A democratic suggestion

0.23 MB·

by  A. Kleppe

A democratic suggestion
A. Kleppe, SACT, Oslo
[email protected]
6 Abstract
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Withintheframeworkofquarkmassmatriceswithademocratictexture,theunitaryrota-
0
2 tion matrices that diagonalize the quark matrices are obtained by a specific parametrization
of the Cabibbo-Kobayashi-Maskawa mixing matrix. Different forms of democratic quark
n
mass matrices are derived from slightly different parametrizations.
a
J

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1 Introduction
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] The Standard Model is flawed by the large number of free parameters, for which there is at
h
p present no explanation. There is no prediction of the family replication pattern, nor of the
-
numberoffamilies. Allthefamiliesarereallytreatedonthesamefooting. MostoftheStandard
p
e Model free parameters reside in “flavour space” - with six quark masses, six lepton masses,
h
four quark mixing angles and ditto for the leptonic sector, as well as the strong CP-violating
[
parameter Θ¯. The structure of flavour space is determined by the fermion mass matrices, i.e.

1
by the form that the mass matrices take in the “weak interaction basis” where mixed fermion
v
8 states interact weakly, in contrast to the “mass bases”, where the mass matrices are diagonal.
8 One may wonder how one may ascribe such importance to the different bases in flavour
9
space, considering that the information content of a matrix is contained in its matrix invariants,
8
0 which in the case of a N ×N matrix M are the N sums and products of the eigenvalues λ ,
j
8. such as traceM, detM,
0
6 I = (cid:80) λ = λ +λ +λ ...
1 1 j j 1 2 3
(cid:80)
v: I2 = (cid:80)jkλjλk = λ1λ2+λ1λ3+λ1λ4+...
i I3 = jklλjλkλl = λ1λ2λ3+λ1λ2λ4+... (1)
X
.
.
.
r
a
I = λ λ ···λ
N 1 2 N
These expressions are invariant under permutations of the eigenvalues, which in the context of
mass matrices means that they are flavour symmetric, and obviously independent of any choice
of flavour space basis.
Eveniftheinformationcontentofamatrixiscontainedinitsinvariants,theformofamatrix
may also carry information, albeit of another type. The idea - the hope - is that the form that
1
the mass matrices have in the weak interaction basis can give some hint about the origin of the
unruly masses. There is a certain circularity to this reasoning; to make a mass matrix ansatz
is in fact to define what we take as the weak interaction basis in flavour space. We denote the
quark mass matrices of the up- and down-sectors in the weak interaction basis by M and M(cid:48),
respectively. We go from the weak interaction basis to the mass bases by rotating the matrices
by the unitary matrices U and U(cid:48),
M → UMU† = D = diag(m ,m ,m ) (2)
u c t
M(cid:48) → U(cid:48)M(cid:48)U(cid:48)† = D(cid:48) = diag(m ,m ,m )
d s b
The lodestar in the hunt for the right mass matrices is the family hierarchy, with two lighter
particles in the first and second family, and a much heavier particle in the third family. This
hierarchy is present in all the charged sectors, with fermions in different families exhibiting very
different mass values, ranging from the electron mass to the about 105 times larger top mass. It
is still an open question whether the neutrino masses also follow this pattern [1].
2 Democratic mass matrices
In the democratic approach [2], [3], [4] the family hierarchy is taken very seriously. It is assumed
that in the weak basis the fermion mass matrices have a form close to the S(3) × S(3)
L R
2
symmetric “democratic” matrix
 
1 1 1
N = k1 1 1 (3)
1 1 1
with the eigenvalues (0,0,3k), reflecting the family hierarchy.
TheunderlyingphilosophyisthatintheStandardModel,wherethefermionsgettheirmasses
from the Yukawa couplings by the Higgs mechanism, there is no reason why there should be a
different Yukawa coupling for each fermion. The couplings to the gauge bosons of the strong,
weak and electromagnetic interactions are identical for all the fermions in a given charge sector,
it thus seems like a natural assumption that they should also have identical Yukawa couplings.
The difference is that the weak interactions take place in a specific flavour space basis, while the
other interactions are flavour independent.
The democratic assumption is thus that the fermion fields of the same charge initially have
thesameYukawacouplings. Withthreefamilies,thequarkmassmatricesintheweakinteraction
basis then have the (zeroth order) form
   
1 1 1 1 1 1
M(0) = ku1 1 1, M(cid:48)(0) = kd1 1 1 (4)
1 1 1 1 1 1
wherek andk havedimensionmass. Thecorrespondingmassspectra(m ,m ,m ) ∼ (0,0,3k )
u d 1 2 3 j
reflect the family hierarchy with two light families and a third much heavier family, a mass hi-
erarchy that can be interpreted as the representation 1⊕2 of S(3). In order to obtain realistic
mass spectra with non-zero masses, the S(3) × S(3) symmetry must obviously be broken,
L R
and the different democratic matrix ans¨atze correspond to different schemes for breaking the
democratic symmetry.
2.1 The lepton sector
We can apply the democratic approach to the lepton sector as well, postulating democratic
(zeroth order) mass matrices for the charged leptons and the neutrinos, whether they are Fermi-
Dirac or Majorana states,
   
1 1 1 1 1 1
Ml(0) = kl1 1 1, Mν(0) = kν1 1 1 (5)
1 1 1 1 1 1
Relative to the quark ratio k /k ∼ m /m ∼ 40−60, the leptonic ratio k /k < 10−8 is so
u d t b ν l
extremely small that it seems unnatural. One way out is to simply assume that k vanishes,
ν
meaning that the neutrinos get no mass contribution in the democratic limit [5]. According to
the democratic philosophy, then there would be no reason for a hierarchical pattern `a la the
one observed in the charged sectors; the neutrino masses could even be of the same order of
magnitude.
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Dataareindeedcompatiblewithamuchweakerhierarchicalstructurefortheneutrinomasses
than the hierarchy displayed by the charged fermion masses.
Unlike the situation for the quark mixing angles, in lepton flavour mixing there are two
quite large mixing angles and a third much smaller mixing angle, these large mixing angles can
be interpreted as indicating weak hierachy of the neutrino mass spectrum. The neutrino mass
spectrumhierarchycouldevenbeinverted; ifthesolarneutrinodoublet(ν ,ν )hasameanmass
1 2
larger than the remaining atmospheric neutrino ν , the hierarchy is called inverted, otherwise it
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is called normal.
Supposing that the neutrino masses do not emerge from a democratic scheme, a (relatively)
flat neutrino mass spectrum could be taken as a support for the idea that the masses in the
charged sectors emerge from a democratic scheme.
3 The democratic basis
In the case that both the up- and down-sector mass matrices have a purely democratic texture,
the quark mixing matrix is V = UU(cid:48)† = U U† = 1, where
dem dem
√ √
 
3 − 3 0
1
Udem = √ √1 √1 √−2 (6)
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2 2 2
is the unitary matrix that diagonalizes the democratic matrix (3).
We use this to define the democratic basis, meaning the flavour space basis where the mass
matrices are diagonalized by (6) and the mass Lagrangian is symmetric under permutations of
the fermion fields (ϕ ,ϕ ,ϕ ) of a given charge sector.
1 2 3
In the democratic basis the mass Lagrangian
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(cid:88)
L = ϕ¯M ϕ = k ϕ¯ ϕ
m (dem) j k
jk=1
is symmetric under permutations of the fermion fields (ϕ ,ϕ ,ϕ ), while in the mass basis with
1 2 3
 
λ
1
M(mass) =  λ2 
λ
3
the mass Lagrangian has the form
L = λ ψ¯ ψ +λ ψ¯ ψ +λ ψ¯ ψ (7)
m 1 1 1 2 2 2 3 3 3
which is clearly not invariant under permutations of (ψ ,ψ ,ψ ).
1 3 3
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We can perform a shift of the democratic matrix, by just adding a unit matrix diag(a,a,a),
M → M ,
0 1
     
1 1 1 a k+a k k
M1 = k1 1 1+ a  =  k k+a k  (8)
1 1 1 a k k k+a
corresponding to the mass spectrum (a,a,3a+3k). The matrix M has a democratic texture,
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both because it is diagonalized by U , and because the mass Lagrangian is invariant under
dem
permutations of the quark fields,
(cid:88) (cid:88)
L = (k+a) ϕ¯ ϕ +k ϕ¯ ϕ (9)
M1 j j j j
j(cid:54)=k
If M and M(cid:48) both have a texture like (8), there is no CP-violation. This is independent of
1 1
how many families there are, because of the degeneracy of the mass values. CP-violation only
occurs once there are three or more non-degenerate families, because only then the phases can
no longer be defined away.
We can repeat the democratic scheme with a number n of families, where the fermion mass
matrices again are proportional to the S(n) ×S(n) symmetric democratic matrix which is
L R
diagonalized by a unitary matrix analogous to U in (6). To the n×n-dimensional democratic
dem
matrix term, we can again add a n×n-dimensional diagonal matrix diag(a,a,...,a), and get a
n×n-dimensional mass spectrum with n massive states, and n−1 degenerate masses. The mass
matrix still has a democratic texture, and there is still no CP-violation.
4 Breaking the democratic symmetry
In order to obtain non-degenerate, non-vanishing masses for the physical flavours (ψ ,ψ ,ψ ),
1 2 3
the permutation symmetry of the fermion fields (ϕ ,ϕ ,ϕ ) in the democratic basis must be
1 2 3
broken. The proposal here is to derive mass matrices with a nearly democratic texture, not
by explicitly perturbing the assumed initial democratic form (3), but instead by perturbing
the matrix U which diagonalises the democratic mass matrix. This is done by deriving the
dem
unitary rotation matrices U, U(cid:48) for the up- and down- sectors, from a specific parameterisation
of the weak mixing matrix V = UU(cid:48)†.
The idea is to embed the assumption of democratic symmetry into the Standard Model
mixing matrix, by expressing the mixing matrix as a product
V = UU(cid:48)† = (U˜U )(U† U˜(cid:48)†) (10)
dem dem
Since both the mixing matrix and its factors, according to the standard parameterisation [6],
are so close to the unit matrix, the rotation matrices U, U(cid:48) are effectively perturbations of the
democratic diagonalising matrix (6). In this way, the weak interaction basis remains close to
the democratic basis.
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4.1 Factorizing the mixing matrix
TheCabbibo-Kobayashi-Maskawa(CKM)mixingmatrix[7]canofcoursebeparametrized-and
factorized - in many different ways, and different factorizations correspond to different rotation
matricesU andU(cid:48). Themostobviousand“symmetric”factorizationoftheCKMmixingmatrix
is, following the standard parametrization [6] with three Euler angles α, β, 2θ,
 c c s c s e−iδ
β 2θ β 2θ 2θ
V = −cβsαs2θeiδ −sβcα −sβsαs2θeiδ +cβcα sαc2θ  = UU(cid:48)† (11)
−c c s eiδ +s s −s c s eiδ −c s c c
β α 2θ β α β α 2θ β α α 2θ
with the diagonalizing rotation matrices for the up- and down-sectors
1 0 0 e−iγ  cosθ 0 sinθ√1 −√1 0 
2 2
U = 0 cosα sinα 1  0 1 0 √1 √1 −√2  (12)
 6 6 6
0 −sinα cosα eiγ −sinθ 0 cosθ √1 √1 √1
3 3 3
and
cosβ −sinβ 0e−iγ cosθ 0 −sinθ√1 −√1 0 
2 2
U(cid:48) = sinβ cosβ 0 1  0 1 0 √1 √1 −√2 ,
 6 6 6
0 0 1 eiγ sinθ 0 cosθ √1 √1 √1
3 3 3
respectively, where α, β, θ and γ correspond to the parameters in the standard parametrization
in such a way that γ = δ/2, δ = 1.2±0.08 rad, and 2θ = 0.201±0.011◦, while α = 2.38±0.06◦
and β = 13.04±0.05◦.
FromtherotationmatricesU andU(cid:48)wethenobtainthemassmatricesM = U†diag(m ,m ,m )U
u c t
and M(cid:48) = U(cid:48)†diag(m ,m ,m )U(cid:48), such that
d s b
 X +H Mˆ Z +W 
1 12
M =  Mˆ∗ X −H Z −W  (13)
6 12
Z∗+W∗ Z∗−W∗ 6T −2X
√ √ √ √
where T is the trace T = m + m + m , and with D = 3s − 2c , C = 3s + 2c ,
u c t θ θ θ θ
F = c s (m −m ),
α α t c
X = 1(m s2 +m c2 −m )(D2+C2−2)+F(D−C)cosγ +T +3m
2 c α t α u u
H = 1(m s2 +m c2 −m )(D2−C2)+F cosγ(D+C)
2 c α t α u
W = 1(m s2 +m c2 −m )(D2−C2)−F (D+C)e−iγ
4 c α t α u
Z = (m s2 +m c2 −m )(cid:2)2+ 1(D−C)2(cid:3) + F (D−C)(eiγ −2e−iγ)−2T +6m
c α t α u 4 2 u
Mˆ = −(m s2 +m c2 −m )(D C +1)−F (C eiγ −D e−iγ)+T −3m
12 c α t α u u
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Similarly for the down-sector,
 X(cid:48)+H(cid:48) Mˆ(cid:48) Z(cid:48)+W(cid:48) 
1 12
M(cid:48) =  Mˆ(cid:48)∗ X(cid:48)−H(cid:48) Z(cid:48)−W(cid:48)  (14)
6 12
Z(cid:48)∗+W(cid:48)∗ Z(cid:48)∗−W(cid:48)∗ 6T(cid:48)−2X(cid:48)
√ √ √ √
with the parameters T(cid:48) = m +m +m , G = 2s − 3c , J = 2s + 3c ,
d s b θ θ θ θ
F(cid:48) = c s (m −m ), and
β β b s
X(cid:48) = 1(m s2 +m c2 −m )(G2+J2−2)−F(cid:48)(J +G)cosγ +T(cid:48)+3m
2 s β b β d b
H(cid:48) = 1(m s2 +m c2 −m )(G2−J2)+F(cid:48)(J −G)cosγ
2 s β b β d
W(cid:48) = 1(m s2 +m c2 −m )(G2−J2)+F(cid:48)(G−J)eiγ
4 s β b β d
Z(cid:48) = (m s2 +m c2 −m )(cid:2)2+ 1(J +G)2(cid:3)+ F(cid:48)(J +G)(2eiγ −e−iγ)−2T(cid:48)+6m
s β b β d 4 2 b
Mˆ(cid:48) = (m s2 +m c2 −m )(GJ −1)−F(cid:48) (J eiγ −Ge−iγ)+T(cid:48)−3m
12 s β b β d b
Inordertoevaluatetowhatdegreetheseratheropaquematricesaredemocratic,wecalculate
numerical matrix elements by inserting numerical mass values. For the up-sector we get the
(nearly democratic) matrix texture
    
1 1 1 1 1
M = Cu k e−i(µ+ρ) 1 1 1 k ei(µ+ρ) +Λ (15)
kpe−iµ 1 1 1 kpeiµ
where the “small” matrix
 
0 0 0
Λ = 0 ε ε(cid:48)e−iρ,
0 ε(cid:48)eiρ η
with ε ∼ ε(cid:48) (cid:28) η < k,p, is what breaks the democratic symmetry, supplying the two lighter
families with non-zero masses. With mass values calculated at µ = M (Jamin 2014) [8],
Z
(m (M ),m (M ),m (M )) = (1.24,624,171550)MeV, (16)
u Z c Z t Z
we get
µ ∼ 2.7895o, ρ ∼ 2.7852o,C = 54240.36 MeV ≈ m /3, and
u t
k ≈ 1.00438, p ≈ 1.06646,ε(cid:48) ≈ 5.0510−5,
ε ≈ 4.610−5 ≈ 2mu,η = 1.81510−2 ≈ 1mtmc.
Cu 2Cu Cu
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For the down-sector, with
(m (M ),m (M ),m (M )) = (2.69,53.8,2850)MeV (17)
d Z s Z b Z
we get another democratic texture,
X +A Ye−iτ e−iν 
M(cid:48) = Cd Yeiτ X −A (1+2A)eiκ  (18)
eiν (1+2A)e−iκ X +Y −A−1
where
C = 966.5MeV, A = 5.610−3, X = 1.0362, Y = 1.0305, and τ ≤ κ ∼ 0.22o < ν ∼ 0.226o.
d
Just like in the up-sector mass matrix, the matrix elements in M(cid:48) display a nearly democratic
texture. For both the up-sector and the down-sector the mass matrices are thus approximately
democratic.
5 Calculability
Inthemassmatrixliteraturethereisanemphasison“calculability”. Theidealistoobtainmass
matrices that have a manageable form, but there is nothing that forces nature to serve us such
user-friendly formalism. It is however tempting to speculate that there are relations between
the elements that could make the democratic matrices more calculable, and in the search for
matrices that are reasonably transparent and calculable, we look at a more radical factorization
of the mixing matrix, viz.
1 0 0  cosω 0 sinω e−iδ√1 −√1 0 
2 2
U = 0 cosα sinα 0 1 0 √1 √1 −√2  (19)
 6 6 6
0 −sinα cosα −sinω eiδ 0 cosω √1 √1 √1
3 3 3
and
cosβ −sinβ 0√1 −√1 0 
2 2
U(cid:48) = sinβ cosβ 0√1 √1 −√2 
 6 6 6
0 0 1 √1 √1 √1
3 3 3
where, as before, δ = 1.2 ± 0.08 rad, and ω = 2θ = 0.201 ± 0.011◦, while α = 2.38 ± 0.06◦,
and β = 13.04 ± 0.05◦. These rotation matrices are still “perturbations” of the democratic
diagonalizing matrix (6), and the up-sector mass matrix has a texture similar to (13),
R+Q+S cosδ R−Q−iS sinδ A−Be−iδ 
1
M = R−Q+iS sinδ R+Q−S cosδ A+Be−iδ  (20)
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A−Beiδ A+Beiδ T −2(R+Q)
where T is the trace, T = m +m +m , and
u c t
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√
R = N (2c2 −1)+T −2 2c F, Q = 3s2 N +3m ,
ω ω ω u
√ √
S = −2 6c s N +2 3s F
ω ω ω
√ √ √
A = N (2c2 +2)−2T + 2c F +6m , B = 6c s N +2 3F s
ω ω u ω ω ω
with N = m s2 +m c2 −m , F = c s (m −m ). This matrix can be reformulated in a
c α t α u α α t c
form similar to (15),
    
1 1 1 1 1
Mu = Cu k e−iµ 1 1 1 k eiµ +Λ
kpe−i(µ−ρ) 1 1 1 kpei(µ−ρ)
where C = R+Q+Scosδ, µ = arctan[Ssinδ/(Q−R)], ρ = arctan(Bsinδ/(A+Bcosδ)),
u
and
 
0 0 0
Λ = 0 ε ε(cid:48)e−iρ
0 ε(cid:48)eiρ η
with
|R−Q−iSsinδ| |A−Be−iδ|
k = |M |/M = , p = |M |/|M | = ,
12 11 R+Q+Scosδ 13 12 |R−Q−iSsinδ|
ε = (|M ||M |−|M |2)/|M |2 = 4RQ−S2 ,
22 11 12 11 |R+Q+Scosδ|2
ε(cid:48) = (|M ||M |−|M ||M |)/|M |2,
23 11 13 12 11
η = (|M ||M |−|M |2)/|M |2
33 11 13 11
Inserting the mass values (16) gives
C = 53723.5MeV,k = 1.00318, p = 1.0828, and
u
ε ≈ 4.6510−5 ≈ 2mu, ε(cid:48) ≈ 4.4410−5, η ≈ 1.8510−2 ≈ 1mtmc
Cu 2Cu Cu
For the down-sector, with
cosβ −sinβ 0√1 −√1 0 
2 2
U(cid:48) = sinβ cosβ 0√1 √1 −√2 ,
 6 6 6
0 0 1 √1 √1 √1
3 3 3
the mass matrix U(cid:48)†diag(m ,m ,m )U(cid:48) reads
d s b
 
X +A Y 1
M(cid:48) = Cd Y X −A 1+2A 
1 1+2A X +Y −A−1
where
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√
C = 2(m c2 +m s2)−2 3c s (m −m ))+2(m −m −m )
d d β s β β β s d b s d
√
X = (2m +m +m +2(m c2 +m s2)+2 3c s (m −m ))/C
b s d d β s β β β s d d
Y = (2m +m +m −4(m c2 +m s2))/C ,
b s d d β s β d
√
A = 2 3c s (m −m ))/C .
β β s d d
Inserting the mass values (17) we moreover get the numerical values
C = 926.448MeV ≈ m /3,X = 1.0375,A = 710−3,Y = 1.0318.
d b
6 Conclusion
By including the democratic rotation matrix in the parametrization of the weak mixing matrix,
we obtain mass matrices with specific democratic textures. In this way we make contact be-
tweenthedemocratichypothesisandtheexperimentallyderivedparametersoftheCKMmixing
matrix, avoiding the introduction of additional concepts.
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