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Standard Model, Higgs bososn, Skyrme model, topological quantum number, QCD, hadron physics
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\begin{center}
\Large{\bf What has been discovered at 125 GeV by the CMS and the ATLAS experiments?} 
\end{center}
\vspace{1in}
\begin{center}
{\bf Syed Afsar Abbas} \\
Department of Physics\\
Aligarh Muslim University, Aligarh - 202002, India\\
e-mail: drafsarabbas@yahoo.in\\
\end{center}
\vspace{1in}
\begin{center}
{\bf Abstract}
\end{center}
\vspace{.5in}

While looking for the putative Higgs boson of the Standard Model of particle physics,
recently, the CMS and the ATLAS experiments at CERN have found strong signals
of a new particle at about 125 GeV. However the decay channels of this particle
had some unexpected and puzzling anomalies not explainable by the Standard Model.
Here we show that what they have found at 125 GeV is the long sought for and the missing
ingredient of the strong interaction  - the sigma-meson of the Chiral Sigma Model,  
within the framework of the Skyrme model with a topological interpretation
of the baryons. Just as a massless gauge boson is a requirement and hence a prediction
of the local gauge theories; in the the same manner, a very heavy scalar meson is a
requirement and hence a prediction of the Skyrme model of the hadrons. The 125 GeV
particle discovered by the CMS and the ATLAS groups is an experimental confirmation
of this unique prediction of the topological Skyrme model.

\newpage

Recently at CERN, the CMS Group [1] and the ATLAS Group [2] announced the discovery of a 
new boson of mass 125 GeV. As the Higgs boson of the Electro-Weak (EW) model was expected to
be seen in this region, most of the physicists thought that this was it. Except for the
mass of the Higgs boson itelf, the EW model is quite restrictive as to how this 
putative Higgs will decay in which channels and as to what fractions of the total decay 
in each channel. However both the groups discovered some anomalous signals which cannot be
explained on the basis of the EW model. These anomalies were firstly too much
decay into the gamma-gamma channel and secondly no signals for the tau-antitau where many 
more were expected. Hence this new particle does not appear to be the putative
Higgs boson of the Standard Model (SM). 

However, both the groups are claimimg strong signals of five-sigma confidence level
for a new particle at 125 GeV. So if not the expected Higgs boson, then what is 
it?

Several papers have already been written to explain these anomalous signals within the 
framework of models which may generically be called "non-conservative".
In this paper we shall however adopt a strict "conservative" approach. This means that
we ask whether it is still possible to seek solution for this new puzzle strictly within 
the ambit of already successful and empirically varified model frameworks of 
particle physics? A careful scrutiny of the latest "Review of Particle Properties" [3]
indicates that there is indeed an entity which fills the bill.

There is this rather unconfirmed particle ${f_0}(500)$ which is identified with the
sigma meson $\sigma$ of the Sigma-Meson model of the hadron physics.
The Particle Data Group [3] themselves state that "the interprettion of this entity
as a particle is controvertial".
Most of the hadron theorists have in the past, to account of the fact that the
$\sigma$ - meson was so reluctant to show up in the laboratory, 
that they started taking seriously
the so called non-linear Sigma-Meson model (see below) where the 
$\sigma$ - meson remains hidden inside a so called, chiral circle.
Hence still adopting a conservative approach and agreeing with a big majority of 
physicists, we believe that in spite of the intense search for the last fifty years or so,
the $\sigma$ - meson of the Sigma-Meson model has not yet been obsreved in the laboratory.
Hence this should be treated as the "missing link" of the hadron physics.

Also as pointed out by the Particle Data Group [3], in the hadron physics jargon, this
missing $\sigma$ particel is called "the Higgs boson of the strong interaction".
So actually there are TWO missing Higgs bosons - one is the Higgs boson of the EW 
model ( which both the CMS and the ATLAS groups were actually looking for at 125 GeV 
and did not find !)
and the other one is this "Higgs boson of the strong interaction" ( which they were not 
even aware of, and which as I shall show below, is what they have actually discovered! ).
I shall therefore show that with this, now the long sought for and the missing link of
the strong interaction has atlast been discovered.

A doubt may arise in the mind of some readers, and that may be related with the high scale
of 125 GeV for the $\sigma$ - meson. So the question is - is it a basic requirement of
hadron physics that the $\sigma$ - meson should be light, say of the scale of the other
hadrons at about 1 GeV? This was actually the point of view which has made people view the
very broad  ${f_0}(500)$ [3] as a possible though weak candidate for the same.

On the basis of the original Gell-Mann-Levy model [4, p. 186],
where the baryon masses are generated through the SSB of the chiral
$SU(2)_R  \otimes SU(2)_L$ model, it would be a natural expectation that the 
$\sigma$ meson would have a mass of the typical hadronic scale of 1 GeV or so.
So also would be the situation for the large number of variants of the Linear
Sigma-Model and the Non-linear Sigma-Models which are popularly being used in hadron 
physics today [4,5,6]. In fact that the $\sigma$ meson has been so reluctant to show up
in the laboratory at these low energies that it was found pragmatic to have it as a
"hidden" state inside the so called chiral circle [4,5,6]. So it is true that on the basis 
of chiral sigma models and their various extensions [4,5,6], it is a natural physical 
expectation that the sigma meson should have a mass of the order of the strong interaction
and that is about 1 GeV or so. It is because of this that one still hears now and then 
that this sigma meson may actually be the ${f_0}(500)$  or   ${f_0}(1370)$ particles,
although as discussed above there are serious doubts abouts these weak claims.

Whitin the area of hadron physics there is one narrow window which does allow one ot look
beyond the above constraints within the chiral Sigma-Model framework.
Herein the sigma meson is expected, or in fact demanded, to be very heavy, or even
infinetly heavy. Though in the original chiral Sigma.Model of Gell-Mann-Levy, 
the degrees of freedom 
were an isotriplet of pions, a scalar sigma-meson and an isodoublet of nucleons.
However the presence of the fermionic nucleons is not absolutely essential for the 
analysis and indeed the non-linear Sigma-Model without the initial fermions has been 
used to generate the nucleon as
a topological soliton of the interacting Nambu-Goldstone pion fields 
[4, p.186] in the Skyrme Model. 

The Linear Sigma-Model in the absence of nucleons is given by

\begin{equation}
L =
{1 \over 2} [{(\partial_\mu \sigma)}^2  + {(\partial_\mu \pi)}^2] -
 V( \sigma^2 + \pi^2 ) 
\end{equation}

where
\begin{equation}
V( \sigma^2 + \pi^2 ) = {\mu^2 \over 2}(\sigma^2 + \pi^2) + 
                        {\lambda^2 \over 4} {(\sigma^2 + \pi^2)}^2 ;
                        \mu^2 < 0  
\end{equation}


The SSB of this 
 $SU(2)_R \otimes SU(2)_L$
model occurs because $\mu^2 < 0$ so that the minimum of the potential is at 


\begin{equation}
\sigma^2 + \pi^2 = f^2 ; f= {(- {\mu^2 \over \lambda})}^{1 \over 2} 
\end{equation}

If we take 
$<0 \mid \sigma \mid 0> = f$ and $<0 \mid \pi \mid 0> = 0$
and working with the shifted fields one finds that the isosinglet mass is
$m_\sigma = \sqrt{2} \mid \mu \mid$ and the isotriplet pion mass is zero
that is that these are the Nambu-Goldstone bosons.

From this linear Sigma-Model one obtains the non-linear Sigma-Model in the standard
methods [5,6]. This involves taking the limit $m_\sigma \rightarrow \infty$
and placing $\pi$ in a non-linear representation of the group SU(2). 
We thus get the non-linear Sigma-Model Lagrangean [4, p. 638]. Supplemented with
the Skyrme stabilizing term it looks as follows [7]:

\begin{equation}
L_S = {{f_\pi}^2\over 4} Tr (L_\mu L^\mu) +
{1\over {32 e^2}} Tr {[L_\mu,L_\nu]}^2
\end{equation}

where $L_\mu = U^\dag {\partial}_\mu U$ . The U field for the
three flavour case for example is

$ U(x) = exp [{{i \lambda^a \phi^a (x)} \over {f_\pi}}]$

with $\phi ^a$ the pseudoscalar octet of $\pi$, K and $\eta$
mesons. In the full topological Skyrme this is supplemented
with a Wess-Zumino effective action

\begin{equation}
\Gamma_{WZ} = {-i\over {240 \pi^2}} \int_{\Sigma} d^5 x 
\epsilon^{\mu \nu \alpha \beta \gamma }
Tr [ L_\mu L_\nu L_\alpha L_\beta L_\gamma ]
\end{equation}

on surface $\Sigma$. Let the field U be transformed by the 
charge operator Q as 

$ U(x) \rightarrow e^{i \Lambda Q} U(x) e^{-i \Lambda Q}$.

where all the charges are counted in units of the absolute value
of the electronic charge.

Note that the above Skyrme Lagrangean demands the existence of a very heavy scalar
sigma-meson. This demand is of the same nature as the corresponding requirement
of a massless gauge boson in a local gauge theory.

Making $\Lambda = \Lambda (x)$ a local transformation the Noether
current is

\begin{equation}
{J_\mu}^{em} (x) = {j_\mu}^{em} (x) + {j_\mu}^{WZ} (x)
\end{equation}
where the first one is the standard Skyrme term and the second
is the Wess-Zumino term

\begin{equation}
{j_\mu}^{WZ} (x) = { N_c \over {48 \pi^2}} \epsilon _{\mu
\nu \lambda \sigma} Tr L^\nu L^\lambda L^\sigma 
( Q + U^\dagger Q U )
\end{equation}

For the hypercharge we take Y = $ N_3 \over 3 $ [7] and demanding that
the proton charge be unit for any arbitrary value of $N_c$ we find all
the chargs. Hence as per the Skyrme Model the electric charges
are [7]

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

These electric charges and their colour dependence should be viewed as a
unique prediction of the Skyrme model. 
Also as we shall discuss below these charges from the Skyrme model
have the correct color dependence as demanded by the SM as well.
So the colour dependence of the electric charge as required by the structure
of the SM as shown below are exactly 
reproduced by the Skyrme model. 
Hence it is heartening to conclude
that the Skyrme model is fully consistent with the Standard Model.
This should be taken as an indication that the Skyrme model should be taken
as a good model to study hadrons at low energies [7].

It is well known that in SU($N_{c}$) Quantum Chromodynamics
in the limit of $N_{c}$  going to infinity the baryons behave
as solitons in an effective meson field theory [7,8]. A
popular candidate for such an effective field theory is the
topological Skyrme Model [4,5,6]. It has been extensively studied for
two or more flavours and it has been shown that the resemblance
of the topological soliton to the baryon in the quark model
in the large $N_{c}$ limit is very strong [4,6]. It's baryon number
and the fermionic character is also well understood [8].

Theoretically the most well studied and experimentally the
best established model of particle physics is the Standard Model
( SM ) based on the group $SU( 3_c ) \otimes SU(2)_L \otimes U(1)_Y$

Also analytically the author obtained the color dependence
of the electric charge in the SM as given above in eqns 8 and 9.
for $N_c$ = 3 this gives the correct charges. It was also demonstrated by the 
author [8] that these were the correct
charges to use in studies for QCD for arbitrary $N_c$.
This was contrary to many who had been using static ( ie. independent
of color ) charges 2/3 and -1/3 [8].

Hence in addition to the other well known properties of the SM,
I would like to stress that the quantization of the electric charge
and the structure of the electric charge arising therein,
especially its color dependence, should be treated as an intrinsic
property of the SM. A consistency with the SM should be an essential 
requirement for phenomenological models which are supposed to
work at low energies and for any extensions of the SM which should
be relevant at high temperatures especially in the context of the
early universe.

So the above Skrme Model which finds justification as a good description of
the QCD at low energies, not only because it has the right symmetries
[4,5,6] of the QCD but also as emphasized here, has the right structure
of the electric charge ( as especilly the electric charge surprisingly has 
colour dependence arising from  the structure of the Standard Model) is 
the correct model to study the low energy properties of the hadrons.
As shown above  this particular model demands a very heavy sigma meson.
Hence what has been seen at 125 GeV by the CMS and the ATLAS experiments
is this particluar scalar particle. As per Skyrme model as above
in fact may even be that $m_\sigma \rightarrow \infty$. We treat 125 GeV being
close to fulfilling this condition as the nucleon mass is about 1 GeV,
much smaller that 125 GeV. Note that the situation here is analogous
to the other complementary case wherein canonically the pion mass of 140 Mev being 
considerably smaller than the scale of 1 GeV,
is taken as effectively being massless [4,5,6].

Next as obseverd by the CMS [1] and the ATLAS [2] experiments they have
observed two puzzling anomalies in the decay of the 125 GeV new particle.
Firstly as per the electro-weak Higgs particle sector a good 6 percent of 
these decay should have occurred in the tau-antitau channels, and both these 
two expereiments found none. Secondly much smaller gamma-gamma channel was 
expected and they obtain many more of these events.

These anomalies indicate that it would be unjustified to associate this
new particle at 125 GeV with the EW Higgs particle. However both these
anomalies are naturally understood as arising from a sigma-meson.
As this sigma-meson arises fron the strong interaction, it will not
couple to the leptons.
So this sigma-meson should not decay into a tau-antitau pair. And this is exactly
what has been observed by the CMS and the ATLAS groups.
This lack of tau-antitau decay is a clear distinguishing fetaure of the sigma-meson model
in contrats to the EW Higgs boson where this channel should exist on fundamental
grounds. On this account itself the sigma-model inetrpretation for the 125 GeV
boson is clearly winning out.


Also the Vector Dominance Model of the Skyrme Model [6] would predict
stronger gamma-gamma decays of the sigma meson in our model, akin to what was observed
by the CMS and th ATLAS eperiments.
Hence clearly these so called anomalies are a clean signature of
this new particle at 125 GeV interpreted as the sigma-meson of the chiral Sigma-Model
forming the basis of the Skyrme model of the strong interaction.

Now a look at the scalar particles which are known to exist as per the Particle Data
Tables [3] and their observed decay properties, we find the follwing examples:
1. ${f_0}(500)$ with m=(400-500) MeV and full width $\Gamma$ = (400-700) MeV,
2. ${f_0}(980)$,
3, ${f_0}(1370)$,
4. ${f_0}(1500)$ and 
5.  ${f_0}(1760)$ 

Taking their decays as a guide, we notice that there are hints of the Vector Domination
in the gamma-gamma channels in these scalar particles, but these seem to defy any 
simplistic pattern. So to predict more precise numbers for the various
decay channels for a very heavy sigma-meson at 125 GeV would require
careful modeliing. We intend to do so in the future.
But that it is indeed the sigma-meson of the strong interaction that has been obserevd
by the 
CMS and the ATLAS experimenbst is quite clear on the basis of the arguments presented above.


In summary, we have demostrated here, quite convincingly, that what has been observed at
125 GeV by the CMS and the ATLAS experiments recently [1,2] is the long sought for and 
the missing link of the strong interaction , the "Higgs boson of the strong interaction"
[PDT, 2012, Ref 3]. This arises because though most of the variants of the 
chiral Sigma-Meson model of Gell-Mann-Levy require a lighter sigma-meson [4],
the non-linear Sigma-Meson model required as a base for the topological
structure model of Skyrme, demands a very heavy sigma-meson. 
This demand for a very heavy sigma-meson mass in the Skyrme Lagrangean is of the same nature
as that of a massless gaige boson in a local gauge theory.
We have also shown why 
this topological interpretation of baryons should be taken seriously as a model 
of the strong interaction of baryons. Hence the Skyrme model uniquely predicts a very heavy 
sigma-meson and the 125 GeV particle disovered at CERN is a confirmation 
of this prediction.

\vskip 1 cm
%\newpage
\begin{center}
{\bf References}
\end{center}
\vskip .5 cm


1. CMS Collaboration, Phys. Lett. {\bf B 716} (2012) 30-61 ("Observation 
   of a new boson at a mass of 125 GeV with the CMS experiment at LHC") 

2. ATLAS Collboration, Phys. Lett. {\bf B 716} (2012) 1-29 ("Observation 
   of a new particle in the search for the Standard Model Higgs Boson with
   the ATLAS detector")

3. J. Beringer et.al. (Particle Data Group), Phys. Rev. {\bf D 86} (2012) 010001
   ("Review of particle properties")

4. R. E. Marshak, "Conceptual foundations of modern particle physics", 
   World Scientific, Singapore, 1993

5. R. Rajaraman, "Solitons and instantons", North-Holland, Amsterdam, 1982

6. A. Hosaka and H. Toki, "Quarks, baryons and chiral symmetry", World Scientific,
   Singapore (2001)

7. A. Abbas, {\bf B 503} (2001) 81-84 ("Does Skyrme model give a consistent 
   description of hadrons?") 

8. A. Abbas, Phys.Lett. {\bf B 238} (1990) 344-347 ("Anomalies and charge
   quantization in the Standard Model with arbitrary number of colours")


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