Weak Basis Transformations Research Articles

Probing quark mass matrices in the flavor basis

In this paper, structural features of quark mass matrices have been analyzed in the flavor basis which is allowed by weak basis transformations in the Standard Model. Using the latest precised data pertaining to quark mixing matrix and small quark masses, we find that the texture structure of these cannot be simplified further. However, we find enormous amount of simplification is possible wherein we have found that a single phase in the mass matrix is adequate to accommodate the CP violating phase of CKM matrix.

CKM mixings from mass matrices with five texture zeros

In this work we carry out an exhaustive study to find quark mass matrices in the Standard Model~(SM), with the maximum number of texture zeros consistent with the experimental data. We found four viable configurations of five texture zeros that adjust the quark masses, the mixing angles and the CP violation phase, with deviations below $1\sigma$ level respect to the current SM best fit values. One of the most important aspects of this work is an economic procedure to find the texture zeros: we resort to the weak basis transformation method, which, as we will show, exhaustively search every possible configuration. We report various leading order relations between the mixing angles and the quark masses for each case.

DsixTools 2.0: the effective field theory toolkit

DsixTools is a Mathematica package for the handling of the standard model effective field theory (SMEFT) and the low-energy effective field theory (LEFT) with operators up to dimension six, both at the algebraic and numerical level. DsixTools contains a visually accessible and operationally convenient repository of all operators and parameters of the SMEFT and the LEFT. This repository also provides information concerning symmetry categories and number of degrees of freedom, and routines that allow to implement this information on global expressions (such as decay amplitudes and cross-sections). DsixTools also performs weak basis transformations, and implements the full one-loop Renormalization Group Evolution in both EFTs (with SM beta functions up to five loops in QCD), and the full one-loop SMEFT-LEFT matching at the electroweak scale.

Five texture zeros for Dirac neutrino mass matrices

In this work, we propose new five texture zeros for the mass matrices in the lepton sector in order to predict neutrino masses. In our approach, we extend beyond the standard model (SM) by assuming Dirac masses for the neutrinos, a feature which allows us to make a theoretical prediction for the lightest neutrino mass in the normal ordering. The textures that were analyzed have enough free parameters to adjust the V PMNS mixing matrix including the CP-violating phase, the neutrino mass squared differences , and the three charged lepton masses. In order to obtain reliable results, we used two different procedures that are based on the weak basis transformation, a well known technique to analyze textures and their implications for flavor physics. The first method was based on a least-squares analysis to theoretically fit the lepton masses and the mixing parameters to their corresponding experimental values; for this case, the best fit obtained for the lightest neutrino mass was eV. The second approach was algebraic, resulting in a lightest neutrino mass consistent with the experimental values and the restrictions arising from the five texture zeros of the mass matrices, and the lightest neutrino mass obtained was (3.5 ± 0.9) × 10−3 eV.

Neutrino mass matrices with three or four vanishing cofactors and nondiagonal charged lepton sector

We investigate the texture structures of lepton mass matrices with four (five) non-zero elements in the charged lepton mass matrix and three (four) vanishing cofactors in the neutrino mass matrix. Using weak basis transformations, all possible textures for three and four vanishing cofactors in $M_{\nu}$ are grouped into 7 classes and predictions for the unknown parameters such as the Dirac CP violating phase and the effective Majorana mass for the phenomenologically allowed textures have been obtained. We, also, illustrate how such texture structures can be realized using discrete Abelian flavor symmetries.

Texture zero mass matrices and CP violation

Within the Standard Model, using the facility of making Weak Basis transformations, attempt has been made to examine the most general mass matrices within the texture zero approach. For the case of quarks, interestingly, one finds a particular set of texture four zero quark mass matrices emerging out to be a unique viable option for the description of quark mixing data as well as for accommodation of CP violation. Similarly, general lepton mass matrices, essentially considered as texture zero mass matrices, yield interesting bounds on the CP violating Jarlskog’s rephasing invariant parameter in the leptonic sector.

Implications of general lepton mass matrices in the standard model on mee

Within the framework of the standard model (SM), using the facility of weak basis (WB) transformations, the general Dirac neutrino mass matrix and the charged lepton mass matrix can essentially be considered as texture two zero mass matrices. Using type I seesaw formula for Majorana neutrino mass matrix, our analysis yields lower bounds ${m}_{ee}\ensuremath{\gtrsim}0.001\text{ }\text{ }\mathrm{eV}$ for normal mass ordering and ${m}_{ee}\ensuremath{\gtrsim}0.08\text{ }\text{ }\mathrm{eV}$ for inverted mass ordering, the latter being tantalizingly close to the expected outcome of the ongoing experiments. Interestingly, for inverted mass ordering, ${m}_{ee}$ is largely independent of variation of mass ${m}_{3}$, whereas, for normal mass ordering with ${m}_{1}$ in the range 0.0001 eV--0.01 eV, the bound on parameter ${m}_{ee}$ gets further sharpened and one obtains ${m}_{ee}$ within the band 0.014--0.042 eV. Further, noting that a particular set of texture four zero quark mass matrices has been shown to be a unique viable option for the description of quark mixing data, an analysis of similar mass matrices in the lepton sector has also been carried out to obtain bounds for the parameter ${m}_{ee}$ with interesting consequences.

More about unphysical zeroes in quark mass matrices

We look for all weak bases that lead to texture zeroes in the quark mass matrices and contain a minimal number of parameters in the framework of the standard model. Since there are ten physical observables, namely, six nonvanishing quark masses, three mixing angles and one CP phase, the maximum number of texture zeroes in both quark sectors is altogether nine. The nine zero entries can only be distributed between the up- and down-quark sectors in matrix pairs with six and three texture zeroes or five and four texture zeroes. In the weak basis where a quark mass matrix is nonsingular and has six zeroes in one sector, we find that there are 54 matrices with three zeroes in the other sector, obtainable through right-handed weak basis transformations. It is also found that all pairs composed of a nonsingular matrix with five zeroes and a nonsingular and nondecoupled matrix with four zeroes simply correspond to a weak basis choice. Without any further assumptions, none of these pairs of up- and down-quark mass matrices has physical content. It is shown that all non-weak-basis pairs of quark mass matrices that contain nine zeroes are not compatible with current experimental data. The particular case of the so-called nearest-neighbour-interaction pattern is also discussed.

Constraining the texture mass matrices

The present work attempts to provide an overview of texture specific mass matrices, which, along with Weak Basis (WB) transformations, emerge as a viable framework for explaining the fermion masses and mixing data in the quark as well as lepton sector.

Fermion mass matrices, textures and beyond

The issue of texture specific fermion mass matrices have been examined briefly from the “bottom-up” perspective. In case no conditions are imposed, the texture ansatze leads to a large number of viable possibilities. However, besides textures, if in case one incorporates the ideas of “natural mass matrices” and uses the facility of Weak Basis Transformations, then one is able to arrive at a minimal finite set of viable mass matrices in the case of quarks.

Seeking Texture Zeros in the Quark Mass Matrix Sector of the Standard Model

Here we show that the Weak Basis Transformation is an appropriate mathematical tool that can be used to find texture zeros in the quark mass matrix sector of the Standard Model. So, starting with the most general quark mass matrices and taking physical data into consideration, it is possible to obtain more than three texture zeros by any Weak Basis transformation. The general quark mass matrices considered, were obtained through a special weak basis transformation wherein the up-quark mass matrix (down-quark mass matrix ) has been taken to be diagonal and the down-quark mass matrix (up-quark mass matrix), in general form, includes the Cabibbo-Kobayashi-Maskawa matrix.

Finding a unique texture for quark mass matrices

Within the Standard Model, starting with the most general mass matrices, we have used the facility of making weak basis transformations and have imposed the condition of ``naturalness'' to carry out their analysis within the texture-zero approach. Interestingly, our analysis reveals that a particular set of texture-4 zero quark mass matrices can be considered to be a unique viable option for the description of quark mixing data.

Comment on “Texture zeros and weak basis transformations in the quark sector of the standard model”

Recently, using specific weak basis transformations, Y. Giraldo [Phys. Rev. D 86, 093021 (2012)] constructed some texture 5 and 4 zero quark mass matrices and examined their compatibility with the quark mixing data. In this comment, we have reanalyzed these to bring forth certain important issues regarding their viability which need to be taken note of.

Reply to “Comment on ‘Texture zeros and weak basis transformations in the quark sector of the standard model”’

We respond to the comments on my recent paper, Y. Giraldo [Phys. Rev. D 86, 093021 (2012)]. In their comments, they begin discussing a particular class of five-zero texture non-Fritzsch-like quark mass matrix, which was proposed by me, and question its validity. Then, they put in doubt the phases included in the unitary matrices used for diagonalizing the quark mass matrices, from which they claim that the $CP$ violating parameter ${\ensuremath{\epsilon}}_{k}$ obtained does not agree with the experimental value. Because of these comments, finally, they recommend being careful while analyzing the implications of weak basis transformations on textures. Other minor points are also discussed by them. In the following, I will show that the mentioned five-zero texture non-Fritzsch-like quark mass matrices are completely valid and generate all the physical quantities involved, including the $CP$ violating parameter ${\ensuremath{\epsilon}}_{k}$, for which is required the inclusion of phases in the unitary matrices used for diagonalizing the quark mass matrices in order to put the Cabibbo-Kobayashi-Maskawa matrix in standard form. These phases can be rotated away so they do not have any physical meaning. Finally, the relevance of the weak basis transformation is appreciated: it is complete and facilitates calculations, especially of texture zeros.

Clues towards unified textures

The issue of texture specific mass matrices has been discussed by incorporating Weak Basis transformations and the concept of "naturalness." Interestingly, we find that starting from the most general mass matrices, one can arrive at texture four zero mass matrices which can fit both quark as well as lepton mixing data and are similar to the original Fritzsch ansatze.

Parallel texture structures with cofactor zeros in lepton sector

In this paper we investigate the parallel texture structures with cofactor zeros in the charged lepton and neutrino sectors. The textures cannot be obtained from arbitrary leptonic matrices by making weak basis transformations, which therefore have physical meaning. The 15 parallel textures are grouped as 4 classes where each class has the same physical implications. It is founded that one of them is not phenomenologically viable and another is equivalent to the texture zero structures extensively explored in previous literature. Thus we focus on the other two classes of parallel texture structures and study the their phenomenological implications. The constraints on the physical variables are obtained for each class, which is essential for the model selection and can be measured by future experiments. The model realization is illustrated in a radiated lepton mass model.

Lower bound on neutrino mass and possibleCPviolation in neutrino oscillations

The phenomenology of the most general lepton mass matrices obtained through weak basis transformations has been discussed. Using a hierarchical parametrization of these mass matrices, the exact relations for lepton mixing angles have been obtained assuming normal neutrino mass hierarchy and natural structures of lepton mass matrices. The recent three neutrino oscillation data together with the condition of naturalness on the Dirac lepton mass matrices provide a lower bound on the lightest neutrino mass of ${\mathrm{m}}_{\mathrm{v}1}>3\text{ }\text{ }\mathrm{meV}$ along with a nonvanishing Dirac $CP$ violating phase.

Minimal weak basis textures and quark mixing data

The phenomenology of the most general quark mass matrices with a minimum of three texture zeros obtained through weak basis (WB) transformations has been discussed within the framework of the standard model. The relationship between the textures, naturalness and WB transformations has also been studied. Adopting a ‘hierarchical’ parameterization for the quark mass matrix, we have detailed exact relations for the various elements of the quark mixing matrix along with the CP asymmetry angle β, illustrating the explicit relationship between the observed hierarchies among the WB quark mass matrices and the quark mixing matrix. Besides exploring the complete parameter space available to the (Mq)11, (q = u, d) elements in these mass matrices, their implications for quark mixing data and CP violation asymmetries have also been explored.

Quark sectorCPviolation of the universal seesaw model

We study the CP violation of universal seesaw model, especially its quark sector. The model is based on SU(2)_L \times SU(2)_R \times U(1)_{Y^\prime}. In order to count the number of parameters in quark sector, we use the degree of freedom of weak basis transformation. For N(3)-generation model, the number of CP violating phase in quark sector is identified as 3N^2-3N+1 (19). We also construct nineteen CP violating weak basis invariants of Yukawa coupling matrices and SU(2) singlet quark mass matrices in the three-generation universal seesaw model. The quark interaction terms induced by neutral currents are given as an exact formula. Both of the charged current and the neutral current are expressed in terms of the mass basis by finding the transformations from weak basis to mass basis. Finally, we calculate the mixing matrix element approximately assuming that the SU(2)_R breaking scale v_R is much larger than the electro-weak breaking scale v_L.

Texture zeros and weak basis transformations in the quark sector of the standard model

Stimulated by the recent attention given to the texture zeros found in the quark mass matrices sector of the Standard Model, an analytical method for identifying (or to exclude) texture zeros models will be implemented here, starting from arbitrary quark mass matrices and making a suitable weak basis transformation, we are be able to find equivalent quark mass matrix. It is shown that the number of non-equivalent quark mass matrix representations is finite. We give exact numerical results for parallel and non-parallel four-texture zeros models. We find that some five-texture zeros Ans$\ddot{\textrm{a}}$tze are in agreement with all present experimental data. And we confirm definitely that six-texture zeros of Hermitian quark mass matrices are not viable models anymore.