An Asymmetric TBM Texture

We construct a texture where the Seesaw matrix is diagonalized by the TriBiMaximal (TBM) matrix with a phase. All CKM and PMNS angles are within their pdg values, and the mass relations of quarks and charged leptons extrapolated to GUT scale are satisfied, including the Gatto relation. The novel ingredient is the asymmetry of the down-quark and charged lepton Yukawa matrices. Explaining the reactor angle requires a $\require{cancel} \cancel {CP}$ phase in the TBM matrix, resulting in the Jarlskog-Greenberg invariant at $|J|=0.028$, albeit with an undetermined sign. While $SO(10)$ restrains the right-handed neutrino Majorana matrix, the neutrino masses are left undetermined.


Introduction
The Standard Model evocates simplicity at smaller scales, both in gauge couplings (grand-unified theories), and in the mass patterns of down-quarks and charged leptons.Yet the mixings of leptons and quarks are starkly different: neutrino oscillations [1] require two large lepton mixing angles.
Quark-lepton mixing disparity, anticipated in the SOp10q-inspired seesaw mechanism [2], yields tiny neutrino masses through the ratio of electroweak to grand-unified scales.
A pretty matrix (with an ugly name Tri-Bi-Maximal) [3], diagonalizes the seesaw matrix.Its two large angles near their pdg values suggest a discrete crystalline flavor symmetry at GUT scales (see [4] for recent reviews).
TBM simplicity comes at a cost: the reactor angle θ 13 generated from the seesaw is zero.The PMNS lepton mixing matrix is an overlap of seesaw and charged lepton mixing matrices.The latter, derived from the charged lepton Yukawa matrix, may generate enough "Cabibbo Haze" [5] to explain the data, but symmetrical Yukawa matrices do not [6] generate enough Cabibbo haze to explain the reactor angle.Indeed, authors who assume symmetrical Yukawa matrices explain the data with seesaw diagonalization beyond TBM [7].
In this paper we argue for seesaw simplicity and seek textures where the value of the reactor angle [8] is fully explained by Cabibbo haze.
We construct asymmetric Yukawa matrices which satisfy all experimental constraints: the CKM matrix, the Gatto relation [9], the down quark to charged lepton mass relations at GUT scales , and the values of the three lepton mixing angles [10] with TBM diagonalization.Our bottom-up approach makes extensive use of the patterns suggested by the SU p5q and SOp10q grand-unified groups.
After systematically exploring different ways to introduce the asymmetry, a unique TBM texture with a simple asymmetry emerges; it fits all data only when a ✟ ✟ CP phase is added, with Jarlskog-Greenberg invariant [11] |J| " 0.028.Its sign comes from the seesaw; it is not specified by our texture.
After a review of the salient Yukawa patterns suggested by SU p5q and SOp10q, we discuss the symmetric Georgi-Jarlskog texture to motivate our procedure.We then construct our asymmetric texture.A discussion of its implications for further theoretical construction follows.The uniqueness of the asymmetry in explaining the data are extensively discussed in the appendices.

The Electroweak Sector
Quarks and charged lepton masses and mixings stem from the Standard Model's Yukawa matrices.We use the basis where the up-quark matrix is diagonal Y p2{3q " m t diagpǫ 4 , ǫ 2 , 1q.The down-quark and charged lepton Yukawa matrices are diagonalized by At GUT scales (10 15 GeV), simplicity emerges, with the diagonal mass matrices given by, up to signs, and m b " m τ .All entries are expressed in terms of λ, the tangent of the Cabibbo angle θ c .The Gatto relation tan θ c « a m d {m s , that links a mixing to a ratio of eigenvalues is explicit.Also from The large top and charm quark masses produce a steeper hierarchy with ǫ « λ 2 .
In SU p5q, the particles of each family are assigned to 5 " r d, pν e , eq s, 10 " r pu, dq, ū, ē s so that the up-quark masses reside in 10 ¨10 " 5s `50 s `45 a while the charged lepton and down-quark masses are in 5 ¨10 " 5 `45.
There are four Yukawa matrices Y 5, Y 5 , Y 45 , Y 45 , so that Y p´1{3q and Y p´1qT are related to Y 45 and Y 5 .
A simple combination of vacuum values due to Georgi and Jarlskog [12] yields the Gatto relation and all at Grand-Unified scales.
In SOp10q a right-handed neutrino s N is appended to each family, fitting in its spinor representation 16 " 5 `10 `1.Masses are generated by three couplings since 16 ¨16 " 10 s `126 s `120 a , and three Yukawa matrices Y 10 , Y 126 , Y 120 .The new features are -a p∆I w " 0q Majorana mass matrix M with couplings M ¨s N s N -a p∆I w " 1 2 q Yukawa matrix Y p0q for neutrino Dirac masses.The simplest BEH SOp10q coupling yields Y p2{3q " Y p0q .
The resulting mass structures are summarized in the following table

Masses
SOp10q Ą SU p5q ˆU p1q 10 126 120 We end this general analysis of the electroweak input to the flavor jungle with a brief discussion of a symmetric texture that shows its inconsistency with TBM mixing.

The Georgi-Jarlskog Symmetric Texture
It is a bottom-up approach that relies heavily on the grand-unified structures evocated by the Standard Model.All parameters are expressed à la Wolfenstein in terms of the Cabibbo angle λ.
In SU p5q thre are two types of BEH Yukawa couplings, 5 and 45.In appendix A, we derive their form where the prefactors a, b, c, g " Op1q.The down-quark and charged lepton couplings follow, The prefactors (neglecting for now the ✟ ✟ CP phase) are identified with the Wolfenstein parameters [13], a " in such a way as to reproduce the CKM matrix, the Gatto relation and the GUT scale mass ratios Eq.( 4).
Since the Yukawa matrices are symmetric, the mixing matrix of the lefthanded charged lepton is closely related to the CKM matrix, according to SU p5q.The lepton mixing angles of the PMNS matrix are now extracted, assuming TBM seesaw diagonalilzation where, neglecting the ✟ ✟ CP phase, Here c ij " cos θ ij and s ij " sin θ ij .The use of Eq.( 8) and ( 9) yields, one third of its pdg value 0.145 1 .Symmetric Yukawa matrices and TBM seesaw diagonalization are incompatible with data.

Asymmetric Textures
Seesaw TBM diagonalization requires asymmetric couplings in the input Yukawa matrices.Eq. (11)  describes mixing between the two lightest families that is already large, so that increasing U p´1q 31 is most likely to yield the desired effect in θ 13 .
The link between U p´1q and the CKM matrix of Eq.( 8) must be loosened.This readily occurs for asymmetric matrices, with unknown V p´1{3q .
1 The addition of the CKM phase will give an Opλ 5 q correction.
The asymmetry may be in the 45 and/or 5 couplings.
-The analysis of appendix A indicates that the 45 coupling in p22q position of Y 45 leads us to the correct mass ratios and CKM angles at GUT scale, as in the Georgi-Jarlskog construction.45 couplings in different places fail in one way or another, in particular for symmetric or antisymmetric off-diagonal couplings.
-The asymmetry is in the 5 coupling.
Asymmetries split into three generic cases, along the p12q ´p21q, p23q ´p32q and p13q ´p31q axes.Assume for simplicity that it appears in only one.In appendix B we show that an asymmetry along p12q ´p21q or p23q ´p32q does not alleviate the θ 13 deficiency.The asymmetry must then reside in the p13q ´p31q axis of the 5 couplings.
To make it as large as possible, we insert a term of Opλq in the 31 position, where now a, b, c, d, g are Op1q prefactors.Note that the 11 term is explicitly inserted as it is of lower order than in the symmetric case, and to make the determinant with cofactor in the 22 position vanish.The Yukawa determinant equality Eq.( 3) is now satisfied.The Yukawa matrices of the down-quarks and charged leptons follow, The prefactors are expressed in terms of the Wolfenstein parameters so as to reproduce the CKM matrix, the GUT scale mass ratios, and the Gatto relation, The new charged lepton mixing matrix, has extra elements of Opλq, which bring the reactor angle to a new value that is above its pdg value by 2.26 ˝2.
2 An asymmetry of Opλ 2 q leaves the reactor angle well below its pdg value.
The other two lepton mixing angles are also off their pdg values, θ 12 " 39.81 ˝p6.16 ˝above pdgq, θ 23 " 42.67 ˝p2.90 ˝below pdgq (19) The distinguishing feature of this asymmetry is a reactor angle above its experimental value.The addition of a ✟ ✟ CP phase [14] in the TBM matrix can be used to lower [15] θ 13 to its pdg value.
What makes this particular texture noteworthy is that by lowering the reactor angle to its experimental value, we not only find an amount of CP-violation that is consistent with experiment, but also align both solar and atmospheric angles to their pdg values.
We do not need to include the Majorana phases [16] which enter only in total lepton-number violating physics.Of the many ways to insert phases in the TBM matrix, we choose Neglecting the CKM phase, The value of θ 13 is lowered by the TBM phase3 to, or in terms of the Wolfenstein parameters, We fit θ 13 to its central pdg value by using Eq.( 23), and find A straightforward computation yields the remaining PMNS angles, θ 12 " 34.16 ˝p0.51 ˝above pdgq, θ 23 " 44.91 ˝p0.66 ˝below pdgq (25) to be compared with Eq.( 19).One phase, consistent with the Jarlskog-Greenberg invariant's pdg absolute value, reconciles all lepton mixing angles with experiment.
A numerical summary of the texture can be found in Appendix C

Theoretical Outlook
The asymmetric TBM texture we just constructed provides an experimentally successful link between the electroweak Yukawa matrices and the seesaw scale Majorana mass matrix of the right-handed neutrinos.Both structures present new theoretical patterns which we briefly address below.

Yukawa Couplings
The crucial ingredient is an asymmetric Opλq term in the 31 element of the SU p5q quintet Yukawa matrix Y 5.
It can arise from the vacuum value of one BEH boson H 5, with the symmetric and antisymmetric couplings canceling (adding) in the 13 (31) positions.However this is not technically natural in the absence of further symmetries.
One simple remedy is to introduce two BEH bosons H 5 and H 1 5, with a Z 2 exchange symmetry H 5 ÐÑ H 1 5.This insures equality between the symmetric and antisymmetric couplings.The desired cancellation occurs when the two vacuum values respect the Z 2 symmetry.
The next step is to single out the ( 13)-(31) axis in the Yukawa matrix.One can simply add only this specific coupling to the Lagrangian, or seek a symmetry-based explanation which points those BEH bosons in the right flavor direction.
A possible understanding appears natural with a T 7 discrete symmetry [17]: the three families form a T 7 triplet, and both H 5 and H 1 5 transform as an antitriplet.The details are beyond the scope of this paper and will be discussed elsewhere.

Seesaw Sector
In the TBM texture, the seesaw neutrino mass formula becomes where M is the Majorana mass matrix of the right-handed neutrinos, and D ν " diagpm 1 , m 2 , m 3 q is the diagonal light neutrino mass matrix.The numerator Y p0q is the neutral lepton Dirac Yukawa matrix which, in SOp10q, is most simply related to the up-quark Yukawa matrix Y p0q " Y p2{3q .
Y p0q inherits the large hierarchy of the up-quark sector4 .This hierarchy is not replicated by the light neutrino data, and Eq.( 26) implies a correlated squared ǫ hierarchy in the Majorana matrix.We therefore separate out the hierarchy from the Majorana matrix By using Eq.( 26), we can express the Majorana matrix in terms of neutrino masses and the ✟ ✟ CP-phase, The light neutrino masses are not yet known, although they are bounded by cosmology [18] and oscillations, m 1 ď 71.17 with 58.9 ď m 1 `m2 `m3 ď 230, pmeVq, for the normal hierarchy.
Eq.(28) yields It depends on the phase and its sign, although its matrix elements are not yet fixed by experiment.Eq.(30) shows that less than one order of magnitude improvement on the cosmological bound will (hopefully soon) result in an actual measurement.
It is a challenge to theories to predict the neutrino masses.For example, all it takes is a Gatto-like relation between the solar angle and m 1 {m 2 [19] to determine that physics.

Conclusion
This paper has presented a grand-unified asymmetric texture for the Yukawa matrices of the Standard Model.It is designed to reproduce the CKM angles, the Gatto relation and the GUT scale relations between down-quark and charged lepton masses.
Here neutrino masses are generated by the seesaw.In the belief that gauge simplicity at GUT scale should be matched by "seesaw simplicity" where only large angles appear, we assume TBM diagonalization of the seesaw neutrino matrix.Seesaw simplicity requires the small PMNS reactor angle θ 13 to be generated through the charged lepton mixings.
Symmetric electroweak textures fall short of "seesaw simplicity".However, in this asymmetric texture the reactor angle θ 13 exceeds its pdg value, while the charged lepton mixing contribution to the solar and atmospheric angles yield values outside their pdg allowances.

A ✟ ✟
CP-phase in the TBM matrix reduces the reactor angle value and affects the solar and atmospheric angles; it is noteworthy that it provides one solution for three problems: -A ✟ ✟ CP-phase with δ " ˘78 ˝lowers θ 13 to its experimental value.This corresponds to the PMNS phase of either 1.32π or 0.68π and Jarlskog-Greenberg invariant J " ¯0.028, which is consistent with present experimental bounds.
-The very same ✟ ✟ CP-phase adjusts the solar and atmospheric neutrino angles to within one degree of their pdg values.
The sign of the phase is a property of the Majorana mass matrix of the right-handed neutrinos, and not determined by the texture.
We expect that the electroweak side of our texture can be applied to Golden Ratio [20] seesaw diagonalization as well.The next step is to find a common organizing principle that relates the seesaw Majorana matrix to the Standard Model Yukawa matrices.We hope to address this question in a future work.

A Symmetric-Antisymmetric Textures
We first consider textures with only symmetric and/or antisymmetric 5 and/or 45 couplings.Our objective is to find out textures that can reproduce mass relations and mixing angles between down-quarks and charged leptons.
For simplicity, -consider all couplings are real -let a single parameter c 1 denote one diagonal or a pair of off-diagonal 45 coupling(s), all other couplings in 5, denoted by a 1 , b 1 , d 1 , g 1 etc.Off-diagonal symmetry/antisymmetry is denoted by sign parameters ς a 1 " ˘1 etc.All couplings will be expressed in integer powers of the Wolfenstein parameter λ with a prefactor: a 1 " aλ n etc.
-taking hint from m b « m τ at GUT scale, the p33q coupling is assumed to be 5 and all other couplings are normalized by this.
An important observation is, det Y p´1{3q should be independent of c 1 so that it approximates det Y p´1q at GUT scale.
Classify these textures as: (i) 45 couplings in off-diagonal entries, (ii) 45 coupling in diagonal entry.

Off-diagonal 45 couplings
Consider a pair of off-diagonal 45 couplings, either symmetric or antisymmetric.There can be three such textures.
p12q ´p21q 45 texture This can not be made independent of c 1 ; thus this texture cannot yield correct mass relations and will not be pursued further.
The eigenvalues of Y p´1{3q Y p´1{3qT are the mass-squared of the down-quarks: m 2 d , m 2 s and m 2 b .These are related by Interestingly, Eqs.(A.8)-(A.10)do not contain any sign ambiguity, therefore, irrespective of sign, we derive Eigenvalues of Y p´1q Y p´1qT , labelled by m 2 e , m 2 µ , m 2 τ , can be derived from those of Y p´1{3q Y p´1{3qT by replacing c Ñ ´3c.This predicts m 2 d `m2 s " m 2 e `m2 µ from Eq.(A.12), which is unsatisfactory.Therefore, this texture with off-diagonal 45 couplings in p13q´p31q position doesn't yield correct masses for charged leptons and down-quarks.
p23q ´p32q 45 texture Proceeding as the previous case, this texture has the following form subject to the constraints pς c 1 , ς b 1 , ς d 1 q " p1, ˘1, ¯1q or p´1, ˘1, ˘1q.Here c, d " Op1q and g, b À Op1q.Solving the eigenvalues of Y p´1{3q Y p´1{3qT yields This leaves The dominant term in Eq.(A.15) is g 2 λ 4 .This suggests that m 2 s « g 2 λ 4 " λ 4 {9 at GUT scale, with g " 1{3.Then, for Y p´1q , we will derive m 2 µ « g 2 λ 4 " λ 4 {9, much smaller than the expected value λ 4 at GUT scale.This shows that the off-diagonal 45 in the p23q ´p32q position also fails to generate the correct mass relations.

Diagonal 45 coupling
Next, we discuss textures with a single 45 coupling in in either p11q or p22q position of Y p´1{3q .(11) 45 texture where b, g " Op1q and a, c À Op1q.
Solving the eigenvalue equations of Y p´1{3q Y p´1{3qT gives irrespective of signs of prefactors.Since a, c À Op1q, Eq.(A.18) is unable to produce m s " λ 2 {3 at GUT scale.This will, in turn, predict lower mass of m µ .Therefore, this texture cannot generate correct masses for down-quarks and leptons.
(22) 45 texture An analysis parallel to the p13q ´p31q 45 texture results in the following form where b, g " Op1q and a, c À Op1q.
Irrespective of the sign of prefactors, this texture produces the same mass relations and mixing angles as the Georgi-Jarlskog texture discussed in section 2.
It should be noted that the PMNS angles θ 23 and θ 12 are not too far off from pdg values in this texture.
The above discussion of this appendix shows that the texture with 45 coupling in the p22q position can, unlike the others, generate the correct mass relations.This implies that the asymmetry lies in the 5 couplings.

B Asymmetric Textures
Following appendix A, we discuss how to introduce asymmetry in the 5 couplings.Decomposing the charged lepton diagonalizing matrix into rotation matrices U p´1q " R 23 pφ 23 q R 13 pφ 13 q R 12 pφ 12 q we recall that in the symmetric texture, , where b " A a ρ 2 `η2 .
Asymmetry can be incorporated by changing these relationships.
For simplicity, let's change one angle at a time, and inspect how the PMNS matrix is affected.
• If we only change φ 12 , θ 13 can be fitted to its pdg value.But then θ 12 is very far away from experiment (7.24 ˝or 11.06 ˝).
• If we only change φ 13 , θ 13 can be fitted to experiment.Choosing φ 13 to be in the first quadrant, θ 12 and θ 23 deviate much less from their pdg values.
None of these seem particularly correct, although the third one looks more promising.
Another way of looking at this phenomenon is to go directly to the Yukawa matrices.There are three generic asymmetries in the Yukawa matrices, and as we will see that changing φ 13 is connected to a particular type, in which Y

Asymmetric Yukawa matrices
For simplicity, consider one asymmetry at a time.
Diagonalizing Y p´1{3q Y p´1{3qT , we find that the Cabibbo angle is given by a c λ m´2 , and the mass squared of down quark is approximately Fitting these to the correct order of λ requires m " n " 3. Now the Yukawa matrix becomes where a ‰ a 1 .Comparing the eigenvalues and eigenvectors of Y p´1{3q Y p´1{3qT with pD p´1{3q q 2 and U CKM , respectively, yield The lepton masses are acquired by c Ñ ´3c.
The eigenvectors of Y p´1q Y p´1qT generate U p´1q , which, with TBM seesaw matrix, yields where a, b, c, g, g 1 " Op1q.
Comparing the eigenvalues and mixing matrix of Y p´1{3q Y p´1{3qT to GUT scale down-quark masses and U CKM yields In the charged lepton sector, diagonalizing Y p´1q Y p´1qT , we get U p´1q , together with TBM seesaw diagonalization which yields still one third of the experimental value, no matter what values g 1 and n take.
p13q ´p31q asymmetry In this texture m " 3 is fixed by the (13) angle of CKM.Furthermore, it can be shown that if n ą 1, the reactor angle is again one third of its pdg value.
Considering m " 3 and n " 1, this texture has been discussed in detail in section 3.
None of these three types of asymmetries yields satisfactory values for the PMNS angles.However there are important differences.
In the first two cases, when the asymmetries are along p12q ´p21q or p23q ṕ32q, the reactor angle is much lower than its pdg value.In these cases TBM diagonalization does not agree with experiment, unless we deviate from it by introducing a new parameter.However, when the asymmetry is along p13q ´p31q, θ 13 is larger than its experimental value.As showed in section 3, introducing a phase in TBM reduces θ 13 , while bringing the other two angles even closer to their pdg central values.