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Quark and Lepton Mass Matrices, Textures, Horizontal Symmetries

  1. Jul 11, 2012 #1
    Does anyone have any good introduction to theories of the quark and lepton mass matrices? Theories like textures and horizontal symmetry. My understanding of research into textures is that it often involves trying to make zero as many entries as possible in the mass matrices. Is that a fair statement?

    Though the quarks have a rather strong mass hierarchy, their mixing matrix, the Cabibbo–Kobayashi–Maskawa matrix is close to the identity matrix.

    The charged leptons also have a rather strong mass hierarchy, and likely also the neutrinos. However, the lepton mixing matrix, the Pontecorvo–Maki–Nakagawa–Sakata matrix, is rather far from the identity matrix.

    Part of the reason for the difference may be that the neutrinos' masses may be due to a [http://en.wikipedia.org/wiki/Seesaw_mechanism seesaw mechanism]. In its simplest form, right-handed neutrinos have Majorana masses close to GUT energy scales, while left-handed and right-handed ones are coupled by Higgs-generated Dirac masses, like the charged leptons and the quarks.

    So all the Dirac mass matrices could be nearly orthogonal, while the right-handed-neutrino mass matrix could be far from orthogonal to them.

    GUT's? The Georgi-Glashow SU(5) GUT is the simplest, and its only mass-matrix constraint is for down-like quarks and charged leptons to have equal mass matrices. Strictly speaking, couplings to the Higgs particle.

    SO(10)? All the Higgs couplings are equal, and right-handed neutrinos cannot have a Majorana mass. Thus, all the mass matrices are orthogonal, making mass unification too successful. The non-orthogonality and the Majorana masses must come from symmetry breaking.

    Its supersets, like E6, also have that problem.

    Are there any halfway-simple symmetry-breaking scenarios that can start with SO(10) or E6 and get the Standard Model's masses and mixing? It may be asking too much to ask for something as simple as the Standard-Model Higgs mechanism, I will concede.
  2. jcsd
  3. Jul 11, 2012 #2


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  4. Jul 13, 2012 #3

    The lepton mixing matrix
    is approximately
    {{sqrt(2/3), sqrt(1/3), 0}, {-1/sqrt(6), 1/sqrt(3), 1/sqrt(2)}, {1/sqrt(6), -1/sqrt(3), 1/sqrt(2)}}

    which is rather far off from the identity matrix.
    That approximation is from sin(a12) = 1/sqrt(3), sin(a13) = 0, sin(a23) = 1/sqrt(2).

    From Particle Data Group, the most recent results:
    m2^2 - m1^2 = (0.0085 - 0.0088 eV)^2
    m3^2 - m2^2 = +- (0.047 - 0.049 eV)^2
    a12 = 33 - 35 d (35 d), a13 = 8.5 - 9.7 d (0d), a23 = 39 - 45 d (45 d)
    In ()'s are the angles that give that nice-looking mixing matrix.

    K. S. Babu's paper has a nice table (Table 1) of the running masses of the quarks and charged leptons. Here are some highlights:

    m(up) = 0.0022 (2 GeV) = 0.0011 (1 TeV) = 0.00049 / 0.00048 (GUT)
    m(down) = 0.005 (2 GeV) = 0.0025 (1 TeV) = 0.00070 / 0.00051 (GUT)
    m(strange) = 0.095 (2 GeV) = 0.047 (1 TeV) = 0.013 / 0.010 (GUT)
    m(charm) = 1.25 (self) = 0.532 (1 TeV) = 0.236 / 0.237 (GUT)
    m(bottom) = 4.20 (self) = 2.43 (1 TeV) = 0.79 / 0.61 (GUT)
    m(top) = 162.9 (self) = 150.7 (1 TeV) = 92.2 / 94.7 (GUT)
    m(electron) = 0.0005110 (self) = 0.000496 (1 TeV) = 0.000284 / 0.000206 (GUT)
    m(muon) = 0.1057 (self) = 0.105 (1 TeV) = 0.0599 / 0.0435 (GUT)
    m(tau) = 1.777 (self) = 1.78 (1 TeV) = 1.022 / 0.773 (GUT)

    Mass values are in GeV, self = mass at the particle's mass scale. The GUT values are for 2*10^(16) GeV, the MSSM, and tan(beta) = 10 / 50.

    For tan(beta) large, m(bottom) and m(tau) are close at GUT energies, as expected from GUT mass unification.

    The quark mixing matrix: {{ud, us, ub}, {cd, cs, cb}, {td, ts, tb}}
    Values from PDG (absolute values):
    {{0.97, 0.23, 0.004}, {0.23, 0.97, 0.04}, {0.009, 0.04, 1.00}}

    The rest of Babu's paper described various model-building efforts for accounting for the elementary-fermion masses and mixings, like mass matrices with forms {{0,A},{A,B}} and {{0,A,0},{A,0,B},{0,B,C}} and {{0,A,0},{A,B,0},{0,0,C}}.
  5. Jul 13, 2012 #4


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    Actually, with the names and references of these models and some help from SPIRE/INSPIRE and KEK, it is possible to do a nice time travel to the golden age of CKM and quark phenomenology, in the late seventies. I guess that the discoverty of the 3rd generation was quite an excitement.
  6. Jul 14, 2012 #5
    What are SPIRE/INSPIRE and KEK? Do they have databases of papers? Or at least abstracts.

    I've tried to calculate whether Higgs-particle interactions could help disambiguate the mass matrices, and I've concluded that they can't. All I get is what one gets at low energies: mass eigenvalues and up/down and electron/neutrino mixing matrices. So MSSM heavy Higgs particles are not likely to help.

    Considering horizontal discrete symmetries like S3 and A3, I find

    S3: {{A,B,B},{B,A,B},{B,B,A}}
    A3: {{A,B,C},{C,A,B},{B,C,A}}
    S2: {{A,B,C},{B,A,C},{D,D,E}}

    The masses are the square roots of the eigenvalues of M = m.(hermitian conjugate of m)
    where m is the Dirac mass matrix.

    For S3, I get 2 separate eigenvalues in the real and complex cases.

    For A3, I get 2 separate eigenvalues in the real case and 3 separate eigenvalues in the complex case.

    For S2, I get 3 separate eigenvalues in the real and complex cases.

    This would seem to rule out the S3 horizontal symmetry for the elementary-fermion mass matrices.
  7. Jul 15, 2012 #6
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