Neutrino mixing sum rules and the Littlest Seesaw

In this work, we study the neutrino mixing sum rules arising from discrete symmetries, and the class of Littlest Seesaw (LS) neutrino models. These symmetry based approaches all offer predictions for the cosine of the leptonic CP phase $\cos \delta$ in terms of the mixing angles, $\theta_{13}$, $\theta_{12}$, $\theta_{23}$, while the LS models also predict the sine of the leptonic CP phase $\sin \delta$ as well as making other predictions. In particular we study the \textit{solar} neutrino mixing sum rules, arising from charged lepton corrections to Tri-bimaximal (TB), Bi-maximal (BM), Golden Ratios (GRs) and Hexagonal (HEX) neutrino mixing, and \textit{atmospheric} neutrino mixing sum rules, arising from preserving one of the columns of these types of mixing, for example the first or second column of the TB mixing matrix (TM1 or TM2), and confront them with an up-to-date global fit of the neutrino oscillation data. We show that some mixing sum rules, for example an \textit{atmospheric} neutrino mixing sum rule arising from a version of neutrino Golden Ratio mixing (GRa1), are already excluded at 3$\sigma$, and determine the remaining models allowed by the data. We also consider the more predictive LS models (which obey the TM1 sum rules and offer further predictions) based on constrained sequential dominance CSD($n$) with $n\approx 3$. We compare for the first time the three cases $n=2.5$, $n=3$ and $n=1+\sqrt{6}\approx 3.45$ which are favoured by theoretical models, using a new type of analysis to accurately predict the observables $\theta_{12}$, $\theta_{23}$ and $\delta$. We study all the above approaches, \textit{solar} and \textit{atmospheric} mixing sum rules and LS models, together so that they may be compared, and to give an up to date analysis of the predictions of all of these possibilities, when confronted with the most recent global fits.


Introduction
Neutrino mass and mixing represents the first and so far the only new physics beyond the Standard Model (SM) of particle physics.We know it must be new physics because its origin is unknown and it is not predicted by the SM.Independently of the whatever the new (or nu) SM is, we do know that the minimal paradigm involves three active neutrinos, the weak eigenstates ν e , ν µ , ν τ (the SU (2) L partners to the left-handed charged lepton mass eigenstates) which are related to the three mass eigenstates m 1,2,3 by a unitary PMNS mixing matrix [1].
Various simple ansatzes for the PMNS matrix were proposed, the most simple ones involving a zero reactor angle and bimaximal atmospheric mixing, s 13 = 0 and s 23 = c 23 = 1/ √ 2, leading to a PMNS matrix of the form, where the zero subscript reminds us that this form has θ 13 = 0 (and θ 23 = 45 • ).
The first approach, which leads to solar sum rules, is to assume that the above patterns of mixing still apply to the neutrino sector, but receive charged lepton mixing corrections due to the PMNS matrix being the product of two unitary matrices, which in our convention is written as V eL V † ν L , where V † ν L is assumed to take the BM, TB or GR form, while V eL differs from the unit matrix.If V eL involves negligible 13 charged lepton mixing, then it is possible to generate a non-zero 13 PMNS mixing angle, while leading to correlations amongst the physical PMNS parameters, known as solar mixing sum rules [13][14][15][16].This scenario may be enforced by a subgroup of A 4 , S 4 , S 5 which enforces the V ν structure [12] while allowing charged lepton corrections.
In the second approach, which leads to atmospheric sum rules, it is assumed that the physical PMNS mixing matrix takes the BM, TB or GR form but only in its first or second column, while the third column necessarily departs from these structures due to the non-zero 13 angle.Such patterns again lead to correlations amongst the physical PMNS parameters, known as atmospheric mixing sum rules.This scenario may be enforced by a subgroup of A 4 , S 4 , S 5 which enforces the one column V ν structure [12] while forbidding charged lepton corrections.
Apart from the large lepton mixing angles, another puzzle is the extreme lightness of neutrino masses.Although the type I seesaw mechanism can qualitatively explain the smallness of neutrino masses through the heavy right-handed neutrinos (RHNs), if one doesn't make other assumptions, it contains too many parameters to make any particular predictions for neutrino mass and mixing.The sequential dominance (SD) [17,18] of right-handed neutrinos proposes that the mass spectrum of heavy Majorana neutrinos is strongly hierarchical, i.e.M atm ≪ M sol ≪ M dec , where the lightest RHN with mass M atm is responsible for the atmospheric neutrino mass, that with mass M sol gives the solar neutrino mass, and a third largely decoupled RHN gives a suppressed lightest neutrino mass.It leads to an effective two right-handed neutrino (2RHN) model [19,20] with a natural explanation for the physical neutrino mass hierarchy, with normal ordering and the lightest neutrino being approximately massless, m 1 = 0.
In this paper we study neutrino solar and atmospheric mixing sum rules arising from discrete symmetries, and also discuss the class of Littlest Seesaw (LS) models corresponding to CSD(n) with n ≈ 3. The motivation is to study all the above symmetry based approaches, namely solar and atmospheric mixing sum rules and LS models, together in one place so that they may be compared, and to give an up to date analysis of the predictions of all of these possibilities, when confronted with the most recent global fits.All these approaches offer predictions for the cosine of the leptonic CP phase cos δ in terms of the mixing angles, θ 13 , θ 12 , θ 23 , which can be tested in forthcoming high precision neutrino experiments.In particular we study the solar neutrino mixing sum rules, arising from charged lepton corrections to TB, BM and GR neutrino mixing, and atmospheric neutrino mixing sum rules, arising from preserving one of the columns of these types of mixing, for example the first or second column of the TB mixing matrix (TM1 or TM2), and confront them with an up-to-date global fit of the neutrino oscillation data.We show that some mixing sum rules, for example all the atmospheric neutrino mixing sum rule arising from a Golden Ratio mixings are already excluded at 3σ a part from GRa2, and determine the remaining models allowed by the data.We also give detailed comparative results for the highly predictive LS models (which are special cases of TM1).These models are highly predictive with only two free real parameters fixing all the neutrino oscillation observables, making them candidates for being the most minimal predictive seesaw models of leptons still compatible with data.This is the first time that the three LS cases corresponding to CSD(n) with n = 2.5, n = 3 and n = 1 + √ 6 ≈ 3.45 have been studied together in one place, using the most up to date global fits.These three cases are predicted by theoretical models.In particular n = 3 was studied in a flavon model based on S 4 [22][23][24][25][26], n = 2.5 was introduced in the tri-direct CP approach based on the flavour symmetry S 4 × Z 5 × Z 8 [30], and n = 1 + √ 6 ≈ 3.45 derived in the modular symmetry framework with three S 4 groups [31][32][33][34].We also propose a new way of analysing these models, which allows accurate predictions for the least well determined oscillation parameters θ 12 , θ 23 and δ to be extracted.
The layout of the remainder of the paper is as follows.In Chapter 2 we introduce the notation for the PMNS matrix and discuss the symmetries of the leptonic Lagrangian.In Chapter 3 and 4 we introduce the atmospheric and solar sum rules for the different models we are studying and confront them with the up-to-date neutrino data global fit.We proceed in Chapter 5 discussing the CDS and the Littlest Seesaw model, showing its high predictivity and the viable parameter space given the experimental data and its fit.Finally we conclude in Chapter 6.

Lepton mixing and symmetries
The mixing matrix in the lepton sector, the PMNS matrix U PMNS , is defined as the matrix which appears in the electroweak coupling to the W bosons expressed in terms of lepton mass eigenstates.With the mass matrices of charged leptons M e and neutrinos M ν LL written as and performing the transformation from flavour to mass basis by the PMNS matrix is given by Here it is assumed implicitly that unphysical phases are removed by field redefinitions, and U PMNS contains one Dirac phase and two Majorana phases.The latter are physical only in the case of Majorana neutrinos, for Dirac neutrinos the two Majorana phases can be absorbed as well.
According to the above discussion, the neutrino mass and flavour bases are misaligned by the PMNS matrix as follows, where ν e , ν µ , ν τ are the SU (2) L partners to the left-handed charged lepton mass eigenstates and ν 1,2,3 are the neutrinos in their mass basis.Following the standard convention we can describe U PMNS in terms of three angles, one CP violation phase and two Majorana phases matrix is diagonal and we notice that for 3 generations we have that Z T 3 is a symmetry of the Lagrangian where T = diag 1, ω 2 , ω and ω = e i2π/3 .The light Majorana neutrino mass matrix is invariant under the Klein symmetry: This can be seen taking the diagonal neutrino mass matrix and performing the transformations and M ν is left invariant with where this result follows from the fact that, in the charged lepton mass eigenstate basis, the neutrino mass matrix is diagonalised by U PMNS as in Eq. (2.2), where any two diagonal matrices commute.Then Eq. (2.13) shows that the matrices S, U are both diagonalised by the same matrix U PMNS that also diagonalises the neutrino mass matrix.Given this result, we can always find the two matrices S, U for any PMNS mixing matrix, and hence the Klein symmetry is present for any choice of the PMNS mixing.However not all Klein symmetries may be identified with finite groups of low order.This description is meaningful if the charged leptons are diagonal (T is conserved) or approximately diagonal (T is softly broken).We are therefore interested in finite groups that are superset of Z U 2 × Z S 2 and Z T 3 and have a triplet representation.Groups of low order that satisfy these constraints are given in Figure 1.
One simple example is the group G = S 4 , of order 24, which is the group of permutation of 4 objects.The generators follow the presentation rules [12] The two possible S 4 triplet irreducible representations with a standard choice of basis [36], gives the generators explicit expression where again ω = e i2π/3 and the sign of the U matrix corresponds to the two different triplet representation.The group S 4 predicts a TB mixing [11], see Figure 2.This can be checked by the fact that S and U are diagonalised by U TB , see Eqs. (2.13).Another commonly used group is A 4 , which has two generators S and U that follow the same presentation rules as in Eq. (2.14) and in a standard basis [37], the generators have the same form as in Eq. (2.15).In order to explain the experimental results G needs to be broken and generate a non-zero (13) PMNS element.This will lead to corrections to the leading order PMNS predictions from the discrete group G.In Figure 3 we illustrate two possible direction we can proceed to do that.The first one is to break the T generator while the Klein symmetry in the neutrino sector is exact (left hand side).This means that the charged lepton matrix is approximately diagonal.In the mass basis we will have then a correction to the neutrino mixing matrix by a unitary matrix V e and the PMNS is now U PMNS =  13) PMNS element, one or more of the generators S, T, U must be broken.In the left panel we depict T breaking leading to charged lepton mixing corrections and possible solar sum rules.In the right panel, U is broken, while either S or SU is preserved leading to neutrino mixing corrections and atmospheric sum rules.
V † e V ν .Applying this to a group G will lead to solar sum rules.The second direction is to preserve Z T 3 but breaking Z U 2 while keeping either Z SU 2 or Z S 2 unbroken (right hand side).This leads to corrections to the prediction of G within the neutrino mixing and to atmospheric sum rules.It is convenient to introduce small parameters that can simplify the sum rules expressions and help us understand their physical behaviour since both in solar and atmospheric sum rules we implement a small deviation from the prediction of the exact finite discrete symmetries.We can consider the deviation parameters s, r, a [38] sin that highlight the differences from TB mixing.Given the latest fit the 3σ allowed range for the solar, reactor and atmospheric deviation are respectively −0.0999 < s < 0.0117, 0.20146 < r < 0.21855, −0.0985 < a < 0.1129. (2.17) This shows that the reactor angle differs from zero significantly (r ̸ = 0), but the solar and atmospheric angles remain consistent with TB mixing (s = a = 0) at 3σ.From a theoretical point of view, one of the goals of the neutrino experiments would be to exclude the TB prediction s = a = 0 [39], which is so far still allowed at 3σ.

Solar mixing sum rules
The first possibility to generate a non-zero reactor angle, whilst maintaining some of the predictivity of the original mixing patterns, is to allow the the charged lepton sector to give a mixing correction to the leading order mixing matrix U ν .This will lead to the socalled solar sum rules, that are relations between the parameters that can be tested.This operation is equivalent to considering the T generator of the S 4 symmetry which enforces the charged lepton mass matrix to be diagonal (in our basis) to be broken.When the T generator is broken, the charged lepton matrix is not exactly diagonal and it will give a correction to the PMNS matrix predicted by the symmetry group G.For example for the S 4 , U PMNS is not exactly U TB but it receives a correction that we will compute.The fact that S and U are preserved leads to a set of correlations among the physical parameters, the solar sum rules which are the prediction of the model.For the solar sum rules we can obtain a prediction for cos δ as we shall now show.
For example consider the case of TB neutrino mixing with the charged lepton mixing corrections involving only (1,2) mixing, so that the PMNS matrix in Eq. ( 2.3) is given by, The elements of the PMNS matrix are clearly related by [16,40] This relation is easy to understand if we consider only one charged lepton angle to be nonzero, θ e 12 then the third row of the PMS matrix in Eq. (3.1) is unchanged, so the elements U τ i may be identified with the corresponding elements in the uncorrected mixing matrix in Eq.(1.1).Interestingly, the above relation still holds even if both θ e 12 and θ e 23 are non-zero.However it fails if θ e 13 ̸ = 0 [41].The above relation in Eq. (3.2) can be translated into a prediction for cos δ as [40]   is the allowed region of the exact TB solar sum rules using the 3σ range of r (i.e. the deviation of sin θ 13 from the TB value), it is plotted in the 3σ range of s (i.e. the deviation of sin θ 12 from the TB value) and using the best fit value a = 0.071.The exact sum rules corresponds to Eq. (3.3).
Similarly blue band is the linearised sum rule allowed region which is given in Eq. (3.4).In the right panel the blue band is the second order expansion sum rule prediction, Eq. (3.5), it matches the exact sum rule. is the allowed region of the exact BM solar sum rules using the 3σ range of r (i.e. the deviation of sin θ 13 from the TB value), it is plotted in the 3σ range of s (i.e. the deviation of sin θ 12 from the TB value) and using the value a = −0.1.The exact sum rules corresponds to Eq. (3.3).Similarly blue band is the linearised sum rule allowed region which is given in Eq. (3.4).In the right panel the blue band is the second order expansion sum rule prediction, Eq. (3.5), it matches the exact sum rule.
but it does not describe adequately the exact sum rules as shown in the left panel of Figure 4. Therefore we can go to the second order expansion, which is and it matches the exact sum rule behaviour as seen on the right panel in Figure 4.
Similarly we can obtain higher order expansion for the other cases and check them against the data, like for the BM case showed in Figure 5.In this case we did not choose the best fit value for a because otherwise it would fall out of the physical range of cos δ since BM  is almost excluded by the data.The approximated expression for the sum rules can help us understand its behaviour and the dependence of cos δ on the other parameters that are in general non-linear and assess the deviation from the non-corrected PMNS mixing.We then expect for the exact sum rules a first order linear dependence on s.
In Figure 6 we present the exact sum rules prediction from Eq. (3.3) for TB, BM, GRa, GRb, GRc and HEX and the constraints from the fit of the neutrino oscillation data [35].We require cos δ to fall in the physical range −1 < cos δ < 1 and we present it in the y-axis.
In this section we discuss the second possibility, that is to have the T generator unbroken, therefore the charged lepton mixing matrix is exactly diagonal.In this case the correction to the PMNS matrix predicted from the group G comes from the neutrino sector and it provides a non zero reactor angle.For each group there are two possible corrections achieved either breaking U and preserving S or with S and U broken and SU preserved.Therefore for each discrete symmetry we will study two mixing pattern [43][44][45].
Let us consider again G = S 4 and the TB mixing in Eq. (1.3) as an example.If we break S and U but preserve SU the first column of the TB matrix is preserved and we have the so-called TM1 mixing pattern [46,47] if instead S is unbroken the second column is preserved and we have the second mixing pattern TM2 We can explicitly check this noticing that The red band is the allowed region of the exact TM2 sum rules using the 3σ range of r and a (i.e. the deviation of sin θ 13 and sin θ 23 from the TB value), and it corresponds to Eq. (4.8).The blue band is given by the linearised sum rule which is given in Eq. (4.10).On the right we zoom on the region −0.1 < a < 0.
meaning that the second column of the TB mixing matrix is an eigenvector of the S matrix.
Similarly for the first column with the SU matrix.In this second case where the second column of TB matrix is conserved we have The red band is the allowed region of the exact TM2 sum rules using the 3σ range of r and a (i.e. the deviation of sin θ 13 and sin θ 23 from the TB value), and it corresponds to Eq. (4.8).The blue band is given by the second order sum rule which is given in Eq. (4.11).On the right we zoom on the region −0.1 < a < 0.
For the other models the discussion is similar where we call X 1 and X 2 the atmospheric sum rules respectively derived by preserving the first and second column of the unbroken group with mixing X.In terms of the deviation parameters for TM2 we have the sum rule We can expand this expression for small deviation parameters and at the zero-th order we have [43] cos δ = − 2a r (4.10) and in Figure 7 we test this approximation against the exact sum rules using the experimental constraint in (2.9).We can see that given the updated data the linear approximation is now insufficient to describe the exact expression as it was instead in previous studies [43].
Similarly for TM1, as seen in Figure 8.This is true for the other model we will discuss later and therefore we provide the higher order expansions that agrees with the exact sum rule in Eq. (4.9) given the current data and is For the TM2 example we see in Figure 7 that the second order expansion is a good description of the exact sum rule.For TM1 instead, as shown in Figure 8 the third order expansion is needed.Since the second exact sum rules are quite involved having an approximated expression is of help to understand the physical meaning of it and to understand the difference with respect to the TB model.We present in Table 1 GRa2 cos δ = (1−tan 2 θ 23 ) csc θ(1−3 sin 2 θ 13 +(1+sin 2 θ 13 ) cos 2θ)   1.
We present with the blue band the exact sum rule prediction for TM2 for cos δ letting sin θ 13 vary in its 3σ range.In orange and purple we present the exact the sum rule predictions for GRa2 and TM1.The yellow and gray regions are respectively the 1σ range of sin θ 23 and cos δ, while the plot covers the whole 3σ range.
in orange and GRb1 in black.
In Figure 10 we show the exact atmospheric sum rules (Table 1) and the corresponding equations for other models that are still allowed from Figure 9.We plot cos δ against sin θ 23 and letting sin θ 13 vary in its 3σ range, this gives the width of the different bands, in yellow and gray respectively are the 1σ band for sin 2 θ 23 and cos δ.The GRb1 mixing do not appear in the plot because it lays in unphysical values of cos δ.In purple, blue and orange we present TM1, TM2 and GRa2.We can see that given the 1σ bands, the GRa2 mixing is favoured when considering normal ordering and without the SK data, since TM2 is allowed only on a small portion of the parameter space as shown in Figure 9.

Littlest Seesaw
There are many mechanism proposed to explain the smallness of the neutrino masses and that remain consistent with the data.For example the type I seesaw mechanism can address the problem through the introduction of heavy right-handed neutrinos.However in general it contains too many parameters to make any predictions for the neutrino mass and mixing.The constrained sequential dominance (CSD) model is a very predictive minimal seesaw model with two right-handed neutrinos and one texture zero [13,[21][22][23][24][25][26][27][28][29].As discussed in the Introduction the CSD(n) scheme assumes that the two columns of the Dirac neutrino mass matrix are proportional to (0, 1, −1) and (1, n, 2 − n) or (1, 2 − n, n) respectively in the RHN diagonal basis (or equivalently (0, 1, 1) and (1, n, n − 2) or (1, n − 2, n)) where the parameter n was initially assumed to be a positive integer, but in general may be a real number.For example the CSD(3) (also called Littlest Seesaw model) [22][23][24][25][26] can give rise to phenomenologically viable predictions for lepton mixing parameters and the two neutrino mass squared differences ∆m 2  21 and ∆m 2  31 , corresponding to special constrained cases of lepton mixing which preserve the first column of the TB mixing matrix, namely TM1 and hence satisfy atmospheric mixing sum rules.
The Littlest Seesaw (LS) mechanism is one of the most economic neutrino mass generation mechanism that is still consistent with the experimental neutrino data [22][23][24].We will show that after the choice of a specific n value, all the neutrino observables are fixed by two free parameters.Different values of n can be realised by different discrete symmetry groups.The LS introduces two new Majorana right-handed (RH) neutrinos N atm R and N sol R that will be mostly responsible for providing the atmospheric and solar neutrino mass respectively and the lightest SM neutrino is approximately massless; this is the idea of sequential dominance (SD) of RH neutrinos combined with the requirement for the N atm R ν e interaction to be zero [48].The Majorana neutrino mass matrix is given by the standard type I seesaw equation where the RH neutrino mass matrix M R is a 2 × 2 diagonal matrix where the convention for the heavy Majorana neutrino mass matrix corresponds to the Lagrangian term ) and the convention for the light Majorana neutrino mass matrix corresponds to the Lagrangian term − 1 2 ν L M ν ν c L as in Eq. (2.2) which follows after performing the seesaw mechanism in Eq. (5.1) [12]. 3he Dirac mass matrix in Left-Right (LR) convention is a 3 × 2 matrix with arbitrary entries where the entries are the coupling between the Majorana RH neutrinos and the SM neutrinos.The first column describe the interaction of the neutrinos in the flavour basis with the atmospheric RH neutrino and the second with the solar RH neutrino.The SD assumptions are that d = 0, d ≪ e, f , and these, together with the choice that of the almost massless neutrino to be the first mass eigenstate m 1 , leads to m 3 ≫ m 2 and therefore a normal mass hierarchy.This description can be further constrained choosing exactly e = f , b = na and c = (n − 2)a giving a simplified Dirac matrix that is called constrained dominance sequence (CSD) for the real number n [13,21,22].It has been shown that the reactor angle is [23] θ 13 ∼ (n − 1) therefore this can provide non-zero and positive angle for n > 1 and also excludes already models with n ≥ 5 since they do not fit the experimental value.The choice n ≈ 3 provides good fits to the data as we shall discuss.Following the literature we will refer to CSD(n) models with n ≈ 3 as Littlest Seesaw (LS) models [23].
The LS Lagrangian unifies in one triplet of flavour symmetry the three families of electroweak lepton doublets while the two extra right-handed neutrinos, N atm R and N sol R are singlets and reads [23] which can be enforced by a Z 3 symmetry and where ϕ atm and ϕ sol can be either Higgs-like triplets under the flavour symmetry or a combination of Higgses electroweak doublets and flavons depending on the specific choice of symmetry to use.In both cases the alignment should follow (5.9) We will refer to the first possibility in Eq. (5.8) as the normal case [22,23] and the second, in Eq. (5.9) as the flipped case [24].The predictions for n in the flipped case are related to the normal one by ) therefore we will discuss them together as one single n case.
There is an equivalent convention that can be found in the literature [33], where the alignment is chosen to be (5.12) that leads to the same results as the previous two cases respectively.In the neutrino mass matrix there will appear a (−1) factor that is only a non-physical phase that can therefore be neglected.In particular the case n = 1 + √ 6 that can be obtained with modular symmetry in [33] 4 is still n = 1 + √ 6 in our convention using the Eq.(5.8).Meaning that the case n = 1 − √ 6 is just the flipped of n = 1 + √ 6 and not a new LS model.We will follow the derivation in [23] and using Eq.(5.8) derive the flipped result with Eq. (5.10).We will consider LS models corresponding to CSD(n) models with n ≈ 3, in particular n = 2.5, 3 and 1 + √ 6 ≈ 3.45, together with their flipped cases.For the normal cases of CSD(n) the mass matrix in the diagonal charged lepton basis is given by where we used Eqs.(5.1), (5.2) and (5.5) and the only relevant phase is η = arg(a/e).At this point we notice that, in the diagonal charged lepton mass basis which we are using, the PMNS mixing matrix is fully specified by the choice of n and the parameters m b /m a and η.Indeed it is possible to derive exact analytic results for the masses and mixing angles [23], and hence obtain the LS prediction for the neutrino oscillation observables.We first observe that where the vector ( 2 3 , − 1 6 , 1 6 ) T is the first column of the TB matrix in Eq. (1.3) and is then an eigenvector of the neutrino mass matrix with eigenvalue 0 and it corresponds to the massless neutrino eigenstate.This means that for a generic n we get a TM1 mixing, Eq. (4.1), where the first column of the TB matrix is preserved and the other two can change.Therefore we can think of the LS as a special case of the atmospheric sum rules for the TB mixing.Since the atmospheric sum rules were derived only using the fact that the first column of the TB matrix is preserved all LS implementations also follow the TM1 sum rules in Eq. (4.1).Once we have noticed this it is clear that m ν can be block diagonalised using the TB matrix with (5.17) Finally we diagonalise m ν block to obtain a matrix of the form diag (0, m 2 , m 3 ) where the matrix including the phases are and the angle we use to diagonalise is with the angle being fully specified by the free parameters m b /m a and η, given by where and Recall that the PMNS matrix is the combination of the charged lepton and neutrino mixing matrices with The neutrino masses can be computed from m ν block and they are and after diagonalisation we can extract the eigenvalues as a function of the LS model parameters and finally For the CP phase δ we have the cosine sum rule that is the same as for the TM1 mixing in Table 1.This can be understood since the LS is a subset of TM1 as we noticed before when we showed that the first column of the TB matrix is an eigenvector of the LS neutrinos mass matrix.Notice that for the flipped case cos δ changes sign (because θ 23 → π − θ 23 ).Further information on the CP phase can be extracted from the Jarlskog invariant, which has been computed for the LS models [23,24]: where the negative sign corresponds to the normal case and the positive sign to the flipped.This leads to the sum rules for sin δ for the respective cases Notice that in this case the model is more predictive than the discrete symmetries and it predicts both sine and cosine fixing unambiguously the CP phase δ.Both sin δ and cos δ change sign going from the normal to the flipped cases meaning δ → π + δ as anticipated before.
The above analytic results emphasise the high predictivity of these models which, for a given choice of n, successfully predict all the nine neutrino oscillation observables (3 angles, 3 masses, 3 phases) in terms of three input parameters namely the effective real masses m a , m b and the phase η, which are sufficient to determine the neutrino mass matrix in Eq. (5.13), where these parameters appear in the above analytic formulas.However one neutrino mass is predicted to be zero (m 1 = 0), corresponding to a predicted normal hierarchy, so one Majorana phase is irrelevant.For the remaining seven observables (3 angles, 2 masses, 2 phases) the overall neutrino mass scale may be factored out, and the Majorana phase is hard to measure, so that in practice we shall focus on the five observables, namely the 3 angles θ 13 , θ 12 , θ 23 , the mass squared ratio ∆m 2  21 /∆m 2 31 = m 2 2 /m 2 3 and the CP violating Dirac phase δ, which are fixed by the two input parameters, the phase η and the ratio of the masses r = m b /m a , In practice, we shall take the two most accurately determined observables, ∆m 2  21 /∆m 2 31 and θ 13 to fix the input parameters η and r = m b /m a within a narrow range, resulting in accurate predictions for the remaining observables θ 12 , θ 23 and the Dirac phase δ.In addition we could add the input parameter n as a free parameter, but this, together with the constrained form of mass matrices, will eventually be determined by the flavour model.In particular successful LS model structure corresponding to CSD(n) can emerge from a theory of flavour as has been discussed in the literature for n = 3 [24], n = 2.5 [50] and more recently n = 1 + √ 6 ≈ 3.45 [30][31][32][33][34].In Figure 11 we consider the LS results for the above three cases with n ≈ 3 and the corresponding flipped cases, which are all realised successfully via S 4 symmetry [23].When we plot the experimental ranges of θ 13 and the mass squared ratio m 2 2 /m 2 3 in the r − η plane, it is clear that only two small allowed parameter regions are allowed, which determine the maximal and minimal values of r and η as the intersection of the blue and orange bands.Once we have the ranges of r and η for each value of n, thanks to the high  3: The LS predictions for n = 3 where the two most accurately measured observables, θ 13 and the mass squared ratio m 2 2 /m 2 3 , are used to accurately determine the two input parameters r = m b /m a = 0.100 ± 0.008 for two η ranges as shown above, corresponding to the centre panel of Fig. 11.This then leads to highly constrained predictions for the less accurately determined observables θ 12 , θ 23 and δ, which may be compared to the current experimental ranges as shown in the table.All results are given to 3σ accuracy.
predictivity of the model we can derive all the physical parameters and we can test them against the observed values.We do this for each value of n = 3, 1 + √ 6 ≈ 3.45 and 2.5 in Tables 3 to 5. We do not present the plot for the flipped cases since they are exactly the same.In fact they involve only the mass ratio and θ 13 .
In Table 3 we focus on the originally studied n = 3 and its flipped case.We present the theoretical prediction and its uncertainty coming from the allowed region in Figure 11 (centre panel) and the experimental bound.Since the theoretical prediction is exact given η and r we are allowing two significant figure for the theoretical errors.We notice that θ 12 and θ 23 fall well within the experimental range for all the cases and that even if δ is still not measured very precisely it allows us to exclude one of the two possible η both in the normal and flipped case.In fact only the η = 2.11 normal case and η = 4.17 flipped case The LS predictions for n = 2.5 where the two most accurately measured observables, θ 13 and the mass squared ratio m 2 2 /m 2 3 , are used to accurately determine the two input parameters r = m b /m a = 0.15 ± 0.01 for two η ranges as shown above, corresponding to the left panel of Fig. 11.This then leads to highly constrained predictions for the less accurately determined observables θ 12 , θ 23 and δ, which may be compared to the current experimental ranges as shown in the table.All results are given to 3σ accuracy.are within the 3σ experimental range.
In Table 4 we focus on n = 1 + √ 6 ≈ 3.45, which can be realised with a modular symmetry [33], we notice that for the normal case both η values are still allowed but with the δ prediction for η = 3.87 that lie at the edge of the allowed experimental range.For the flipped case instead η = 2.42 is excluded, thanks again to the bound on δ.As before, in going from n = 1 + √ 6 to the flipped only changes the sign of t in Eq. (5.21).The prediction for the mass ratio, θ 13 and θ 12 are independent of this sign while θ 23 and δ are affected by it, as we can see in Eqs.(5.26) and as discussed above for δ.The predictions are thus related by tan θ 23 → cot θ 23 (or θ 23 → π − θ 23 ) and δ → δ + π.
In Table 5 we focus on n = 2.5 and notice that, given the δ values, η = 4.7 is excluded for the normal case while for the flipped both η values are allowed.Finally, θ 23 lies in the higher and lower end of the experimental range respectively for the normal and flipped case making the n = 2.5 disfavoured given the current data.This case is also known in the literature as n = −1/2 using the convention in Eq. (5.12).But it is more consistent to refer to it as n = 2.5 in our notation.
In summary, we see that most of the LS models with n ≈ 3 are still allowed by current data.We have considered the cases n = 2.5 and n = 1+ √ 6 ≈ 3.45 and compared the results to n = 3 which was the originally proposed CSD(3).We emphasise the high predictivity of the LS models which have three input parameters describing nine neutrino observables.We have presented a new method here to present the results, namely to use the two most accurately measured observables, θ 13 and the mass squared ratio ∆m 2  2 /∆m 2 3 = m 2 2 /m 2 3 , to accurately constrain the two input parameters r = m b /m a and η.This then leads to highly constrained predictions for the less accurately determined observables θ 12 , θ 23 and δ, which can be tested by future neutrino oscillation experiments.Indeed already some of the possible LS cases are excluded by current data.In addition all these LS cases predict zero lightest neutrino mass m 1 = 0, with a normal neutrino mass hierarchy, and the neutrinoless double beta decay parameter m ββ equal to m b , which is just the first element of the neutrino mass matrix in Eq. (5.13).Indeed m ββ = m b can be readily determined from ∆m 2 2 = m 2 2 , but its value is too small to be measured in the near future so we have not considered it here.On the other hand, a non-zero measurement of m 1 or m ββ in the inverted mass squared ordering region would immediately exclude the LS models.

Conclusions
In the past decades many attempts have been made to explain the flavour structure of the PMNS matrix by imposing symmetry on the leptonic Lagrangian.These symmetries imply correlations among the parameters that are called sum rules.We have studied two types of sum rules: solar and atmospheric mixing sum rules.Then we have studied the littlest seesaw (LS) models which obey the TM1 atmospheric mixing sum rule but are much more predictive.The goal of this paper has been to study all these approaches together in one place so that they may be compared, and to give an up to date analysis of the predictions of all of these possibilities, when confronted with the most recent global fits.
In the case of solar mixing sum rules, the T generator of a given symmetry group is broken in the charged lepton sector in order to generate a non-zero reactor angle θ 13 .This leads with prediction for cos δ that can be tested against the experimental data.These in turn show a preference for GRa and GRb mixing while BM and GRc are constrained to live in a very small window of the parameter space of current data.Future high precision neutrino oscillation experiments will constrain solar mixing sum rules further as discussed elsewhere [40].
The atmospheric mixing sum rules instead come from either the breaking of both S and U in the neutrino sector while preserving SU or by breaking S and preserving U .In this case we have two relations among the parameters that can be tested.We noticed that only TM1, TM2 and GRa2 are still allowed by the neutrino oscillation data with a preference for GRa2 and with TM2 very close to be excluded.Future high precision neutrino oscillation experiments will constrain atmospheric mixing sum rules further as discussed elsewhere [43].
We have also considered the class of LS models that follow the constrained sequential dominance idea, CSD(n) with n ≈ 3. The LS models obey the TM1 atmospheric mixing sum rule, but have other predictions as well.We have compared the cases n = 2.5, n = 3 and n = 1 + √ 6 ≈ 3.45 which are predicted by theoretical models.These models are highly predictive with only two free real parameters fixing all the neutrino oscillation observables, making them candidates for being the most minimal predictive seesaw models of leptons still compatible with data.This is the first time that all three n values above, both normal and flipped cases, have been studied together in one place, using the most up to date global fits.We have also proposed a new way of analysing these models, which allows accurate predictions for the least well determined oscillation parameters θ 12 , θ 23 and δ which we have shown to lie in relatively narrow 3σ ranges, much smaller than current data ranges, but (largely) consistent with them, allowing these models to be decisively tested by future neutrino oscillation experiments, as has been discussed elsewhere [25].In our analysis we have ignored the model dependent renormalisation group (RG) corrections to LS models which have been shown to be generally quite small [51].
In conclusion, we have shown that the recent global fits to experimental data have provided significantly improved constraints on all these symmetry based approaches, and future neutrino oscillation data will be able to significantly restrict the pool of viable models.In particular improvements in the measurement of the leptonic CP violating Dirac phase δ will strongly constrain all these cases.This is particularly true in LS models which provide very precise theoretical predictions for δ, as well as θ 12 and θ 23 , consistent with current global fits.Future precision neutrino experiments are of great importance to continue to narrow down the choice of possible PMNS flavour models based on symmetry and lead to a deeper understanding of the flavour puzzle of the SM.

Figure 2 :
Figure 2: A schematic diagram that illustrate the way that the two subgroups Z U 2 × Z S 2 and Z T 3 of a finite group work in the charged lepton and neutrino sectors in order to enforce a particular pattern of PMNS mixing.In this example, the group S 4 leads to TB mixing.

Figure 3 :
Figure 3: In order to generate a non-zero (13) PMNS element, one or more of the generators

Figure 4 :
Figure 4: Solar mixing sum rule predictions for TB neutrino mixing.In both panels the red band

Figure 5 :
Figure 5: Solar sum rule predictions for BM neutrino mixing.In the both panels the red band

Figure 6 :
Figure6: Summary of exact solar sum rule predictions for different types of neutrino mixing.In the top left hand panel we present with the different colored band the sum rule prediction for TB for cos δ letting sin θ 12 vary in its 3σ range, the different color denoted different choice of sin θ 23 given in the legend, in its 3σ range and the width of the band is given by the 3σ range in sin θ 13 .The green and yellow band are the 1σ range for respectively cos δ and sin θ 23 .Similar plots for BM, GRa, GRb, GRc and HEX are presented respectively on the top right, center right, center left, bottom left, bottom right panels.The exact sum rules for the different models are derived from Eq. (3.3).

Figure 7 :
Figure7: The red band is the allowed region of the exact TM2 sum rules using the 3σ range of r and a (i.e. the deviation of sin θ 13 and sin θ 23 from the TB value), and it corresponds to Eq. (4.8).The blue band is given by the linearised sum rule which is given in Eq. (4.10).On the right we zoom on the region −0.1 < a < 0.

1 3 13 2 − 3s 2 13 .Figure 8 :
Figure8: The red band is the allowed region of the exact TM2 sum rules using the 3σ range of r and a (i.e. the deviation of sin θ 13 and sin θ 23 from the TB value), and it corresponds to Eq. (4.8).The blue band is given by the second order sum rule which is given in Eq.(4.11).On the right we zoom on the region −0.1 < a < 0.

2 (θ 23 )Figure 10 :
Figure10: Summary of exact atmospheric sum rule predictions which predict cos δ in terms of the other mixing angles for different types of lepton mixing corresponding to a preserved column of the PMNS matrix.The corresponding Eqs. are collected in Table1.We present with the blue band the exact sum rule prediction for TM2 for cos δ letting sin θ 13 vary in its 3σ range.In orange and purple we present the exact the sum rule predictions for GRa2 and TM1.The yellow and gray regions are respectively the 1σ range of sin θ 23 and cos δ, while the plot covers the whole 3σ range.

Figure 11 : 20 −
Figure 11: The results for the LS models with n ≈ 3. The input parameters η and r = m b /m a are constrained to a good degree of accuracy by only two experimental observables, namely θ 13 and the mass ratio m 2 2 /m 2 3.The 3σ allowed region for θ 13 and the mass ratio are respectively the blue and orange band.The area of intersection is the allowed parameter space for η and r.From the left to the right we assume, n = 2.5, 3 and 1 + √ 6 ≈ 3.45.
12 c 23 − c 12 s 13 s 23 e iδ c 12 c 23 − s 12 s 13 s 23 e iδ c 13 s 23 s 12 s 23 − c 12 s 13 c 23 e iδ −c 12 s 23 − s 12 s 13 c 23 e iδ c 13 c 23 Subgroups of SU (3) with triplet representations.The smaller of two groups connected in the graph is a subset of the other.Figure from [12].
the exact and approximated second

Table 1 :
Exact and approximated sum rules for the experimentally viable models, where θ = arctan1ϕ and ϕ = 1+ Summary of exact atmospheric sum rule predictions which predict the solar angle for different types of lepton mixing corresponding to a preserved column of the PMNS matrix, with only a mild dependence on the reactor angle.The corresponding Eqs. are collected in Table2.The pink, blue, red, orange and black curves are respectively the predictions for TM1, TM2, GRa1, GRa2 and GRb1 mixing patterns.The 3σ allowed region is in green.

Table 4 :
The LS predictions for n = 1 + √ 6 ≈ 3.45 where the two most accurately measured observables, θ 13 and the mass squared ratio m 2 2 /m 2 3 , are used to accurately determine the two input parameters r = m b /m a = 0.072 ± 0.004 for two η ranges as shown above, corresponding to the right panel of Fig.11.This then leads to highly constrained predictions for the less accurately determined observables θ 12 , θ 23 and δ, which may be compared to the current experimental ranges as shown in the table.All results are given to 3σ accuracy.