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electric charge, Standard Model, spontaneus symmetry breaking, Higgs mechanism, anomalies
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%%%%%%    charge quantization in SM - I   %%%%%%%%%%%%%%
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\documentstyle[12pt]{article}
\begin{document}
\vskip 1 cm
\begin{center}
\Large{\bf Electric charge in the Standard Model of particle physics}
\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 }

At present the Standard Model is empirically the most successful model of particle physics.
However, the electric charge which is a most basic and fundamental physical quantity,
is popularly believed, not to be quantised in the Standard Model. 
This is what all the text books say.
We look into the basis of the above statement and show that this arises from a gross misunderstanding
of the mathematical structure of the Standard Model. 
We show that in contrast to the popularly held above view, the electric 
charge is consistently and fully quantized in the Standard Model.
This arises from a consistent physical interpretation of the mathematical structure of the Standard Model.
Lack of this understanding is being detrimental to further appreciation of the real
significance of the Standard Model in the broad framework of particle physics.

\end{centerline}


\vskip 1 cm
{\bf Keywords:} electric charge, Standard Model, spontaneus symmetry breaking, Higgs mechanism, anomalies
\vskip .2 cm
{\bf PACS clssification:} 12.15.-y, 11.15.Ex, 11.30.Ly

\newpage

\vskip .2 cm

{\bf (I) An historical introduction to electric charge in modern particle physics}

\vskip .2 cm

Shortly after the discovery of neutron, Heisenberg in 1932 [1] put proton and neutron in
together as a doublet of the group SU(2) ( now called isospin group ), to explain the charge
independence of the strong interaction. Taking proton having eigenvalue $T_3$=1/2
and neutron as $T_3$ = -1/2 and given the fact that one can associate a baryon number B value of
magnitude one, the electric charge for these can be defined as


\begin{equation}
Q = T_3 + {B \over 2}
\end{equation}

This worked well for
($\Delta^{++}, \Delta^{+},\Delta^{0}, \Delta^{-}$) as T=3/2 quadruplet,
($\pi^{+}, \pi^{0}, \pi^{-}$) and   ($\rho^{+}, \rho^{0}, \rho^{-}$) as T=1 triplet
mesons and $\omega^{0}$ as T=0 singlet meson.

The above expression for the electric charge worked for all the particles known until the
late 40s, when strange particles started to make their presence felt [2]. 
When the $\Lambda$ and K particles
were discovered in the 1950s they were found to be produced 
copiusly but were observed to decay slowly
with a long lifetime. It was postulated [2, p. 257] that they had a new quantum number, 
called Strangeness, associated with them, and which was conserved in the strong interaction
to explain their copious production, but was violated in their weak decay.
Let this new quantum number, Strangeness, be associated with a 
new U(1) group with one generator.
The isospin group SU(2) above is now expanded to the group $SU(2) \otimes U(1)$
Therafter the Gell-Mann-Nishijima relation [3,4] was suggested for the electric charge of
all the old and new particles in 1953 as

\begin{equation}
Q = T_3 + {Y\over 2}
\end{equation}

where Y = B + S and where Y is called the Hypercharge as a generator of the U(1) group in the 
larger group $SU(2) \otimes U(1)$.

As we shall show below it was this expression for the $SU(2) \otimes U(1)$ of the strong interaction
which was arbitrarily picked up by Glashow in 1961 for the weak interaction also. 
However in the meantime
though (p,n) were providing the lowest dimensional fundamental representation of the isospin
SU(2) group, it was $SU(2) \otimes U(1)$ of the strong interaction 
which was the basis of the Gell-Mann-Nishijima expression as given above.
The straightfoward generalization of (p,n) the isospin group SU(2) to (p,n,$\Lambda$) for the group
SU(3) within the so called Sakata model, failed to describe the baryons known at that time. 
Hence Gell-Mann [5] and Ne'eman [6] 
suggested late in 1961, that it was the octet of the mesons and the baryons, 
christened as the Eightfold-Way [7] 
which was what actually represented the larger group SU(3)s reality.  
Note that the fundamental representation of SU(3) played no role in the Eightfold-Way picture and it was
the tensor reprentation that gave physical reality to the octet representation.
That it was actually the three quarks that 
formed the three dimensional fundamental reprsentation
came later in 1964 [8,9]. Gell-Mann however consisdered the Eightfold-Way to be 
the only physical reality while
right until his Nobel Speech in 1969 he maiantained that the quarks were mere 
mathematical artefacts/fiction.
He accepted the quarks as being physically real
only after later complelling evidences from 
deep-inelastic scattering data from SLAC. 
So it was only later in 1964 that truly the Gell-Mann-Nishijima expression for the
electric charge got associated with the larger group SU(3) of the 
strong interaction and the two terms 
got related the two diagonal generators of the group SU(3).

\vskip .2 cm

{\bf (II) Electric charge in the Standard Model} 

\vskip .2 cm

But in early 1961 it was the Gell-Mann-Nishijima expresion for the group $SU(2) \otimes U(1)$.
And the Eightfold way was stll to come. It was that then 
Glashow while studying the weak interaction
sought to incorporate electric charge in a larger electro-weak group as $SU(2)_W \otimes U(1)_W$
where the subscript W refers to the fact that one was not talking of the strong interaction 
( as what Gell-Mann
and Nishijima were doing as above ) but a different groups defining the Weak 
and Electromagnetic interactions
in a partially unified manner. Glashow in early 1961 suggested that it is possible to take ($\nu$, e)
and (p,n) (note that neither the Higgs mechanism nor the quarks were known at that time) 
as left-handed and their corresponding right-handed partners
(we shall discus these in more detail below) and he just picked up 
(actually at random) the Gell-mann-Nishijima
definition of the electric chage for the weak group $SU(2)_W \otimes U(1)_W$ as

\begin{equation}
Q = {T_3}^W + {Y_W \over 2}
\end{equation}

Here $Y_W$ is called weak-hypercharge indicating its differetnt origin from the strong hypercharge above.

He then fixed the values of the weak-hypercharge $Y_W$ to fit the various charges 
to the first generation of matter
particles $(\nu,e)$ and (p,n). For example to get the correct electric charge 
for the left-handed electron and the
corresponding $\nu$ he fixed $Y_W = -1$ and so on.
Glashow was thus able to identify the group structure for the unified electro-weak theory in 1961. 
The use of the Higgs mechanism in this electroweak group came 
much later in 1964 through the work of Salam amd Weinberg.

Note that the above definition of the electric charge as defind by 
Glashow is arbitrary, mainly because of the 
fact that the weak-hypercharge defined as a generator of the group U(1) is not constrained 
to give any quantized value to $Y_W$. It can be any number and as indicatd above 
Glashow had to fix it by hand to give
proper charges to the matter particles. Hence as per the above definition, 
electric charge cannot be said to be
properly quantized in the Standard Model [11, p. 346]. 
Glashow arbitaraily just picked up the Gell-Mann-Nishijima charge definition
from strong interaction. His convention still is the most popular one today and a 
large number of text-books today are using the Glashow convention 
[ref 11, p. 346 and  Ref 12 for a sample of books using Glashow convention]
We may say about "half the particle physics world" uses the above convention to define electric
charge is the Standard Model.

Note that because the above definition is just a convention, 
and also there is no reason why one should have
the "ugly" factor of 1/2 attached to $Y_W$, and  
perhaps aesthetically it would be more satisfying to remove this 
factor of 1/2 and define the electric charge in the 
Electro-Weak group $SU(2)_W \otimes U(1)_W$ as
per a new convention as

\begin{equation}
Q = {T_3}^W + {Y_W}
\end{equation}

Hence the values of weak-hypercharges change with respect to the Glashow definition, 
e.g, for the
$(\nu, e)$ now the value of $Y_W$ = -1/2. Again all the values of hypercharges are fixed 
as per this new convention.
We may say that the other "half of the particle physics world" prefers to use this other convention
and we have listed several books doing so in Ref [13].
We have listed a large number of books and the reader is invited to pick up a book not listed here
and find out which "half of the particle physics world" does it belong to.

If you do not want be part of the above two halves, you are welcome to join me in defining a new
minority convention of electric charge as

\begin{equation}
Q = {T_3}^W + {Y_W \over 1947}
\end{equation}

Now the value of the weak-hypercharge for the $(\nu, e)$
left handed doublet is $Y_W$ = - 1947/2 and similarly for the other entities in the first generation
of particles in the Standard Model.

So now we have defined three conventions - call 
the first one of Glashow as Convention I, the next one 
of Lee, Weinberg and others as Convention II and the last odd one as given above as Convention III.
How is it possible? The reason is that first we are fixing the weak-hypercharges in each case
to get the proper charges for the matter fields in each generation. The next important fact is that
the anomalies in the Standard Models cancel generation by generation [11] and that these hypecharge 
assignments ensure that as they necessarily amount to Tr(Q) = 0 for each generation [11, p. 348],
and thus these are always true for each definition separately.

However for a model as successful as the Standard Model of particle physics, 
it appears a a major weakness
that electric charge is not quantized in it [11] and 
that the definition of the electric charge itself
is conventional. As scientists, we would like our definitions of quantities, 
and that too as basic as the electric charge, not to be arbitray and 
conventional. So let us try to see, as to in what manner 
can we go beyond the limitations that
we have imposed upon ourselves, as to the electric charge in the Standard Model.

\vskip .2 cm

{\bf (III) Going beyond conventions in the Standard Model}

\vskip .2 cm

How do we go beyond the limits of the above conventions as to the electric 
charge in the Standard Model?
A hint arises from the fact that all the three 
conventionlal definitions we had used only for the matter fields
in each generation. 
Glashow had not known of the Higgs mechanism of sponatneous symmetry breaking in 1961.
Let us now include and Higgs field also in the standard way as

\begin{equation}
\Phi = \left( \begin{array}{ccc}
\phi^+ \\
\phi^0 
\end{array} \right)
\end{equation}

With an unknown weak-hypercharge defined as $Y_\phi$.
Now to get the correct charges for the Higgs field above we need

Convention I :  $Y_\phi = 1$

Convention II:  $Y_\phi = 1/2$

Convention III: $Y_\phi = 1947/2$

So with all these different definitions with fantastically different weak-hypecharges 
we still manage to get the correct electric charges for the Higgs fields.
Since Higgs is the uniform and ubiquitous field providing all the well known properties to the 
Standard Model, is it possible that the above three results for the Higgs hypercharges are trying
to tell us something.

A moments thought tells us that indeed it is so. All the three conventions basically
for the the electric charge can be generalised to  
 
\begin{equation}
Q = {T_3}^W + b {Y_W}
\end{equation}

where b is fixed so that the upper and lower components of the Higgs doublet get the correct charges.
Immediately we see that it is when we take b = $1 \over {2 {Y_\phi}}$.
So the above electric charge becomes

\begin{equation}
Q = {T_3}^W + ({1 \over {2 {Y_\phi}}}) {Y_W}
\end{equation}

Now we see that for $Y_W = Y_\phi$ the correct Higgs doublet charges arise.
So clearly hidden within the above arbitarainess of the three conventions, atleast for the
Higgs doublet, is an exact definition of the electric charge which is not arbitrary or conventional.
It is exact and gives correct quantized charges to the higgs doublet.

How about the matter fields? No problem, as long as we take their weak hypecharges to be proportinal to
the Higgs Weak-hypercharge. For example for the first generation 

\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} ({4\over 3}) ;
                    Q(u_R) ={1\over 2} ({4\over 3}) ;
                     d_R, Y_d = {Y_\phi} ({-2\over 3}) ;
                    Q(d_R) ={1\over 2} ({-2\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}

Thus only by including the Higgs doublet in a consistent manner and by looking deeply into the
so called arbitrariness involed in the defintion of the electric charge in the Standard Model we
find entirely on the basis of internal consistency and logic that the electric charge 
is unambigously defined in the Standard Model as
$Q = {T_3}^W + ({1 \over {2 {Y_\phi}}}) {Y_W}$.
This gives correct charges to the Higgs doublet and also with proper 
definitions of the weak-hypercharges for the
matter field in each generation - which are always proportional to the Higgs weak-hypercharges, give the 
correct quantized electric charges in the Standard Model.
Hence we state that contrary to the popular belief [11,12,13], the 
electric charge is consistently quantised in the Standard Model. 
Also we note that this quantization never fixes the Higgs weak-hypercharge.

So the arbitaraines in the definition of the electric charge, was trying to tell us somethig, in particluar
about the Higgs doublet and also from there about the matter field charges. Once we had the correct
definition of the electric charge in the Standard Model
$Q = {T_3}^W + ({1 \over {2 {Y_\phi}}}) {Y_W}$,
though we had to guess the other charges for the matter fields, the charges do came out exact and fully
quantized. Hence our simple arguments show that electric charge is actully self-consistently
quantized in the Stanadard Model and that it happens for each generation separately and that
the Higgs weak-hypecharge never gets constrained to any value.

The above amazingly follows from simple consistency arguments within the Standard Model.
Next, is it possible to obtain the above electric charge expressions in the Standard Model
in a more mathematical and rigorous manner?
Or is it possible that we may understand the 
various hypercharges defined as being proportional to the
Higgs hypercharge and the correct definition of the electric charge in terms of the Higgs hypercharge
as $Q = {T_3}^W + ({1 \over {2 {Y_\phi}}}) {Y_W}$ in a
mathematically consistent manner? 
It is heartening to note that indeed it is so. Below we do that.

\vskip .2 cm
{\bf (IV) Mathematical charge quantization in the Standard Model}
\vskip .2 cm 


Let us start by looking at the first generation of quarks and leptons
(u, d, e,$\nu$ )  and assign them to
$SU(3)_{c} \otimes SU(2)_L \otimes U(1)_Y$ representation as follows 
[14,15]. 


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

\begin{equation}
 e_R; (1,1,Y_e)
\end{equation}

To keep things as general as possible this brings in five unknown 
hypercharges.


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'= a'I_3 + b'Y \end{equation}

We can always scale the electric charge once as $Q={Q'\over
a'}$ and hence ($b={b'\over a'}$)

\begin{equation} Q = I_3 + bY \end{equation}


In the Standard Model $ SU(3)_{c} $ $\otimes$ $ SU(2)_{L}$ $\otimes$
$U(1)_{Y}$ is spontaniously broken through the Higgs mechanism to the
group $ SU(3)_{c} $ $\otimes$ $U(1)_{em}$ . In this model the Higgs is
assumed to be doublet $ \phi $ with arbitrary hypercharge $ Y_{\phi}$.
 The isospin $I_3 =- {1\over2}$ component of the
Higgs 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}

Note that this is exactly the same as the eqn. (8) above.
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 Standard Model. One has to provide new physical inputs to
proceed further.
Here one demands that fermions acquire masses through Yukawa coupling in
the Standard Model. 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
Standard Model [14,15] as follows:


\begin{displaymath}
 q_L = \pmatrix{u \cr d}_L , Y_q = {{Y_\phi} \over{3}},  \end{displaymath}
\begin{displaymath} Q(u) = {2\over 3}, Q(d) = {-1\over 3}
 \end{displaymath}
\begin{displaymath} u_R, Y_u = {3\over{4}} {Y_\phi}, Q(u_R) ={2\over{3}} 
 \end{displaymath}
\begin{displaymath} d_R, Y_d = {-2\over{3}} {Y_\phi}, Q(d_R) ={-1\over3} 
 \end{displaymath}
\begin{displaymath} l_L = \pmatrix{\nu \cr e}, Y_l = -Y_\phi,
Q(\nu) = 0, Q(e) = -1
  \end{displaymath}

\begin{equation}
 e_R, Y_e = -2Y_\phi, Q(e_R) = -1
\end{equation}

It has also been shown [14] that  for arbitrary $ N_{c} $ the colour
dependence of the electric charge as demanded by the Standard Model is 

\begin{displaymath}
\newline Q(u) = {1\over 2}(1+{1\over N_c})
\end{displaymath}

\begin{equation}
\newline Q(d) = {1\over 2}(-1+{1\over N_c})   
\end{equation}

Note that within the Standard Model group 
$ SU(3)_{c} $ $\otimes$ $ SU(2)_{L}$ $\otimes U(1)_Y$ the electroweak sector consists of 
$ SU(2)_{L}$ $\otimes U(1)_Y$. Still the electric charge knows of the colour degree of freedom.
Interestingly though the electromagnetism does not know of the colour, still the electric
charge has colour, existing within its guts, so to say.
This shows that the Standard Model still is more unified than we had visualized so far.

Hence here we have shown that mathematically the structure of the Standard Model is such
that the electric charge is fully quanitized in it. In adition, this structure,
as obtained on the basis of our earlier phenomenological considerations, is fully supported by
our mathemtical analysis here.

\newpage
\vskip .2 cm
{\bf (V) Conclusions}
\vskip .2 cm


We started with the standard textbook statements that in the Standard Model the 
definition of the electric charge 
is conventional and that it is not quantised.
This is the understanding of the electric charge as given in all the text books.
And this also is what is taught in all the graduate schools.
Three conventions of defining the electric charge were discussed and thereafter we showed that
incorporation of the Higgs field pointed to a hidden consistency in the electric charge
in the Standard Model. That led us to the conclusion, on the basis of self-consistency arguments, 
that the electric charge
is actually quantized in the Standard Model. Upto this point, this should be construed to be
true as a phenomenologically consistent statement.

Next we showed that the same can now be put on a more solid mathematical basis.
We have shown that the mathematical structure of the Standard Model is such that the electric
charge if fully quantized in the Standard Model. Hence the phenomenological and the 
mathematical derivations give identical results.
Hence not knowing this basic fact will be detrimental to a proper understanding of particle
physics in the future.
Therefore it is pointed out that all the text books should be corrected to account for this 
new basic reality of the Standard Model. 

Note that in the above quantization of the electric charge, the
Higgs hypercharge $Y_{\phi}$ always cancels 
out and hence remains unconstrained. 
A few others, while looking at the electric charge in the Standard Model, had ignord this fact 
as indicated in Ref [14,15,16].
This is important as not paying attention to this fact
can lead to asking wrong questions, as for example for the possibility of milli-charged 
particles, which are ruled out by the Standard Model as per the above arguments [15, 16].
Note that a repetitive structure gives charges for the other generation
of fermions as well.
Hence the electric charge quantization as indicated here and as discussed in ref [14,15,16]
is based on physically rigorous and mathematically consistent arguments. 

\vskip .2 cm

\newpage

{\bf References}
\vskip .2 cm
(1). W. Heisenberg, " Ueber den bau der atomkerne - I", Zeit Physik {\bf 77}, 1 (1932)
\vskip .2 cm
(2). J. J. Sakurai, "Invariance principle and elementary particles", Princeton University Press,
   Princeton, 1964
\vskip .2 cm
(3). M. Gell-Mann, "Isotopic spin  and new unstable particles", Phys. Rev. {\bf 92}, 833 (1953)
\vskip .2 cm
(4). T. Nakano and K. Nishijima, "Charge independence of V-particles", Prog. Theo. Phys. {\bf 10}, 581 (1953)
\vskip .2 cm
(5). M. Gell-Mann, "A theory of strong interaction symmetry", Caltech Report CTLS-20 (1961)
\vskip .2 cm
(6). Y. Ne'eman, "Derivaton of strong interaction from a gauge invariance", Nucl. Phys. {\bf 26}, 222 (1961)
\vskip .2 cm
(7). M. Gell-Mann and Y. Ne'eman, "The Eightfold Way", W.A. Benjamin Pub. Co., New York, 1964
\vskip .2 cm
(8). M. Gell-Mann, "A schematic model of baryons and mesons", Phys. Lett {\bf B 8}, 214 (1964)
\vskip .2 cm
(9). G. Zweig, CERN Report No. 8182/TH401 (1964)
\vskip .2 cm
(10). S. Glashow, "Partial symmetry of weak interactions", Nucl. Phys. {\bf 22}, 579 (1961)
\vskip .2 cm
(11). T-P. Cheng and L-F. Li, "Gauge theory of elementary particle physics",
    Clarendon Press, Oxford, 1984, p. 346
\vskip .2 cm
(12). L. H. Ryder, "Quantum Field Theory", Cambridge University Press,
    Cambridge, 1986 (reprinted 2006), p. 309; 
    
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    A. Bettini, "Introduction to elementary particle physics", Cambridge University Press, Cambridge, 
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    {\bf 70}, 1019 (2007), p. 1024
\vskip .2 cm
(13). T. D. Lee, "Particle physics and introduction to field theory", Harwood, New York, 1981, p. 674;
    
    S. Weinberg, "The quantum theory of fields", Vol. II, Cambridge University Press, Cambridge, 1996, p. 385;
    
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    K. Huang, "Quarks, leptons, and gauge fields", "World Scientific, Singapore, 1992, p. 114;
    
    D. H. Perkins, "Introduction to High Energy Physics", Cambridge University Press, Cambridge, 2000, p. 350;
    
    M. Srednicki, "Quantum Field Theory", Cambridge University Press, Cambridge, 2007, p. 543;
    
    W. N. Cottingham and D. A. Greenwood, "An introduction to the Standard Model of particle physics",
    Cambridge University Press, Cambridge, 1998, p. 132 
\vskip .2 cm    

(14). A Abbas, "Anomalies and charge quantization in the Standard Model with arbitrary
number of colours", Phys. Lett {\bf B 238}, 344 (1990)
\vskip .2 cm
(15). A Abbas, "Spontaneous symmetry breaking, quantization of the electric charge and 
the anomalies", J. Phys. {\bf G 16}, L163 (1990)
\vskip .2 cm 
(16). A. Abbas, "Standard Model of particle physics has charge quantization", Physics Today,
July 1999, p. 81
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