As shown in Appendix B, due to the unitarity of UR and the special form of the MR mass matrix in Eq. By multiplying this clear form of the phase matrix P " we obtain a mixing matrix that conforms to standard PDG phase conventions. It is useful to compare the mixing matrix TM in Eq. 4.10) with a general parametrization of the mixing matrix PMNS in terms of deviations from the TB mixture [25],.
Consider the size of the quark mixing matrix (CKM) elements [6] and also the neutrino mixing matrix (PMNS) elements [7]. There may be a relationship between the mass values and the values of the mixing matrix elements, but so far no relationship exists other than simple numerology. Because it is not part of the underlying family symmetry, it is therefore clear that the U-symmetry of Eq.
Breakdown of the leading order structure can occur in the charged lepton or neutrino sector. The first involves the charged lepton corrections of simple leading-order mixing patterns that result in the solar mixing sum rules, as discussed in Subsection 3.5. If there is a break in the neutrino sector, it is possible to break one or both of the leading-order Z2 Klein symmetry factors.
T brokenT broken
Derivation of the Master Formula
Without assuming SD, we write the see-saw matrices in simple notation as, mD = . where we have written the complex Dirac masses as a, b, c, d, e, f with d = 0 and the positive real masses of right-handed neutrinos as Y, X. We have labeled the dominant right-handed and Yukawa neutrino couplings Mainly responsible for atmospheric neutrino mass m3 as "atm", subdominant ones mainly responsible for solar neutrino mass m2 as "sol", and quasi-detached (sub-subdominant) ones mainly responsible for m1 as "dec". We will order the right-hand neutrino masses as M1 < M2 < M3, and then identify Matm, Msol, Mdec with M1, M2, M3 in all possible ways.
It is clear that in the limit that m1 ⇥ 0 then the sub-subdominant right-handed neutrino and its associated couplings labeled "dec" are completely decoupled and the above model reduces to a two right-handed neutrino model. In that limit, we simply drop the third term [in square brackets] in Equations 8-13 in anticipation of this. According to quantum mechanics, it is not necessary for the standard model statesνe,νµ,ντ to be identified in a one-one way with the mass eigenstatesν1,ν2 andν3, and the matrix elements of U give the quantum amplitude that a particular standard model state contains a mixture of a particular mass eigenstate.
The probability that a particular neutrino mass state contains a particular SM state can be represented by colors as in Fig. Note that neutrino oscillations are only sensitive to the differences between the squares of the neutrino masses ∆m2ij ≡m2i −m2j, and give no information. about the absolute value of the neutrino mass squared eigenvalues m2i. There are basically two patterns of neutrino mass quadrature curves consistent with the atmospheric and solar data as shown in Fig.
Note that neutrino oscillation experiments only determine the difference between the squared values of the masses. Also, whilem22>m21, it is currently unknown whether m23 is heavier or lighter than the other two, corresponding to the left and right panels of the figure, referred to as normal or inverse mass square order, respectively. Finally, the value of the lightest neutrino mass (sometimes referred to as the neutrino mass scale) is currently unknown and is represented by a question mark in each case.
CSD3”
2RHN
The reactor angle ✓13 has been multiplied by a factor of 5 for visual convenience (there is no physical relevance of the number 5). We now illustrate the success of the scheme by presenting numerical results for the neutrino mass matrix in equation 23 for the specific choice of input parameters, namely n = 3 and . In Table 1 we compare the above numerical benchmark arising from the neutrino mass matrix in Eq. 24 with the global best-fit values from [25] (setting m1 = 0).
In addition, we predict = 71.9 which is not shown in the Table since the neutrinoless double beta decay parameter is mee = mb = 2.684 meV for the above parameter set which is practically impossible to measure in the foreseeable future. These predictions can be compared with the global best fit values from [25] (for m1 = 0), given on the last line. Using the results in Table 1, the baryon asymmetry of the Universe (BAU) due to N1 = Natm leptogenesis was estimated for this model [16]:.
We emphasize that the successful predictions of the model in the lepton sector (respectively the prediction of the PMNS matrix) is independent of the specific values of these mass parameters. The least suppressed connection, the renormalizable H5T3T3 operator responsible for the top quark mass, is the last diagram in Fig. Thus, the hierarchy of up quark masses as well as the CKM mixing angles are given by powers of.
Due to the structure of this matrix, any phase introduced by h⇠i can be reabsorbed by the appropriate redefinition of the three Ti fields, so that Y u does not contain a source of CP violation. When considering the Yukawa structures of bottom quarks and charged leptons, we inevitably have to discuss the triplet flavones A4.6 The assignments of all flavones under the symmetries of the family are shown in Table 1. The field ⇠ which acquires a VEV v⇠ ⇠ 0.06MGUT generates a hierarchy Fermion mass structure in the up-type quark sector through terms such as vuTiTj(v⇠/M)6 i j, where vu is the VEV of Hu.
It also partially contributes to the mass hierarchy for down-type quarks and charged leptons and provides the mass scales for the right-handed neutrinos, as discussed later. For the seesaw mechanism, we will introduce another convention for Yukawa and Majorana masses.
CSD3
Effects of RG running
Having calculated the mass insertion parameters on the GUT scale, it is now necessary to consider their evolution down to the electroweak scale. Only then can we compare the model's predictions with experimental measurements of observable taste. This evolution is described by the RG equations explicitly given in Appendix E in the SCKM base.
Technically, we perform RG, which runs in two stages, first from MGUT to MR, where the right neutrinos are integrated, and then from MR to MSUSY ~ MW. To derive analytical results, we estimate run effects using a leading logarithmic approximation. Since the run also affects the Yukawa matrices themselves, further basis transformations must be applied to the superfields that diagonalize the low-energy Yukawa matrices.
Details of the various steps involved in calculating the low energy mass insertion parameters can be found in Appendix F. For the down-type squarks and the charged dragon, the resulting effects can simply be absorbed into new order one- coefficients. It is interesting to see that this is not the case for the up-type squarks, where the order of the (13) and (23) elements of δLRu is changed.
For completeness, we present the flavor structure of the low-energy δs in terms of their λ suppression, which should be compared with Eq. Despite its tremendous success, the Standard Model of particle physics is widely regarded as the low energy limit of a more fundamental theory. To understand the nature of flavor in such extensions of the SM, it is necessary to answer the following three questions.
Mass insertion parameters in SUSY S 4 xSU(5)xU(1)
They are of the same order of magnitude, but it is the current electron EDM limit that limits our parameter space. We have shown that the model leads to mass insertion parameters in Eqs that correspond very closely to the MFV forms, where LL,RRu,d,e are unit matrices and LRu,d,e are proportional to the Yukawa matrices. This is in clear contrast to the US(3) family symmetry models previously studied, where there were large e↵ects.
Comparing this with the discussion of Section 3.2.3 reveals that, with the actual MEG associated, |( LLe )12|. The red dotted lines show the experimental measurements, while the blue ones correspond to the bounds for the NP contributions as given in Eq. CP-violating e↵ects in the quark sector can manifest through quark EDMs as well as quark Chromo Electric Dipole Moments (CEDM).
In light of the success that led to the discovery of M-Theory, Vafa [2] applied a similar non-perturbative limit to type II-B theory, in which he found that the theory could be e↵ectively described as a 12-dimensional theory, despite the fact that there are no meaningful 12-dimensional supergravity theories. When we refer to M-Theory as a framework for doing phenomenology, we then refer to the 11-dimensional supergravity theory as the starting point and not the unknown full membrane theory. This happens as the compacted space allows a 3-fold singularity with an orbifold singularity supporting the gauge fields, while localized conical singularities on these 3-fold chiral superfields support in irreps of the associated gauge interaction.
An important point of the framework is that if the compaction is fluxless, the moduli fields cannot have a perturbative superpotential due to an exact Peccei-Quinn symmetry. This symmetry forces the axions - which are the real moduli complex partners in the chiral superfield - to have a shear symmetry which, in conjugation with holomorphism, results in non-perturbative contributions to the superpotential [4]. Early semi-realistic constructions involved an SU(5) gauge group, for each the derived model was called G2MSSM since it also had the same particle content as of the MSSM.
The model was based on an idea from Witten [6], where the combination of discrete geometric symmetries and the topological nature of compacted space provide a natural discrete symmetry that does not move with the gauge set. This in turn allows one to allow a GUT scale measure for the triplet color partners of the MSSM Higgses, and hence to solve the doublet-triplet problem of SU(5) SUSY GUTs.
