An expansion for Neutrino Phenomenology

We develop a formalism for constructing the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and neutrino masses using an expansion that originates when a sequence of heavy right handed neutrinos are integrated out, assuming a seesaw mechanism for the origin of neutrino masses. The expansion establishes relationships between the structure of the PMNS matrix and the mass differences of neutrinos, and allows symmetry implications for measured deviations from tri-bimaximal form to be studied systematically. Our approach does not depend on choosing the rotation between the weak and mass eigenstates of the charged lepton fields to be diagonal. We comment on using this expansion to examine the symmetry implications of the recent results from the Daya-Bay collaboration reporting the discovery of a non zero value for theta_{13}, indicating a deviation from tri-bimaximal form, with a significance of 5.2 sigma.

where P = diag(e iα 1 /2 , e iα 2 /2 , 1) is a function of the Majorana phases, present if the right handed neutrinos are Majorana, while δ is a Dirac phase. This latter phase can contribute in principle to neutrino oscillation measurements, while the Majorana phases cannot. Recent global fit results on neutrino mass differences and measured mixing angles using old/new reactor fluxes are given in Ref. [3], (with new reactor flux results in brackets): The error is the reported 1σ error. Note that ∆m 2 32 ≡ m 2 3 −(m 2 1 +m 2 2 )/2 and ∆m 2 32 > 0, (< 0) corresponds to a normal (inverted) mass spectrum. This pattern of experimental data is perhaps suggestive of a PMNS matrix that has at least an approximate "tri-bimaximal" (T B) form [4]. Fixing sin 2 (θ 12 ) = 1/3 and θ 13 = 0 the T B form is for a particular phase convention.
The structure of this matrix could be fixed by underlying symmetries. In attempting to determine such an origin of this matrix, the "flavour" basis where one assumes U(e, L) = diag(1, 1, 1) is frequently used.
When this assumption is employed the relationship between the weak and mass neutrino eigenstates is identified with the U P M N S , i.e. ν i = (U P M N S ) ij ν j for i, j = 1, 2, 3. There have been many attempts to link the approximate T B form of the neutrino mass matrix to symmetries of the right handed neutrino interactions in this basis, see Ref. [5,6] for a review. Recent experimental results provide evidence for deviations from this T B form. Evidence for sin 2 (θ 13 ) = 0 in global fits is reported to be > 3σ in Ref. [3] at this time. As this paper was approaching completion, the discovery of non-vanishing θ 13 was announced by the Daya Bay Collaboration [7] with a reported value of sin 2 (2 θ 13 ) = 0.092 ± 0.016(stat) ± 0.005(syst), (5) corresponding to 5.2σ evidence for non zero θ 13 . It is reasonable to expect further speculation about the origin of the deviation from T B form in light of this result, where again the flavour basis will be frequently assumed.
There is no clear experimental support for assuming that U(e, L) = diag (1,1,1). This choice can be motivated by an ansatz related to the origin of the approximately diagonal structure of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and a further ansatz on the relation between U † (d, L) and U † (e, L), see Ref. [5] for a coherent discussion on this approach. This choice can also be justified with model building in principle, see Ref. [8] for an example. Conversely, in grand unified models frequently associated with the high scale involved in the seesaw mechanism, the quark and lepton mass matricies can be related in such a manner that the flavour basis cannot be chosen, or at least, the choice of the flavour basis can be highly artificial.
It is of interest to have a formalism for neutrino phenomenology that is as robust and basis independent as possible. Clearly linking symmetries to the form of the PMNS matrix requires that the physical consequences of a symmetry are not dependent on a basis that can be arbitrarily choosen for U(e, L), such as the flavour basis. In this paper, we develop a perturbative approach to the structure of the PMNS matrix as an alternative to symmetry studies that are basis dependent. 1 Using this approach we show a basis independent relationship between the eigenvectors that give U(ν, L) and U(e, L) corresponding to T B form at leading order in our expansion. We also examine the prospects of relating patterns in the data in low scale measurements of neutrino properties with high scale flavour symmetry breaking, illustrating how our approach can be used to study the recent reported discovery of non zero θ 13 reported in Ref. [7].

II. CONSTRUCTING A FLAVOUR SPACE EXPANSION FOR NEUTRINO PHENOMENOLOGY
In this section we develop a formalism linking the measured differences in the neutrino mass eigenstates with the structure of the PMNS matrix, which we will refer to as a flavour space expansion (FSE). There is an experimental ambiguity in the measured mass hierarchies at this time. The neutrino mass spectrum can be a normal hierarchy (m 3 > m 2 > ∼ m 1 ) or an inverted hierarchy (m 3 < m 1 < ∼ m 2 ). In our initial discussion we will assume a normal hierarchy. The formalism can be reinterpreted for an inverted hierarchy. 1 This is a modern implementation in the neutrino sector of an old idea of relating the mixing matricies of the standard model (SM) to the measured quark or neutrino masses, for pioneering studies with this aim see Ref. [9,[14][15][16].
Recall the standard seesaw scenario [17], with three right handed neutrinos N Ri . 2 We use the notation = (ν L , e L ) for the left handed SU(2) doublet field and e R for the SU(2) singlet lepton field carrying hypercharge. These fields carry flavour indicies i, j. 3 We also defineH = i τ 2 H , where H is the Higgs doublet of the SM, with H T = (0, v/ √ 2). Then the lepton sector of the Lagrangian in the seesaw scenario is the following here we have defined N c Ri = CN T Ri with C the charge conjugation matrix. There is freedom to rotate to the mass basis by introducing the unitary rotation matricies U(f, L/R), so that The kinetic terms are unchanged by these rotations, and one is free to further rotate to a basis in the flavour space of M, y E , y ν defined by the set of all possible M, y E , y ν related through We choose to make the initial rotation to a basis where M is diagonal real and nonnegative. This fixes U(N, R). Initially the Majorana mass matrix has three complex eigenvalues M a,b,c = η a,b,c |M a,b,c | = η a,b,c (µ a,b,c ), here η a,b,c are the Majorana phases. Working in this basis [18][19][20] shifts any Majorana phases into the y ν matrix and N i = √ η i N Ri + η i N c Ri . Integrating out the heavy N i one obtains the dimension five operator [21] with the matrix Wilson coefficient C ij = (y T ν ηM −1 y ν ) ij . The key observation that we use in this work is that when integrating out the N i in sequence 4 a perturbative expansion for neutrino phenomenology that is 2 Our initial discussion will largely follow Ref. [18][19][20]. 3 These flavour indices will run over a, b, c for the three N i . We use this notation to distinguish these flavours from the measured mass differences of the physical eigenstates (m1,2,3) and also the index on the Yukawa vectors of each N i , which run over 1, 2, 3. 4 We distinguish in this paper right handed neutrinos that are in a mass diagonal basis, reserving the notation Ni for such states, compared to NRi states, which are not mass diagonal in general.
related to a hierarchy in the magnitude of the contributions to C ij can be constructed. Note that this is also the key point in the sequential dominance idea of Refs. [10][11][12][13], however, the formalism we will develop is distinct from these past results.
B. Perturbing the Seesaw

Model Dependence of Perturbing the Seesaw
The FSE we develop depends on the mass spectrum and the Yukawa couplings of the N i . Also, in principle, the flavour orientation 5 effects the quality of the FSE. Experimentally, at this time, all of these Lagrangian parameters are individually unknown, thus we will be forced to assume that a perturbative expansion of the form we employ exists. Our approach is to view the seesaw Lagrangian given in Eq.
(6) as an effective theory, and we do not attempt to uniquely identify and restrict ourselves to a particular UV physics that dictates the low energy parameters in this effective theory in this paper. However, the formalism we develop allows perturbative investigations of the neutrino mass spectrum and mixing angles in many scenarios and is in fact quite general. It would be surprising if all of the unknown parameters conspired to forbid an expansion of this form from being present. We seek to illustrate this point in this section, by demonstrating a simple scenario where the FSE would be of some utility.
There are of course cases when the formalism we outline cannot be used. When the µ a,b,c and Yukawa couplings of the N i are both highly degenerate a FSE cannot be used. 6 This is also the case if a nondegenerate pattern of the µ a,b,c and the Yukawa couplings are such that the perturbations to the neutrino masses are similar in magnitude. However, as the µ a,b,c depend on the matching of the effective theory to UV physics, while the Yukawa couplings are dimension four and are not UV sensitive, this would require tuning the physics of different energy scales. Unless model building is used to relate the masses and Yukawa couplings of the N i so that the contributions to the ν L masses are the same, a FSE is expected quite generally, and can be related to neutrino mass differences, as we will show.
One can study the expected spectrum of left handed masses due to Eqn. (6) in generic scenarios. Without assuming other interactions beyond the SM, the N Ri are not distinguished by any quantum number. As a result, the non-diagonal mass matrix that is the coefficient of the operator matrix O ij = N Ri N Rj given by M ij is naively expected to have entries that are all similar in magnitude. As the operator is dimension three, the mass matrix is expected to be proportional to the highest scale UV physics (violating lepton number) that was integrated out leading to this effective theory. We denote this scale by M 0 and the naive non-diagonalized mass matrix in this case as Logarithmic corrections of this form are required to cancel the renormalization scale dependence of the pole masses of the N Ri . Including such corrections leads to M ij M 0 (1 + ij ). These corrections split the mass spectrum; diagonalizing one finds A hierarchical spectrum of µ i is expected with the pattern (µ 1 , µ 2 , µ 3 ) M 0 (1, , 2 ).
In the case of degenerate µ i due to the M ij mass matrix not conforming to these generic expectations, the expansion can follow from a hierarchy in the Yukawa couplings of the N i -such as the hierarchical pattern of Yukawa couplings in the SM. Lastly, the suitability of the expansion can follow only from the orientation in flavour space of the Yukawa coupling vectors of the N i . In summary, the appropriateness of the FSE is clearly model dependent, but we expect it to be broadly applicable in realistic models.

C. Developing the Seesaw Perturbations
Consider supplementing the SM field content by a single right handed neutrino 7 N a and an interaction term that couples it into a linear combination of β L . The coupling is fixed by the complex Yukawa vector x T = {x 1 , x 2 , x 3 } in flavour space whose form can be constrained by a flavour symmetry, but is here left arbitrary. We absorb the overall Majorana phase into this vector. The relevant Lagrangian terms 8 are given by The right handed neutrino can be integrated out, giving for the left handed neutrino mass matrix a nonzero The with m real and non-negative. 9 These vectors are also eigenvectors of also with eigenvalues m 2 . One finds the leading eigenvector and Here µ a is real and non-negative due to the initial flavour basis choice that fixed U(N, R), while U(ν, L) and x are in general complex. 10 The remaining two eigenvectors ρ b,c are such that ρ a | ρ b ≡ ρ a † ρ b = 0, ρ a | ρ c = 0. These eigenvectors will lead to the remaining (smaller) mass eigenvalues, and will perturb the leading eigenvalue and eigenvector and thus the U(ν, L) matrix. This is the expansion we seek to exploit. Consider the perturbation that generates the second eigenvalue of the neutrino mass matrix due to a second right handed neutrino N b , which we define to couple into a linear combination of the L given by y T = {y 1 , y 2 , y 3 }. One obtains a second eigenvalue in the left handed neutrino mass matrix so long as y x. The perturbation of M M † is given by At leading order the ρ b,c have degenerate (vanishing) eigenvalues. One is free to rotate to a chosen basis in these vectors. We rotate to a basis in these vectors such that the following perturbation vanishes for ρ c With this choice ρ c retains a vanishing eigenvalue when N b is integrated out. The eigenvector ρ c should be orthogonal to ρ a ∝ x. A normalized eigenvector basis at leading order is then Now we can determine the perturbation on the leading order eigenvectors and eigenvalues when N b is integrated out. The corrections to the eigenvalues and eigenvectors using perturbation theory are given by Nonzero eigenvalues are obtained for m b,c at second order in the expansion due to the orthogonal basis vectors causing the leading perturbations to each of these masses to vanish. The leading perturbation to the eigenvectors and m a is first order in the expansion. We retain the leading perturbation on the eigenvectors and the leading and subleading effects on the masses to obtain nonzero eigenvalues. We find the following for the perturbations Here δm 2 ij = m 2 i − m 2 j , µ 2 ij = v 4 /(4 µ i µ j ) and cos θ xy = | x * · y|/| x|| y| is a measure of (the cosine of) the angle between the x, y Yukawa vectors. The masses are evaluated to the appropriate order in the perturbative expansion and the eigenvector perturbations are in general complex. Note that for the phenomenology of the U P M N S matrix that we will study it will be sufficient to only retain the leading perturbation, while when studying the mass spectrum the leading and sub-leading perturbations should be retained.
Finally integrate out the third right handed neutrino N c with Majorana mass µ c , which couples into a linear combination of the β L dictated by z T β = {z 1 , z 2 , z 3 }. The resulting eigenvector perturbations are The mass perturbations are The measured masses of the neutrino's are related to these perturbations as It is interesting to note that a normal hierarchy emerges quite naturally from the FSE as the leading neutrino mass m a receives corrections at linear order to its mass, while the remaining masses only receive corrections at second order in the perturbations.
It is also important to note that this formalism does not require a hierarchy of the form δ 2 m 2 i δm 2 i , only δ 2 m 2 i , δm 2 i m 2 a is required. Expanding on this important point in more detail, it is not required that the perturbation due to integrating out N b is larger than the perturbation due to integrating out N c . Only that the effect of integrating out each of these right handed neutrinos perturbs the initial mass matrix -which is dominated by integrating out the initial right handed neutrino N a . The existence of these perturbations are not necessarily direct statements on the relative size of the µ i as we discuss in more detail in the next section. where the perturbations are dictated primarily by a hierarchy in µ a,b,c . We will then discuss the case where the expansion arises primarily due to the orientation of the Yukawa coupling vectors in flavour space.

The expansion without flavour alignment
We can examine the quality of the expansion by comparing to the measured mass differences in neutrinos. Consider the case that the size of the perturbations is generic in the sense that the Yukawa coupling vectors are taken to be O(1) with the orientation in flavour space not significantly effecting the quality of the expansion. In this case, the expansion follows from the relative size of the µ i , i.e. µ a < µ b < µ c .
Consider the generic case in a normal hierarchy. Using the FSE and retaining the dominant term while in the same manner δm 2 b ∼ ∆m 2 21 and the expansion requires δ 2 m 2 c < ∆m 2 32 , ∆m 2 21 . Due to this, the expansion requires v 2 | z| 2 /(2 µ c ) < ∆m 2 32 ∼ 0.05 eV. This condition is consistent with current bounds on the absolute neutrino mass scale, with a 95% C.L. bound of m ν = 0.28 eV quoted in Ref. [22], assuming ΛCDM cosmology. It is also consistent with current bounds from Tritium β decay experiments [23] which quote m(ν e ) < 2 eV at 95% C.L.
Expressing this condition in terms of the high mass scale of the N c integrated out, µ c /| z| 2 > ∼ 10 14 GeV.
Generically one expects the mass scale of the right handed neutrino operator to be the largest scale integrated out that violated L number, and this condition for the lightest neutrino is clearly consistent with naive expectations of M 0 ∼ M pl .

The expansion with flavour alignment
Now consider the case where the perturbative expansion follows from the flavour orientation of the x, y, z vectors primarily. An example where this is the case is when the threshold matching onto the UV physics is such that the Wilson coefficient matrix of O ij yields a mass matrix with nearly degenerate eigenvalues. This occurs for example when In this case, a nearly degenerate mass spectrum of the N R follows, µ a µ b µ c . Generating a normal or inverted hierarchy if one also has | x| ∼ | y| ∼ | z| requires more precise alignments in flavour space and allows a geometric interpretation of the measured neutrino mass spectrum. In the FSE, the tree level masses of the SM neutrinos are m 2 A = µ 2 aa | x| 4 + 2µ 2 ab | x| 2 | y| 2 cos 2 θ xy + 2µ 2 ac | x| 2 | z| 2 cos 2 θ xz + µ 2 bb | y| 4 cos 2 θ xy + µ 2 cc | z| 4 cos 2 θ xz , Consider the case that all of the µ 2 ij are similar in magnitude and | x| > ∼ | y| ∼ | z| so that the mass spectrum is primarily dictated by the orientation of the Yukawa vectors in flavour space. An example of an inverted or normal hierarchy is shown in Fig. (1) in this case. Normal hierarchy on the left, inverted on the right.

E. U P M N S and Flavour Space
Now, assuming that a FSE expansion exists, let us consider its utility in examining the form of U P M N S .
The rotation matrix U(ν, L) with v i = ( ρ i + δ ρ i + δ 2 ρ i ) is given by U(ν, L) = ( v 3 , v 2 , v 1 ). The charged lepton mass matrix after electroweak symmetry breaking is given by M e = v y E / √ 2 and diagonalized by 11 11 The hierarchy of the charged lepton masses can be used to organize an expansion of U(e, L) in the same manner by constructing M † e Me in principle. Conversely, in principle, one could employ an ansatz that the hierarchy in these mass eigenvalues could be related to an expansion of U(e, R). As such we do not employ a FSE on U(e, L) † . This ambiguity also limits the application of a FSE to the CKM matrix. In this manner, the expansion we employ is most useful for expanding U(ν, L) when a seesaw mechanism is the origin of the smallness of the neutrino masses.
The expanded U P M N S is of the form This result makes clear that the first two right handed neutrinos that were integrated out in the FSE contribute to the leading order structure of the PMNS matrix. The leading order of the ρ i eigenvectors only depended on x, y. As we have discussed, these neutrinos can be integrated out in sequence, and for Yukawa couplings   Trivially one finds It follows that in the flavour basis x · y = 0 so δm a = 0 and (δ ρ a , δ ρ b , δ ρ c ) = (0, 0, 0). Now consider including a third neutrino eigenvalue, retaining the required perturbation. For a solution, z 2 = z 3 is required, and consequently δ 2 m 2 a = 0 as x· z = 0. As a result the leading eigenvector ρ a is unperturbed by integrating out both N b , N c in the flavour basis, one finds A µ ↔ τ symmetry implemented on y and z is consistent with T B form of the PMNS matrix, as expected.

A. Perturbative breaking of T B form
Now consider the breaking of T B form. The value of θ 13 measured by the DAYA-Bay collaboration is in agreement with the global fit values given in Ref. [3], as such, we use the fit results to find the measured pattern of deviations in T B form. It is instructive to construct the following ratios of experimental values characterizing the deviations of T B form in each mixing angle. Using the small angle approximation tan 2 (δθ 12 ) sin 2 (δθ 13 ) = 0.02 ± 0.32 tan 2 (δθ 23 ) sin 2 (δθ 13 ) = 0.01 ± 0.10 tan 2 (δθ 12 ) tan 2 (δθ 23 ) = 2.0 ± 36.
error. Various breaking of T B form can be studied using the FSE and compared to these results. Consider the case that N b retains a µ ↔ τ flavour symmetry in its couplings to the charged leptons, but N c breaks such a symmetry so that deviations in T B form are expected. Fix z T = z T + (0, ∆ 1 , ∆ 2 ) with ∆ 1 = ∆ 2 and treat this breaking as a perturbation using the FSE. We use ∆m 2 AB ∆m 2 AC assuming a normal hierarchy. At leading order in the FSE the breaking of T B has the pattern tan 2 (δθ 12 ) sin 2 (δθ 13 ) = 0, tan 2 (δθ 23 ) sin 2 (δθ 13 ) = 2 cos 2 θ ρ b z + cos 2 θ ρcz cos 2 θ ρ b z + 2 cos 2 θ ρcz , tan 2 (δθ 12 ) tan 2 (δθ 23 ) = 0. We emphasize however that the relationships between the eigenvectors determine the form of the PMNS matrix in general. This is easy to demonstrate in more detail. Consider retaining a µ ↔ τ symmetry imposed on y and z but deviating from the flavour basis, choosing x, y, z as above but unfixing σ i . At leading order in the expansion of U P M N S one can solve for the condition that T B form is still obtained for general σ i . One finds that only the flavour basis for σ i gives a valid solution at leading order in the expansion. This makes clear that µ ↔ τ symmetry imposed on the Lagrangian alone is not related to T B form in a basis independent manner.
As a further example that µ ↔ τ symmetry is also not unique or of particular physical significance in allowing T B form (with an appropriate choice on the σ i eigenvectors), consider the following procedure. Choose a (e, µ, τ ) symmetry on the first interaction eigenvector, ie x T = (1, 1, 1)/ √ 3, and y T = (1, 1, 0)/ √ 2 as a simple interaction eigenvector for N b leading to an orthonormal eigenbasis at leading order, finding This procedure can be used for any flavour symmetry chosen to fix x, y in the FSE for the N i .
We also note that the impact of sterile neutrinos weakly coupled to the SM on neutrino phenomenology can be systematically studied with this approach. For example, one can relate any measured value of a deviation from T B form to the particular Yukawa coupling vector of a single sterile neutrino, which can be shown to accommodate the value of θ 13 reported by the Daya-Bay collaboration while the three right handed neutrinos partners of the SM fields give an exact T B form of the U P M N S matrix.

IV. CONCLUSIONS
Flavour symmetries that are related to the structure of the U P M N S matrix only in a particular basis choice of U(e, L) can lead to suspect physical conclusions. As an alternative to basis dependent symmetry studies, we have developed a perturbative expansion relating the measured masses of the neutrinos to the form of the PMNS matrix. This expansion offers a promising framework for broadly understanding the implications of the systematically improving experimental neutrino data, particularly in a normal hierarchy.
We have illustrated our approach in an example where the flavour basis was chosen, for the sake of familiarity, and then shown how the expansion can control the predictions of T B form being broken. However, the approach we outline can accommodate any basis choice. Indeed, it is the relationships between the eigenvectors that dictate the form of the U P M N S matrix in a U(e, L) basis independent manner. This formalism can be employed in model building to attempt to determine a compelling origin of the eigenvector relationship that corresponds to T B form at leading order in the FSE. Further, as the breaking of T B form is now experimentally established due to the discovery of a non zero θ 13 in Ref. [7], we expect the FSE to be of some phenomenological utility in falsifying mechanisms of the breaking of the T B form of the U P M N S matrix, as the pattern of this breaking is further resolved experimentally in the years ahead.