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standard model, neutrino, topology, gauge symmetry, space-time
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%%%%      June 3 2012                       %%%%%%%%%%%%%%
%%%%%%    Neutrino - faster than light   %%%%%%%%%%%%%%
%%%%%%
\documentstyle[12pt]{article}
\begin{document}
\vskip 1 cm
\begin{center}
\Large{\bf The enigmatic neutrinos}
\end{center}
\vskip 1.5 cm
\begin{center}
{\bf Syed Afsar Abbas} \\
Department of Physics\\
Aligarh Muslim University, Aligarh - 202002, India\\
(e-mail : drafsarabbas@yahoo.in)
\end{center}
\vskip 1 cm  
\begin{centerline}
{\bf Abstract }
\end{centerline}
\vskip 3 mm

We do a careful and consistent study of what masses and mass-mixing 
between different generations actually means in the Standard Model. 
We are thus led to a new complementary 
"Pure Weak Group" basis structure. This allows one to obtain a more basic understanding of 
wherefrom the wave-corpuscle duality actually arises in quantum mechanics.
We show that it is the gauge symmetries, which determine not only the fundamental interactions in particle 
physics, but is is these that are also the cause of the wave-corpuscle duality in quantum mechanics.
We also find a mathematical reason as to why only three generations of particles exist in nature. 
The wave-corpuscle duality manifests itself differently for the neutrinos.
Their corpuscular character demands that they have zero inertial mass and travel with the velocity of 
light and simultaneously and dually to it, as a wave, they oscillate and have a non-zero and a
different "gravitational" mass and which makes them travel superluminally.
Hence it is a clear prediction of our model that neutrinos, in a pure oscillation mode,
travel with velocites greater than that of light.
Also the neutrinos, having a non-zero gravitational mass and zero-inertial mass, become an ideal
candidate to be the Dark Matter of the universe.

\vskip 1 cm
{\bf Keywords:} Neutrino physics, Standard Model

\newpage

{\bf (I) Introduction}

\vskip .2 cm

True to the title of this paper, "the enigmatic neutrinos", there was some recent excitement
as to their true nature.
First in September 2011 ( revised in November 2011 ) the OPERA Collaboration at Gran Sasso, 
Italy announced the amazing discovery of neutrinos going faster than light [1].
Next the ICARUS group from the same laboratory announced in March 2012 that they only got 
neutrinos going at the speed of light [2]. 
In the meantime the OPERA group continued to critically look at their own 
results, and now it appears that further analysis of new systematic erros in their 
experiments, casts doubt on their above claimed result.
However, better and more thorough
experiments both by OPERA and ICARUS and others are on the way.
In this paper we study these enigmatic neutrinos more carefully. And it is a 
clear prediction of our model, that specifically in the oscillatory phase, the neutrinos
should travel faster than light. We discuss this below. 

In this paper we shall look at this problem  within the framework of the highly successful 
Standard Model (SM) of particle physics. The SM requires a zero mass 
neutrino. However, neutrino oscillations have clearly demonstrated experimentally 
this demands a 
non-zero mass for the same. This indicates that one has to go beyond the SM in some manner to 
understand this non-zero mass for the neutrinos.
This is conventionally done in some framework of going beyond the SM 
like the GUTS, SUSY etc. We do not do that in this paper. Instead we actually look inside the 
SM - in a way looking into the "guts" of the SM. This means studying at a fundamental manner 
as to what the SM is actually doing and more importantly what it is not doing - nay, 
actually what it cannot do!
As per a careful study of  masses it emerges that what is needed is a new 
"Pure Weak Group" basis 
structure ( to be discussed below ) which basically solves the problem. 

We shall show that there are thus two kind of masses. The SM provides one kind of mass which 
we show is actually an "inertial mass" and which requires a corpuscular aspect  of the 
particles.  Within the "Pure Weak Group" (PW) basis there arises another independent kind of 
mass which is logically associated with the "gravitational mass" of the particle and 
which requires a 
wave character for the particle. It appears that these two pictures: the SM on the one 
hand and the Pure Weak Group (PW) basis on the other hand, actually 
represent a fundamental 
duality of nature. Thus these two are a fundamental and simultaneously existing aspects of 
particle physics. Next we show that the equivalence principle i.e. 
"the inertial mass" = "the gravitational mass" provides the underlying basis for the 
"wave-corpuscle" aspect of massive particles in quantum mechanics. ( Note: we are using the word 
corpuscle in "wave-corpuscle" to avoid ambiguity and reserve the word "particle" as a 
generic name to identify say an electron or a proton).  

We shall show that the wave-corpuscle character ( i.e. their inertial and the gravitational 
masses being the the same ) holds for all the particles known to us except the 
neutrinos. Neutrinos are unique and different from all the other particles. We shall show 
that for the neutrinos the inertial mass is zero ( and this make it travel with the velocity 
of light ) and that the gravitational mass is positive and 
non-zero. Its gravitational mass is what allows it to oscillate and is the basis of its wave 
nature. This also makes it travel faster than light. Thus the wave-corpuscle duality of 
quantum mechanics shows that the neutrinos are different. Due to its corpuscular  character 
and its mass being zero, it travels 
with the velocity of light and simultaneously due to its wave character, 
having a non-zero 
positive gravitational mass, it travels with a velocity greater than that of light 
and also displays oscillations in this process.

Photon displays its wave nature through say interference and diffraction experiments and 
its corpuscular nature through say the photoelectric effects. In the same 
manner neutrinos show 
up its corpuscular nature through direct experiments like 
$\nu_{\mu} + n \rightarrow p + \mu^{-}$
and its wave character through oscillations experiments (so one has to make sure that the 
detected neutrinos should be the one partaking in oscillations and that it is not a direct 
one!).
The difference with respect to photon here is that for the photon, the corpuscular-wave 
property holds for the entity existing at the same point. While for the neutrinos, the two 
aspects are showing up at different locations at different space points. 
Clearly non-locality is fundamentally 
required for a proper understanding of what a neutrino really is and as to how it will 
show up in our instruments.

As to the experimental detection of the velocities of the neutrinos, 
one has to carefully distinguish between the direct detection
of the neutrinos (where they travel with the velocity of light)
and the neutrinos in the oscillatory phase wherein they travel with a velocity greater 
than that of light.
The OPERA experiment is unique in that they are actually detecting 
tau-neutrinos in their beam and for a beam, which started off as a muon-neutrino beam, these 
tau-neutrinos do necessarily belong to the oscillating neutrinos. This is thus 
an unambiguous signature of oscillating character and the tau-neutrinos 
had to be arising from the oscillating beam.
And as per our model we predict that these neutrinos shall be found to be travelling 
faster than light.

Also as the neutrinos have zero inertial mass and a positive non-zero 
gravitational mass,
hence it will interact gravitationally only ( besides its weak interaction )
with all the other particles and thus be an ideal candidate to be the Dark Matter of the 
Universe.

\vskip .2 cm

{\bf (II) Masses and mass mixing in the SM}

\vskip .2 cm

The hugely successful SM is based on the group structure
${SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}$
with three repetitive  set of elements called the generations of the fermions. This symmetry is 
spontaneously broken by a complex scalar Higgs doublet to
${SU(3)_{c}} \otimes {U(1)_{em}}$.
The Higgs is basically required to ensure renormalizability of the theory and to generate proper 
masses for the gauge particle $ W^{\pm}, Z^0 $ while leaving photon as massless.
We state a few salient points as per popular understanding of the SM 
to allow for a more focussed discussion later.

\vskip .2 cm

{\it (a). Electric charge in the SM} 

\vskip .2 cm

Electric charge is defined in the SM as

\begin{equation}
Q = I_3 + {Y \over 2}
\end{equation}

where $T_3$ is the diagonal generator of 
$SU(2)_{L}$ and Y that of  ${U(1)_{Y}}$. The hypercharge Y is arbitrarily fixed to
provide correct charges for the fermions. And because of this arbitrariness, it is 
believed that the electric charge is not quantized in the SM ( e.g. see 
[3] or any other more recent textbook ).

\vskip .2 cm

{\it (b). Mass in the SM}
 
\vskip .2  cm

For each generation, masses for the fermions are generated through Yukawa coupling with 
the Higgs. 
However this mass generation is believed to be arbitrary and inelegant and it has 
been famously dubbed as "Deux ex mochima" ( literal translation: 
God from the machine; meaning out of the blue to overcome a seemingly difficult problem ). 

Thus lack of proper charge quantization  and arbitrary Yukawa masses are considered as
two shortcomings of the SM.

\vskip .2 cm

{\it (c). Mass mixing across different generations}

\vskip .2 cm
 
However as per Cabibbo we know that quarks mix across generations. So to understand the 
mass problem one has to expand the language to properly incorporate the three families.
"We simplified our construction by taking one family of fermions. In this 
{\it{approximation}} there is no distinction between gauge eigenstates and mass eigenstates" 
[3]. Hence a generalization for mass mixing 
across generations is made. Thus 
Yukawa terms, mixing mass terms across different generations are obtained [3]. 

Hence on obtains a 
3x3 mass matrix which in general may be complex and need not be symmetric or hermitian.
Note the important point that the individual mass terms in this
3x3 matrix are Yukawa like mass terms linking fermions across different generations [3].
Clearly the diagonal terms correspond to Yukawa mass term for the same flavour.
However in the non-diagonal terms the left-handed fermion would be from one family while the 
corresponding right-handed terms in the Yukawa would be from a different family [3].

That such terms are fundamentally permitted in the SM have been uncritically accepted.
We examine the basis of this assumption below.
 
\vskip .2 cm

{\it (d). Biunitary transformation} 

\vskip .2 cm

We know from a general theorem from Algebra that any complex non-singular matrix can be 
written as a product of an unitary and a hermitian matrix [3]

\begin{equation}
M = H V
\end{equation}

Note that $M M^{\dagger}$ is hermitian and positive and thus can be diagonalized
by a unitary matrix S

\begin{equation}
S^\dagger ( M M^\dagger ) S = {M_d}^2
\end{equation}

where ${M_d}^2$ are positive and real.

Given another independent unitary matrix T, the above can be split down to $M_d$
through a biunitary transformation.

\begin{equation}
S^\dagger M T = M_d
\end{equation}

where at last $M_d$ is diagonal and positive mass eigenvalue.

Such biunitary transformation are obtained by letting the two unitary matrix operators act 
on the left-handed and the right-handed components of the wave function separately as

\begin{displaymath}
\Psi_L  \rightarrow  S \Psi_L \\  
\end{displaymath}
\begin{equation}
\Psi_R  \rightarrow  T \Psi_R
\end{equation}

\vskip .2 cm

{\it (e). Neutrino oscillations} 

\vskip .2 cm

As the neutrino oscillation is an experimentally well established fact, this demands
that neutrino have a non-zero mass. Taking cue from the CKM mechanism one demands similar 
terms 
for the neutrino masses also, that is mass mixing matrix across the three generations
for the neutrinos as well [3]. So given a mass eigenstate and another gauge eigenstates as

\begin{equation}
 \left( \begin{array}{clc}
\nu_{e} \\
\nu_{\mu} \\
\nu_{\tau}
\end{array} \right)_{gauge} = S
% \left( \begin{array}{clc}
%S
%\end{array} \right)
 \left( \begin{array}{clc}
\nu_1 \\
\nu_2 \\
\nu_3
\end{array} \right)_{mass}
\end{equation}

where S is CKM kind of 3x3 matrix. Then this allows for the quantum mechanical oscillations of 
the neutrinos [3].
 
\vskip .2 cm

{\bf (III). Description of our model} 

\vskip .2cm

{\it (a) Electric charge quantization in the SM}

\vskip .2 cm

As given in Section II part (A) in a canonical manner the electric charge is defined as 
in eqn. (1). Historically this was useful in identifying the unification aspect of the
weak and the electromagnetic interaction and also helped in identifying neutral weak current 
where both the weak and the electromagnetic parts play a role in an unified manner. 

However a major fault in 
this definition is that it makes an unambiguous statement that electric charge is not quantized 
in the SM. As we shall show below that this is wrong. And this fact in itself is highly 
nontrivial. 
Hence sticking to this wrong definition of the electric charge in the 
SM may obfuscate the physics and may be detrimental 
to our understanding as when we are 
forced to go deeper into other issues like, in particular as to what mass actually is in the 
SM, especially with regard to the neutrinos. 
So first we have to get rid of the wrong definition of 
electric charge given in eqn. (1).

To do so let us define our model [4,5].
To start with 
let us first ignore the right-handed neutrino ( note that we shall include it consistently 
below ) in say the first generation
of particle. In the SM this is represented in the group
${SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}$ as

\begin{displaymath}
q_L = \pmatrix{u \cr d}_L \sim (3, 2, Y_q) ; 
u_R \sim (3, 1, Y_u) ; d_R \sim (3, 1, Y_d)  \end{displaymath}
\begin{equation} l_L = \pmatrix{\nu  \cr e}_L \sim (1, 2, Y_l) ;  
e_R \sim (1, 1, Y_e)  \end{equation}

Let us now define the electric charge in the most general way in terms of 
the diagonal generators of ${SU(2)_{L}} \otimes {U(1)_{Y}}$ as
\begin{equation}
Q = I_3 + b Y
\end{equation}
where b is an arbitrary parameter.

In the SM 
${SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}$
is spontaneously broken through a Higgs mechanism to the group
${SU(3)_{c}} \otimes {U(1)_{em}}$.
Here the Higgs is assumed to be a doublet $\phi$ with arbitrary hypercharge 
$Y_{\phi}$. 
The isospin $I_3 =- {1\over2}$ component of the
Higgs field develops a nonzero vacuum expectation value $<\phi>_o$. Since 
we want
the $U(1)_{em}$ generator Q to be unbroken we require $Q<\phi>_o=0$. This
right away fixes b in (3) and we get

\begin{equation} 
Q = I_3 + ({1 \over 2Y_\phi}) Y 
\end{equation}

To proceed further one imposes the anomaly cancellation conditions
to establish constraints on the various hypercharges above.
First ${[SU(3)_c]}^2 U(1)_Y$ gives $2 Y_q = Y_u + Y_d$
and  ${[SU(2)_L]}^2 U(1)_Y$ gives $3 Y_q = - Y_l$. Next 
${[U(1)_Y]}^3$ does not provide any new constraints. 
So the anomaly 
conditions themselves are not sufficient to provide quantization of 
electric charge in the SM. One has to provide new physical inputs to 
proceed further. There are two independent ways to do so.

{\bf Method 1}:
Here one demands that fermions acquire masses through Yukawa coupling in 
the SM. This brings about the following constraints:

\begin{equation}
Y_u = Y_q + Y_{\phi} ; Y_d = Y_q - Y_{\phi} ; Y_e = Y_l - Y_{\phi}
\end{equation}

Note that  $2 Y_q = Y_u + Y_d$ from the anomaly cancellation condition for
${[SU(3)_c]}^2 U(1)_Y$ is automatically satisfied here from the Yukawa
condition above. Now using $3 Y_q = - Y_l$ from anomaly cancellation
along with Yukawa terms above in 
${[U(1)_Y]}^3$ does provide a new constrains of $Y_l = - Y_{\phi}$. 
Putting all these together one immediately gets charge quantization in the 
SM [4,5] as follows:

\begin{displaymath}
q_L = \pmatrix{u \cr d}_L , Y_q = {{Y_\phi} \over 3} ; 
Q(u) = {1\over 2} ({1+{1\over 3}}) ; Q(d) = {1\over 2} ({-1+{1\over 3}})
\end{displaymath}
\begin{displaymath} u_R, Y_u = {Y_\phi} ({1+{1\over 3}}) ;
                    Q(u_R) ={1\over 2} ({1+{1\over 3}}) ; 
                     d_R, Y_d = {Y_\phi} ({-1+{1\over 3}}) ;
                    Q(d_R) ={1\over 2} ({-1+{1\over 3}}) 
\end{displaymath}
\begin{equation} l_L = \pmatrix{\nu \cr e}_L ; Y_l = -Y_\phi ;
Q(\nu) = 0, Q(e) = -1 ; e_R, Y_e = -2Y_\phi ; Q(e_R) = -1
\end{equation}

Note that in the above quantization of the electric charge, 
Higgs hypercharge $Y_{\phi}$ always cancels 
out and hence remains unconstrained. This is important as not paying attention to this fact
can lead to asking wrong questions, as for example on the possibility of milli-charged 
particles, which are ruled out by the SM as per the above arguments [6].
A repetitive structure gives charges for the other generation
of fermions as well.

{\bf Method 2}:
Next we ignore Yukawa coupling and impose the vector nature
of the electric charge [4,5] which means that photon couples identically to 
the left handed and the right handed charges. That is $Q_L = Q_R$

\begin{displaymath}
{1\over 2} ({1+{{Y_q}\over {Y_\phi}}}) =  
{1\over 2}  {Y_u \over {Y_\phi}}                   
 ; giving:  Y_u = Y_q + Y_{\phi}
\end{displaymath}
\begin{displaymath}
Q(d) = {1\over 2} ({-1+{{Y_q}\over {Y_\phi}}})=
{1\over 2}  {Y_d \over {Y_\phi}}
 ; giving: Y_d = Y_q - Y_{\phi}
\end{displaymath}
\begin{equation} 
{1\over 2} ({-1+{{Y_l}\over {Y_\phi}}})=
{1\over 2}  {Y_e \over {Y_\phi}}
 ; giving: Y_e = Y_l - Y_{\phi}
\end{equation}

And thereafter charge quantization as in method 1.

Next, note that for the left handed charge of u- ( similarly for the 
d-quark) one obtained: 
${1\over 2} ({1+{{Y_q}\over {Y_\phi}}}) = 
{1\over 2} ({1+ {1\over 3}})$. Note that $1 \over 3$ is the baryon number 
[4,5].
We suggest that this is a general property and that we should define the 
baryon number in SM as  

\begin{equation}
{Y_q \over Y_\phi} = B
\end{equation}

Here in the SM the baryon number is arising as the ratio of the 
hypercharge of the left handed quark with respect to the Higgs 
hypercharge.
Since Higgs is providing the ubiquitous background uniform structure within
which the particles exist, this is a consistent definition of the baryon  
charge for the left handed quarks.

By definition the baryon number is arising for the left handed quark 
representation. 
It is important to note that this so called global 
quantum number is chiral in nature. 
This is fine, as the weak interaction is only left-handed anyway.

Similarly for the left handed electron the electric charge is
${1\over 2} ({-1+{Y_l\over {Y_\phi}}})$
We now associate a lepton number with

\begin{equation} 
{Y_l \over {Y_\phi}} = - L 
\end{equation}

which gives the correct charges. Notice that
this is a natural definition of the lepton number. Just as for baryon number
in the SM the lepton number arises as the ratio of the hypercharge of the 
left handed lepton with respect to the Higgs hypercharge and as such
is natural to treat this as the lepton number. This more so, as the Higgs
hypercharge remains unconstrained by the theory and the lepton number is 
thus fixed by the background Higgs.
And as for the case of the baryon number, the lepton number is also only 
defined from the left handed representation and is chiral in nature.

So the SM model is showing clearly that the electric charge is consistently quantized in the 
SM and also that it also throws up the baryon number and the lepton numbers and 
both of which are 
rooted on to the Higgs's hypercharge.

Note that because of the different lepton numbers arising internally for each generation
there are three different repetitive generations not knowing about each other. These lepton 
numbers are conserved independently of each other. This is the structure of the SM as found 
here.
All the charges and the masses are generated repeatedly for each generation separately 
and individually and one not knowing about the other.

\vskip .2 cm

{\it (b). Right-handed neutrino}

\vskip .2 cm

Let us now add the right handed neutrino for the first generation 
irreducible representation given in eqn.(7).
Let it be defined as ( in the same notation ): 
\begin{equation}
\nu_R ; (1,1,Y_\nu)
\end{equation}

This brings in additional term from the Yukawa coupling given 
in eqn.(10) as

\begin{equation}
Y_\nu = Y_l + Y_\phi
\end{equation}

Now ${[U(1)_Y]}^3$ anomaly condition, with all the Yukawa couplings and 
the $3 Y_q = - Y_l$ condition, does not provide any new constraint on 
hypercharges. Only that it is consistent with the other conditions.
Without $\nu_R$ it was this anomaly cancellation condition that gave 
crucial information which ensured charge quantization. Now with the 
incorporation of $\nu_R$, the property of electric 
charge quantization is lost.

Thus, let us now impose an empirical constraint. Let us assume that the
the electric charge of this new entity - $\nu_R$ is zero. This would be 
consistent with the overall empirical reality, as any charged $\nu_R$ 
would have 
made its presence felt in the laboratory or in the cosmological data.
Thus we are demanding it to be inert. We find that
 
\begin{equation}
Q(\nu_R) = {{Y_\nu} \over {2 {Y_\phi}}}
\end{equation}

Note that demanding that
$Q(\nu_R) = 0$ means that it is
${{Y_\nu} \over {Y_\phi}} = 0$. This means that either $Y_\nu = 0$
or ${Y_\phi} = \infty$. Now  ${Y_\phi} = \infty$ is ruled out as
we saw earlier that all the electric charges had factors like
${Y \over {Y_\phi}}$ with Y's for different representations of fermions,
and which will get messed up with this value of $Y_\phi$. 
Hence necessarily:
$Y_\nu = 0$.

What is the significance of this result. We saw earlier that it is the 
ratio of hypercharges with that of the Higgs hypercharge, that the baryon 
number and the lepton number, and the thereon the electric charges get 
defined in the SM. So $Y_\nu = 0$ is special. 
So right away one sees that we cannot define any lepton 
number for this $\nu_R$. So though the left handed neutrino exists and is
identified by its lepton number which puts it in the left handed doublet 
with the electron, the so called right handed neutrino
is completely unlike it, and has no associated lepton number.
This $\nu_R$ is colourless, massless, chargeless, 
weak-isospin-less, hypercharge-less and lepton-number-less. 

Once we have realized that $\nu_R$ has no lepton number, one sees that
hence there cannot be any Yukawa coupling mass 
for the left handed neutrino. 

To understand it we go to Wigner's analysis [7] of the irreducible 
representation of massless entities for the Poincare group.
For massless fields, he showed that in  addition when parity 
is conserved (so for photon), that there are two states of 
polarization +h and -h, where h stands for helicity. But for massless 
entities, when parity is not
conserved ( as in the case of weak interaction ), the two states
+h and -h are actually two independent and different irreducible representations
of the Poincare group. The left handed neutrino gets lumped with the
left handed electron by virtue of having a lepton number.
And therefore the other entity, the so called right handed neutrino, 
being of a different representation, is actually another independent entity altogether.
And this is exactly what we have found here.
Thus our work is a confirmation of Wigner's work on Poincare group.
Clearly so far, the so called right handed neutrino has been 
misunderstood completely.
This was shown earlier in ref [8]. The point is that 
the left handed neutrino has zero inertial mass. And this is true for each generations 
separately.

Note that primarily because of its inertial nature inclusion of the $\nu_R$
had no effect on the analysis done above, as to the masses and the charges of
all the fermions in a particular generation in the SM.

\vskip .2 cm

{\it (c) On mass in SM}

\vskip .2 cm

Note that the vector condition $Q_l = Q_R$ imposes exactly the same 
constraints on hypercharges as does the Yukawa coupling conditions
(the Method I and II above).
Thus the physical information content of these two methods is exactly the 
same. Thus the Yukawa mass and the electric charge of a particle
bootstrap upon each other. One needs the other to establish its own identity.
So the quantized electric charge and Yukawa masses go hand in hand in the SM.
What does it all mean?

We take the vector nature of electromagnetism as obvious. However note that 
this is a highly non-trivial property of electromagnetism.
For example the group $U(1)_Y$ above does not have it.
As we saw above both the left-handed and the right handed charges arise in the SM. 
So the vector nature of electromagnetism arises as a nontrivial and derived 
property in the SM. It arises as a consequence of the spontaneous symmetry breaking by a 
Higgs doublet which also ensures renormalizability.

However note that the Yukawa coupling is no less fundamental than the vector nature of 
the electromagnetism, as to the information content vis-a-vis the electric charge 
quantization. It turns out that the masses through Yukawa coupling for each 
generation in the SM also arise as a derived property in the SM and
is clearly no less basic that the electric charge in it 
and which too arises as a derived quantity.
You cannot dispense with one or the other. Because of this inbuilt characteristic of 
the electric charge and the mass through Yukawa coupling, these should be treated as being 
unique predictions of
the SM itself, very much like as the neutral currents were a clear prediction of the
unification process built into the SM

Also note that as per Wigner's analysis of the Poincare group representation of particles,
a massive particle is completely defined by its mass and its spin. We believe that
all the massive particles observed in nature correspond to this classification. 
However note that though the electric charge is such a 
fundamental property of the particles, it was not included in Wigner's analysis. In spite of 
this shortcoming, how come, all the massive particles observed so far,do follow the 
Wigner's classification faithfully?. 

On the basis of our analysis here, we have an explanation. The particle's irreducible 
representation given in terms of mass is good enough as mass in the SM contains as much 
information about the particle as does the electric charge. 
Their information content is exactly the same. 
Hence getting the result with the mass being the only player is as good as bringing in both
the mass and the charge into play.

Next point is that the Principle of Relativity states that " Physical laws of mechanics and 
electromagnetism are covariant in going from one inertial observer to another",
Note that clearly the word "mechanics" corresponds to to the word "mass: 
and the word "electromagnetism"
pertains to the word "charge". 
Also as per our analysis of the complete equivalence between the mass 
and the charge for a particle 
in the SM, it means that the charge is as much "inertial" as the mass is. \
Clearly also that 
the mass that we obtained in the SM should be termed as "inertial" mass.

We know that mass is an invariant quantity under Loretnz Transformation and electric charge 
is taken as scalar under the same transformation. However it is well known that it is 
difficult to maintain Lorentz Invariance for an extended charged body. Hence to ensure 
Lorentz Invariance for the charges we better stick to the notion of a "point particle". 
Hence the massive and charged body 
that we have obtained in the SM necessarily describes a "corpuscular" entity.

Note that all this happens for each generation separately. Both the masses and the charges of 
each generation are not aware of the presence of the other generations.

\vskip .2 cm
{\it (d) On mass mixing in SM?}
\vskip .2 cm

We have seen above that very strictly the charge quantization and and its one-to-one 
association with the mass generation of fermions through Yukawa coupling occurs 
generation by generation. And each generation being independent of the others.
There are clearly no Yukawa terms across two different generations  within the SM. 
All such terms are zero in the 
SM. It is not an "approximation" but an inherent property of the structure of the SM
that the charges are quantized and the Yukawa masses obtained are for each generation 
and independent of the other generations.
 
How and why does this come about?
We know from Cabibbo that mixing does occur between quarks (d,s). 
So one may ask as to why not the quarks in the same doublet mix as 

\begin{equation}
 \left( \begin{array}{clc}
u^\prime \\
d^\prime 
\end{array} \right)_{gauge} =
 \left( \begin{array}{clc}
m_{11} & m_{12}\\
m_{21} & m_{22}
\end{array} \right)
 \left( \begin{array}{clc}
u \\
d
\end{array} \right)_{mass}
\end{equation}

An immediate answer is that the charge conservation prevents this from happening. 
But let us go  
above as to when charge had not yet been created ( as per Method I above )
but Yukawa masses did exist. So mass mixing or lack of it should arise from how the above 
matrix will behave with respect to the Yukawa masses in the SM.

We know that as per the Irreducible Representation idea of the Poincare groups, each particle 
should have its own mass, independent of all others. So clearly if these u-and d- quarks are 
IR of the Poincare group then they should have masses independent of any mixing.
But if they are not IR of the Poincare then this does not hold. How can we be sure that these 
quarks correspond to the IR of the Poincare group. If they are not, then of course they may
mix. That they do not actually mix is because of the following reason.

Note that the charge changing weak currents require two independent flavours 
say (u d) to exist in the 
left handed mode only and that their Lorentz structure is of the pure (V-A) kind.
This is incorporated in the SM in terms of the groups structure
${SU(2)_L} \otimes {U(1)_Y}$. Now chiral this group is spontaneously broken   
to the non-chiral ${U(1)_{em}}$ group. This requires currents of vector V nature.
Note that a property of the neutral currents ( both weak and electromagnetic ) is that they 
require only one flavour. So how does one make sure that this V-A to V transition and 2-flavor 
to 1-flavour aspect is properly implemented in the SM. 
We have seen that the mass should be treated as being more primitive than 
the charge. So this is done by making sure that the mass matrix in eqn. (18) is 
diagonal. This is done to 
ensure that the two-flavour to the one-flavour property, as to its mass, 
is properly maintained. 
So we demand that in the SM


\begin{equation}
 \left( \begin{array}{clc}
u^\prime \\
d^\prime 
\end{array} \right)_{gauge} =
 \left( \begin{array}{clc}
m_{11} & 0 \\
0  & m_{22}
\end{array} \right)
 \left( \begin{array}{clc}
u \\
d
\end{array} \right)_{mass}
\end{equation}


Hence now one has ensured that these particles are actually an IR of the Poincare group, 
i.e. obtaining their own mass and 
not bothering about the others. 
The mass then gives parity conserving electric charge of the electromagnetism 
( as done by Method I above )
and its single flavour character to obtain Lorentz structure of "V" nature.
Implementation of the proper spontaneous breaking of the 
chiral group ${SU(2)_L} \otimes {U(1)_Y}$   
to the non-chiral ${U(1)_{em}}$ group was part of the kinematic structure of the SM. 
The implementation of the same done dynamically, is to 
ensure that the reduction 2-flavour to the 1-flavour is 
properly maintained through the mass matrix diagonalization.

Thus only two flavours at a time, and that is what constitutes a single generation of the
SM, is all that is needed. All other generations are spurious as to the inbuilt mass and 
charge structure of the SM for a particular generation is concerned. 
Next in a repetitive manner, take another generation and that one is immune to 
the other generations in as much as the mass and the charge are concerned.

Clearly this is the reason that the anomaly cancellations in the SM  
are well known to be fulfilled generation wise. One would have  expected 
that all the fermions in all the families 
should have together ensured the cancellation of the anomalies.
But it is only generation-wise that the anomalies cancel. 
And the reason is as above i.e.
to ensure that the fermions in the SM form the IR of the Poincare group. 

Therefore the complex mass matrix mixing with Yukawa mass terms across different
generations as discussed above in the SM (eg see [3]), 
{\it does not exist in the SM}. It is completely spurious to the SM.

So if such a mass mixing matrix does exist, 
on the one hand, it should be outside the framework of 
the SM and on the other hand, the individual mass terms would have nothing to do 
with the Yukawa 
mass terms. Clearly as neutrinos do have some kind of a mass (?) 
as it is known to  oscillate, how 
such a matrix arises will be discussed below.
 
So all the massive particles for each generation have inbuilt constraints arriving from the above
highly non-trivial structures. Whatever masses and whatever charges we obtain are due to the 
consistency of the SM. Now along with all this is arising an entity - neutrino, which 
of course naturally is also  part of 
the above structures also which holds for all the massive particles. But 
neutrino is massless ( besides being chargeless ) and this masslessness ( and mass here means 
that it has no inertial mass from the Yukawa coupling in the SM ) demands that it take part in 
charge changing weak currents with a pure (V-A) structure. This additional demand does not 
exist for any other fermion as they are all massive and charged. 
So neutrino is unique due to this 
additional requirement. Hence it says that though neutrino arise in the SM as chargeless and 
massless particles, it should manifest its pure (V-A) character, as an extra property 
somehow.

This above fact is very revealing and points to a new and independent structure, which 
should be dual to the SM structure and also be coexisting and simultaneous to it.
We call it "Pure Weak Group" basis structure
named as such to incorporate the pure (V-A) structure. 
It will be this new basis which will provide 
neutrinos with a new kind of non-inertial and non-Yukawa type of mass.

\vskip .2 cm
{\it (d) Pure Weak Group (PW) basis}
\vskip .2 cm

First note that as neutrinos in the SM have zero 
(inertial) mass for each generation 
and with the charge changing current that it participates in, it has pure (V-A) left-handed 
structure. 
So it is a pure $\nu_{gauge}$ left-handed state.
Now if there arises some different mass matrix ( different from the Yukawa coupling  
terms )
herein, then for example as in eqn (2) which is complex and non-singular,
then for these basis' ${\nu_{mass}}$ and ${\nu_{gauge}}$ are
connected by a unitary transformation as in eqn (6).
Then $({\nu_{gauge}} ( M M^\dagger ) {\nu_{gauge}})$ gives
$({\nu_{mass}} (S^\dag ( M M^\dagger ) S) {\nu_{mass}}) = {M_d}^2$
which is already diagonal and positive in mass-squared. So a single unitary transformation 
on a pure left-handed basis gives a chiral diagonal mass-squared. 
If there is no right handed part to 
give a biunitary transformation, we have a pure new mass-squared which is non -inertial and non 
-Yukawa kind of mass. So right away one can see that at the least, we get a mass-squared for 
the neutrino and this is of an entirely new kind. 
This is true for each generation of neutrino separately in SM.

So as the neutrino is massless ( i.e. its Yukawa mass in SM is zero ) then to generate 
another kind of independent mass for it, we are suggesting a new Pure Weak Group (PW) 
structure for it. Note that when the exact symmetry
${SU(3)_{c}} \otimes {U(1)_{em}}$ of the SM is restored to
${SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}$
then all the matter particles become massless, i.e. they do not have any 
inertial and Yukawa kind of mass.
So for the unbroken
${SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}$
all the fermions are massless and these are all neutrino-like. So what we saw for each neutrino as 
massless particles in SM, here now all the fermions are massless and all are neutrino -like. So 
how should we treat them so that these all would have mass-squared associated with them. 
This is done in the new Pure Weak Group basis below.

For this structure, we assume that in addition to the existing  SM structure as 
above, there
exists a parallel and concurrent to it another new "Pure Weak Group" (PW) basis 
structure. This is based on another
group ${SU(3)_{c}} \otimes {SU(2)_{LP}} \otimes {U(1)_{YP}}$
where the subscript LP labels the Pure Weak Groups and distinguishes this from the
corresponding SM group structure. In the SM, the group is broken by a complex Higgs doublet. 
Below we shall assume that this new PW group is broken by a real three dimensional SO(3) 
Higgs. So we assume that at the basic and primitive level we have a 
${SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}$ group
structure. There exist thereafter two different co-existing and dual paths through which this 
symmetry is broken. One is our old friend  - the SM. The next one is this new PW path.
This new kind of duality is essential to our understanding and will actually provide the 
fundamental reason for 
other dualities like wave-corpuscle duality in quantum mechanics to exist.
So schematically we have the following structures:

\begin{equation}
\begin{array}{ccc}
{SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}} & &  \\
\swarrow &  \searrow \\
{SU(3)_{c}} \otimes {SU(2)_{LP}} \otimes {U(1)_{YP}} & 
{SU(3)_{c}} \otimes {SU(2)_{L}} \otimes {U(1)_{Y}}
\end{array}
\end{equation}

Note that the defining Higgs in one is SU(2) complex and the other a real SO(3) as below.

We break the PW symmetry group through a triplet of real scalar Higgs field

\begin{equation}
\Phi = \left( \begin{array}{ccc}
\phi_1 \\
\phi_2 \\
\phi_3
\end{array} \right)
\end{equation}

and break only the $SU(2)_{LP}$ group such that the residual symmetry group is U(1).
That means that $W^{\pm}$ become massive and $W^3$ remains massless.
We start with a scalar potential [3,7]

\begin{equation}
V(\Phi) = - \mu^2 \Phi^2 + \lambda {(\Phi^2)^2} 
\end{equation}

Note that minimization of $V(\Phi)$ only determines the magnitude

\begin{equation}
\vert {\langle \Phi \rangle}_0 \vert = \frac{v}{\sqrt{2}}
\end{equation}

One is free to choose the vacuum state which we take as

\begin{equation}
{\langle \Phi \rangle}_0  = 
\frac{1}{\sqrt{2}} 
\left( \begin{array}{ccc}
0\\
0\\
v
\end{array} \right)
\end{equation}

i.e. in the z-direction.

We demand that as the symmetry SO(3) is spontaneously broken we define the charge Q such that 
only one generator as above is not broken

\begin{equation}
Q {\langle \Phi \rangle}_0 = 0 
\end{equation}

which means that charge gets defined as the diagonal generator of SO(3) [7]

\begin{equation}
Q = T_3 =
\left( \begin{array}{ccc}
1 &  & \\
 & -1 & \\
 &  & 0
\end{array} \right)
\end{equation}

which ensures that it is $W^3$ which remains massless and that $W^{\pm}$ become massive
and the residual group is $U(1)_z$ which show that the conserved charge is along the 
z-direction. This is for one-generation. 

To understand the role of other role of other generations 
let us now define the representations of the three 
generations as per PW group
${SU(3)_{c}} \otimes {SU(2)_{LP}} \otimes {U(1)_{YP}}$
as
 
\begin{displaymath}
\pmatrix{u^i \cr d^i}_L \sim (3, 2, 0) ; {u^i}_R \sim (3, 1, Y_{u^i}) ;
{d^i}_R \sim (3, 1, Y_{d^i})  \end{displaymath}
\begin{equation}
\pmatrix{\nu^i \cr e^i}_L \sim (1, 2, 0) ; {\nu^i}_R \sim (3, 1, Y_{\nu^i}) ;
{e^i}_R \sim(3, 1, Y_{e^i})  
\end{equation}

where i=1.2.3 correspond to the three generations.
So eg.

\begin{equation}
\left( \begin{array}{ccc}
\nu_2\\
\nu_2
\end{array} \right)_L =
\left( \begin{array}{ccc}
\nu_\mu\\
\mu\\
\end{array} \right)_L
\end{equation}

As above we had broken the SO(3) symmetry so that the third axis was the unbroken one and 
thereby the conserved electric charge was $T_3$. Now as SO(3) is isomorphic to SU(2) 
we make an additional demand 
that the conservation between the algebra also extend to the conservation of the charges,
thereby translating 
to the diagonal generator of SU(2) as defining the charge for the fermions as

\begin{equation}
Q = \tau_3
\end{equation}

and thus

\begin{equation}
Q 
\left( \begin{array}{ccc}
\nu_e\\
e
\end{array} \right)_L =
\left( \begin{array}{ccc}
+ \frac{1}{2} \nu_e\\
- \frac{1}{2} e
\end{array} \right)_L
\end{equation}

Note that the charges here in the group $SU(2)_{LP} \otimes U(1)_{YP}$
are very different from the electric charges we found in the SM.
In fact here even the neutrinos are charged! Note also that
$Q(e) - Q(\nu_e)$ = -1. So the left handed weak vertex connects properly to
$W^\pm$ gauge bosons. And so all is well.

Note that in SM the lepton number arose from the ratio of the fermions hypercharge 
and the Higgs hypercharge. Here that freedom does not exist, as the electric charge is 
defined entirely within the group $SU(2)_{LP}$ itself with no contribution from the
$U(1)_{LP}$.
So how does one generation distinguishes itself from the others? 
Well, the freedom to choose the 
vacuum direction, which for the first generation we chose it to be the z-direction, provides 
the distinction between the different generations. 
Hence for the second generation let us choose the vacuum as 

\begin{equation}
{\langle \Phi \rangle}_0  = 
\frac{1}{\sqrt{2}} 
\left( \begin{array}{ccc}
0\\
v\\
0
\end{array} \right)
\end{equation}

that is now the ground state $\Phi$ points in the y-direction.Thus now it is $T_2$
which defines the electric charge for the second generation. 
Next then $\tau_2$ gives charges to
the fermions. This distinguishes the second generation from the first one.
Next choosing the vacuum along the x-direction gives uniqueness to the third generation.

Right away we see that now we have an explanation of as to why there are three generations of 
fermions in particle physics. 
This is so because in the PW group the Higgs with three real fields in 
$\Phi$ provides the three directions: z-, y- and x- along which the vacuum may be chosen
and which provide different charges to different generations and distinguishes them.

Also this is not just a chance happening or coincidence. 
It is providing us with a deep and fundamental 
understanding of the structure of space-time and matter in the universe.

Note that all the fermions in this PW group basis are massless (i.e. have zero inertial mass 
of Yukawa-type from the SM). However only neutrinos continue to be massless in the SM itself
( while all the other particles acquire non-zero inertial Yukawa-kind of mass ).
As such for the neutrinos the standard analysis of massless particles as representation of the 
Poincare group would hold [7]. Thus for the massless particle the little group is the 
Euclidean group in two dimensions E(2).
E(2) group is generated by the group of rotations ( generated by one say $T_z$ ) and 
translations ( generated by $L_y$ and $L_x$ ) in the plane. Thus when we choose $T_z$ as the 
rotation directions ( along which the helicity gets defined ) 
then this is exactly is the same as 
the $T_z$ symmetry breaking direction of SO(3) in the PW group. There is one-to-one 
correspondence between the two. Similarly for the other two generations,. 
So the freedom to choose
the E(2) group for the massless neutrinos in the SM also provides consistent reason 
as to why there are three generations of particles.

Now what is the scale of the massive $W^\pm$ bosons in the PW group of
${SU(3)_{c}} \otimes {SU(2)_{LP}} \otimes {U(1)_{YP}}$. 
We know the scale of the 
Eletro-weak group in the SM to be about 250 GeV. 
Now the only scale left and known to us ( and which has always been begging for a $\it proper$ 
explanation ) is of course our well known friend, the Planck Mass 
${(\frac{h c}{2 \pi G})}^\frac{1}{2}$ = 1.2 x ${10}^{19} \frac{GeV}{c^2}$. 
So the gauge bosons 
here in PW group may be as heavy as that. One may protest that such a mass is beyond the 
reach of our accelerators. But this is not relevant here. 
What is relevant is the Planck length
of ${10}^{-33} cm$ which is the range of the Planck gauge particle  as 
weak force mediator over this range. 
This length is of course real and relevant. Also note 
that the Planck mass is anyway much smaller than the infinite mass of gauge boson required to 
justify the successful Fermi model of the weak interaction!

Now clearly as per the PW model the Planck mass is the reasonable mass for the PW 
interaction.
Now it is a folklore in particle physics that as to gravity, if nothing else, at the least,
it would demand 
quantization definitely at the Planck scale. That is at this mass scale quantization of gravity 
would become inevitable. 
Now we reverse the argument. We already have here a quantized theory of the 
weak interaction which demands a Planck scale mediator of force. 
Hence as Planck scale is arising consistently in a quantized model here, 
so there must be lurking the force of gravity in the background somewhere.
We have not yet shown fully that indeed there is gravity in this model, 
but only an intriguing hint that 
it should be so in this new PW model. Further as discussed above, as to the three generations 
based on the three space coordinates z-, y-, and x-, consolidates this expectation.

So all said and done, the pattern of the total symmetry breaking in our PW model here is

\begin{equation}
\begin{array}{clc}
 & {{SU(3)_{c}} \otimes {SU(2)_{LP}} \otimes {U(1)_{YP}}} & \\
 & \downarrow                                           & \\
 & {{SU(3)_{c}} \otimes [ {U(1)_z} \oplus {U(1)_y} \oplus {U(1)_x} ] \otimes {U(1)_{YP}}} & 
\\
 & = {{SU(3)_{c}} \otimes [{\bf U(1)}] \otimes {U(1)_{YP}}} &
\end{array}
\end{equation}

And thus $Q = Q_z + Q_y + Q_x$
with

\begin{equation}
Q 
\left( \begin{array}{clc}
\left( \begin{array}{clc}
u\\
d 
\end{array} \right)_z \\  
\left( \begin{array}{clc}
c \\
s 
\end{array} \right)_y \\
\left( \begin{array}{clc}
t\\
b 
\end{array} \right)_x
\end{array} \right)_L = 
\left( \begin{array}{clc}
Q_z  &     &     \\
     & Q_y &     \\
     &     & Q_x 
\end{array} \right)
\left( \begin{array}{clc}
\left( \begin{array}{clc}
u\\
d 
\end{array} \right)_z  \\
\left( \begin{array}{clc}
c \\
s 
\end{array} \right)_y \\
 \left( \begin{array}{clc}
t\\
b 
\end{array} \right)_x
\end{array} \right)_L
\end{equation}

This gives vectors of identical charges as

\begin{equation}
\left( \begin{array}{clc}
u\\
c\\
t 
\end{array} \right)_L ;
\left( \begin{array}{clc}
d\\
s\\
b
\end{array} \right)_L ;
\left( \begin{array}{clc}
\nu_e\\
\nu_\mu\\
\nu_\tau
\end{array} \right)_L ;
\left( \begin{array}{clc}
e\\
\mu\\
\tau
\end{array} \right)_L
\end{equation}

Where the charges Q are $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$ and $-\frac{1}{2}$
respectively for the above vectors which are all left-handed.
Let us look at the neutrino vector and call it $L_\nu$. Recall our discussion in the
biunitary transformation section and the discussion at the beginning of the current 
section. 
We assume that due to the Higgs spontaneous symmetry breaking there develops a complex chiral 
mass matrix which defines the chiral field $L_\nu$. 
The mass matrix product $M {M^\dagger}$ is hermitian and so for the left 
handed neutrinos we obtain $\langle {L_\nu} \vert M {M^\dagger} \vert {L_\nu} \rangle$
for the left handed gauge state.

There is another way of showing, as we have asserted above, that there are no Yakawa-mass
mixing terms across generations in the SM, that if we do insist for a term like the one here, 
then the nondiagonal terms connecting proper hypercharges of different generations of the 
left-handed doublets, do not exist.

Next we assume that first the spontaneous symmetry breaking generates a mass vector basis.
Next as there also arises a charge defined by the diagonal generator of $SU(2)_{LP}$,
that these states may be connected by a unitary transformation $L_\nu = S {L_{mass}}$
which arises from the mass basis the gauge basis.
Then $({\nu_{gauge}} ( M M^\dagger ) {\nu_{gauge}})$ gives

\begin{equation}
\langle {L_{mass}} \vert {S^\dagger} M {M^\dagger} S \vert {L_{mass}} \rangle = 
\left( \begin{array}{clc}
m_{\nu_{e}}  &             &            \\
           & m_{\nu_\mu} &            \\
           &             & m_{\nu_\tau}
\end{array} \right)
\end{equation}

where we see that for each flavour these mass-squared are real and positive. So 
mass-squared comes as a fundamental chiral quantity from the PW group structure.
Clearly this mass is non-inertial and non-Yukawa kind.

Note that all the fermions have this diagonal mass squared for each flavour which arises 
from the mixing of generations and is different from the inertial mass.

What is this mass-squared and where is it coming from? Obviously it is coming from the 
structure of this new Pure Weak Group symmetry that we have discussed here. It is clearly 
chiral in nature in contrast to the scalar inertial mass obtained from the SM.

It may be pointed out that such a mass squared had already been labelled as  
"topological mass' by the author [9]. The word topology therein refers to a study of complementarity 
between what one identifies as "geometric" in nature and dual to it as what may be referred to 
as "topological" as to the space-time and matter in the nature. Here we shall not go too deep 
into that issue and the reader is invited to read that paper for information [9].

Hence it should be clear now that all the particles in addition to their inertial-Yukawa kind 
of mass arising from the SM have in addition another independent topological mass-squared 
arising from our new PW structure. 
These two masses are fundamentally different from each other, i.e. the chiral $m^2$ is 
different from the non-chiral ${m_I}^2$ of the inertial masses.

Note that the mass -squared and the charges for all the fermions as per these 
representations have arisen purely from the spontaneous symmetry breaking of 
$SU(2)_{LP}$ only, That is the group $U(1)_{LP}$ has played so far no role in the analysis.
This is because these left-handed entities were singlet of this group.
However from the same representations one sees that there do exist the correponding 
right-handed 
entities which are singlet of $(SU(2)_{LP}$ and have undefined, unquantized and unspecified 
$U(1)_{LP}$
quantum numbers. For different generations these are singlets of different
$SU(2)_{z,y,x}$ groups and thus they know which generation they belong to. So the non-zero and 
unspecified hypercharge numbers that they have, should be identified with some kind of a 
generation number. Thus we place all the right handed entities of a particular left -handed 
entity as a vector, say for the neutrino as

\begin{equation}
\vert {R_\nu} \rangle = 
\left( \begin{array}{clc}
\nu_{e} \\
\nu_\mu \\
\nu_\tau
\end{array} \right)_R
\end{equation}

Now this right-handed vector can be inserted in the mass-squared expression to break it up 
into the individual mass terms as stated in biunitarty transformation section as

\begin{equation}
\langle {L_\nu} \vert M \vert {R_\nu} \rangle \langle {R_\nu} \vert 
{M^\dagger} \vert {L_\nu} 
\rangle  \rightarrow {m.m}
\end{equation}
 
So the topological mass squared splits up into the single power mass.
Note that in the above equation the unspecified hypercharge of various right-handed 
components 
cancel and thus though they define a generation, are never actually pinned down as 
a specific 
number. It is not quantized and
in fact it can be any arbitrary real number rooted on the vacuum.
Their main role here is to act on the vacuum 
to primarily reduce the mass-squared to a simple mass "m". Now what is this mass?
It is logical to associate this with the "gravitational mass". Since time immemorial
we have been aware of a mass different from the inertial mass and that has been the 
"gravitational mass". 
Identifying this new 
mass with the "graviational mass ' is logical, consistent and seems to be completing the 
physical reality.
Hence as per what we have done so far all the particles have an inertial mass 
and a different
gravitational mass.

But first to the neutrinos. Now neutrino as per what has been discussed here have zero 
inertial 
mass and non-zero gravitational mass. So having a gravitational mass, they will partake in 
gravitational interactions with all the other particles 
( besides their weak interaction ) and thus be a 
suitable candidate to be the Dark Matter particle of the universe. Earlier one thought that 
these Dark Matter particles would be some WIMPS arising from some supersymmetric 
scenario. But now all that is irrelevant on the basis of what we have seen here, It is 
our good old friend, the neutrino, which is the Dark Matter of the Universe. And so all the 
effects that 
we had visualized for the DM, affecting life on on the the Earth [10] and as a new source of 
energy as proposed 
for the first time [11], are relevant and valid.

Before we end this section let us point out the interaction Lagragean for the $SU(2)_{LP}$ 
group is

\begin{equation}
{\cal{L}} = 
\frac{1}{2} [ \sqrt{2} \overline{\nu_{e}} {\gamma_\mu} {e_{L}} {W_\mu}^{+}
           + \sqrt{2} \overline{e_{L}} {\gamma_\mu} {\nu_{L}} {W_\mu}^{-} 
    + ( \overline{\nu_{e}} {\gamma_\mu} {\nu_{e}} 
    -  \overline{e_{L}} {\gamma_\mu} {e_{L}} )  {W_\mu}^{3} ]
\end{equation}

Now ${W_\mu}^{3}$ couples to the neutral currents of $\nu_{e}$ and e where each has a charge 
of 
$\frac{1}{2}$ and -$\frac{1}{2}$ respectively and which are not electromagnetic charge or the 
neutral charges of the SM.
So how this new PW neutral current would behave has to be figured out along with the manner 
that the symmetry is broken as given in eqn. (32) above.
We could guess that this should have something to say about the Dark Energy 
problem - in 
particular the cosmological constant.

\vskip .2 cm
{\it (e) Superluminal Neutrinos}
\vskip .2 cm

Since in the PW model we necessarily obtained the composite topological mass-squared,
we should use the relation

\begin{equation}
{E^2} = {p^2} {c^2} + {{m_g}^2} {c^4}
\end{equation}

and thus $E = p {v_p}$ where

\begin{equation}
{v_{p}} = c \sqrt{1 + \frac{{m_g}^2 {c^4}}{p^2}}
\end{equation}

This is the phase velocity of the wave and the group velocity is

\begin{equation}
{v_{g}} = {\frac{\partial E}{\partial p}} =
        {\frac{c}{\sqrt{1 + \frac{{m_g}^2 {c^4}}{p^2}}}} 
\end{equation}

such that

\begin{equation}
{v_g}{v_p} = {c^2}
\end{equation}

So necessarily the neutrinos are travelling with the above phase velocity.
Note that this is necessarily as per the wave motion. This thus gives wave character to the 
neutrino and to all the other entities for
which the above equations holds.

Note that in addition the inertial mass equations hold

\begin{equation}
E = m_{I} \gamma c^2  ;   p = m_{I} \gamma v
\end{equation}

where $m_I$ is the inertial mass. Putting this in the above equations we get
for the phase velocity $v_{p} = \frac{c}{\beta} \rangle c$
and the group velocity $v_{g} = v$ where "v" is the particle velocity when the inertial mass 
of the entity is non-zero. Thus all the matter particles with non-zero inertial mass 
travel with the group velocity and for them as de Broglie had pointed out the corresponding 
phase 
velocity was fictitious, as that did not carry any physical information. That is for them the 
group velocity, which was as above always smaller than the velocity of light, was only 
physical.

For neutrinos as the inertial mass was zero it would act as a corpuscle 
and it will travel with the velocity of light.
But the wave character neutrino travels with the phase velocity arising from the physically relevant
oscillation phenomenon, and that is greater than that of light. 

The wave-corpuscle duality of quantum mechanics requires the neutrinos to display both these 
characteristics simultaneously 
but exclusively - that is when it behaves like a corpuscle and travels 
with the velocity of light, it will not be acting as a wave. But when it behaves as a wave, 
it oscillates and travels with a velocity greater than that of 
light. Mass oscillation is a physically relevant and physically measurable phenomenon and 
thus the phase velocity does carry physical signal for the neutrino.

However for the particle which had non-zero inertial mass and which in the SM behaved
as a corpuscle. Next, these same particles have another gravitational mass arising from the 
new Pure Weak Group (PW) model proposed here and which as seen above, conforms to the wave 
character of the entity. Hence using the equivalence principle that

\begin{equation}
m_{I} = m_{g}
\end{equation}

This immediately sets up a basis for the wave-corpuscle duality of quantum mechanics.
So now we know where the corpuscle-wave duality of quantum mechanics comes from. This is
because underlying there are two dual and complementary realities, the SM and the PW models 
which are the ones which act simultaneously and in a complementary manner.
One is corpuscular in character and the other wave like. Hence the equivalence principle 
for massive particles brings about the wave-particle duality of quantum mechanics.

This also explains why this wave-particle duality requires an exclusiveness as per Bohr. 
This is because 
these two i.e. the SM and the PW are orthogonal and different systems providing 
basic structures to the gauge and the matter particles. 
One is corpuscular in nature with velocities always less that of light 
and the other has a wave nature and with phase velocities always greater than that of light.
But as they stand for the same particle, they have the same mass (for massive particles) 
and thus have to act exclusively.

However for massive particles this makes sure that we are talking of one single entity at a 
single space-time point whether they act as a corpuscle or they act as a wave. 
But the neutrino is unique, 
as to the fact that as it has different inertial mass ( actually zero ) and  a non-zero 
gravitational mass, it has non-locality built into it intrinsically.

Note the standard picture of a superluminal particle having imaginary mass
is unacceptabel. Here the superluminal neutrino has positive mass and has a positive 
energy and they also oscillate.

Note the meaning of eqn. (42) of the three velocities therein is that only two of these are 
independent
and only two may be physically relevant. The third one then is automatically determined 
and is therefore "fictitious" as de Broglie would have put it! So for inertially massive 
particles 
it is $v_{g} = v$ as the velocity of the particle and "c" is the other physical velocity.
For neutrino it is "c" and $v_p$ which are physically relevant and $v_g$ is fictitious. 

Hence it is the oscillating part of the neutrino which is superluminal 
So when looking for a superluminal neutrino it is 
essential that one makes sure that the neutrino that one sees in ones detector is actually 
the oscillating one. 
Remember that dual to it is another neutrino, which depending upon the 
technique 
employed for detection, may be the direct one, travelling with the velocity of light.
The OPERA experiment is in a privileged position that they are detecting the tau-neutrinos
in an appearance experiment. For a beam which started as a $\mu$-neutrino, detecting a 
$\tau$-neutrino is the clearest and unambiguous signal that what they see
is actually from an oscillating neutrino mode.
Hence we feel that at presnt the OPERA collaboration has the best chance of detecting
a superluminal neutrino. The ICARUS collaboration, once they can identify the tau-neutrinos in
the oscillation mode, may also detect the superluminal neutrinos.

The issue of the helicity of the neutrino is also an important issue.
Note that one advantage of the neutrino picture 
here is that they have a well defined helicity. First they have as a 
corpuscles, the velocity of light, and hence never at rest and thus helicity is a good 
quantum number. But one knows well that massive particles do not have well defined 
helicity as they have a rest frame and thus the massive neutrino would be overtaken and thus 
helicity reversed. But the catch is that this is true for neutrino which move with
$v_g$. In our picture the mas-squared neutrinos are moving with the velocity $v_p$ and are
thus superluminal for which a rest frame does not exist and thus helicity continues to be a 
good quantum number!


\vskip .2 cm 
{\bf (IV) Conclusions}
\vskip .2 cm 

We found here that the SM gives inertial mass as Yukawa coupling and electric charge 
quantization on one-to-one basis. 
It is this which makes these follow the Principle of Relativity for mass and charge and also 
thus provide them with a corpuscular character.

Next as SM only allows for three different repetitive generations of fermions, it 
allows for no mass mixing matrix across generations. This led us to propose here a new
Pure Weak Group basis (PW) which has the freedom to generate a new and different kind of
topological mass-squared and thus the gravitational mass from there. 
These are chiral in character.
This also demands a wave character for interpretation of this mass.
All this is non-inertial. These basically obey eqns. (39) and (40) and not eqn (43) above.

We also obtain a new understanding as to why there are three generations of fermions in 
particle physics. It arises from the fact that there are only three space coordinates
z,y, and x. This itself arose from the manner that spontaneous symmetry breaking occurs
in the new PW model.

Through the equivalence principle $m_{I} = m_{g}$, we obtain a more basic understanding of 
where the wave-corpuscle duality of quantum mechanics arises for the massive (inertail mass)
particles. When neutrinos are included, then we find that at the base of the apparent
wave-corpuscle duality in quantum mechanics lies two fundamental, dual and simultaneously 
existing symmetries of the 
SM and the PW models. This in turn leads to this quantum mechanical property.
So we see that the gauge symmettris, not only dictate the fundamental interactions 
but also determine this fundamental underlying quantum mechanical reality of quantum mechanics. 

Now neutrinos have zero inertial mass and non-zero gravitational mass, and hence they are 
ideal candidates for the being the Dark Matter particle of the Universe. 

Also as zero mass particles they travel with the velocity of light in the corpuscular mode. 
Simultaneously they also, having non-zero gravitational mass, partake in oscillations and 
travel with a velocity greater than that of light while displaying a wave character.
Thus it acts nonlocally at two different places and may thus solve the horizon 
problem of cosmology.
Thus this prediction of superluminal neutrinos should be confirmed by the experimentalists.

Though not defined explicitly here, the presence of the Planck mass scale in PW model, plus 
the explicit role of the z-, y- and x- spacial coordinates, and the interplay of continuous
hypercharge parameters apparently exploring the global structure of the vacuum, that one feels 
that gravity should be in here somewhere. 
That has to be demonstrated next. In addition the 
unexplained Dark Energy problem or the Cosmological Constant problem should be amenable to 
the new neutral current inherent in the PW structure of the group $SU(2)_{LP}$.

\newpage
\begin{center}
{\bf References }
\end{center}
\vskip .5 cm

1. The OPERA collaboration: T. Adam et al., "Measurement of the neutrino velocity with 
the OPERA detector in the CNGS beam", arXiv: 1109.4897

\vskip .2 cm

2. The ICARUS collaboration: M. Antonello et al., "Measurement of the neutrino velocity
with the ICARUS detector at the CNGS beam", arXiv: 1203.3433 

\vskip .2 cm

3. T-P. Cheng and L-F. Li, "Gauge theory of elementary particle physics",
Clarendon Press, Oxford, 1984

\vskip .2 cm

4. A Abbas, "Anomalies and charge quantization in the Standard Model with arbitrary
number of colours", Phys. Lett B 238 (1990) 344

\vskip .2 cm

5. A Abbas, "Spontaneous symmetry breaking, quantization of the electric charge and 
the anomalies", J. Phys. G 16 (1990) L163

\vskip .2 cm 

6. A. Abbas, "Standard Model of particle physics has charge quantization", Physics Today,
July 1999, 81

\vskip .2 cm 

7. L. H. Ryder, "Quantum Field Theory", Cambridge University Press, 
Cambridge, 1986

\vskip .2 cm

8. A. Abbas, "What the right handed neutrino really is?", 
arXiv: 0912.3077

\vskip .2 cm

9. A. Abbas, " An alternative framework of geometry and topology in 
relativity", published in "Physical interpretation of relativity theory"" 
Ed M. C. Duff, P. Rowlands and V. Gladyshev, Proc. International Meeting, 
Moscow, Russia, 2009, published by BMSTU, Moscow, 
ISBN: 978-5-7038-3394-0 AND arXiv: 0903.5532

\vskip .2 cm

10. Samar Abbas and Afsar Abbas, "Volcanogenic dark matter and mass 
extinctions". Astropart. Phys. 8 (1998) 317

\vskip .2 cm

11. S. Abbas and A. Abbas and S. Mohanty, "A new signature of Dark Matter"
arXiv: hep-ph/9709269

\vskip .2 cm

\end{document}
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