Flavor Mixing of Quarks and a NewTexture

Harald Fritzsch
Department für Physik

Universität München

Theresienstraße 37

D-80333 München, Germany

###### Abstract

We discuss a new mass matrix with specific texture zeros for the quarks. The
three flavor mixing angles for the quarks are functions of the quark masses
and can be calculated. The following ratios among CKM matrix elements are
given by ratios of quark masses:

$|V_{td}/V_{ts}|\simeq\sqrt{m_{d}/m_{s}}$ and
$|V_{ub}/V_{cb}|\simeq\sqrt{m_{u}/m_{c}}$.

Also we can calculate two CKM matrix elements:

$|V_{cb}|\simeq|V_{ts}|\simeq 2\left(m_{s}/m_{b}\right)$.

This relation as well as the relation
$|V_{td}/V_{ts}|\simeq\sqrt{m_{d}/m_{s}}$ are in good agreement with the
experimental data. There is a problem with the relation
$|V_{ub}/V_{cb}|\simeq\sqrt{m_{u}/m_{c}}$, probably due to wrong estimates of
the quark masses $m_{u}$ and $m_{c}$.

The flavor mixing of the quarks and leptons is still a mystery in physics, and
thus far it is not possible to calculate the flavor mixing angles. These
angles have been measured in many experiments. For the six quarks the flavor
mixing is described by the CKM matrix:

$\displaystyle V_{CKM}=\left(\matrix{V_{ud}&V_{us}&V_{ub}\cr
V_{cd}&V_{cs}&V_{cb}\cr V_{td}&V_{ts}&V_{tb}\cr}\right).$ (1)

The absolute values of the nine matrix elements have been measured to a high
degree of accuracy, as given by the Particle Data group [1]:

$\displaystyle\left|V_{\rm CKM}\right|=\left(\matrix{0.97401\pm
0.00011&0.22650\pm 0.00048&0.00361^{+0.00011}_{-0.00009}\cr 0.22636\pm
0.00048&0.97320\pm 0.00011&0.04053^{+0.00083}_{-0.00061}\cr
0.00854^{+0.00023}_{-0.00016}&0.03978^{+0.00082}_{-0.00060}&0.999172^{+0.000024}_{-0.000035}\cr}\right)\;.$
(2)

The matrix elements $|V_{us}|$ and $|V_{cd}|$ are approximately equal, also
the matrix elements $|V_{cb}|$ and $|V_{ts}|$. The following two ratios of the
CKM matrix elements are particularly interesting and will be discussed later
in more detail:

$\displaystyle\left|\frac{V_{ub}}{V_{cb}}\right|=0.0891^{+0.0041}_{-0.0040}\;,\quad\left|\frac{V_{td}}{V_{ts}}\right|=0.2147^{+0.0092}_{-0.0083}\;.$
(3)

There are several ways to describe the CKM-matrix in terms of three angles and
one phase parameter. I prefer the parametrization, which Z. Xing and I
introduced years ago [2], given by the angles $\theta_{u}$, $\theta_{d}$,
$\theta$ and a phase parameter $\phi$, which describes CP violation:

$\displaystyle V_{\rm CKM}$ $\displaystyle=$
$\displaystyle\left(\matrix{c_{\rm u}&s_{\rm u}&0\cr-s_{\rm u}&c_{\rm u}&0\cr
0&0&1\cr}\right)\times\left(\matrix{e^{-{\rm i}\phi}&0&0\cr 0&c&s\cr
0&-s&c\cr}\right)\times\left(\matrix{c&-s_{\rm d}&0\cr s_{\rm d}&c_{\rm
d}&0\cr 0&0&1\cr}\right)$ (4) $\displaystyle=$
$\displaystyle\left(\matrix{s_{\rm u}s_{\rm d}c+c_{\rm u}c_{\rm d}e^{-{\rm
i}\phi}&s_{\rm u}c_{\rm d}c-c_{\rm u}s_{\rm d}e^{-{\rm i}\phi}&s_{\rm u}s\cr
c_{\rm u}s_{\rm d}c-s_{\rm u}c_{\rm d}e^{-{\rm i}\phi}&c_{\rm u}c_{\rm
d}c+s_{\rm u}s_{\rm d}e^{-{\rm i}\phi}&c_{\rm u}s\cr-s_{\rm d}s&-c_{\rm
d}s&c\cr}\right)\;.$

Here we used the short notation: $c_{u,d}\sim\cos\theta_{u,d}$,
$s^{~{}}_{u,d}\sim\sin\theta_{u,d}$, $c\sim\cos\theta$ and $s\sim\sin\theta$.

These four parameters, three angles and one phase parameter, can be determined
from the experimental data, given in eqs. (2) and (3). Their central values
are:

$\displaystyle\theta_{u}\simeq 5.0^{\circ}\;,\quad\theta_{d}\simeq
12.3^{\circ}\;,\quad\theta\simeq 2.4^{\circ}\;,\quad\phi\simeq 90^{\circ}\;.$
(5)

Relations between the quark masses and the mixing angles can be derived, if
the quark mass matrices have ”texture zeros”, as shown by S. Weinberg and me
in 1977 [3].

In 1978 I studied the flavor mixing of six quarks. Here I used a mass matrix
with four ”texture zeros” [4]:

$\displaystyle M_{\rm q}=\left(\matrix{0&A_{\rm q}&0\cr A^{*}_{\rm q}&0&B_{\rm
q}\cr 0&B^{*}_{\rm q}&C_{\rm q}\cr}\right)\;.$ (6)

Here q stands for ”u” (the ”upsector”) or ”d” ( the ”down sector”). This
specific mass matrix has been studied later by many physicists.

Now the two mass matrices are diagonalized by the unitary transformations
$V^{\dagger}_{\rm u}M_{\rm u}V_{\rm u}={\rm diag}\\{m_{u},-m_{c},m_{t}\\}$ and
$V^{\dagger}_{\rm d}M_{\rm d}V_{\rm d}={\rm diag}\\{m_{d},-m_{s},m_{b}\\}$.

The CKM matrix is given by the product of the two unitary matrices: $V\equiv
V^{\dagger}_{\rm u}V_{\rm d}$. Thus one can calculate the CKM mixing angles
$\theta_{u}$, $\theta_{d}$ and $\theta$ as functions of the quark masses. In a
good approximation we find:

$\tan\theta_{d}\simeq|V_{td}/V_{ts}|\simeq\sqrt{m_{d}/m_{s}},\hskip
28.45274pt\tan\theta_{u}\simeq|V_{ub}/V_{cb}|\simeq\sqrt{m_{u}/m_{c}}.$ (7)

$\sin\theta\simeq|V_{cb}|\simeq|\sqrt{m_{s}/m_{b}}-e^{{\rm
i}\varphi}\sqrt{m_{c}/m_{t}}|.$ (8)

In the last relation a phase parameter appears - the phase difference between
the off-diagonal matrix elments B and B*.

The values of the six quark masses have been measured in many experiments.
These values depend on the energy scale. They have recently been calculated at
the energy scale, given by the mass of the Z-boson: $M_{Z}\simeq 91.2~{}{\rm
GeV}$ ( [5], [6] ):

$m_{u}=1.24\pm 0.22~{}{\rm MeV},m_{c}=0.62\pm 0.02~{}{\rm GeV},m_{t}=168.26\pm
0.75~{}{\rm GeV}.$

$m_{d}=2.69\pm 0.19~{}{\rm MeV},m_{s}=53.5\pm 4.6~{}{\rm MeV},m_{b}=2.86\pm
0.03~{}{\rm GeV}.$

Thus the relative uncertainty of a quark mass is larger for the small quark
masses - the mass of the t-quark has the smallest relative error - only about
$0.5\%$. For the u-quark the relative error is about 17 percent.

Using these quark masses and eq.(8), one finds, if the phase parameter is
zero:

$|V_{cb}|\simeq 0.076.$ (9)

But the eperiments give (see eq. (2)):

$|V_{cb}|\simeq 0.041.$ (10)

If the phase parameter is not zero, $|V_{cb}|$ is even larger. Thus the mass
matrix with four texture zeros, given in eq. (6), is not correct.

I was searching for a mass matrix with a specific texture, which allows to
calculate $V_{cb}$ and $V_{ts}$ correctly. Now I found the following mass
matrix with three texture zeros:

$\displaystyle M_{\rm q}=\left(\matrix{0&A&0\cr A^{*}&|B/2|&B\cr
0&B^{*}&C\cr}\right)\;.$ (11)

Although a nonzero (2,2) matrix element has been introduced to modify my
original texture (eq. (6)) ( see e.g.[7, 8]), it is the first time that the
specific texture in eq. (11) is considered.

This matrix can be diagonalized by an orthogonal transformation, which is
similar to the transformation, used for the mass matrix, given in eq. (6).
Using the parameters of this transformation, one can calculate the mixing
angles as functions of the quark masses:

$\tan\theta_{d}\simeq|V_{td}/V_{ts}|\simeq\sqrt{m_{d}/m_{s}},\hskip
28.45274pt\\\ \tan\theta_{u}\simeq|V_{ub}/V_{cb}|\simeq\sqrt{m_{u}/m_{c}}.$
(12)

These two relations did not change, if the mass matrix is changed from (6) to
(11). But the third relation changes:

$\displaystyle\sin\theta\simeq|V_{cb}|\simeq|V_{ts}|\simeq
2\left|\frac{m_{s}}{m_{b}}-e^{{\rm i}\varphi}\frac{m_{c}}{m_{t}}\right|\simeq
2\frac{m_{s}}{m_{b}}\;.$ (13)

Here again a phase parameter $\varphi$ appears, since the mass matrices have
complex off-diagonal matrixelements. However in a good approximation this
phase can be neglected, since the mass ratio (${m_{c}}/{m_{t}})$ is very small
- it contributes only about $0.4\%$. If the ratio (${m_{c}}/{m_{t}}$) is
neglected, we obtain:

$\displaystyle|V_{cb}|\simeq|V_{ts}|\simeq 2\frac{m_{s}}{m_{b}}\;.$ (14)

Now we compare the experimental value of $|V_{cb}|$ and the estimated value of
${m_{s}}$/${m_{b}}$:

$\displaystyle\left|V_{cb}\right|:0.0395=>0.0415,$ (15) $\displaystyle
2\frac{m_{s}}{m_{b}}:0.034=>0.042.$ (16)

In a good approximation both numbers are equal. Thus the third angle of the
CKM matrix is also given by a ratio of quark masses: (${m_{s}}/{m_{b}}$).

Now we consider the ratio:

$\displaystyle\left|\frac{V_{td}}{V_{ts}}\right|=0.2147^{+0.0092}_{-0.0083}\;.$
(17)

This ratio can be calculated - it is given by the ratio of $m_{d}$/$m_{s}$:

$|V_{td}/V_{ts}|\simeq\sqrt{m_{d}/m_{s}}\simeq 0.224.$ (18)

Thus the experimental number and the calculated value agree very well. But
there is a problem with the second relation:

$\displaystyle\left|\frac{V_{ub}}{V_{cb}}\right|=0.0891^{+0.0041}_{-0.0040}\;.$
(19)

Again this ratio can be calculated. It is given by the ratio
($m_{u}$/$m_{c}$). If the quark masses, which are quoted above, are used, one
finds:

$|V_{ub}/V_{cb}|\simeq\sqrt{m_{u}/m_{c}}\simeq 0.045.$ (20)

The experimental value is almost twice as large as the value, given by the
ratio ($m_{u}$/$m_{c}$), and the question arises: Is this mass ratio correct?

The ratio ($m_{d}$/$m_{s}$) can be calculated in chiral perturbation theory -
thus it is better known than the quark masses $m_{d}$ or $m_{s}$.

However the ratio ($m_{u}$/$m_{c}$) cannot be calculated in chiral
perturbation theory. It can only be calculated by using the actual values of
the quark masses. The errors of the quark masses, quoted above, are probably
much larger than indicated. For example, the up quark mass could be 3.8 MeV
and the mass of the charm quark about 600 MeV. In this case we would obtain:

$|V_{ub}/V_{cb}|\simeq\sqrt{m_{u}/m_{c}}\simeq 0.08.$ (21)

Thus there would be no problem with the experimental value.

It seems that the quark mass matrices with the texture, given in eq. (11),
describe very well the flavor mixing of the quarks. The quark flavors mix,
since the mass matrix of the quarks has this specific texture. This texture
might be due to a new discrete symmetry, which is beyond the Standard Model of
particle physics. A deeper understanding of this texture might open the road
to the physics beyond the Standard Model.

## References

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H. Fritzsch and Z. Z. Xing, Phys. Rev. D 57 (1998) 594.

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S. Weinberg, Trans. New York Acad. Sci. 38, 185 (1977).

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H. Fritzsch. Nucl. Physics B 155 (1979) 189.

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G. Y. Huang and S. Zhou, Phys. Rev. D 103 (2021), arXiv:2009.04851.

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