The predictions are obtained by a simple diagonalization of the resulting mass matrix under the convention that the smallest mass squared difference defines the solar oscillation frequency and the next largest the atmospheric. The largest mass squared difference is the sum of the smaller two and converges to when is small, as required by data. Degenerate eigenvalues are treated as if they possessed small splittings induced by symmetry breaking effects in anticipation of Subsection II.2. The split levels are then associated with the solar mixing sector and interpreted as such to make mass hierarchy predictions. Even then, the neutrino mass ordering can only be predicted when the lightest eigenvalue vanishes. This occurs in all classes except , and . In these cases, one may derive relationships between the mixing parameters and discrete hierarchy choices. Degeneracies leading to invariant matrix subspaces in classes and yield additional freedom corresponding to an arbitrary rotation within the invariant subbasis. This is parameterized by a mixing angle in Table 2 that may take on any value. It should be noted that symmetry breaking effects of , which select a definite mass basis, destroy this freedom by selecting a particular value of . Here, the symmetry breaking mechanism yields discrete variable changes and is therefore more important than in the other cases where deviations from the predictions of Table 2 are parametrically small, or proportional to the symmetry breaking order parameter. The goal of this exercise is to understand how close each symmetry structure comes to reproducing the current neutrino oscillation data, and in what ways they tend to fail.

Class Charge Matrix Predictions
, (Normal Hierarchy) , (Inverted Hierarchy) Can tune ’s to fit data. See text for details.
, , , Normal Hierarchy
, , , Any Hierarchy
, , , Normal Hierarchy
, , , Inverted Hierarchy
, , , Any Hierarchy
, , , Inverted Hierarchy,
, , , Inverted Hierarchy,
, , , Inverted Hierarchy
, , , Inverted Hierarchy
, , , Inverted Hierarchy
Table 2: Exhaustive summary of neutrino mass matrices with protected by Abelian flavor symmetries. Column one defines the symmetry class as referred to throughout the text. Columns two and three list representative charge assignments to the neutrino flavor eigenstates and the implied mass matrix, respectively. The final column shows oscillation parameter predictions for each class. None of these are consistent with current neutrino data. See the text for more information.

Most predictions of Table 2 are well defined and need little explanation. , however, is more involved and deserves a separate discussion for clarity. Changing the parametrization of the mass matrix for convenience to


I find that the lightest mass eigenvalue is zero, while the absolute mass squared differences are and . The associated atmospheric and solar parameters depend on their relative splitting sizes. We are left with two cases defined by their predicted mass hierarchy. For the inverted hierarchy, and , while all mixing angles vanish except , which is given by . For the normal hierarchy, the mass squared differences are reversed and all mixing angles vanish except , which is given by . It is interesting that this is the only case, due to availability of three free mass matrix parameters, that allows for a nonzero solar mass squared difference. Consequently, this class is well-suited to fit the neutrino data with only minor modifications. While neither case is consistant with neutrino oscillation phenomenology, it is clear that the normal hierarchy choice can be pushed “closer” to the observed form. For nearly maximal atmospheric mixing, must be small. For the mass squared differences to work out, the parameters and must be unnaturally tuned to . Hence, this scenario is far from ideal when considered with universally parameter values. This mass matrix texture is theoretically motivated by variants of symmetries and is commonly found in the literature. See for example (de Gouvea and Jenkins, 2008a; Frigerio and Smirnov, 2002; Merle and Rodejohann, 2006) and references therein.

Each entry of Table 2 defines a class of models with similar characteristic predictions. It is important to note that none of these classes fit the neutrino mixing data. This is another reiteration of the fact that for Majorana neutrinos, , which implies a nonzero rate. In particular, none of the classes predict realistic mixing angles. This can be seen by inspection, as most cases predict some combination of maximal or vanishing solar/atmospheric angles. Classes and are less trivial but still ultimately fail. fails due to the relationship between the angles , implying that a small must be accompanied by either a small or . Similarly, in must be close to to insure a small reactor angle. Additionally, all classes except the first yield degenerate eigenvalues implying , which further contradicts observation.

ii.2 Broken Abelian Flavor Symmetries

Exact may be excluded by this line of reasoning, but it still may be unobservably small. In this case, the symmetries of Table 2 should still be approximately valid, only broken by some small amount . These broken scenarios should still retain some of the features of their parent classes, such as mass hierarchy or large/small mixing angles. Hence, one would expect that classes such as , , and with multiple predictions that can be made consistant with data, will be broken far less than, say, classes and that are far from data.

I parameterize this symmetry breaking with the introduction of a spurious scalar field , charged under , that acquires a nonzero vev. For this spurion analysis, is just a mathematical construct used to understand the pattern and size of symmetry breaking. Specifically, I assume that has charge and acquires a real vev . With this, small mass term corrections are induced to help fit the data. The form of the resulting mass matrix may now be described by a 4-tuple . For example, the symmetry of class may be broken to yield




depending on the spurion charge assignment. In what follows, I will refer to these broken structures by integer valued 4-tuples. In the previous example, this would be and , respectively. These are not unique but adequately represent a broken class. In parent classes , , and with a single assigned charge, the non-trivialNontrivial in this context refers to an assignment that yields perturbed mass matrices distinct from the parent class structure. spurion charges are always multiples of . These cases yield between two and three perturbed mass matrix structures for each class. For the remainder of the parent classes, with two distinct charge values, non-trivial spurion charges must be multiples of either neutrino charge value or their sum. This freedom leads to a proliferation of perturbed mass matrices - between 37 and 52 for each class. For a general charge assignment, the lowest order mass matrix elements are given by


where are complex constants for the terms. Once again, it is reasonable to assume that all nonzero are of the same order of magnitude, since they all arise as coupling constants within the spurion included invariant Lagrangian. Clearly, the delta function vanishes for all but a single value. From here, it is trivial to build up the matrix to higher order. If the operator is some invariant function of the neutrino and spurion fields, of order after flavor symmetry breaking, then the operator is also invariant of order after acquires a vev. This process may be continued indefinitely to yield the full matrix. Thus, each nonzero matrix element is expressable as a power series in . For most cases of interest, where is sufficiently small, all but the leading terms described by Eq. (II.7) may be neglected. This new broken matrix may be diagonalized perturbatively to yield predictions for oscillation and kinematic parameters as a function of . Due to the number of charge assignments and breaking patterns, it is most efficient to study this numerically.

ii.2.1 Numerical Results

I now numerically survey all mass matrices generated from a broken flavor symmetry to order in and in all other elements. The unequal treatment of the mass matrix is due to the search goal of small . Corrections higher than make little difference to the fit, but one must be able to distinguish cases with different large suppressions, even when the rest of the matrix is identical. It will turn out that no structure with suppressed by more then is consistant with data. Thus, truncating the search at is safe. Using the notation of Eq. (II.7), I set in order to remove the ambiguity of simultaneous rescalings of all the and . Furthermore, I rescale , where stands for the set of all nonzero powers of in the original mass matrix and is the greatest common denominator function. This removes another unphysical ambiguity. To see this, suppose that two different charge assignments yield the same perturbed matrix structure up to an overall rescaling of . In this case, the best fit to the data would select out preferred values related by the rescaling, but the same mass matrix elements. Thus, from the neutrino mass generation standpoint, both charge assignments yield the same predictions and one should be selected to represent the systemThis is not true when a spurion analysis beyond neutrino mass is performed. In that case, each assignment would yield distinct predictions for other processes.. After all rescalings and truncations, there are 230 distinct possibilities that may be categorized into one of the eleven classes of Table 2. In each case, I scan the symmetry breaking parameter , the mass scale and coupling constants to fit current neutrino constraints listed in Table 1 and extract the smallest allowed .

Specifically, for each model I perform a fit to minimize the function


where the primed quantities are evaluated from the broken mass matrix and the unprimed best fit values and uncertainties are taken from Table 1. Here, only and are varied to yield the one parameter function . When falls below a critical value, the broken flavor symmetry is allowed at a specified confidence. To be conservative, I use confidence limits throughout this analysis. The critical value depends on the number of degrees of freedom within the system and is therefore different for each case. The allowed domain where falls below its critical value may then easily be scanned for the smallest value. If no such region exists, the charge assignment and breaking structure is disfavored by current data at confidence. To maintain the perturbativity of the system, I hold ; in which case, the largest possible (fourth order) corrections are only . Typically, corrections will be much smaller than this since goes like some power of and small values favor small . For this procedure to make sense, the constants must be constrained by some naturalness criterion, else the small values may be compensated by large coupling constants resulting in the loss of algebraic structure information. In this spirit, I only allow the constants to vary symmetrically about unity, in a sense, by a small amount. That is, for some small number . A one and two order of magnitude spread is defined by and , respectively. Ideally, the relative size of each matrix element should be determined by alone, so it is clear that should not be much greater than for a typical . Even this range is dangerous near the upper limit where order terms can easily be larger than order terms. To remove this problem completely for the full range, one needs which offers very little parameter freedom and is not realistic. I present data for , , and in the attempt to cover a comprehensive range of naturalness criterion.

Figure 1: Histogram of minimal values extracted from data allowed models, color coded by model class. From top to bottom, the panels represent the , , and cases. defines fine tuning sensitivity by limiting the relative magnitudes of the free matrix parameters as . A larger value represents more parameter freedom, which leads to a greater number of allowed models and smaller , as is clear from the plot. See the text for more details.

Figure 1 histograms the allowed flavor structures by minimum value for , , and on a scale. These are color-coded by parent class. Reference lines indicating the current bound (solid black) and future reach (broken black) are included for reference as well as the anarchy bound (solid gray). Due to increasing parameter freedom, the number of allowed models increases and broadens as the value is pushed higher. For example, the models of span less than four orders of magnitude, while the models of span seven. The smallest values for , , and are , , and , respectively. Optimally minimized charge assignments along with their relevant parameters are summarized in Table 3. The mean values for each is also shown for easy comparison. The broken symmetry that yields the smallest rates are for the , and cases and for the case. The last symmetry, optimal for , only beats out by . These are members of very similar parent classes and . It is not a coincidence that these both contain invariant subspaces that allow for additional parameter freedom and predict the normal mass hierarchy. is so small in these cases because it is suppressed by . The reason there is so great a difference between them is due solely to the increased parameter freedom of higher values allowing for smaller . Even these tiny variations in are amplified in the relations and can easily yield order of magnitude differences.

Case Class Charge Hierarchy
Table 3: Summary of models with the smallest values for the , , and cases, as listed in column one. Columns two, three and four list the model’s parent class, a representative charge assignment and derived . Column five shows the optimal amount of symmetry breaking via the order parameter to be compared with the mean value in column six. The last four columns give mixing angle and mass hierarchy predictions. Two items with similar are listed for case for easy comparison. See the text for details.

Inspection of the class descriptions of Table 2 reveals that both classes and have two distinct problems that must be solved by symmetry breaking. Specifically, it must induce a nonzero , as well as push and , respectively, up to allowed levels. Additionally, the symmetry breaking mechanism must be able to explain the values required by data, namely and for classes and , respectively. Looking at the broken case as a representative example, it is easy to see how these are solved. An allowed real valued matrix corresponding to the case is


where and . The is included in the matrix for illustrative purposes. Some predictions of this structure are , and up to modifications. Taking the allowed and neutrino mass scale range in the last relation implies between and as observed in the fit. The large solar mixing angle data selects the upper part of this range which leads to a large prediction. Once the numerical factors are accounted for, this matrix structure is very similar to that of class , with the lower right block elements of the same order, disagreeing at most by . As previously discussed, is a popular texture that is allowed, provided small symmetry breaking. The charge assignment of class fits the data in the scenario with , yielding the suppressed . It turns out that the majority of the minimized matrices with acquire this form. This is optimal from the small perspective, since all data can be accommodated with a vanishingly small element. That is, the matrix element does not significantly contribute to the fit. I find that no broken symmetry with an exact texture zero can yield this approximate structure, but highly suppressed terms are possible. For these cases, the size of depends most critically on the power suppression as opposed to the amount of symmetry breaking.

A handful of these, particularly those of class , will be probed by next generation experiments. For example, in the representative case, the class charge assignment will be explored. The breaking is optimized with , leading to an suppressed with the inverted mass hierarchy. It should be noted that null result bounds set by experiments do constrain those models with higher minimal values. However, a measurement of does not pick out a particular broken flavor model since all that is being plotted is the minimum value. In other words, all unconstrained models should be considered equal candidates. Furthermore, bounds or measurements of small only indicate a flavor symmetry when they fall below the anarchy bound near . Above that value suppressed could simply be a random fluctuation of an anarchical mass matrix.

Figure 2: Scatter plot projections of neutrino oscillation parameter predictions taken from data allowed models. and parameter confidence limits are indicated by the dark and light gray regions, respectively.

Each of these models, taken at their minimum values, make predictions for other observables. These are found by diagonalizing the perturbed mass matrix, together with the minimization parameters, to obtain the mass eigenvalues and mixing angles. They may also be combined, as prescribed in Table 1, to yield the mass squared differences as well as the effective neutrino mass relevant to tritium beta decay and the cosmological neutrino mass sum. This prescription also selects the neutrino mass ordering predicted by each model. Figure 2 displays scatter plot predictions of the models allowed by data shown as projections onto the neutrino oscillation parameter space. The dark and light gray shaded rectangles illustrate the and allowed regions, respectively. These are not true confidence contours, as they do not take correlations into account, but adequately reflect parameter regions allowed by data for the purposes of this analysis. Most of the scattered points are found to possess the normal mass ordering. There are cases predicting normal and cases predicting inverted mass orderings. Generally, the normal hierarchy cases yield a lower . A quick inspection of this plot reveals that the mass squared differences, particularly , have little spread and are thus not suited to distinguish between the symmetry structures. may be tuned to fit data in even the unbroken cases enumerated in Table 2 with minimal effects on remaining parameters. Additionally, is a relatively small splitting that arises when is broken in all classes but , where it is present from the beginning. For models that allow such small splittings, it is easy to adjust parameters to fit the data. Mixing angles, however, turn out to be a much better diagnostic tool. The unbroken classes of Table 2 predict vanishing or maximal angles for the majority of entries. If these fit the data as given, say, maximal and vanishing , symmetry breaking is likely to drive these away from the preferred values. In the opposite limit, when predictions are far from data, the symmetry breaking must drive the angle significantly to reach the allowed range. This results in a large spread in predicted angles. I find that and are the best laboratories for constraining the model classes. The points in the projection onto this parameter plane spans almost the entire allowed region. The triangular shape of the scatter profile in this panel is an artifact of the fitting procedure. If a large deviation from best fit is found in one parameter, there is little room left for deviations in any other parameter.

Figure 3: Scatter plot projections of neutrino kinematic and absolute mass predictions (, and ) taken from data allowed models. Current confidence limits are indicated by the gray region, and the solid black contour line illustrates the next generation experimental reach.

In a similar way, Figure 3 displays prediction scatter plots in the space of absolute mass observables , and . The direction is histogrammed in the third panel of Figure 1 and is included here to help visualize the relationship among the parameters. Once again, light gray illustrates the allowed region. The solid black contour line represents the reach of next generation experiments. These contours only approximate the present and future experimental bounds, as there are large correlations among , and under the three light Majorana neutrino assumption employed here. See for example Pascoli et al. (2006); Barger and Whisnant (1999); Pascoli and Petcov (2003); de Gouvea and Jenkins (2005); Fogli et al. (2004) and references therein for a discussion of these correlations. The smallest inverted hierarchy is for the charge structure . This is the only class model consistent with data, and thus requires a relatively large amount of symmetry breaking, with . The remaining six inverted hierarchy cases are of classes , and , consistent with expectations from the Table 2 predictions. Within these, the hierarchy determination depends heavily on the specific charge structure. Those predicting normal spectra generally have larger symmetry breaking in order to overcome the influence of their parent classes. Nevertheless, these can have as small as due to large power suppressions. Still, it is gratifying that the inverted cases are all above the confidence lower limit defined in terms of the entry of Table 1 with and . Evaluated, this is . As expected, I only find models predicting normal mass hierarchies below this bound. Future experiments will constrain many of these symmetry structures. Cosmological observations aimed at measuring the observable seem to have the best prospects. They have the potential to probe all of the inverted hierarchy models as well as many normal ones. Additionally, a measurement of the neutrino hierarchy could have a great impact on flavor symmetry models. Next generation neutrino oscillation experiments are expected to provide non-trivial information regarding the mass spectra. The majority are based on neutrino/anti-neutrino asymmetries via Earth matter effects Mohapatra et al. (2007); de Gouvea (2004a, b); de Gouvea et al. (2005); de Gouvea and Jenkins (2005); de Gouvea and Winter (2006), but depend strongly on large values. The small scenario is explored in de Gouvea and Winter (2006); de Gouvea and Jenkins (2005); de Gouvea et al. (2005) considering oscillation and non-oscillation searches.

It is important that these results and predictions are interpreted correctly. The nature of the fitting procedure naturally selects the lowest value allowed by the data in order to obtain the smallest possible . With this in mind, at least one predicted parameter should sit at the edge of its allowed region for each model. This does not mean that all of the explored symmetry structures are on the verge of exclusion. I point out that these are only predictions for the matrix that yield the smallest value, and which is also consistent with data. If future experimental constraints tighten, one must simply redo the fit with the new data. Generally, larger symmetry breaking would be needed which would push , and by extention , higher. In terms of the histograms of Figure 1, the net result would be a general movement of upward. The lowest values of are typically suppressed by the highest powers of and will be affected the most in this transformation. It would lead to a narrower distribution. In this process, some models may be excluded, but it is not guaranteed.

Figure 4: Plot of minimization results as a function of the fractional neutrino oscillation parameter uncertainty within the case. The upper panel shows the variation in the smallest extracted values while the lower panel depicts the variation in the number of allowed models. The black curves let all parameter uncertainties vary about the current best fit points (solid) and alternative central point (dashed). This point is defined by and with all other parameters held at their best fit values. This choice is consistent with data at the level. For the solid colored curves, only one uncertainty is allowed to vary at a time about the best fit point with the others held at their current values.

In Figure 4, I show the movement of the numerical results as a function of parameter uncertainty, taken as a fraction of the current deviation, as given in Table 1. Here, is defined by


The reduced is used as input in the minimization of Eq. (II.8) and simulates future improvements in neutrino parameter measurements. This is shown in a scale down to . The black curves are obtained by varying the uncertainty on all of the oscillation parameters. The solid and dashed lines use central values equal to the best fit parameters of Table 1 and equal to an alternative allowed point, respectively. The latter was chosen as a logical possibility with large deviations from maximal atmospheric mixing () and vanishing (). All other parameters were held at the current best fit. The solid colored curves were obtained by varying the parameter uncertainties individually, assuming the current best fit central values. The upper and lower panels show the variation of the smallest derived values and the total number of allowed models respectively with parameter uncertainty. The general trend is as expected. As the uncertainty is decreased, is pushed larger while the number of allowed models decrease. In terms of individual parameter curves, , and to a much lesser extent , show the largest variations. All others induce very little deviation and almost sit directly on top of each other in the lower panel. Thus, it is clear that improved measurements of are essential to bounding within this framework.

The slope changing curve features of Figure 4 are not numerical artifacts. In the upper panel, they correspond to changes in the charge structure of the minimal model, each of which has its own vs. slope. Taking the solid black total variation curve as an example, I find that the six largest slope changes all correspond to optimal charge assignment changes. The exact models involved here are not very enlightening. As decreases, the optimal model class jumps from to to . If the parameter best fit points remain at their current level, the bound on the minimum value will triple, provided a two order of magnitude improvement in oscillation measurements. Even these enhanced values are beyond future experimental prospect, so the more significant result is related to the number of allowed models. In this scenario, the number of allowed broken models is reduced by roughly a factor of three to . Such a small number of possibilities would constrain neutrino mass ultraviolet completions and help guide model builders in further constructions. Of course, this all depends on the exact neutrino parameter values. For example, the variations are not so drastic under the alternative best fit point. This is so because the majority of the smallest structures predict large and deviations from maximal due to symmetry breaking effects. This point may be taken to an extreme by postulating a best fit point directly at the predictions of the minimal model; in which case, the lowest will not vary with , although the available model space might. It is not clear if the opposite extreme exists. Namely, is there a best fit choice that can push into the next generation reach or that will narrow down the model space to a single, or even zero, symmetry structure? Based on current results, the former case seems doubtful, but the latter model reduction remains a serious possibility. This speculation may be verified by a scan over possible best fit parameter points, but such an analysis is not warranted without experimental direction in terms of improved bounds and parameter measurements.

Iii Concluding Remarks

While exact symmetries generating are excluded by current data, slightly broken symmetry structures are still allowed that result in well beyond the reach of future experiments. I systematically study these broken Abelian symmetries via a spurion analysis to obtain the smallest allowed values. I find that, allowing reasonable coupling constants, many models are excluded by data and that the smallest is constrained to be larger than about at confidence. The structures yielding the smallest allowed values imply general predictions for neutrino experiments, including large deviations from vanishing and maximal , in addition to the normal mass spectra. Improvements in future data could increase these limits and perhaps, depending on the central parameter values, single out a handful of allowed models that may be explored in more depth. Qualitatively, the main result is that there is currently a small but non-zero lower bound to that may be improved by future precision measurements.

At face value, these results are only valid under specific circumstances. Namely, I explore models containing three light Majorana neutrinos subject to a broken Abelian flavor symmetry with a non-trivial transformation in the absence of fine tuning. These happen to predict small bounds. How far can this be pushed? Abelian symmetries are better suited to this task than non-Abelian ones, but one may still consider discrete flavor groups. See for example Ma (2007) and references therein. The fact that all broken Abelian flavor models predicting are excluded helps motivate that no exact Abelian discrete symmetry (a discrete subgroup of ) can be allowed. This says nothing of non-Abelian discrete groups, which is therefore a limitation of the analysis. Next, relaxing the naturalness requirement, allowing the parameters to take on any value, will push down the allowed bounds, but will not admit solutions with vanishing . The introduction of new degrees of freedom is more complicated, but potentially testable by other means depending on the masses and couplings involved. New physics above the scale will decouple from the system and can be ignored for these purposes de Gouvea and Jenkins (2008a). In the intermediate range between and , heavy particle mediation of via contact effective operators can become important and influence the relationship between and the effective measured in . These quantities should converge for small , as motivated in Section I, but the detailed rate depends on the new physics model. Here, I only consider the mass matrix element, which will deviate from at sufficiently large . Hence, in the presence of TeV scale new physics, the large models of Figure 1 may not be mapped onto in the standard way. Less caution is needed with the smallest value models, which comprise the main object of this study. Moving down in mass scale below approximately , the characteristic momentum transfer, new physics can mediate the decay and interfere with the light neutrinos to suppress , regardless of the derived value Wolfenstein (1981); de Gouvea et al. (2007); de Gouvea (2005); Kayser (1984). This was discussed in de Gouvea et al. (2007) in terms of the eV scale type-I Seesaw mechanism, where the effect is particularly clear and can yield vanishing rates. In this case, for singlet neutrinos, the extended mass matrix is relevant to the system and the upper diagonal block, which includes , is zero by construction. Thus, the relation between the light neutrino defined in the last entry of Table 1 and is completely spoiled. However, one could hope to observe these light degrees of freedom in sterile neutrino oscillation searches or astrophysical phenomenon. The last consideration is the presence of exact at some high scale where neutrino masses are generated. In the absence of a flavor symmetry to protect it, a non-zero will be generated via two loop renormalization group effects, as motivated in Davidson et al. (2007). Details of this generation mechanism depend on new, intermediately scaled physics and has yet to be calculated for the element of the mass matrix. Therefore, the qualitative existence of a lower bound seems robust under the introduction of arbitrary new physics at scales greater than . This offers a much needed handle, or at least a conceptual proof of principle, on means of selecting the Dirac neutrino nature.

This analysis is best suited to describe the theoretical possibilities of a scenario in the distant future where null results push bounds below the anarchy limit at . Above this value, small could be the result of mass matrix statistical fluctuations. Thus, subject to the above testable qualifications, it would be safe to assume that one of the flavor symmetries explored here is at work to suppress or that neutrinos are Dirac particles. Detailed speculations on such broad experimental improvements are beyond the scope of this work, but it is reasonable that these strong bounds would be accompanied by similar enhancements in other neutrino related parameters. In such a world, this analysis would guide model builders toward the complete and correct neutrino mass model. Currently, bounds are over two orders of magnitude away from this situation. Still, until a positive signal is detected, this scenario remains a logical possibility that should be explored to properly understand the options open to nature. In the meantime, experimental bounds may be used to constrain the flavor symmetry model space.

Special thanks to Andre de Gouvea and Alex Friedland for useful discussions on this topic and comments on the original manuscript. This work was performed under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. This work was also funded in part by the US Department of Energy Contract No. DE-FG02-91ER40684.


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