Bled Workshops in Physics Vol. 18, No. 2
A
Proceedings to the 20th Workshop What Comes Beyond ... (p. 72) Bled, Slovenia, July 9-20, 2017
6 Phenomenological Mass Matrices With a Democratic Texture
A. Kleppe * SACT, Oslo
Abstract. Taking into account all available data on the mass sector, we obtain unitary rotation matrices that diagonalize the quark matrices by using a specific parametrization of the Cabibbo-Kobayashi-Maskawa mixing matrix. The form of the resulting mass matrices is consistent with a democratic scheme with a well-defined, stepwise breaking of the initial flavour symmetry.
Povzetek. Avtorica izbere parametrizacijo mešalne matrike Cabibba, Kobayashija in Mas-kawe, poišče zanjo unitarne rotacijske matrike, ki pri tej parametrizaciji diagonalizirajo masne matrike kvarkov. Izmerjene mase kvarkov zavrti v startni masni matriki, ki sta skladni z demokraticno shemo matrik z dobro definirano in postopno zlomljeno zacetno simetrije.
Keywords: Mass matrices, CKM matrix, Democratic texture 6.1 Mass states and flavour states
In this work, we take a very phenomenological approach on the fermion mass matrices, by assuming that the quark mass matrices can be derived from a (naive) factorization of the Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix V [1], which appears in the charged current Lagrangian
Lcc = -	W, + h.c.	(6.1)
where 9 and 9' are quark fields with charges Q and Q — 1, correspondingly.
From the perspective of weak interactions, Lcc describes an interaction between left-handed flavour states. From the point of view of all other interactions, the interaction takes place between mixed physical particle states - where "physical particles" refer to mass eigenstates of the mass matrices M and M' appearing in the mass Lagrangian
Lmass = fMf + f'M 'f'
where f, f' are fermion flavour states of charge 2/3 and -1/3, respectively, with the corresponing mass matrices denoted as M = M(2/3) and M' = M'(—1/3). Our
* E-mail: kleppe@nbi.dk
6 Phenomenological Mass Matrices With a Democratic Texture
73
dilemma is in a way how to understand the relation between physical particles and flavour states.
We imagine that all the flavour states live in the same "weak basis" in flavour space, while the mass states of the 2/3-sector and the -1/3-sector live in their separate "mass bases". We go between the weak basis and the mass bases of the two charge sectors by rotating with the unitary matrices U and U', which are factors of the CKM-matrix, V = UU'*.
M -> UMU1" = D = diag(mu,mc,mt)	(6.2)
M' -> U'M'U/f = D' = diag(md,ms,mb)
Since V = 1, the up-sector mass basis is different from the down-sector mass basis, the CKM matrix thus bridges the two mass bases.
The mass Lagagrangian reads
Lmass = fMf + f'M'f' = f Df + if'D 'f'	(6.3)
where f, f' are the flavour states and f, f' are the mass states. We of course know the diagonal mass matrices D(2/3) and D '(-1/3), it is M(2/3) and M'(-1/3) that we are looking for, in the hope that their form can shed light on (the mechanism behind) the mysterious, hierarchical fermion mass spectra.
Whereas the quark mass eigenstates are perceived as "physical", and the weakly interacting flavour states are percieved as mixings of physical particles, in the lepton sector the situation is somewhat different, due to the fact that neutrino mass eigenstates don't ever appear in interactions - they merely propagate in free space. In the realm of neutral leptons it is actually the flavour states ve
74 A. Kleppe
that we perceive as "physical", since they are the only neutrinos that we "see", as they appear together with the charged leptons. As the charged leptons e, t are assumed to be both weak eigenstates and mass eigenstates, the only mixing matrix that appears in the lepton sector is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix U which only operates on neutrino states,
'Ve \	/V
V|| I = U(PMNS) I V2
\VJ	\V3,
where (vi , v2, v3) are mass eigenstates, and ( Ve, V|, vt) are the weakly interacting "flavour states". In the lepton sector, the charged currents are thus interpreted as charged lepton flavours (e, t) interacting with the neutrino flavour states
(Ve,V|,VT).
6.2 Factorizing the weak mixing matrix
The usual procedure in establishing an ansatz for the quark mass matrices is based on some argument or model. Here we follow a rather phenomenaological approach, looking for a factorization of the Cabbibo-Kobayashi-Maskawa mixing matrix, which would give the 'right' mass matrices. The CKM matrix can of course be parametrized and factorized in many different ways, and different factorizations correspond to different rotation matrices U and U', and correspondingly to different mass matrices M and M'.
We choose what we perceive as the most obvious and "symmetric" factorization of the CKM mixing matrix is, following the standard parametrization [2] with three Euler angles a, 29,
/	cpC2e	spC2e	S2ee-l6\
V = I -cpsas2eel6 - spca -spsas2eel6 + cpca saC2e I = UU'f
\-CpCaS2eel6 + spsa -spcas2eei6 - cpsa cac2e J with the diagonalizing rotation matrices for the up- and down-sectors
'e-iY
(6.4)
U =
'10 0 0 cos a sin a 0 - sin a cos a
1
cos 9 0 sin 9 0 10 I W(p) =
eiY/ V-sin9 0 cos9,
cee-iY 0 see-iY -saseeiY ca saCeeiY |w(p) -caseeiY sa caCeeiY -
(6.5)
and
U' =
^cos | - sin | 0\ /e-iY sin | cos | 0I 0 0 1J \	eiY,
'CpCee-iY -sp -cpsee-1^
cos 9 0 - sin 9
0 10 1 W(p) sin 9 0 cos 9
= | spcee see
-iY
Cp -spsee
-iY
|w(p)
(6.6)
iY
0
cee
iY
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75
respectively, where W(p) is a unitary matrix which is chosen is such a way that y is the only phase in either of the mass matrices,
cos p ± sin p^ 1 0 0 v 0 t sin p cos p
cos p 0 ± sin p 0 1 0 yT sin p 0 cos p
cos p ± sin p 0 0 0 1 iTsin p cos p 01
Here p is unknown, whereas a, p, 9 and y correspond to the parameters in the standard parametrization, with y = 6/2,6 = 1.2±0.08 rad,and 29 = 0.201 ±0.011°, while a = 2.38 ± 0.06° and p = 13.04 ± 0.05°. In this factorization scheme, a and p are rotation angles operating in the up-sector and the down-sector, respectively.
With the rotation matrices U(a, 9, y, p) and U'(p, 9, y, p), we obtain the the up- and down-sector mass matrices
M = U^diag(mu,mc,mt)U and M' = U'^diag(md,ms,mb)U',
such that
/M,, MT2 MTSN M = I M2I M22 M23 I VMa, M32 M33/
Xc2e + Ys2
= Wt(p) I Zse eiY
Zse e-iY (X - Y)cese\ Y - 2Z cot 2a -Zce eiY I W(p)
JX - Y)cese -Zce e-iY Xs2 + Yc2 /
(6.7)
where X = mu, Z = (mt — mc) sin a cos a and Y = mc sin2 a + mt cos2 a; and
Mi, Mi, M
Mi =
Lii
M21 ^M31
12
M2 2 M3 2
13
M2 3 M3 3/
= W t(p)
X isd + y ^
Zice e-
iY
Zice eiY (Xi - Yi)ceseN Yi + 2Zi cot 2ß -Zise e-iY
V(Xi - Yi)cese -Zise eiY
Xi c2 + Y is2
W(p) (6.8)
where X' = mb, Z' = (ms — md) sin p cos p and Y' = md cos2 (3 + ms sin2 p. The two mass matrices thus have similar textures.
From Y = mc sin2 a+mt cos2 a, Z = (mt — mc) sin a cos a, Y' = md cos2 p + ms sin2 p and Z' = (ms — md) sin p cos p, we moreover have
= X,
= Y - Z cot a, mt = Y + Z tan a
md = Yi - Zi tan ß, ms = Yi + Zicot ß, mb = Xi
(6.9)
m
m
u
c
6.3 The matrix W
We choose the matrix W(p) as
(cos p — sin p 0\ 0 0 1 | , (6.10) sin p cos p 0
76 A. Kleppe
which gives the up-sector mass matrix
/ Xc2 + Ysg Zse e-iY (X - Y)cese\ M = Wf I Zse eiY Y - 2Z cot 2a -Zce eiY I W = V(X - Y)cese -Zce e-iY Xs2 + Yc2 /
( A Zse e-iY H \ = Wf I Zse eiY F -Zce eiY I W = V H -Zce e-iY K J
Acp + Ksp + H sin2p 1 (K - A) sin2p + H cos2p -Ze-iY sin(p - 9)N 2(K - A) sin2p + H cos2p As? + KcP - H sin2p -Ze-iY cos(p - 9)
-ZeiY sin(p - 9)	-ZeiY cos(p - 9)	F
(6.11)
With
A = Xcg + Ysg, H = (X - Y)cese and K = Xs^ + Yc^, we get
^X cos2 |i + Y sin2 |i (Y — X) sin |i cos |i —Z sin |i e-iyN M = ( (Y - X) sin | cos |X sin2 | + Y cos2 | -Z cos | e-iY |	(6.12)
-Z sin | eiY	-Z cos | eiY	F
— y
where | = p-9, and as before, X = mu, Z = (mt -mc) sin a cos a, Y = mc sin2 a+ mt cos2 a, and F = Xsg + Ycg = Y - 2Zcot 2a = trace(M) - X - Y.
Now, depending on the value of | = p - 9, we get different matrix textures,
e.g.
1 = P - 9	0 or n	n/4	n/2
M11 = Xc2 + Ys2	X	(X + Y )/2	Y
M!2 = 2 (Y - X)s22	0	(Y - X)/2	0
M13 = -Zs2 e-iY	0	-Ze-iY/V2	-Ze-iY
M22 = Xs2 + Yc2	Y	(X + Y )/2	X
M23 = -Zc^ e-iY	-Ze-iY	-Ze-iY/V2	0
M33 = F	Y - 2Z cot 2a	Y - 2Z cot 2a	Y - 2Z cot 2a
So for p - 9 = 0 or n, we get the simple form
/X 0 0 \
M(0,n) = I 0 Y -Ze-iY I ,	(6.13)
\0 -ZeiY F J
and for p - 9 = n/2, equally simple
/ Y 0 -Ze-iy\ M(n/2) = ( 0 X 0 | \-ZeiY 0 F /
(6.14)
6 Phenomenological Mass Matrices With a Democratic Texture
77
Applying the same procedure on the down-sector, we get the down-sector mass matrix
/ X'sC + Y'c2 Z'ce eiY (X' - Y')cese\ M' = W(p)f I Z'ce e-iY Y' + 2Z'cot2p -Z'se e-iY I W(p) = \(X' - Y')ccsc -Z'se eiY X'cC + Y's2 /
/X' sin2 ^' + Y' cos2 ^' (X' - Y') sin ^' cos ^' Z' cos ^' eiY \ = I (X' - Y') sin ^' cos ^' X' cos2 ^' + Y' sin2 ^' -Z' sin ^' eiY I (6.15) \ Z' cos ^' e-iY	-Z' sin ^' e-iY	F' J
where ^' = p + 0, and as before, X' = mb, Z' = (ms - md) sin p cos p, Y' = md cos2 p + ms sin2 p, and F' = Y' + 2Z' cot 2p = trace(M') - X' - Y'. Depending on the value of ^' = p + 0, we get different matrix textures.
h ' = p + e	0 or n	n/4	n/2
Mi ! = X, + Y,	Yi	(Xi + Y i)/2	Xi
Mi2 = 2(Xi - Yi)s2^,	0	(Xi - Yi)/2	0
Mi3 = Z/ eiY	Z iety	Z ieiy/V2	0
M12 = X ^, + Y ^,	Xi	(Xi + Y i)/2	Yi
M23 = -Z %/ eiY	0	-Z ieiY/V'2	-Z ieiY
M2 3 =Fi	Yi + 2Zi cot 2ß	Yi + 2Zi cot 2ß	Yi + 2Zi cot 2ß
So for ^' = p + 0 = 0 or n, we get
/ Y' 0 Z'eiY\
M '(0, n) = I 0 X' 0 I	(6.16)
\Z 'e-iY 0 F ' /
and for ^' = p + 0 = n/2, we get
/X' 0 0 \
M'(n/2) = I 0 Y' -Z'eiY I	(6.17)
\0 -Z'e-iY F' J
6.4 Texture Zero Mass Matrices
The textures (6.13) and (6.14), as well as (6.16) and (6.17), make us wonder if our scheme implies quark mass matrices of texture zero.
Texture zero matrices can be said to have come about because of the need to reduce the number of free parameters, since the fermion mass matrices are 3x3 complex matrices, which without any constraints contain 36 real free parameters. It is however always possible to perform a unitary transformation that renders an arbitrary mass matrix Hermitian [5], so there is no loss of generality to assume that the mass matrices be Hermitian, reducing the number of free parameters to 18. This is still a very big number, which in the end of the 1970-ies prompted Fritzsch
78 A. Kleppe
[4], [6] to introduce "texture zero matrices", mass matrices where a certain number of the entries are zero.
Since then, a huge amount of articles have appeared, with analyses of the very large number of (different types of) texture zero matrices and their phenomenology. In the course of this work, a number of of texture zero matrices has been ruled out, singling out a smaller subset of matrices as viable [7]. Among the texture 4 zero matrices the only matrices that are found to be viable are:
'A B 0N B* DC 0 C* 0,
ABC B* D0 ,C* 0 0,
A0B 0 0 C VB* C* D,
0 C 0N C* AB 0 B* D/
0 0 C 0 A B
C* B* D
'D C BN C* 0 0 B* 0 A
while
A00 0 C B .0 B* D,
and
A0B 0 C 0 B* 0 D
are among the matrices that are ruled out. In our scheme this precisely corresponds to the matrices (6.13), (6.14), (6.16) and (6.17), which means that our mass matrices M and M' are not of texture zero. This can be expressed as a constraint on the values of the angle p,
p = ^Nn± 9	(6.18)
where N G Z, ruling out the matrices M(2Nn — 9) and M'(1 Nn + 9), so our mass matrices M and M' are not of texture zero. Instead, they display a democratic texture.
6.5 Democratic mass matrices
Initially, we were looking for mass matrices with a democratic structure [3], where the assumption is that both the up- and down-sector mass matrices start out from a form of the type M0 = kN and M0 = k'N where
/i i r
N = I 1 1 1 iii
The underlying philosophy is that in the Standard Model, where the fermions get their masses 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.
A matrix of the form M = kN moreover has the mass spectrum (0,0,3k), reflecting the phenomenology of the fermion mass spectra with one very big, and two much smaller mass values. In the weak basis M = kN is however totally
6 Phenomenological Mass Matrices With a Democratic Texture
79
flavour symmetric, which means that the (weak) flavours ft are indistinguishible ("absolute democracy").
In the assumed initial stage, since the up-sector mass matrix and the down sector mass matrix are identical except for the dimensional coefficients k and k', the mixing matrix is equal to unity, so there is no CP-violation. In order to obtain the final mass spectra with the three hierarchical non-zero values, the initial flavour symmetry displayed by the matrices M0 and M0 must be broken, in such a way that the mixing matrix becomes the observed CKM matrix (with a CP-violating phase).
An "ansatz" within the democratic scenario then consists of a specific choice of a flavour symmetry breaking scheme. And it is precisely what we are looking for: a credible flavour symmetry breaking scheme that gives the observed mass spectra.
Our initial assumption is that the rotation matrices (6.5), (6.6) which diagonal-ize the up-sector and down-sector mass matrices, are given by the factorization of the Cabibbi-Koabayashi-Maskawa matrix (6.4), with well-known angles. The only "steering-parameter parameter" is then p, in the sense that different values of p correspond to mass matrices of different form.
6.5.1 A democratic substructure
We now reparametrize the mass matrices (6.12) and (6.15),
M =
Xc| + Ys2 (Y -	-Zs
(Y - X)S|xC|x Xs| + Ycj2
~ZS|x eiY
-ZC| eiY

-ZC| e-iY
and
X 's|, + Y ' c|,
M' = I (X' - Y')s|,C|, Z'c2/ e-iY
(X' - Y')s|,c|, X 'c|, + Y 's|, -Z's,,/ e-iY
— y
'2 '
Z 'c2/ eiY -Z's2/ eiY
in a way that reveals their "democratic substructure":
e
|
F
F
and
M = | R |Mn||R J + ( X I	(6.19)
SeiV \1 1 1 / V Se-iV V Q,
r ' M11^ r ' \ Ix '
M ' = 1 R '	1 1 1 1 1 1 1 R ' 1 + 1 X ' 1 (6.20)
V S 'e-iV \1 1 y \ S'eiyJ \ Q '
where
P = VÎY-Xsin(p - 9), R = VÏY-Xcos(p - 9), S =	, Q = F - S2,
and
80 A. Kleppe
P' = V|Y' - Xcos(p + 9), R' = -VlY' - Xsin(p + 9), S' = x, Q' =
F' - S'2.
These matrices can in their turn be rewritten as
/sin y	\ /1 1 1\ /sin y	\ /X \
M = B I cos y )(l1l)| cos y	I + I X I (6.21)
V	Geiy/ \1 1 1J \	Ge_iy/ \ Qj
where
y = p - 9, B = Y - X, G = -Z/(Y - X), Q = F - BG2. Likewise,
/cos y'	\ /1 1 1\ /cos y'	\ /X'
M' = B' I - sin y'	) (111)1 - sin y' ) + ( X'
V	G 'e-iV \1 1 y \	G 'eiy/ \ Q'
(6.22)
where
y' = p + 9, B' = Y' - X', G' = Z'/(Y' - X'), Q' = F' - B'G'2.
So without any assumptions about an initial democratic texture, we get a mass matrix structure that can be interpreted as originating from a democratic mass matrix, where the flavour symmetry has subsequently been broken in a very specific manner.
6.6 Flavour symmetry breaking mechanisms
The goal of our investigation is to get some hint about the form that the quark mass matrices take in the weak basis - and the hint we get from the matrices (6.21) and (6.22) is that the mass matrices come about from a kind of democratic scenario where the initial flavour symmetry is broken in a stepwise fashion.
Flavour symmetries relate the different flavours fj, and in the democratic scenario, where the initial form of the mass matrices is taken to be
I111)
M0 = kN = k I 1 1 1 I ,	(6.23)
111
the mass Lagrangian reads
3
Lmass = kfNf = X kfi fj i=1,j = 1
This means that in the democratic scheme, all the flavours f j are initially indistin-guishible, with the same Yukawa coupling for all the flavours: a totally flavour symmetric situation.
6 Phenomenological Mass Matrices With a Democratic Texture
81
Following the hint given by our approach, we now postulate that the mass matrices originate from a democratic form (6.23), and that the initial overall flavour symmetries have subsequently undergone a stepwise breaking. To show how this works, we start with a generic matrix Mo, and take the first symmetry breaking step to be
Mo = kN ^ Mi = I E I I 11 1 ll I E I	(6.24)
Here the mass spectrum is basically unchanged even though the flavour symmetry is partially broken, with the mass Lagrangian
Lmass = kfMi f = E2XX + EJ(xf3 + f3X) + J2ff3
where x = fi + f2; thus the flavour symmetry f ^ f2 is still unbroken. In the next step, we lift the remaining flavour symmetry by rotating the two equal terms,
(E, E) —» (Lsinn, Lcosn),
which gives
(sin n	\ /1 1 1\ /sin n
cosn I 11 1 1 I I cosn T/\l 1 l/V	T
where L2 is the only dimensional parameter, and T = J/L. In order to account for CP-violation, we moreover introduce a phase y, in a way that reflects that CP-violation is connected to the presence of three families (with only two families there is no CP-violation):
(sin n	\ /1 1 1\ /sin n	\
cos n I I 1 1 1 I I c°sn I , (6.26)
Teiy/ \1 1 1J \	Te-iYJ
where the CP-breaking phase is connected to the third family, as it should. We know nothing about the values of L, n, T, but by the assumption that the trace of the mass matrix is constant through all the flavour symmetry breaking steps, we get
L2 + T2 = 3k
But the matrix M3 still has determinant zero, and a mass spectrum with two vanishing and one non-zero mass value. We therefore add an extra term to M3, which like in (6.21) and (6.22), is of diagonal form. This gives us the final mass matrix
(sin n	\ /1 1 1\ /sin n	\
cos n II 1 1 1 II cos n I +A (6.27)
TeiV \1 1 1/ \	Te-iY/
where A is a diagonal matrix. Our scheme thus reads:
(6.25)
82 
A. Kleppe
•	We start with the democratic matrix M0 = kN, k = Trace(M)/3, with total flavour symmetry f ^ f2 & f3 in the weak basis.
•	Assumption: the trace of the matrix M is constant throughout every flavour symmetry breaking step.
•	First flavour breaking step (6.24): M0 —> Mi. The flavour symmetry f i ^ f2 still remains, and there is still only one non-zero mass value, but f3 is singled out.
•	Next flavour breaking step (6.25): M1 —» M2, lifting the flavour symmetry fi f2.
•	Introducing a CP-violating phase (6.26): M2 —> M3.
•	Last step (6.27): adding a diagonal matrix to M3, M3 —» M4 = M3 + A, whereby we get the three observed non-zero mass values.
6.7 Conclusion
Without introducing any new assumptions, by just factorizing the "standard parametrization" of the CKM weak mixing matrix in a specific way, we obtain mass matrices with a specific type of democratic texture and a well-defined scheme for breaking the initial flavour symmetry. Our approach thus hints at a democratic scenario, which comes from the formalism without any other assumptions than a very natural and straightforward way of factorizing the weak mixing matrix.
References
1.	M. Kobayashi, T. Maskawa; Maskawa (1973), "CP-Violation in the Renormalizable Theory of Weak Interaction", Progress of Theoretical Physics 49 (2): 652657.
2.	L.L. Chau and W.-Y. Keung (1984), "Comments on the Parametrization of the Kobayashi-Maskawa Matrix", Phys. Rev. Letters 53 (19): 1802.
3.	A. Kleppe, "A democratic suggestion", hep-ph/1608.08988.
4.	H. Fritzsch, Phys. Lett. B 70, 436 (1977), Phys. Lett. B 73, 317 (1978).
5.	D. Emmanuel-Costa and C. Simoes, Phys. Rev. D 79, 073006 (2009).
6.	H. Fritzsch, "Texture Zero Mass Matrices and Flavor Mixing of Quarks and Leptons", hep-ph/1503.07927v1.
7.	S. Sharma, P. Fakay, G. Ahuja, M. Gupta, Phys. Rev. D 91, 053004 (2015).