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CarlB
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Abstract: Since 1982 the Koide mass relation has provided an amazingly
accurate relation between the masses of the charged leptons. In
this note we show how the Koide relation can be expanded to cover
the neutrinos, and we use the relation to predict neutrino masses.
Full paper attached, this version is slightly abbreviated.
In 1982, Yoshio Koide [ http://www.arxiv.org/abs/hepph/0506247 ], discovered a formula relating the
masses of the charged leptons:
[tex]\frac{(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2}
{m_e+m_\mu+m_\tau} = \frac{3}{2}.[/tex]
Written in the above manner, this relation removes one degree of
freedom from the three charged lepton masses. In this paper, we
will first derive this relation as an eigenvalue equation, then
obtain information about the other degrees of freedom, and finally
speculatively apply the same techniques to the problem of predicting the neutrino masses.
If we suppose that the leptons are composite particles made up of
colorless combinations of colored subparticles or preons, we expect
that the three colors must be treated equally, and therefore a
natural form for a matrix operator that can cross generation
boundaries is the \emph{circulant}:
[tex]\Gamma(A,B,C) = \left( \begin{array}{ccc} A&B&C\\C&A&B\\B&C&A
\end{array}\right).[/tex]
where A, B and C are complex constants. Other authors have
explored these sorts of matrices in the context of neutrino masses
and mixing angles. Such matrices have eigenvectors of the form:
[tex]n\ra = \left(\begin{array}{c}
1\\e^{+2in\pi/3}\\e^{2in\pi/3}\end{array}\right), \;\;\;\;n=1,2,3.
[/tex]
If we require that the eigenvalues be real, we obtain that $A$ must
be real, and that B and C are complex conjugates. This reduces
the 6 real degrees of freedom present in the 3 complex constants
A, B and C to just 3 real degrees of freedom, the same as the
number of eigenvalues for the operators. In order to parameterize
these sorts of operators,let us write:
[tex]\Gamma(\mu,\eta,\delta) = \mu\left(\begin{array}{ccc}
1&\eta\exp(+i\delta)&\eta\exp(i\delta)\\
\eta\exp(i\delta)&1&\eta\exp(+i\delta)\\
\eta\exp(+i\delta)&\eta\exp(i\delta)&1
\end{array}\right),[/tex]
where we can assume eta to be non negative. Note that while
eta and delta are pure numbers, mu scales with the
eigenvalues. Then the three eigenvalues are given by:
[tex]\begin{array}{rcl}
\Gamma(\mu,\eta,\delta)\;n\ra &=& \lambda_n \;n\ra,\\
&=&\mu(1+2\eta\cos(\delta+2n\pi/3))\;n\ra.
\end{array}[/tex]
The sum of the eigenvalues are given by the trace of Gamma:
[tex]\lambda_1+\lambda_2+\lambda_3 = 3\mu,[/tex]
and this allows us to calculate mu from a set of eigenvalues. The
sum of the squares of the eigenvalues are given by the trace of
Gamma^2:
[tex]\lambda_1^2+\lambda_2^2+\lambda_3^2 = 3\mu^2(1+2\eta^2),
[/tex]
and this gives a formula for eta^2 in terms of the eigenvalues:
[tex]\frac{\lambda_1^2+\lambda_2^2+\lambda_3^2}
{(\lambda_1+\lambda_2+\lambda_3)^2} = \frac{1+2\eta^2}{3},[/tex]
The value of delta is then easy to calculate.
We have restricted Gamma to a form where all its eigenvalues are
real, but it is still possible that some or all will be negative.
For situations where all the eigenvalues are non negative, it is
natural to suppose that our values are eigenvalues not of the
Gamma matrix, but instead of Gamma^2.
The masses of the charged leptons are positive, so let us compute
the square roots of the masses of the charged leptons, and find the
values for mu_1, eta_1^2 and delta_1, where the subscript
will distinguish the parameters for the masses of the charged
leptons from that of the neutral leptons.
Given the latest PDG data (MeV):
[tex]\begin{array}{rcccl}
m_1 &=& m_e &=& 0.510998918 (44)\\
m_2 &=& m_\mu &=& 105.6583692 (94)\\
m_3 &=& m_\tau &=& 1776.99 +0.290.26
\end{array}[/tex]
and ignoring, for the moment, the error bars, and keeping 7 digits
of accuracy, we obtain
[tex]\begin{array}{rcl}
\mu_1 &=& 17.71608 \;\;{\rm MeV}^{0.5}\\
\eta_1^2 &=& 0.5000018\\
\delta_1 &=& 0.2222220
\end{array}[/tex]
The fact that eta_1^2 is very close to 0.5 was noticed in 1982
by Yoshio Koide at a time when the
tau mass had not been experimentally measured to anywhere near
its present accuracy. That delta_1 is close to
2/9 went unnoticed until this author discovered it in
2005 and announced it here on Physics Forums.
The present constraints on the electron, muon and tau masses exclude
the possibility that \delta_1 is exactly 2/9, while on the other
hand, \eta_1^2 = 0.5 fits the data close to the middle of the
error bars. If we interpret \Gamma as a matrix of coupling
constants, \eta^2 = 1/2 is a probability. Thus if the preons are
to be spin1/2 states, the P = (1+\cos(\theta))/2 rule for
probabilities implies that the three coupled states are
perpendicular, a situation that would be more natural for classical
waves than quantum states.
By making the assumption that \eta_1^2 is precisely 0.5, one
obtains a prediction for the mass of the tau. Since the best
measurements for the electron and muon masses are in atomic mass
units, we give the predicted tau mass in both those units and in MeV:
[tex]\begin{array}{rcll}
m_\tau &=& 1776.968921(158) \;\; & {\rm MeV}\\
&=& 1.907654627(46) \;\; & {\rm AMU}.
\end{array}[/tex]
The error bars in the above, and in later calculations in this
paper, come from assuming that the electron and muon masses are
anywhere inside the measured mass error bars.
This gives us the opportunity to fine tune our estimate for \mu_1
and \delta_1. Since the electron and muon data are the most
exact, we assume the Koide relation and compute the tau mass from
them. Then we compute \mu_1 and then \delta_1 over the range of
electron and muon masses, obtaining:
[tex]\begin{array}{rcll}
\delta_1 &=&.22222204715(311)\;\;\;&\textrm{from MeV data}\\
&=&.22222204717(48)\;\;\;&\textrm{from AMU data}.
\end{array}[/tex]
If \delta_1 were zero, the electron and muon would have equal
masses, while if \delta_1 were \pi/12, the electron would be
massless. Instead, \delta_1 is close to a rational fraction,
while the other terms inside the cosine are rational multiples of
pi. Using the more accurate AMU data, the difference
between \delta_1 and 2/9 is:
[tex]2/9  \delta_1 = 1.7505(48)\;\times10^{7},[/tex]
and we can hope that a deeper theory will allow this small
difference to be computed. This difference could be written as
[tex]1.75\;\times 10^{7} = \frac{4\pi}{3^{12}}\left(\alpha +
\mathcal{O}(\alpha^2)\right)[/tex]
for example.
Note that \delta_1 is close to a rational number, while the other
terms that are added to it inside the cosine are rational multiples of pi. This
distinction follows our parameterization of the eigenvalues in that
the rational fraction part comes from the operator Gamma, while
the 2n\pi/3 term comes from the eigenvectors. Rather than
depending on the details of the operator, the 2n\pi/3 depends only
on the fact that the operator has the symmetry of a circulant
matrix.
We will use m_1, m_2 and m_3 to designate the masses of the
neutrinos. The experimental situation with the neutrinos is
primitive at the moment. The only accurate measurements are from
oscillation experiments, and are for the absolute values of the
differences between squares of neutrino masses. Recent 2\sigma
data are:
[tex]\begin{array}{rcll}
m_2^2m_1^2 &=& 7.92(1\pm.09) \times 10^{5}\;\; & {\rm eV}^2\\
m_3^2m_2^2 &=& 2.4(1+0.210.26) \times 10^{3}\;\; & {\rm eV}^2
\end{array}[/tex]
Up to this time, attempts to apply the unaltered Koide mass formula
to the neutrinos have
failed, but these attempts have assumed that
the square roots of the neutrino masses must all be
positive. Without loss of generality, we
will assume that \mu_0 and \eta_0 are both positive, thus there
can be at most one square root mass that is negative, and it can be
only the lowest or central mass.
Of these two cases, having the central mass with a negative square
root is incompatible with the oscillation data, but we can obtain
\eta_0^2 = 1/2 with masses around:
[tex]\begin{array}{rcl}
m_1 &=& 0.0004\;\; {\rm eV},\\
m_2 &=& 0.009\;\; {\rm eV},\\
m_3 &=& 0.05\;\; {\rm eV}.
\end{array}[/tex]
These masses approximately satisfy the squared mass differences of
as well as the Koide relation as
follows:
[tex]
\frac{m_1+m_2+m_3}{(\sqrt{m_1}+\sqrt{m_2}+\sqrt{m_3})^2}
\;=\;\frac{2}{3}.[/tex]
The above neutrino masses were chosen to fix the value of eta_0^2
= 1/2. As such, the fact that this can be done is of little
interest, at least until absolute measurements of the neutrino
masses are available. However, given these values, we can now
compute the mu_0 and delta_0 values for the neutrinos. The
results give that, well within experimental error:
[tex]
\begin{array}{rcl}
\delta_0 &=& \delta_1\;\; +\;\; \pi/12,\\
\mu_1 / \mu_0 &=& 3^{11}.
\end{array}[/tex]
Recalling the split between the components of the cosine in the
charged lepton mass formula, the fact that pi/12 is a rational
multiple of pi suggests that it should be related to a symmetry of
the eigenvectors rather than the operator. One possible explanation
is that in transforming from a right handed particle to a left
handed particle, the neutrino (or a portion of it) pick up a phase
difference that is a fraction of pi. As a result the neutrino
requires 12 stages to make the transition from left handed back to
left handed (with the same phase), while the electron requires only
a single stage, and it is natural for the mass hierarchy between the
charged and neutral leptons to be a power of 121 = 11.
We will therefore assume that the relations given above
are exact, and can then use the
measured masses of the electron and muon to write down predictions
for the absolute masses of the neutrinos. The parameterization of
the masses of the neutrinos is as follows:
[tex]m_n = \frac{\mu_1}{3^{11}}(1+\sqrt{2}\cos(\delta_1+\pi/12 +
2n\pi/3)),[/tex]
where mu_1 and delta_1 are from the charged leptons.
Substituting in the the measured values for the electron and muon
masses, we obtain extremely precise predictions for the neutrino
masses:
[tex]\begin{array}{rcll}
m_1 &=& 0.000383462480(38) \;\; & {\rm eV}\\
&=&0.4116639106(115)\times 10^{12} \;\; & {\rm AMU}
\end{array}[/tex]
[tex]\begin{array}{rcll}
m_2 &=& 0.00891348724(79)\;\; & {\rm eV}\\
&=&9.569022271(246)\times 10^{12} \;\; & {\rm AMU}
\end{array}
[/tex]
[tex]\begin{array}{rcll}
m_3 &=& 0.0507118044(45) \;\; & {\rm eV}\\
&=&54.44136198(131)\times 10^{12} \;\; & {\rm AMU}
\end{array}[/tex]
Similarly, the predictions for the differences of the squares of the
neutrino masses are:
[tex]\begin{array}{rcll}
m_2^2m_1^2 &=& 7.930321129(141)\times 10^{5} \;\; & {\rm eV}^2\\
&=&.913967200(47)\times 10^{24} \;\; & {\rm AMU}^2
\end{array}
[/tex]
[tex]\begin{array}{rcll}
m_3^2m_2^2 &=& 2.49223685(44)\times 10^{3} \;\; & {\rm eV}^2\\
&=&2872.295707(138)\times 10^{24} \;\; & {\rm AMU}^2
\end{array}[/tex]
As with the Koide predictions for the tau mass, these predictions
for the squared mass differences are dead in the center of the error
bars. We can only hope that the future will show our calculations
to be as prescient as Koide's.
That the masses of the leptons should have these sorts of
relationships is particularly mysterious in the context of the
standard model. It is
hoped that this paper will stimulate thought among theoreticians.
Perhaps the fundamental fermions are bound states of deeper objects.
The author would like to thank Yoshio Koide and Alejandro Rivero for
their advice, encouragement and references, and Mark Mollo
(Liquafaction Corporation) for financial support.
Carl
accurate relation between the masses of the charged leptons. In
this note we show how the Koide relation can be expanded to cover
the neutrinos, and we use the relation to predict neutrino masses.
Full paper attached, this version is slightly abbreviated.
In 1982, Yoshio Koide [ http://www.arxiv.org/abs/hepph/0506247 ], discovered a formula relating the
masses of the charged leptons:
[tex]\frac{(\sqrt{m_e} + \sqrt{m_\mu} + \sqrt{m_\tau})^2}
{m_e+m_\mu+m_\tau} = \frac{3}{2}.[/tex]
Written in the above manner, this relation removes one degree of
freedom from the three charged lepton masses. In this paper, we
will first derive this relation as an eigenvalue equation, then
obtain information about the other degrees of freedom, and finally
speculatively apply the same techniques to the problem of predicting the neutrino masses.
If we suppose that the leptons are composite particles made up of
colorless combinations of colored subparticles or preons, we expect
that the three colors must be treated equally, and therefore a
natural form for a matrix operator that can cross generation
boundaries is the \emph{circulant}:
[tex]\Gamma(A,B,C) = \left( \begin{array}{ccc} A&B&C\\C&A&B\\B&C&A
\end{array}\right).[/tex]
where A, B and C are complex constants. Other authors have
explored these sorts of matrices in the context of neutrino masses
and mixing angles. Such matrices have eigenvectors of the form:
[tex]n\ra = \left(\begin{array}{c}
1\\e^{+2in\pi/3}\\e^{2in\pi/3}\end{array}\right), \;\;\;\;n=1,2,3.
[/tex]
If we require that the eigenvalues be real, we obtain that $A$ must
be real, and that B and C are complex conjugates. This reduces
the 6 real degrees of freedom present in the 3 complex constants
A, B and C to just 3 real degrees of freedom, the same as the
number of eigenvalues for the operators. In order to parameterize
these sorts of operators,let us write:
[tex]\Gamma(\mu,\eta,\delta) = \mu\left(\begin{array}{ccc}
1&\eta\exp(+i\delta)&\eta\exp(i\delta)\\
\eta\exp(i\delta)&1&\eta\exp(+i\delta)\\
\eta\exp(+i\delta)&\eta\exp(i\delta)&1
\end{array}\right),[/tex]
where we can assume eta to be non negative. Note that while
eta and delta are pure numbers, mu scales with the
eigenvalues. Then the three eigenvalues are given by:
[tex]\begin{array}{rcl}
\Gamma(\mu,\eta,\delta)\;n\ra &=& \lambda_n \;n\ra,\\
&=&\mu(1+2\eta\cos(\delta+2n\pi/3))\;n\ra.
\end{array}[/tex]
The sum of the eigenvalues are given by the trace of Gamma:
[tex]\lambda_1+\lambda_2+\lambda_3 = 3\mu,[/tex]
and this allows us to calculate mu from a set of eigenvalues. The
sum of the squares of the eigenvalues are given by the trace of
Gamma^2:
[tex]\lambda_1^2+\lambda_2^2+\lambda_3^2 = 3\mu^2(1+2\eta^2),
[/tex]
and this gives a formula for eta^2 in terms of the eigenvalues:
[tex]\frac{\lambda_1^2+\lambda_2^2+\lambda_3^2}
{(\lambda_1+\lambda_2+\lambda_3)^2} = \frac{1+2\eta^2}{3},[/tex]
The value of delta is then easy to calculate.
We have restricted Gamma to a form where all its eigenvalues are
real, but it is still possible that some or all will be negative.
For situations where all the eigenvalues are non negative, it is
natural to suppose that our values are eigenvalues not of the
Gamma matrix, but instead of Gamma^2.
The masses of the charged leptons are positive, so let us compute
the square roots of the masses of the charged leptons, and find the
values for mu_1, eta_1^2 and delta_1, where the subscript
will distinguish the parameters for the masses of the charged
leptons from that of the neutral leptons.
Given the latest PDG data (MeV):
[tex]\begin{array}{rcccl}
m_1 &=& m_e &=& 0.510998918 (44)\\
m_2 &=& m_\mu &=& 105.6583692 (94)\\
m_3 &=& m_\tau &=& 1776.99 +0.290.26
\end{array}[/tex]
and ignoring, for the moment, the error bars, and keeping 7 digits
of accuracy, we obtain
[tex]\begin{array}{rcl}
\mu_1 &=& 17.71608 \;\;{\rm MeV}^{0.5}\\
\eta_1^2 &=& 0.5000018\\
\delta_1 &=& 0.2222220
\end{array}[/tex]
The fact that eta_1^2 is very close to 0.5 was noticed in 1982
by Yoshio Koide at a time when the
tau mass had not been experimentally measured to anywhere near
its present accuracy. That delta_1 is close to
2/9 went unnoticed until this author discovered it in
2005 and announced it here on Physics Forums.
The present constraints on the electron, muon and tau masses exclude
the possibility that \delta_1 is exactly 2/9, while on the other
hand, \eta_1^2 = 0.5 fits the data close to the middle of the
error bars. If we interpret \Gamma as a matrix of coupling
constants, \eta^2 = 1/2 is a probability. Thus if the preons are
to be spin1/2 states, the P = (1+\cos(\theta))/2 rule for
probabilities implies that the three coupled states are
perpendicular, a situation that would be more natural for classical
waves than quantum states.
By making the assumption that \eta_1^2 is precisely 0.5, one
obtains a prediction for the mass of the tau. Since the best
measurements for the electron and muon masses are in atomic mass
units, we give the predicted tau mass in both those units and in MeV:
[tex]\begin{array}{rcll}
m_\tau &=& 1776.968921(158) \;\; & {\rm MeV}\\
&=& 1.907654627(46) \;\; & {\rm AMU}.
\end{array}[/tex]
The error bars in the above, and in later calculations in this
paper, come from assuming that the electron and muon masses are
anywhere inside the measured mass error bars.
This gives us the opportunity to fine tune our estimate for \mu_1
and \delta_1. Since the electron and muon data are the most
exact, we assume the Koide relation and compute the tau mass from
them. Then we compute \mu_1 and then \delta_1 over the range of
electron and muon masses, obtaining:
[tex]\begin{array}{rcll}
\delta_1 &=&.22222204715(311)\;\;\;&\textrm{from MeV data}\\
&=&.22222204717(48)\;\;\;&\textrm{from AMU data}.
\end{array}[/tex]
If \delta_1 were zero, the electron and muon would have equal
masses, while if \delta_1 were \pi/12, the electron would be
massless. Instead, \delta_1 is close to a rational fraction,
while the other terms inside the cosine are rational multiples of
pi. Using the more accurate AMU data, the difference
between \delta_1 and 2/9 is:
[tex]2/9  \delta_1 = 1.7505(48)\;\times10^{7},[/tex]
and we can hope that a deeper theory will allow this small
difference to be computed. This difference could be written as
[tex]1.75\;\times 10^{7} = \frac{4\pi}{3^{12}}\left(\alpha +
\mathcal{O}(\alpha^2)\right)[/tex]
for example.
Note that \delta_1 is close to a rational number, while the other
terms that are added to it inside the cosine are rational multiples of pi. This
distinction follows our parameterization of the eigenvalues in that
the rational fraction part comes from the operator Gamma, while
the 2n\pi/3 term comes from the eigenvectors. Rather than
depending on the details of the operator, the 2n\pi/3 depends only
on the fact that the operator has the symmetry of a circulant
matrix.
We will use m_1, m_2 and m_3 to designate the masses of the
neutrinos. The experimental situation with the neutrinos is
primitive at the moment. The only accurate measurements are from
oscillation experiments, and are for the absolute values of the
differences between squares of neutrino masses. Recent 2\sigma
data are:
[tex]\begin{array}{rcll}
m_2^2m_1^2 &=& 7.92(1\pm.09) \times 10^{5}\;\; & {\rm eV}^2\\
m_3^2m_2^2 &=& 2.4(1+0.210.26) \times 10^{3}\;\; & {\rm eV}^2
\end{array}[/tex]
Up to this time, attempts to apply the unaltered Koide mass formula
to the neutrinos have
failed, but these attempts have assumed that
the square roots of the neutrino masses must all be
positive. Without loss of generality, we
will assume that \mu_0 and \eta_0 are both positive, thus there
can be at most one square root mass that is negative, and it can be
only the lowest or central mass.
Of these two cases, having the central mass with a negative square
root is incompatible with the oscillation data, but we can obtain
\eta_0^2 = 1/2 with masses around:
[tex]\begin{array}{rcl}
m_1 &=& 0.0004\;\; {\rm eV},\\
m_2 &=& 0.009\;\; {\rm eV},\\
m_3 &=& 0.05\;\; {\rm eV}.
\end{array}[/tex]
These masses approximately satisfy the squared mass differences of
as well as the Koide relation as
follows:
[tex]
\frac{m_1+m_2+m_3}{(\sqrt{m_1}+\sqrt{m_2}+\sqrt{m_3})^2}
\;=\;\frac{2}{3}.[/tex]
The above neutrino masses were chosen to fix the value of eta_0^2
= 1/2. As such, the fact that this can be done is of little
interest, at least until absolute measurements of the neutrino
masses are available. However, given these values, we can now
compute the mu_0 and delta_0 values for the neutrinos. The
results give that, well within experimental error:
[tex]
\begin{array}{rcl}
\delta_0 &=& \delta_1\;\; +\;\; \pi/12,\\
\mu_1 / \mu_0 &=& 3^{11}.
\end{array}[/tex]
Recalling the split between the components of the cosine in the
charged lepton mass formula, the fact that pi/12 is a rational
multiple of pi suggests that it should be related to a symmetry of
the eigenvectors rather than the operator. One possible explanation
is that in transforming from a right handed particle to a left
handed particle, the neutrino (or a portion of it) pick up a phase
difference that is a fraction of pi. As a result the neutrino
requires 12 stages to make the transition from left handed back to
left handed (with the same phase), while the electron requires only
a single stage, and it is natural for the mass hierarchy between the
charged and neutral leptons to be a power of 121 = 11.
We will therefore assume that the relations given above
are exact, and can then use the
measured masses of the electron and muon to write down predictions
for the absolute masses of the neutrinos. The parameterization of
the masses of the neutrinos is as follows:
[tex]m_n = \frac{\mu_1}{3^{11}}(1+\sqrt{2}\cos(\delta_1+\pi/12 +
2n\pi/3)),[/tex]
where mu_1 and delta_1 are from the charged leptons.
Substituting in the the measured values for the electron and muon
masses, we obtain extremely precise predictions for the neutrino
masses:
[tex]\begin{array}{rcll}
m_1 &=& 0.000383462480(38) \;\; & {\rm eV}\\
&=&0.4116639106(115)\times 10^{12} \;\; & {\rm AMU}
\end{array}[/tex]
[tex]\begin{array}{rcll}
m_2 &=& 0.00891348724(79)\;\; & {\rm eV}\\
&=&9.569022271(246)\times 10^{12} \;\; & {\rm AMU}
\end{array}
[/tex]
[tex]\begin{array}{rcll}
m_3 &=& 0.0507118044(45) \;\; & {\rm eV}\\
&=&54.44136198(131)\times 10^{12} \;\; & {\rm AMU}
\end{array}[/tex]
Similarly, the predictions for the differences of the squares of the
neutrino masses are:
[tex]\begin{array}{rcll}
m_2^2m_1^2 &=& 7.930321129(141)\times 10^{5} \;\; & {\rm eV}^2\\
&=&.913967200(47)\times 10^{24} \;\; & {\rm AMU}^2
\end{array}
[/tex]
[tex]\begin{array}{rcll}
m_3^2m_2^2 &=& 2.49223685(44)\times 10^{3} \;\; & {\rm eV}^2\\
&=&2872.295707(138)\times 10^{24} \;\; & {\rm AMU}^2
\end{array}[/tex]
As with the Koide predictions for the tau mass, these predictions
for the squared mass differences are dead in the center of the error
bars. We can only hope that the future will show our calculations
to be as prescient as Koide's.
That the masses of the leptons should have these sorts of
relationships is particularly mysterious in the context of the
standard model. It is
hoped that this paper will stimulate thought among theoreticians.
Perhaps the fundamental fermions are bound states of deeper objects.
The author would like to thank Yoshio Koide and Alejandro Rivero for
their advice, encouragement and references, and Mark Mollo
(Liquafaction Corporation) for financial support.
Carl
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