TIFR/TH/23-3

The Higgs Boson and its Physics — an Overview

Sreerup Raychaudhuri

Department of Theoretical Physics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400 005, India.

E-mail<EMAIL_ADDRESS>

Abstract

> The Higgs boson plays a central role in the Standard Model, as well as in
> theories which go beyond it. This article is therefore divided into two
> parts. The first takes s historical approach and shows how the mass problem
> entered weak interaction theory from the beginning and how it was solved by
> invoking the Higgs boson. This is followed by a construction of the Glashow-
> Salam-Weinberg model, again stressing the role played by the Higgs doublet.
> This part culminates in the Higgs boson discovery of 2012. The second part
> first discusses the shortcomings of the Standard Model and then touches upon
> the major theories which try to improve upon it, mostly with profound
> consequences on the Higgs sector. This is followed up by short descriptions
> of a number of popular extensions of the Higgs sector, and culminates in a
> brief introduction to effective field theory approaches to studying the
> Higgs sector.

Invited article to appear in a Special Issue of the Indian Journal of Physics

Part 1 : Roads to the Higgs boson

The 2012 discovery of the Higgs boson created a worldwide sensation and was
followed a year later by the award of a much-belated Nobel Prize to two of the
seven scientists who had originally worked out the basic theory of this
elusive particle in 1964. While it was widely reported that the search for the
Higgs boson was the culmination of an effort which had lasted the 48 years
from 1964 to 2012, the real journey had started much earlier. In fact, it
started in 1934 when Enrico Fermi made a facile assumption in an effort to
explain beta decay. The Higgs boson discovery was, therefore, the culmination
of a 78 year-long series of developments.

In order to appreciate the proper significance of the Higgs boson as the
keystone, as it were, of the electroweak theory, it is necessary to retrace
this long path and see how the current formalism was arrived at. The first
part of this article, therefore, does just this. It starts with the Fermi
theory of weak interactions and explains successive developments in a part-
historical part-pedagogic manner and ends with the Higgs boson discovery.

However, the Higgs boson, now that it is known to exist, throws up almost as
many questions as it answers. Is it an elementary particle or a composite? Are
there more scalars in the theory? Can we understand why at high energies, it
behaves as if tachyonic? All of this has led to intense speculation about the
nature and properties of this particle, and suggestions that it is just a part
of some additional structures in the electroweak sector – the tip of the
iceberg, as it were. The second part of this article will discuss these ideas
in some detail.

This article will deal mostly with theoretical and phenomenological aspects of
the Higgs story. The purely experimental aspects are undoubtedly fascinating,
but they call for a different article and a different author with the
appropriate expertise.

The titles of the different sections invoke different evolutionary stages in
the historical development of Western art. This is not whimsical but a serious
attempt to draw parallels with parallel fields of human creativity, of which
science forms just one, even though it is expressed in a different form.

##  1 Beginnings: Gauge Theory and Electroweak Unification

The concept of a symmetry, though understood in geometry from ancient times,
and intuitively used by Newton and his followers, finds its precise expression
in the variational formulation of classical mechanics. A symmetry is really a
no-go theorem, for it means that we cannot physically distinguish between two
different configurations of the dynamical variables describing a system, i.e.
the action remains invariant. It was Emmy Nöther who famously discovered in
1918 that there is an intimate connection between symmetries of the action and
conserved quantities [1], but the fact that there is an intimate connection
between symmetries and the fundamental forces of Nature was really an epiphany
central to Einstein’s 1915 theory of general relativity [2]. The equivalence
principle, the foundation on which general relativity is based, is simply the
statement that general coordinate invariance – a symmetry – becomes manifest
in experiments as a gravitational force [3]. General coordinate invariance is,
of course, just a no-go theorem for attaining an absolute frame of reference
$\grave{\rm a}$ la Newton. Soon, Hermann Weyl [4] discovered another symmetry
of the free Einstein equations, viz., under dilatations, to which he gave the
name ’gauge’ symmetry.

The advent of quantum mechanics [5] in 1925-26 led to a similar idea being
applied by V. Fock and, independently, by F. London [6] in the context of the
Schrödinger equation. The force, in this case, turned out to be another well-
known one, viz., electromagnetism. The name given by Weyl to a dilatation,
i.e., gauge transformation, was now re-purposed to describe a phase
transformation. In modern parlance, it can be a combination of the two.
However, since the core idea invokes special relativity, it is better
expressed in a fully relativistic formalism.

We start, therefore, with the Lorentz-invariant action for a free complex
scalar field

$S\stackrel{{\scriptstyle\text{def}}}{{=}}\int
d^{4}x\left[\partial^{\mu}\varphi^{*}(x)\
\partial_{\mu}\varphi(x)+M^{2}\varphi^{*}(x)\varphi(x)\right]$ (1)

and note that it is invariant under a phase transformation

$\varphi(x)\to\varphi^{\prime}(x)=e^{-ie\alpha}\;\varphi(x)$ (2)

if $\alpha$ is independent of spacetime coordinates $x$, but not otherwise.
Here $e$ is a book-keeping parameter, which can be different for different
fields. However, a spacetime-independent $\alpha$ would mean the same phase
transformation at all points in spacetime — which would require a signal to
propagate instantaneously. Relativity forbids this and therefore, demands that
$\alpha$ should be spacetime-dependent, i.e. $\alpha=\alpha(x)$. On the other
hand, the equation of motion, which arises from this action is

$\left(\Box+M^{2}\right)\;\varphi(x)=0$ (3)

indicating that $\alpha$ is the phase of the scalar wavefunction, when we make
the transition from a field theory to a single-particle equation. In quantum
mechanics, the probability interpretation tells us that the phase of this
wavefunction does not affect any physics, and this means that there must exist
a symmetry under phase transformations — another no-go theorem that the phase
cannot be measured. Translated to the field theory we have a paradox, viz., if
$\alpha$ is local, then this phase invariance is lost. This may be compared to
the loss of local Lorentz invariance of the spacetime derivative
$\partial_{\mu}$ in a curved space, and it has the same solution as in general
relativity, viz., to add a field of force $A_{\mu}(x)$ which will nullify the
effect and restore the symmetry. Thus, we rewrite the action as

$S\stackrel{{\scriptstyle\text{def}}}{{=}}\int
d^{4}x\left[D^{\mu}\varphi^{*}(x)\
D_{\mu}\varphi(x)+M^{2}\varphi^{*}(x)\varphi(x)\right]$ (4)

where $D_{\mu}=\partial_{\mu}+iA_{\mu}(x)$ is called a (gauge) covariant
derivative and we require that the transformations

$\varphi(x)\to\varphi^{\prime}(x)=e^{-i\alpha}\;\varphi(x)\qquad\qquad
A_{\mu}(x)\to A^{\prime}_{\mu}(x)=A_{\mu}(x)+\partial_{\mu}\alpha(x)$ (5)

occur simultaneously. Taken together, we refer to this as a local gauge
transformstion. The new field $A_{\mu}(x)$ – which is known as the gauge field
– must have its own dynamics, which involves the construct
$\partial_{\mu}A_{\nu}(x)$. The minimal Lorentz and gauge invariant construct
with this object is $F^{\mu\nu}F_{\mu\nu}$, where
$F_{\mu\nu}(x)=\partial_{\mu}A_{\nu}(x)-\partial_{\nu}A_{\mu}(x)$. Adding this
to the action, we obtain

$S\stackrel{{\scriptstyle\text{def}}}{{=}}\int
d^{4}x\left[D^{\mu}\varphi^{*}(x)\
D_{\mu}\varphi(x)+M^{2}\varphi^{*}(x)\varphi(x)-\frac{1}{4}\;F^{\mu\nu}(x)\,F_{\mu\nu}(x)\right]$
(6)

from which the equations of motion become

$\displaystyle\left(D_{\mu}+iA_{\mu}\right)^{2}\varphi(x)$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\partial_{\mu}F^{\mu\nu}$ $\displaystyle=$
$\displaystyle J^{\nu}(x)$ (7)

where

$J_{\mu}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}i\left[\varphi^{*}(x)\;\partial_{\mu}\;\varphi(x)-\varphi(x)\;\partial_{\mu}\;\varphi^{*}(x)\right]$
(8)

is a conserved Nöther current corresponding to the phase symmetry. These
correspond so perfectly to the equations of a charged scalar $\varphi(x)$ in
an electromagnetic field $A_{\mu}(x)$ that it is quite natural to identify the
gauge field $A_{\mu}$ with the force field of Maxwellian electrodynamics. This
is the essence of Fock and London’s understanding of the electromagnetic field
as a consequence of the phase symmetry of quantum mechanics.

We note that the addition of a mass term

$S_{m}\stackrel{{\scriptstyle\text{def}}}{{=}}\int d^{4}x\
M_{A}^{2}\,A^{\mu}(x)A_{\mu}(x)$ (9)

would result in a mass $M_{A}$ for the quanta of the electromagnetic field,
i.e. the photon. However, such a term is clearly not gauge invariant and hence
the photon must perforce remain massless — as indeed, it is.

It is somewhat mind-boggling to think that all electromagnetic phenomena –
from lightning bolts in the sky to the tiny currents in our brains – arise
from an inability to measure the phase of so tenuous a thing as a wavefunction
111It certainly failed to convince Einstein himself, whose original idea it
stemmed from. This led to Einstein attacking the very basis of phase
invariance, viz., the probability interpretation of quantum mechanics, coining
the dictum ’God does not play dice’. But we now have plenty of evidence that,
in fact, He does.. However, this is no more surprising than that a universal
force like gravity, felt so acutely on the Earth’s surface, should stem from
our inability to detect the state of acceleration of an observer. A more
pedantic statement would be that our mathematical understanding of forces in
terms of fields of force lend themselves to an understanding in terms of
symmetry [7].

The discovery of the Dirac equation in 1928 [8] immediately led to a
reformulation of the above theory for a charged fermion field, and this led to
a description of quantum elctrodynamics, or QED, which has the classical
action

$S_{\rm QED}\stackrel{{\scriptstyle\text{def}}}{{=}}\int
d^{4}x\left[i\overline{\psi}(x)\gamma^{\mu}D_{\mu}\psi(x)-m\overline{\psi}(x)\psi(x)-\frac{1}{4}\;F^{\mu\nu}(x)\,F_{\mu\nu}(x)\right]$
(10)

where $D_{\mu}=\partial_{\mu}+ieQ_{e}A_{\mu}$. This has local gauge invariance
under

$\psi(x)\to\psi^{\prime}(x)=e^{-ieQ_{e}\alpha}\;\psi(x)\qquad\qquad
A_{\mu}(x)\to A^{\prime}_{\mu}(x)=A_{\mu}(x)+\partial_{\mu}\alpha(x)$ (11)

where $e$ is the proton charge and $Q_{e}=-1$. This also leads to the
conservation of the Nöther current

$J_{\mu}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}e\overline{\psi}(x)\gamma_{\mu}\psi(x)\qquad\qquad\partial_{\mu}J^{\mu}(x)=0$
(12)

which is nothing but the electromagnetic current familiar to us in everyday
life. It is this formulation which initiated all the triumphs of the 1930s and
1940s in quantum electrodynamics.

Almost simultaneously with the invention of quantum field theory [9], however
came the realisation that there are two more fundamental fields of force.
Chadwick’s discovery of the neutron in 1932 [10] inspired Heisenberg to
postulate the existence of the strong (nuclear) force [11], which was soon
formulated as a field theory by Yukawa [12]. But also, simultaneously, came
the first understanding of weak interactions. As far back as 1899, Rutherford
had identified alpha and beta particles as separate components of radioactive
emanations, with greatly different penetrating power [13]. It was, however,
Becquerel who identified the beta rays as high-energy electrons [14]. The fact
that they have energies around an MeV was a clear proof that they did not come
from among the atomic orbitals which have binding energies of a few tens of
keV at most. However, the question of how high energy electrons can remain
bound inside the positively-charged nucleus remained a mystery till the advent
of quantum mechanics. It was Ambartzumian and Iwanenko who proved [15], using
the Uncertainty Principle, that electrons bound inside nuclei must have
energies close to a GeV222This is a textbook exercise today., whereas the beta
particles have energies an order of magnitude less. It was clear, therefore,
that the beta rays must be produced at the instant of interaction. The
extremely long lives of nuclides undergoing beta decay made it clear that this
interaction was not the strong interaction.

The situation was clarified by the neutron discovery. The mass difference
between a neutron and a proton is just about 1.2 MeV, and therefore, the beta
decay process could be interpreted as a neutron decaying into a proton and an
electron. Pauli (1930) then postulated an additional massless, neutral
particle – the neutrino – to ensure energy-momentum conservation [16]. This
novel object would be a participant only in the weak (nuclear) interaction,
which must be the cause of beta decay processes of the form

${}_{1}n^{0}\to{}_{1}p^{+}+e^{-}+\bar{\nu}_{e}$ (13)

where we have written $\bar{\nu}_{e}$ instead of just $\nu$ with the benefit
of hindsight. The ingredients were all in place now for Fermi to put together
his heuristic theory of beta decay [17], with the famous current-current form
of the interaction333It is worth mentioning that Fermi had submitted his paper
to Nature, which had rejected it ’because it contained speculations too remote
from reality to be of interest to the reader’ [18]., i.e.

$S_{\beta}=\int d^{4}x\ G_{W}\;J^{\mu}_{(h)}(x)\,J_{\mu}^{(\ell)}(x)$ (14)

where the hadronic current is

$J^{\mu}_{(h)}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\overline{\psi}_{p}(x)\gamma^{\mu}\psi_{n}(x)$
(15)

and the leptonic current is

$J_{\mu}^{(\ell)}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\overline{\psi}_{e}(x)\gamma_{\mu}\psi_{\nu}(x)$
(16)

while $G_{W}$ is a (dimensionful) coupling constant.

This is a classic four-fermion interaction, and was clearly inspired [20] by
the form of the effective interaction for an electromagnetic scattering of the
type, say, $e^{+}e^{-}\to\mu^{+}\mu^{-}$, which takes the form

$\displaystyle{\cal L}_{\rm eff}$ $\displaystyle=$ $\displaystyle
e^{2}\,J^{\mu}_{\rm(e)}\frac{g_{\mu\nu}}{E_{\rm cm}^{2}}J^{\nu}_{(\mu)}$
$\displaystyle=$ $\displaystyle\frac{e^{2}}{E_{\rm
cm}^{2}}\;\overline{\psi}_{e}(x)\gamma^{\mu}\psi_{e}(x)\
\overline{\psi}_{\mu}(x)\gamma^{\mu}\psi_{\mu}(x)$

In QED the effective coupling $e^{2}/E_{\rm cm}^{2}$ is energy-dependent.
Whatever was the reasoning which led

Fermi to set this as to a constant for weak interactions eventually proved to
be a very deep insight — and, as a matter of fact, it lies at the very core of
the theme of this article. The choice of vector currents, instead of any other
Lorentz covariant structures, is more understandable as inspired by the
example of QED, but it proved to be an equally brilliant choice444In fact,
when eventually published, Fermi’s work was considered less remarkable for its
form of the interaction than for making a connection between the interaction
action and the decay rate, which he later formulated as the Golden Rule (see
E. Fermi, Nuclear Physics U. of Chicago Press (1950)). The groundwork for
this, however, had already been done by Dirac [21].

Figure 1: Fermie-Kurie plot of $\beta$ decay of Ga-66. Reproduced from Ref.
[23] (1963).

With this new interaction, Fermi was able to predict the energy distribution
of the beta particles. His original graphs were qualitative, but soon they
were found to stand up extremely well to experimental measurements [22]. A
later comparison [23] is shown in Figure 1 which shows a Fermi-Kurie plot in
radioactive Gallium-66, where the horizontal axis is $W$, the energy of the
beta particle, and the vertical axis is the quantity $(N/p^{2}F)^{1/2}$, where
$N$ is the number of beta particles with momentum $p$ and $F$ is the so-called
Fermi factor.

The (solid) straight line is the prediction of the Fermi theory and it may be
seen that the dots, which are the experimental data, correspond very closely
with the theoretical line. Deviations at low energies may be attributed to
scattering of the beta particles in the solid material. Apart from this, the
excellent straight line formed by the data points is a strong vindication of
Fermi’s theory of beta decay. Moreover, the fact that the line remains
straight as it intersects the horizontal axis is a proof that the neutrino
mass is zero — at least so far as this experiment goes. In fact the best upper
bounds on neutrino mass do come from beta-decay data555Though we now know that
neutrinos have tiny but nonzero masses, no beta decay experiment has ever
reached the sensitivity to see a deviation in the Fermi-Kurie line close to
its intersection with the horizontal axis. The latest results are reported by
[24]..

It later turned out that the Fermi theory with vector-vector interactions
cannot explain the energy plots in many cases of beta decay, prompting Gamow
and Teller to add an axial vector-axial vector current to the Fermi
interaction [25]. Ever since, beta decay processes are classified in textbooks
as as Fermi transitions and Gamow-Teller transitions. It was also not long
before the weak interaction was found to be universal, i.e. a similar four-
fermion current- current interaction could correctly predict the muon lifetime
with the same value of $G_{W}$ as required for beta decay [26]. The Fermi
theory remains, to this day, the best explanation of the low energy behaviour
of weakly-decaying particles.

Trouble arises, however, when we go to higher energies. The reason is not
difficult to understand. If we go back to the QED process
$e^{+}e^{-}\to\mu^{+}\mu^{-}$, we can calculate the total cross-section,
neglecting fermion masses, as [27]

$\sigma_{\rm QED}=\frac{1}{12\pi}\left(\frac{e^{2}}{E_{\rm cm}}\right)^{2}$
(18)

which arises (see above) when we have an effective coupling constant
$e^{2}/E_{\rm cm}^{2}$. If we replace this, for a weak interaction between
massless fermions, by $G_{W}$, then we should get

$\sigma_{\rm weak}=\frac{e^{4}}{12\pi}\frac{1}{E^{2}_{\rm
cm}}\times\left(\frac{G_{W}}{e^{2}/E_{\rm
cm}^{2}}\right)^{2}=\frac{G_{W}^{2}}{12\pi}E_{\rm cm}^{2}$ (19)

Unlike the QED case, this grows without limit as $E_{\rm cm}$ grows, and will
sooner or later render the $S$-matrix non-unitary666In technical language, it
will violate perturbative unitarity.. Since this is not acceptable in any
quantum theory, we must conclude that the Fermi theory breaks down at higher
energies. On the other hand, making $G_{W}$ energy-dependent $\grave{\rm a}$
la QED would lead to large deviations in the Fermi-Kurie plot.

The above paradox is nicely solved by the so-called intermediate vector boson
(IVB) hypothesis [28]. For, if the massless photon $\gamma$ is replaced by a
massive vector boson $W$ in the $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process, we will
get a diagram of the form

where the overall factor in the cross-section will be replaced by

$\left(\frac{e^{2}}{E_{\rm cm}^{2}}\right)^{2}\to\left(\frac{g^{2}}{E_{\rm
cm}^{2}-M_{W}^{2}}\right)^{2}$ (20)

where $M_{W}$ is the mass of the $W$ boson and $g$ replaces $e$ as a weak
coupling constant. Now, obviously, in the low- and high- energy limits, we
will have

$\left(\frac{g^{2}}{E_{\rm
cm}^{2}-M_{W}^{2}}\right)^{2}\to\left\\{\begin{array}[]{l}\left(g^{2}/M_{W}^{2}\right)^{2}\
\ \ {\rm when}\ E_{\rm cm}\ll M_{W}\\\ \left(g^{2}/E_{\rm cm}^{2}\right)^{2}\
\ \ {\rm when}\ E_{\rm cm}\gg M_{W}\end{array}\right.$ (21)

The low energy limit is a constant
$G_{W}^{2}=\left(g^{2}/M_{W}^{2}\right)^{2}$ — exactly as guessed by Fermi —
and in the high energy limit we get the suppression factor $E_{\rm cm}^{-4}$,
which is sufficient, as in QED, to ensure perturbative unitarity.

In the above, however, we have cheated, for the amplitude with the full
expression for the massive vector boson propagator is

${\cal L}_{\rm
eff}=e^{2}\,J^{\mu}_{\rm(h)}\frac{g_{\mu\nu}-p_{\mu}p_{\nu}/M_{W}^{2}}{E_{\rm
cm}^{2}-M_{W}^{2}}J^{\nu}_{(\ell)}$ (22)

where $p$ is the four-momentum flowing through the $W$ propagator. While the
first (photon-like) term in the propagator does indeed lead to a good high-
energy behaviour, the second $p$-dependent term again provides a factor of
$E_{\rm cm}^{2}$ in the amplitude. Thus, with this term, the bad behaviour,
viz. unitarity violation, comes back. For this reason, though the IVB
hypothesis was first made in 1949 777The IVB hypothesis was originally
proposed, almost casually, in Ref. [28]., it began to be taken seriously only
after the establishment of the V-A form of the Fermi current [29]. This was
because a solution for the problem had been discovered by Schwinger [30] in
1957. His idea was to point out that the weak amplitude can be written

${\cal L}_{\rm eff}=e^{2}\,J^{\mu}_{\rm(h)}\frac{g_{\mu\nu}}{E_{\rm
cm}^{2}-M_{W}^{2}}J^{\nu}_{(\ell)}-\frac{e^{2}}{M_{W}^{2}}\,\frac{p_{\mu}J^{\mu}_{\rm(h)}\
p_{\nu}J^{\nu}_{(\ell)}}{E_{\rm cm}^{2}-M_{W}^{2}}$ (23)

and this will become exactly like a massive photon if

$p_{\mu}J^{\mu}_{\rm(h)}=p_{\nu}J^{\nu}_{(\mu)}=0$ (24)

which, in position space, becomes

$\partial_{\mu}J^{\mu}_{\rm(h)}=\partial_{\nu}J^{\nu}_{(\ell)}=0$ (25)

Now these are conserved currents, and therefore, by the inverse of Nöther’s
theorem. there must be an underlying symmetry. Taken together with the
existence of a new vector boson, it was not difficult to guess that this would
surely be a gauge symmetry, with the gauge boson as the IVB. In fact, it would
have to be not one gauge boson, but two, for the $W^{-}$ in the figure must
have its antiparticle, the $W^{+}$.

Schwinger actually went further — he assumed that the photon may be a neutral
companion to these two charged gauge bosons and hence what we would then have
is a unified theory of electromagnetism and weak interactions, i.e. an
electroweak theory. Such a triplet would arise as a vector in an isospin-like
gauge theory888In group theoretic language: as the adjoint representation in
an $SU(2)$ gauge theory, just as Yang, Mills and Shaw had shown [31] in 1954\.
It was an exciting idea, and so Schwinger made his student Glashow work out
the phenomenological details of this first idea of electroweak interactions
for his 1959 Ph.D. thesis [32]. A similar idea was also proposed by Salam and
Ward around the same time [33]. In fact, Salam and Ward stuck to this problem,
writing a series of papers, of increasing sophistication, trying to set up a
gauge theory of electroweak interactions. This eventually led to Salam being
included in the Nobel Prize for developing the Standard Model.

The realisation soon dawned on Glashow that if we take the iso-triplet
$(\\!\\!\begin{array}[]{ccc}W^{+}&W^{0}&W^{-}\end{array}\\!\\!)$ then the
group symmetry will compel the $W^{0}$ to couple to a pair of neutrinos. In
fact, the couplings of the $W$ triplet to electrons and (electron) neutrinos
would have the form

$\displaystyle{\cal L}_{W}$ $\displaystyle=$
$\displaystyle\frac{g_{2}}{\sqrt{2}}\;\overline{e}(x)\gamma^{\mu}\nu_{e}(x)W_{\mu}^{-}(x)+\frac{g_{2}}{\sqrt{2}}\;\overline{\nu}_{e}(x)\gamma^{\mu}e(x)W_{\mu}^{+}(x)$
(26)
$\displaystyle\\!\\!\\!\\!-\frac{g_{2}}{2}\;\overline{e}(x)\gamma^{\mu}e(x)W_{\mu}^{0}(x)+\frac{g_{2}}{2}\;\overline{\nu}_{e}(x)\gamma^{\mu}\nu_{e}(x)W_{\mu}^{0}(x)$

If the last term is to be absent, we must put $g_{2}=0$, which will remove all
the interactions giving rise to beta and muon decay. Therefore, one cannot
associate the $W^{0}_{\mu}$ field with the photon as he, Schwinger, as well as
Salam and Ward, had done.

Sticking manfully to his task, however, Glashow in 1961 decided to extend his
model by adding an extra $U(1)$ symmetry as a direct product to the $SU$
symmetry [34]. Thus, there were now two neutral gauge bosons $W^{0}$ and
$B^{0}$.

$\displaystyle{\cal L}_{W,B}^{0}$ $\displaystyle=$
$\displaystyle\frac{g_{2}}{2}\;\overline{\nu}_{e}(x)\gamma^{\mu}\nu_{e}(x)W_{\mu}^{0}(x)-\frac{g_{2}}{2}\;\overline{e}(x)\gamma^{\mu}e(x)W_{\mu}^{0}(x)$
(27)
$\displaystyle\\!\\!\\!\\!-\frac{g_{1}}{2}\;\overline{\nu}_{e}(x)\gamma^{\mu}\nu_{e}(x)B_{\mu}^{0}(x)-\frac{g_{1}}{2}\;\overline{e}(x)\gamma^{\mu}e(x)B_{\mu}^{0}(x)$

Though the $B_{\mu}$ corresponds to a $U(1)$ symmetry in Eqn. (27) it cannot
be the photon, for it will only decouple from neutrinos if $g_{1}=0$, which is
trivial. Glashow then made the heuristic assumption that the $W^{0}$ and
$B^{0}$ are mixed states (for some unknown reason), i.e.

$\displaystyle W_{\mu}^{0}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}$ $\displaystyle
Z_{\mu}(x)\cos\theta-A_{\mu}(x)\sin\theta$ $\displaystyle B_{\mu}^{0}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}$ $\displaystyle
Z_{\mu}(x)\sin\theta+A_{\mu}(x)\cos\theta$ (28)

leading to

$\displaystyle{\cal L}_{W,B}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g_{2}\sin\theta-g_{1}\cos\theta\right)\
\overline{\nu}_{e}(x)\gamma^{\mu}\nu_{e}(x)\ A_{\mu}(x)$ (29)
$\displaystyle\\!\\!\\!\\!\\!-\,\frac{1}{2}\left(g_{2}\sin\theta+g_{1}\cos\theta\right)\
\overline{e}(x)\gamma^{\mu}e(x)\ A_{\mu}(x)\dots$

By tuning the mixing angle $\theta$ such that

$\tan\theta\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{g_{1}}{g_{2}}$ (30)

Glashow was able to ensure that the $A_{\mu}(x)$’s coupling to a neutrino pair
would cancel — and thus the $A_{\mu}(x)$ could be identified with the photon,
with $e=g_{2}\sin\theta$ to get the right QED coupling of electrons to
photons999It is often asked why Glashow chose the direct product group
$SU(2)\times U(1)$ rather than the simpler $U(2)$. The reason is as follows.
In the above, that would mean putting $g_{1}=g_{2}$, i.e. $\theta=\pi/4$. We
would then have $e=g_{2}/\sqrt{2}$ and hence $G_{W}=2e^{2}/M_{W}^{2}$, leading
to $M_{W}\approx 23$ GeV. In Glashow’s time, this would have seemed absurdly
high. Today, on the other hand – with hindsight – we know that this is
actually about 3.5 times too small.. It was the introduction of an additional
$U(1)$ symmetry – and hence an additional IVB – which did the trick, by
cancelling the $W^{0}$ coupling to neutrinos against the extra $U(1)$
contribution to the same. This produced a phenomenologically viable model of
leptons101010It proved ultimately good enough to win Glashow a Nobel Prize
(1979)., albeit with a fine-tuned parameter $\theta$, for which fine-tuning
there was no natural explanation. We may note, however, that if we replace the
lepton doublet
$\overline{L}=(\\!\\!\begin{array}[]{cc}\overline{\nu}_{e}&\overline{e}\end{array}\\!\\!)$
by a nucleon doublet
$\overline{N}=(\\!\\!\begin{array}[]{cc}\overline{p}&\overline{n}\end{array}\\!\\!)$
we will obtain a similar-looking interaction

$\displaystyle{\cal L}_{W,B}^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(g_{2}\cos\theta-g_{1}\sin\theta\right)\
\overline{p}(x)\gamma^{\mu}p(x)\ A_{\mu}(x)$ (31)
$\displaystyle\\!\\!\\!\\!-\frac{1}{2}\left(g_{2}\sin\theta+g_{1}\cos\theta\right)\
\overline{n}(x)\gamma^{\mu}n(x)\ A_{\mu}(x)+\dots$

To make the photon-neutron coupling vanish, however, we must have

$\tan\theta=-\frac{g_{1}}{g_{2}}$ (32)

which would then give the right sign for the photon-proton coupling. But we
cannot have both the fine-tuning relations (30) and (32) simultaneously,
unless $g_{1}=0$, which is trivial. It may have been for this reason that
Glashow’s 1961 paper restricts itself to electroweak interactions of
leptons111111The failure of the $SI(2)\times U(1)$ theory for hadrons has its
explanation in the simple fact that hadrons are not elementary articles. When
the same idea is applied to an $SU(2)$ doublet of quarks, it works perfectly.
However, when Glashow wrote his paper in 1961, composite nucleons would have
seemed too far-fetched an idea. This came in only with the quark model in
1964..

Moreover, in Glashow’s work, no explanation was attempted for the fact that
the photon is massless and the $W^{\pm}$ are massive (and so, presumably is
the $W^{0}$, or its mixed version). This, in fact, was the elephant in the
room, ’a stumbling block we must overlook’, as Glashow, somewhat
apologetically, put it121212Italics by the present author.. However, that it
did bother him is clear from the fact that he modestly put the words ’partial
symmetry’ into the title of his paper, that is to say, a symmetry which would
not apply to the mass terms for the $W$-bosons131313In justification, Glashow
quoted two well-known examples – the scale and chiral invariance of massless
theories. These are discussed in the next section., but to the rest of the
Lagrangian.

Glashow’s tour de force, like many another famous work, ’fell stillborn from
the press’. By the time Weinberg’s classic paper on electroweak interactions
appeared in 1967, Glashow had acquired only 4 (non-self) citations. The
tension between gauge theory and IVB masses proved too much for most
scientists to swallow, and as a result, gauge theory (except QED) was cast
into the shade. The 1960s were a period when current algebras and $S$-matrix
theory dominated the world of quantum field theory, and, on the
phenomenological side, the composite nature of hadrons was postulated and
proved. Only a few intrepid souls were still bothered about electroweak
unification.

Today Glashow’s 1961 paper has well over 9,000 citations and is regarded as
the seminal paper in the development of the Standard Model. What made the
difference? It was the solution of the mass problem, the removal of the
’stumbling block’ — to which we must now turn.

##  2 Dark Ages: The Mass Problem

How does one postulate both a symmetry and also introduce terms that break it?
This is not difficult to envisage. To take a common example, the Earth may be
approximated, for many purposes, as a perfect sphere, i.e. a solid having
rotational symmetry (red curves in Figure 2). However, we are aware of
irregularities (sketched highly exaggerated as the black curve in Figure 2),
which matter only when the Earth’s gravitational field has to be considered to
a high level of accuracy. In technical language, the Earth’s gravitational
potential has a dominant monopole term, which represents the spherical
symmetry. Then there are small dipole terms, quadrupole terms, etc., i.e.
terms which represent the breaking of rotational symmetry.

Figure 2: Sketch of the Earth (left) and Oumuamua (right).

The above is an example of explicit symmetry-breaking, where the higher
multipole terms which do not respect the spherical symmetry are very small.
However, this cannot work in the case when the deviation from the symmetry is
very large. For example, no one would dream of approximating the cigar-shaped
interstellar object Oumuamua (see Figure 2) as a sphere with small
corrections. In particle physics, large mass terms, such as are demanded for
the $W^{\pm}$ bosons, would correspond to these large deviations from gauge
theory. Where could they be coming from?

A similar question had plagued the theory of strong interactions as well. In
the 1950s, the dominant theory was still the 1935 theory of Yukawa [12], with
some group theoretic modifications over the years. This theory of pions and
nucleons has a nucleon $N$ of mass around 938 MeV and a pion $\vec{\pi}$ of
mass around 140 MeV, transforming under $SU(2)$ of isospin as a doublet and a
triplet respectively. However, the free theory has a large global symmetry
$SU(2)_{L}\times SU(2)_{R}$ if it is written in terms of massless left- and
right-handed nucleons, viz.

${\cal L}_{\pi
N\bar{N}}\stackrel{{\scriptstyle\text{def}}}{{=}}i\overline{N}_{L}\not{\\!\partial}N_{L}+i\overline{N}_{R}\not{\\!\partial}N_{R}+\partial^{\mu}\vec{\pi}\cdot\partial_{\mu}\vec{\pi}$
(33)

where
$\overline{N}\stackrel{{\scriptstyle\text{def}}}{{=}}(\\!\\!\begin{array}[]{cc}\overline{p}&\overline{n}\end{array}\\!\\!)$
is the nucleon doublet and
$\vec{\pi}\stackrel{{\scriptstyle\text{def}}}{{=}}(\\!\\!\begin{array}[]{ccc}\pi^{+}&\pi^{0}&\pi^{-}\end{array}\\!\\!)$
is the pion triplet. Obviously, this is invariant under global transformations

$N_{L}\to N_{L}^{\prime}=\mathbb{U}_{L}N_{L}\qquad\qquad N_{R}\to
N_{R}^{\prime}=\mathbb{U}_{R}N_{R}$ (34)

where $\mathbb{U}_{L}$ and $\mathbb{U}_{R}$ are $SU(2)$ matrices. However, a
mass term

${\cal L}_{m}=m_{N}\left(\overline{N}_{L}N_{R}+\overline{N}_{R}N_{L}\right)$
(35)

immediately breaks this symmetry, unless, indeed
$\mathbb{U}_{L}=\mathbb{U}_{R}$, i.e. the global $SU(2)_{L}\times SU(2)_{R}$
breaks down to the diagonal $SU(2)$ of isospin. At low energies, the nucleon
mass $m_{N}\approx 938$ MeV is a large value, which cannot be attributed to
explicit symmetry-breaking.

The solution had, in fact, already been discovered in condensed matter
physics, where the theory of superconductivity incorporates a phenomenon now
called spontaneous symmetry-breaking. It had appeared, in a non-relativistic
form, in the 1950 theory of Landau and Ginzburg [35], where the photon
effectively acquires a mass inside a superconducting medium below the critical
temperature and cannot propagate — making the macroscopic medium at once a
perfect conductor and a perfect diamagnet. This phase transition occurs when a
dynamical variable of the material, called an order parameter by Landau, is
close to a ground state, which does not obey the symmetry of the Lagrangian.
The idea was given a firm basis in the BCS theory of superconductivity [36],
where the hitherto-esoteric order parameter assumes a concrete form as the
density of a Bose-Einstein condensate of paired electrons (Cooper pairs),
forming quasi-bosons. As one would expect, the BCS ground state breaks the
gauge symmetry of electromagnetism, which would otherwise preclude a photon
mass.

Could there be a similar phenomenon happening in the pion-nucleon regime? This
was the question asked by Nambu and Jona-Lasinio in 1960 [37], and they
managed to construct an elaborate theory based closely on the BCS theory, with
a nucleonic condensate $\langle N\bar{N}\rangle$ which would break the chiral
$SU(2)_{L}\times SU(2)_{R}$ down to $SU(2)$ of isospin. Once the symmetry is
broken spontaneously, there is no restriction on the mass of the nucleon, for
it is related to the energy gap between the ground state and the first excited
state of the condensate. Some mechanism was postulated to create the $\langle
N\bar{N}\rangle$ bound state — this would be the analogue of the Fröhlich
effective interaction which creates the Cooper pairs. Inspired by the success
of Nambu and Jona-Lasinio, Gell-Mann and Lévy [38] then created a simpler
model where the condensate $\langle N\bar{N}\rangle$ is replaced by a scalar
field $\sigma$. This came to be known as the (linear) sigma model. In this
model, one creates an $SU(2)$ doublet

$\Phi(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{I}\;\sigma(x)+i\;\vec{\mathbb{T}}\cdot\vec{\pi}(x)$
(36)

and adds to the isospin Hamiltonian a potential energy term of the form

$V(\Phi)\stackrel{{\scriptstyle\text{def}}}{{=}}-\frac{\mu^{2}}{2}\Phi^{\dagger}\Phi+\frac{\lambda}{4!}(\Phi^{\dagger}\Phi)^{4}$
(37)

which is closely analogous to the Ginzburg-Landau Hamiltonian if $\mu$ and
$\lambda$ are real and positive. This potential breaks the $SU(2)$ symmetry
spontaneously and permits the nucleons to have large masses141414Some of the
details are indicated, in the electroweak context, in the next section.. At
the same time, the pions remain necessarily massless, following a 1962 theorem
proved by Goldstone, Salam and Weinberg[39]. As a matter of fact, Goldstone
had discovered [40] in 1961 that theories of the sigma model type always
contain massless bosons, but a year later, he teamed up with Salam and
Weinberg to prove the Goldstone theorem, which states [39] that whenever a
continuous global (or local) symmetry is spontaneously broken, there will be
massless bosons. Today they are called Goldstone bosons and are ubiquitous in
quantum field theories. The pions in the sigma model are Goldstone bosons, and
necessarly massless. However, it is possible to introduce pion mass terms as
explicit symmetry-breaking terms, which necessarily makes the pion masses
small. Here, it seems, lay the explanation of why the pions are so much
lighter than the nucleons.

Nucleons and pions are not, however, the only hadrons. Throughout the 1950s
and early 1960s, new hadrons with exotic properties were being discovered
[41], and these were then postulated by Gell-Mann[42] and Zweig[43] in 1964 to
be composites of a set of three basic quarks. The analysis of deep-inelastic
scattering data in 1969 established the existence of these quarks[44], and a
gauge interaction between them was established by the early 1970s [45]. The
sigma model, therefore, never really achieved the status of a fundamental
theory; however, it proved to be the first and paradigm case for a whole class
of effective field theories. But it was in the theory of electroweak
interactions that the ideas introduced in the sigma model secured their
biggest triumph.

##  3 Early Renaissance: The Higgs Mechanism

It may have seemed natural, in the early 1960s, to create a model of
electroweak interactions on the analogy of the sigma model. This would
certainly have been an excellent means of generating masses for the IVB
particles, and would nicely round off Glashow’s unified electroweak theory.
However, the concept foundered on the rock of the Goldstone theorem. A sigma
model analogue would indeed make the vector bosons massive (except for the
photon), but in their place we would have three massless scalars, with
interactions of the same strength. Such bosons would have surely shown up long
before in cosmic rays, or nuclear reactions, for the weak interactions are
ubiquitous. There were not even light bosons analogous to the pions, to which
one could attach an explicit symmetry-breaking. For this reason, no one
interested in particle physics hastened to create a sigma model for
electroweak interactions.

Once again, the clue came from condensed matter physics. In 1963, Anderson
published a study [46] of superconductivity, in which he suggested that the
Goldstone degree of freedom arising from spontaneous symmetry-breakdown in a
metallic superconductor actually reappears as a longitudinal mode of the
photon151515which corresponds to longitudinal oscillations in the free
electron gas, quantised as ’plasmons’., rather than as an independent massless
boson. The two massless bosons ’cancel’, in Anderson’s language, leaving a
massive (vector) boson. This is, in effect, what is known today as the Higgs
mechanism. There was really no reason for Anderson not to go on to rewrite his
non-relativistic model in a relativistic avatar, since he did mention Yang-
Mills theory in passing. However, he drew back from this step, contenting
himself with the statement that ’it is not at all clear’ that a similar
mechanism would work in Sakurai’s theory [47] of the strong interactions.

In this 1960 paper, Sakurai had tried to write down a Yang-Mills theory for
the $SU(2)_{I}\times U(1)_{Y}$ flavour symmetry of hadrons [47]. This had not
been taken seriously for the same reason as Glashow’s electroweak paper,
because the gauge bosons were necessarily massless, leading to a long-range
interaction; and even if the gauge symmetry could be spontaneously broken,
there would be massless strongly-interacting Goldstone scalars which are known
not to exist. Interestingly, the group symmetry is the same as that discovered
by Glashow, and the mathematical structure of Sakurai’s theory is almost
identical to that of Weinberg’s electroweak model. Of course, Sakurai’s model
did not survive the discovery that flavour symmetries are global in nature —
and inexact to boot. But it is ironic that what was dismissed as ’not at all
clear’ in Anderson’s 1963 paper, was precisely what became crystal clear in
Weinberg’s 1967 paper161616Weinberg later wrote, in a characteristically
insightful article [48]: ”In fact, Anderson was right: the reason for the
exception noted by Higgs et al. is that it is not possible to quantize a
theory with a local symmetry in a way that preserves both manifest Lorentz
invariance and the usual rules of quantum mechanics, including the requirement
that probabilities be positive. In fact, there are two ways to quantize
theories with local symmetries: one way that preserves positive probabilities
but loses manifest Lorentz invariance, and another that preserves manifest
Lorentz invariance but seems to lose positive probabilities, so in fact these
theories actually do respect both Lorentz invariance and positive
probabilities; they just don’t respect our theorem.” Here, by ’our’ theorem,
he meant the Goldstone theorem[39]..

It did not take long for the relativistic generalisation to come. Three groups
arrived at practically the same formulation independently during 1964\. In
those pre-arXiv days, neither of them knew of the others’ work till it was
actually published. The first off the mark were Englert and Brout [49] at
Brussels, followed by Higgs [50] at Edinburgh, and a little later, by
Guralnik, Hagen and Kibble [51] at London. Thus, the Anderson-Brout-Englert-
Guralnik-Hagen-Higgs-Kibble mechanism – or Higgs mechanism for short if we do
not wish to mention all the ’magnificent seven’ – came into being. Its
application to the Abelian gauge theory of Section 1 is outlined below in
terms of three ’miracles’.

We start with the scalar gauge theory of Eqn. (6) keeping the scalar massless
and add to it two self-interaction terms similar to those of the sigma model,
i.e. Eqn. (37), thereby getting

$\displaystyle S$ $\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}$
$\displaystyle\int d^{4}x\left[(D^{\mu}\varphi)^{*}(x)\
D_{\mu}\varphi(x)-\frac{1}{4}\;F^{\mu\nu}(x)\,F_{\mu\nu}(x)-V(\varphi)\right]$
$\displaystyle V(\varphi)$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}}{2}\;\varphi^{*}(x)\varphi(x)+\frac{\lambda}{4!}\left\\{\varphi^{*}(x)\varphi(x)\right\\}^{2}$
(38)

where, as in the sigma model, $\mu$ and $\lambda$ are real and positive
constants. It should be noted that the $\mu$ term has the wrong sign for a
mass term171717If we try to identify it as a mass term, then $\varphi$ would
be a tachyon and the theory would therefore be unphysical. and should be
regarded as an interaction term.

If we plot the potential energy term $V(\varphi)$ as a function of $\varphi$,
we will obtain the so-called ’Mexican hat’ potential shown in Figure 3. Since
the potential depends only on $|\varphi|^{2}$, it will have rotational
symmetry in the complex $\varphi$-plane. Close to the origin, i.e. at small
values of $|\varphi|^{2}$, the quadratic term will dominate, producing
approximately an inverted paraboloid of revolution, but at higher values of
$|\varphi|$ the quartic term will begin to dominate, causing a turnaround and
making the potential positive. These contrary terms will therefore create a
minimum for some value $|\varphi|=v$, which is easily calculated as

$\frac{v}{\sqrt{2}}=\sqrt{\frac{6\mu^{2}}{\lambda}}$ (39)

Figure 3: Spontaneous symmetry-breaking potential in a $U(1)$ gauge theory..

However, there is no restriction on the phase of $\varphi$, which indicates
that in the complex $\varphi$-plane there is a ring of possible minima of the
potential energy with radius $|\varphi|=v$. In the quantum theory, these will
correspond to degenerate vacua $\langle\varphi\rangle=v/\sqrt{2}$. Of these,
only one can be the physical vacuum, and this will be a random choice denoted
in Figure 3 by a red dot. The unstable point $\varphi=0$ is denoted by a white
dot. We now notice that the phase invariance — or $U(1)$ gauge invariance — is
spontaneously broken by the choice of a single vacuum state along the ring.

Recognising that at energies (temperatures) far above $V(v/\sqrt{2})$ the
$U(1)$ symmetry will be there, but it will only be at low energies
(temperatures) close to $V(v/\sqrt{2})$, that the symmetry is broken, we
realise that this is really a phase transition from a symmetric phase to a
broken-symmetric phase. As the system in the symmetric phase cools down, the
false vacuum at $\varphi=0$ will become manifestly unstable, and thus any
quantum fluctuation will cause the system to ’roll down’ to ons of the
degenerate true vacua.

To write down a quantisable field theory, we must now expand $\varphi$ around
the vacuum state. Before we do so, however, it is convenient to choose the
axes in the complex $\varphi$-plane such that the vacuum point lies along the
real axis — this can be done without loss of generality and at the same time
enables us to eliminate an extraneous phase parameter, which would otherwise
have to be carried through the calculation. We now write

$\varphi\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{v}{\sqrt{2}}+\omega(x)$
(40)

where $\omega(x)$ is the scalar degree of freedom which is amenable for a
quantum field theory. In terms of this, the action becomes

$S=\int d^{4}x\left[(\partial^{\mu}+ieA^{\mu})(\frac{v}{\sqrt{2}}+\omega^{*})\
(\partial_{\mu}-ieA_{\mu})(\frac{v}{\sqrt{2}}+\omega)-\frac{1}{4}\;F^{\mu\nu}\,F_{\mu\nu}-V(\omega)\right]$
(41)

with a potential energy term

$V(\omega)=\mu^{2}\left({\rm
Re}\;\omega\right)^{2}+\mu^{2}\;\omega^{*}\omega\;{\rm
Re}\;\omega+\frac{\lambda}{4!}\left(\omega^{*}\omega\right)^{2}$ (42)

where we need to substitute $\mu^{2}=\lambda v^{2}/12$ and a constant term has
been dropped. The kinetic term in Eqn. (41), when expanded, contains a term

${\cal L}_{m}=+\frac{1}{2}\;e^{2}v^{2}\;A^{\mu}A_{\mu}$ (43)

The above is a mass term for the photon, and therefore, we identify
$M_{\gamma}=ev$. This is the first miracle of the sigma model, or more
generally of the spontaneously-broken theory, that the choice of one out of
multiple possible vacua generates a gauge boson mass. If we consider the
quadratic term in Eqn. (42), we will note that it is a mass term with the
correct sign, i.e., the real scalar field ${\rm Re}\;\omega$ has a mass
$\sqrt{2}\mu$. This is the second miracle of Higgs [50], that the tachyonic
scalar theory, once the proper vacuum is chosen, becomes a real scalar theory
with a real mass. This scalar is known as the Higgs boson181818This name was
first used, with justification, by B.W. Lee [52], since it was Higgs who had
pointed out the presence of ’incomplete multiplets’ i.e. the real part without
the imaginary part of the scalar field..

The action in Eqn. (41) still has a problem, since it contains the complex
scalar field $\omega={\rm Re}\;\omega+i{\rm Re}\;\omega$. but only the real
field ${\rm Re}\;\omega$ has a finite mass term. The other real field ${\rm
Im}\;\omega$ remains massless and indeed, it is the Goldstone boson of the
theory. This is where the third miracle comes in, i.e. the one devised by
Anderson and the seven authors of 1964. The trick is go back and parametrise
the complex scalar field $\varphi$ as

$\varphi(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\eta(x)\;e^{i\theta(x)}$
(44)

At this stage, the gauge symmetry is not broken, and hence, with hindsight, we
can perform a $U(1)$ gauge transformation as in Eqn. (2) to obtain

$\varphi(x)\to\varphi^{\prime}(x)=\eta(x)\;e^{ie[\theta(x)+\alpha(x)]}$ (45)

Since we are free to choose the gauge function $\alpha(x)$, we simply choose
it such that

$\alpha(x)\stackrel{{\scriptstyle\text{def}}}{{=}}-\theta(x)$ (46)

in which case,

$\varphi(x)\to\varphi^{\prime}(x)=\eta(x)$ (47)

The physical system remains the same at this stage, because of the gauge
invariance. The action in this so-called unitary gauge now becomes

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left[(D^{\mu}\eta)^{*}(x)\
D_{\mu}\eta(x)-\frac{1}{4}\;F^{\mu\nu}(x)\,F_{\mu\nu}(x)-V(\eta)\right]$
$\displaystyle V(\eta)$ $\displaystyle=$
$\displaystyle-\frac{\mu^{2}}{2}\;\eta^{2}(x)+\frac{\lambda}{4!}\eta^{4}(x)$
(48)

where only the real scalar field $\eta(x)$ appears. This is still tachyonic,
but we now note that the minimum of the potential lies at $\eta$ given by $\pm
v=\pm\sqrt{12\mu^{2}/\lambda}$. Choosing the positive value, and making the
shift $\eta(x)=v/\sqrt{2}+h(x)$, now produces both the desirable features of
the sigma model, viz., a mass term for the photon $A_{\mu}(x)$ and a mass term
of the proper sign for the Higgs boson $h(x)$. Obviously, the degree of
freedom denoted by the field $\theta$ has vanished from the action in Eqn.
(48). It is still there, however — as Anderson had realised — disguised as the
longitudinal polarisation mode of the photon.

For the Higgs mechanism to work, then, we require a tachyonic scalar field
theory with a Mexican hat potential, where we make two successive
reparametrisations of the complex scalar field — the first is a gauge choice
in the unbroken theory, and then, when the symmetry breaks, a constant shift
in the remaining real scalar theory. This produces the three desirable
features (’miracles’) mentioned above. viz.,

  * •

The photon acquires a mass.

  * •

The real scalar mode acquires a mass.

  * •

The massless scalar mode disappears from the theory.

Even today, more than a quarter of a century later, the deep insight into
Nature provided by spontaneous symmetry-breaking and the Higgs mechanism is
quite awe-inspiring. It was soon to provide the key to developing a theory of
the electroweak interactions. We must recognise, however, that the choice of
parameters in the potential is quite arbitrary, and is in no way demanded by
the gauge symmetry. The spontaneously-broken gauge theory, therefore, is a
mixed marriage of a pure gauge theory and a specific potential choice, which
is driven purely by phenomenological demands.

##  4 High Renaissance: the Glashow-Salam-Weinberg Model

Given the breakthrough of 1964, it took surprisingly long for someone to come
up with a generalisation of the Abelian $U(1)$ Higgs model to the non-Abelian
case of an $SU(2)\times U(1)$ local gauge theory à la Glashow or Sakurai.
However, it was elegantly done in a 1967 paper by T.W.B. Kibble, one, indeed,
of the ’magnificent seven’ who had proposed the Abelian Higgs mechanism [53].
The essence of this work is very close to the sigma model191919Actually,
Kibble’s paper first discusses a general non-Abelian gauge group before taking
up the special case of the Glashow-Sakurai group $SU(2)\times U(1)$.. Instead
of one complex scalar $\varphi(x)$, the theory has four complex scalars
$\varphi_{0}(x)$ and
$\vec{\varphi}(x)=\left\\{\varphi_{i}(x)|i=1,2,3\right\\}$ forming a doublet
of $SU(2)$

$\Phi(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{I}\;\varphi_{0}(x)+i\;\vec{\mathbb{T}}\cdot\vec{\varphi}(x)$
(49)

and the potential energy terms $V(\Phi)$ will be identical to those in Eqn.
(37). We start, as before, with the action

$S\stackrel{{\scriptstyle\text{def}}}{{=}}\int d^{4}x\
\left[\left(\mathbb{D}^{\mu}\Phi\right)^{\dagger}\mathbb{D}_{\mu}\Phi-\frac{1}{8}\;{\rm
Tr}\left[\mathbb{F}^{\mu\nu}\,\mathbb{F}_{\mu\nu}\right]-\frac{1}{4}\,B^{\mu\nu}B_{\mu\nu}-V(\Phi)\right]$
(50)

where
$\mathbb{W}_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\vec{\mathbb{T}}\cdot\vec{W}_{\mu}=\frac{1}{2}\vec{\sigma}\cdot\vec{W}_{\mu}$,
and

$\displaystyle\mathbb{D}_{\mu}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}$
$\displaystyle\mathbb{I}\,\partial_{\mu}-ig\,\mathbb{W}_{\mu}-\frac{ig^{\prime}}{2}\,\mathbb{I}\,B_{\mu}$
$\displaystyle\mathbb{F}_{\mu\nu}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}$
$\displaystyle\partial_{\mu}\mathbb{W}_{\nu}-\partial_{\nu}\mathbb{W}_{\mu}-ig\left[\mathbb{W}_{\mu},\mathbb{W}_{\nu}\right]$
$\displaystyle B_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ (51)

Following the Abelian case, we can make a polar parametrisation, writing

$\Phi(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\\!\begin{array}[]{c}0\\\
\eta(x)\end{array}\right)\;e^{i\,\vec{\mathbb{T}}\,\cdot\,\vec{G(x)}}$ (52)

and then apply on it a $SU(2)$ local gauge transformation to get rid of the
exponentiated (Goldstone) fields, leaving

$\Phi(x)=\left(\\!\begin{array}[]{c}0\\\ \eta(x)\end{array}\right)$ (53)

The potential term now becomes identical to that in Eqn. (48) and a shift
$\eta(x)=v/\sqrt{2}+H(x)$ will produce gauge boson mass terms of the form

$\displaystyle S_{m}$ $\displaystyle=$ $\displaystyle\int d^{4}x\
\left(\\!\begin{array}[]{cc}0&v/\sqrt{2}\end{array}\\!\right)\left(\\!g\,\mathbb{W}^{\mu}+\frac{g^{\prime}}{2}\mathbb{I}\,B^{\mu}\\!\right)\left(\\!g\,\mathbb{W}_{\mu}-\frac{g^{\prime}}{2}\mathbb{I}\,B_{\mu}\\!\right)\left(\\!\\!\begin{array}[]{c}0\\\
v/\sqrt{2}\end{array}\\!\\!\right)$ (63) $\displaystyle=$ $\displaystyle\int
d^{4}x\ \left[\
\frac{g^{2}v^{2}}{8}\left(W_{1}^{\mu}-iW_{2}^{\mu}\right)\left(W_{1\mu}+iW_{2\mu}\right)\right.$
$\displaystyle\left.\hskip
43.36243pt+\frac{v^{2}}{8}\left(\\!\begin{array}[]{cc}W_{3}^{\mu}&B^{\mu}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}g^{2}&-gg^{\prime}\\\
-gg^{\prime}&g^{\prime
2}\end{array}\\!\right)\left(\\!\begin{array}[]{c}W_{3\mu}\\\
B_{\mu}\end{array}\\!\right)\right]$

Kibble’s paper stopped at this point, simply stating that the gauge bosons
would have masses202020In fact, Kibble stated that all the gauge bosons have
mass.. However, the thread was brilliantly taken up by Weinberg [54] later the
same year. It was Weinberg who pointed out that Eqn. (63) means that the
$\vec{W}_{\mu}$ and $B_{\mu}$ fields are unphysical because of the presence of
mixing terms. However, this can be dealt with quite easily. From the first two
$W$’s we can define conjugate complex fields

$W_{\mu}^{\pm}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{\sqrt{2}}(W_{1\mu}\mp
iW_{2\mu})$ (64)

and for the $W_{3\mu}$ and the $B_{\mu}$, the mass matrix is diagonalised by
the orthonormal physical states

$Z_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{gW_{3\mu}+g^{\prime}B_{\mu}}{\sqrt{g^{2}+g^{\prime
2}}}\qquad\qquad\qquad
A_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{g^{\prime}W_{3\mu}-gB_{\mu}}{\sqrt{g^{2}+g^{\prime
2}}}$ (65)

corresponding to eigenvalues $g^{2}+g^{\prime 2}$ and $0$ respectively. Eqn.
(63) then reduces to

$S=\int d^{4}x\
\left[\frac{g^{2}v^{2}}{4}W^{+\mu}W^{-}_{\mu}+\frac{v^{2}}{8}(g^{2}+g^{\prime
2})Z^{\mu}Z_{\mu}\right]$ (66)

We thus have charged vector bosons $W^{\pm}_{\mu}$ with mass
$M_{W}=\frac{1}{2}gv$, a neutral vector boson $Z_{\mu}$ with mass
$M_{Z}=\frac{1}{2}v\sqrt{g^{2}+g^{\prime 2}}$ and a massless vector boson
$A_{\mu}$. This certainly fits the pattern of massive weak bosons and a photon
– if the $A_{\mu}$ can be identified with the photon. For this, it must couple
to fermion exactly like the photon. It was here that Weinberg pointed out that
we can recast Eqn. (65) in the exact form of Eqn. (28) in Glashow’s model,
provided we set

$\tan\theta\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{g^{\prime}}{g}$ (67)

Now this, if we write $g=g_{2}$ and $g^{\prime}=g_{1}$, is precisely what
Glashow had obtained — see Eqn. (30) — by fine tuning the mixing so that the
photon coupling to neutrinos vanishes. We thus have a beautiful understanding
of where this particular relation comes from – it arises from the pattern of
spontaneous symmetry-breaking and is not a fine tuning at all! Naturally, this
relation also means that we can identify the $A_{\mu}$ field as Glashow’s
photon. It is for this insight that the mixing angle $\theta$ in Eqn. (30)
has, ever since, been known as the Weinberg angle and denoted $\theta_{W}$. In
terms of this we have the useful relation

$M_{Z}=\frac{M_{W}}{\cos\theta_{W}}$ (68)

Profound as it may be, this $SU(2)\times U(1)$ electroweak model is predicated
on the existence of a bunch of scalar fields, of which — before 2012 — there
was no evidence. What made Weinberg’s model a runaway success was its ability
to explain fermion masses and couplings. Here again, the spadework had been
done by Salam and Ward [55], as early as 1964. They did not, however,
incorporate any solution for the gauge boson masses, contenting themselves
with the comment that the weak bosons must be ’outrageously’ heavy.

Weak interaction theory, in fact, has a serious difficulty, not just with
gauge boson masses, but also with fermion masses. The reason lies in their
parity-violating nature. This had been proposed as early as 1956 by Yang and
Lee[56], and was soon proved in beautiful experiments by Wu et al[57] and by
Goldhaber et al[58]. This was followed by the discovery [29] that the weak
interaction currents are of the $V$-$A$ form, which, in the IVB theory would
mean that the couplings of the $W$ boson are of the form

$\displaystyle S_{W}$ $\displaystyle\sim$ $\displaystyle\int d^{4}x\
\overline{\psi}(x)\gamma^{\mu}\left(\\!1-\gamma_{5}\\!\right)\psi(x)\
W_{\mu}(x)+{\rm H.c.}$ (69) $\displaystyle\sim$ $\displaystyle\int d^{4}x\
\overline{\psi_{L}}(x)\gamma^{\mu}\psi_{L}(x)\ W_{\mu}(x)+{\rm H.c.}$

where we define

$\psi_{L}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{2}\left(\\!1-\gamma_{5}\\!\right)\,\psi(x)\qquad\qquad\psi_{R}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{1}{2}\left(\\!1+\gamma_{5}\\!\right)\,\psi(x)$
(70)

However, the $W$ boson is charged and it couples to a $SU(2)$ doublet of
scalars. Accordingly, we must create an $SU(2)$ doublet of fermions. For the
moment, we just choose the electron and its neutrino, à la Weinberg, and
create a doublet with the left-handed components

$L_{L}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\\!\\!\\!\begin{array}[]{c}\nu_{eL}(x)\\\
e_{L}(x)\end{array}\\!\\!\\!\right)\qquad\qquad\overline{L_{L}}(x)=\left(\\!\\!\\!\begin{array}[]{cc}\overline{\nu_{eL}}(x)&\overline{e_{L}}(x)\end{array}\\!\\!\\!\right)$
(71)

with a gauge-kinetic term

$S_{L}=\int d^{4}x\
i\,\overline{L_{L}}(x)\,\gamma^{\mu}\,\mathbb{D}^{(L)}_{\mu}\,L_{L}(x)$ (72)

where

$\mathbb{D}^{(L)}_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{I}\,\partial_{\mu}+ig\,\mathbb{W}_{\mu}-\frac{ig^{\prime}}{2}Y_{L}\,\mathbb{I}\,B_{\mu}$
(73)

to be added to the rest of the electroweak action. Since the right-handed
components do not interact with the $W$ boson (but the $e_{R}$ most certainly
interacts with the photon), we make them $SU(2)$ singlets, with gauge-kinetic
terms

$S_{R}=\int d^{4}x\
i\,\overline{e_{R}}(x)\,\gamma^{\mu}\,D^{(e)}_{\mu}e_{R}(x)$ (74)

where

$D^{(e)}_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\partial_{\mu}-\frac{ig^{\prime}}{2}Y_{e}\,B_{\mu}$
(75)

also to be added to the rest of the electroweak action. In view of the
experiments of Goldhaber et al [58], no right-handed neutrino field is
included in Eqn. (75).

It may be noted that the gauge-covariant derivatives of Eqns. (73) and (75)
differ from those in Eqns. (51) and (4) respectively because of the presence
of the weak hypercharges $Y_{L}$ and $Y_{e}$. To find these quantum numbers of
the chiral fermions, we need to expand the interaction terms in Eqns. (72) and
(74), thereby obtaining

$\displaystyle S_{\rm int}^{\rm cc}=\int d^{4}x\
\frac{g}{2\sqrt{2}}\;\left[\overline{\nu}_{e}(x)\gamma^{\mu}\left(\\!1-\gamma_{5}\\!\right)e(x)W_{\mu}^{+}(x)+{\rm
H.c.}\right]$ (76)

for the charged current interactions i.e. terms with currents212121These
currents will themselves transform like charged objects under the $U(1)$ of
QED. that couple to the charged $W^{\pm}$. We can now write the Fermi
interaction of Eqn. (14) (with a $V$-$A$ modification) as a second order
effective interaction of this theory in the low-energy limit to get the
identification

$\frac{G_{F}}{\sqrt{2}}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{g^{2}}{8M_{W}^{2}}$
(77)

where $G_{F}$ is the Fermi coupling constant. The presence of $\gamma^{\mu}$
as well as $\gamma^{\mu}\gamma_{5}$ interactions ensures that both Fermi-type
and Gamow-Teller type interactions are taken care of.

The corresponding electromagnetic interactions are now obtained by
substituting Eqn. (28) in Eqns. (72) and (74) to get

$\displaystyle S_{\rm int}^{\rm em}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\ \left[-\frac{1}{2}g\sin\theta_{W}\left(1-Y_{L}\right)\
\overline{\nu_{eL}}\gamma^{\mu}\nu_{eL}A_{\mu}\right.$ (78)
$\displaystyle\hskip
39.74872pt\left.+\frac{1}{2}g\sin\theta_{W}\left(1+Y_{L}\right)\
\overline{e_{L}}\gamma^{\mu}e_{L}A_{\mu}+\frac{1}{2}g\sin\theta_{W}Y_{e}\
\overline{e_{R}}\gamma^{\mu}e_{R}A_{\mu}\right]$

where we also substitute $g^{\prime}=g\tan\theta_{W}$ from Eqn. (67). This
will yield the QED interactions if we put

$Y_{L}=1\qquad\qquad Y_{e}=2\qquad\qquad g\sin\theta_{W}=e$ (79)

Once the quantum numbers $Y_{L}$ and $Y_{e}$ are known, we can determine the
neutral current interactions, i.e. those involving the vector boson $Z_{\mu}$,
as

$S_{\rm int}^{\rm nc}=\int d^{4}x\
\left[\frac{g}{4\cos\theta_{W}}\overline{\nu_{e}}\gamma^{\mu}{\left(1-\gamma_{5}\right)\nu_{e}Z_{\mu}}-\frac{g}{4\cos\theta_{W}}\overline{e}\gamma^{\mu}\left(1-4\sin^{2}\theta_{W}-\gamma_{5}\right)eZ_{\mu}\right]$
(80)

All of this falls into place very nicely, and today it has all been verified
experimentally in a myriad ways.

##  5 Late Renaissance: quarks, mixing and a third generation

Despite the elegance of the Weinberg-Salam construction, in 1967, there still
remained three serious issues with this version of the electroweak model of
leptons.

  * •

The first of these was the same problem which had plagued Glashow’s 1961
model, viz., we cannot write this interaction for a nucleon doublet, for there
would then be a non-vanishing neutron-photon coupling. In fact, the neutron
does have a magnetic moment, and hence it does couple to the magnetic field
components in the photon, but the above form of interaction would couple it to
the electric field.

  * •

The second problem was that the separation of chiral fermions into left-handed
doublets and right handed singlets does not permit the fermions to have a
mass. For example, an electron mass term

${\cal
L}_{m}^{(e)}=m_{e}\,\overline{e}\,e=m_{e}\left(\overline{e_{L}}\,e_{R}+\overline{e_{R}}\,e_{L}\right)$
(81)

will not be gauge invariant, since (i) $e_{L}$ has weak isospin $T_{3}=-1/2$
and $e_{R}$ has $T_{3}=0$, and (ii) $e_{L}$ has $Y=Y_{L}=-1$ and $e_{R}$ has
$Y-Y_{e}=-2$. The only way to retain the gauge invariance would be to set
$m_{e}=0$, whereas the electron is known to have a finite mass222222One might
argue that the electron mass is small and may be treated as an explicit
breaking of the gauge symmetry. However, with hindsight, we know that the same
argument applies to the top quark, and its huge mass çan by no means be
regarded as an explicit symmetry-breaking..

  * •

A more subtle issue attaches itself to the axial vector coupling of the $Z$.
The axial vector current should be conserved in the limit of unbroken gauge
symmetry, but sny axial current is well known to have a chiral anomaly. This
arises from the $B_{\mu}$ part of the interaction and is, therefore,
proportional to the hypercharge $Y$. To have a renormalisable quantum field
theory, we must arrange for the total anomaly to cancel[59, 60], i.e. we
demand $\sum_{i}Y_{i}=0$ for the particles coupling to the $Z$ boson. Clearly,
if the particles in the theory are $\nu_{eL}$, $e_{L}$ and $e_{R}$, then
$\sum_{i}Y=Y_{L}+Y_{L}+Y_{e}=-4$. This non-cancellation of the anomaly was the
third problem.

The first and third problem were solved only with the introduction of quarks,
but the second problem – that of how chiral fermions acquire masses – was
solved by Weinberg [54]. The idea is that the symmetry-breaking mechanism
would generate the fermion masses. This is because the symmetries of the
leptonic theory permit the existence of Yukawa couplings

$\displaystyle S_{\rm Yuk}$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}$ $\displaystyle\int
d^{4}x\left[y_{e}\overline{L_{L}}(x)\,\Phi(x)\,e_{R}(x)+{\rm H.c.}\right]$
(82) $\displaystyle=$ $\displaystyle\int
d^{4}x\left[y_{e}\overline{\nu_{eL}}(x)\varphi^{+}(x)e_{R}(x)+y_{e}\overline{e_{L}}(x)\varphi^{0}(x)e_{R}(x)+{\rm
H.c.}\right]$

After symmetry-breaking, we replace
$\varphi^{0}\to\frac{v}{\sqrt{2}}+\eta^{0}(x)$, and obtain

$\displaystyle S_{\rm Yuk}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left[y_{e}\overline{\nu_{eL}}(x)\varphi^{+}(x)e_{R}(x)+y_{e}\overline{e_{L}}(x)\eta^{0}(x)e_{R}(x)\right.$
(83) $\displaystyle\left.\hskip
122.85876pt+\frac{y_{e}v}{\sqrt{2}}\overline{e_{L}}(x)\,e_{R}(x)+{\rm
H.c.}\right]$

The last of these terms is clearly a mass term. Writing it separately, we
obtain

$S_{\rm m}^{(e)}=\int d^{4}x\
\frac{y_{e}v}{\sqrt{2}}\left[\overline{e_{L}}(x)\,e_{R}(x)+\overline{e_{R}}(x)\,e_{L}(x)\right]=\int
d^{4}x\ m_{e}\overline{e}e$ (84)

where

$m_{e}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{y_{e}v}{\sqrt{2}}$ (85)

The electron mass is thus a product of the vacuum expectation value $v$ with
the Yukawa coupling232323Conversely, we can write the Yukawa coupling
$y_{e}=\sqrt{2}m_{e}/v$. $y_{e}$, and obviously will vanish as $v\to 0$. This
may be regarded as a fourth miracle of electroweak theory. There will,
however, be no neutrino mass, since there is no $\nu_{R}$, as pointed out by
Goldhaber et al[58] and hence, there will be no neutrino Yukawa coupling.

The 1969 discovery of quarks [61] provided a new impetus to attempts to
include strongly-interacting states in the electroweak model [62]. For the $u$
and $d$ quarks, it was a simple matter. They can be combined into a $SU(2)$
doublet for the left-handed components

$Q_{L}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\\!\\!\\!\begin{array}[]{c}u_{L}(x)\\\
d_{L}(x)\end{array}\\!\\!\\!\right)\qquad\qquad\overline{Q_{L}}(x)=\left(\\!\\!\\!\begin{array}[]{cc}\overline{u_{L}}(x)&\overline{d_{L}}(x)\end{array}\\!\\!\\!\right)$
(86)

while the right-handed components $u_{R}$ and $d_{R}$ are singlets under
$SU(2)$. We now write the gauge kinetic term for these quarks as

$S_{q}=\int d^{4}x\
\left[i\,\overline{Q_{L}}(x)\,\gamma^{\mu}\,\mathbb{D}^{(Q)}_{\mu}\,Q_{L}(x)+i\,\overline{u_{R}}(x)\,\gamma^{\mu}\,D^{(u)}_{\mu}u_{R}(x)+i\,\overline{u_{R}}(x)\,\gamma^{\mu}\,D^{(u)}_{\mu}u_{R}(x)\right]$
(87)

where, as in the leptonic case, we have covariant derivatives

$\mathbb{D}^{(Q)}_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{I}\,\partial_{\mu}+ig\,\mathbb{W}_{\mu}-\frac{ig^{\prime}}{2}Y_{Q}\,\mathbb{I}\,B_{\mu}\qquad
D^{(u)}_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\partial_{\mu}-\frac{ig^{\prime}}{2}Y_{u}\,B_{\mu}\qquad
D^{(d)}_{\mu}\stackrel{{\scriptstyle\text{def}}}{{=}}\partial_{\mu}-\frac{ig^{\prime}}{2}Y_{d}\,B_{\mu}$
(88)

Expanding this, we get charged current interactions

$\displaystyle S_{\rm int}^{\rm cc}=\int d^{4}x\
\frac{g}{2\sqrt{2}}\left[\overline{u}(x)\gamma^{\mu}\left(\\!1-\gamma_{5}\\!\right)d(x)\,W_{\mu}^{+}(x)+{\rm
H.c.}\right]$ (89)

and electromagnetic interactions

$\displaystyle S_{\rm int}^{\rm em}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\ \left[\frac{1}{2}e\left(1+Y_{Q}\right)\
\overline{u_{L}}\gamma^{\mu}u_{L}A_{\mu}+\frac{1}{2}eY_{u}\
\overline{u_{R}}\gamma^{\mu}u_{R}A_{\mu}\right.$ (90) $\displaystyle\hskip
32.52127pt\left.-\frac{1}{2}e\left(1-Y_{Q}\right)\
\overline{d_{L}}\gamma^{\mu}d_{L}A_{\mu}+\frac{1}{2}eY_{d}\
\overline{d_{R}}\gamma^{\mu}d_{R}A_{\mu}\right]$

using Eqns. (67) and (91) to replace $g^{\prime}$ and $g$ by $e$. It is now
easy to obtain the proper quark charges by setting

$Y_{Q}=\frac{1}{3}\qquad\qquad Y_{u}=\frac{4}{3}\qquad\qquad
Y_{d}=-\frac{2}{3}$ (91)

leading to the usual electromagnetic interaction

$S_{\rm int}^{\rm em}=\int d^{4}x\
\left[\frac{2e}{3}\,\overline{u}(x)\gamma^{\mu}u(x)A_{\mu}(x)-\frac{e}{3}\,\overline{d}(x)\gamma^{\mu}d(x)A_{\mu}(x)\right]$
(92)

Both these quarks have nonzero charge, but when combined into $n=udd$, of
course, the net charge is zero. This solves the first problem, and also
explains why the neutron has a magnetic (dipole) moment.

Inclusion of these quarks also solves the second problem, but with a twist. If
we naively sum the hypercharges of the neutrino-electron and up-down pairs
(constituting a generation), we will get

$2Y_{L}+Y_{e}+2Y_{Q}+Y_{u}+Y_{d}=-2-2+\frac{2}{3}+\frac{4}{3}-\frac{2}{3}=-\frac{8}{3}$
(93)

indicating that the chiral anomaly does not cancel. However, if we take into
account that every quark comes in 3 colours, then the addition must be

$2Y_{L}+Y_{e}+3\times(2Y_{Q}+Y_{u}+Y_{d})=-2-2+3\left(\frac{2}{3}+\frac{4}{3}-\frac{2}{3}\right)=0$
(94)

We now have almost everything fitting in the Glashow-Salam-Weinberg model,
except for the strange quark $s$ 242424It is an interesting social phenomenon
that the nomenclature of hadronic $SU(2)$ doublets has proceeded from the
classical (proton, neutron), to the colloquial (up, down), to the fanciful
(charm, strange) and finally the downright irreverent (top, bottom). It is,
perhaps, for the best that there are no more.. That it cannot be a singlet of
$SU(2)$ is clear from the fact that we have weak decays like
$K^{+}(u\bar{s})\to\mu^{+}\nu_{\mu}$ which must involve a $\bar{s}dW$ vertex.
This puzzle led to the creation of different models [63], of which Lee [59]
pointed out that the simplest, and, as it later turned out, correct one was
that due to Glashow, Iliopoulos and Maiani (or GIM for short) [64]. This was
simply to postulate the existence of a new quark $c$ — called the charm quark
— which would be the $SU(2)$ partner of the $d$ quark, just as the $u$ is the
$SU(2)$ partner of the $d$ quark. Thus, the GIM model envisages two fermion
generations, viz.,

$\displaystyle L_{L}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\\!\\!\\!\begin{array}[]{c}\nu_{eL}\\\
e_{L}\end{array}\\!\\!\\!\right)\quad\ell_{R}^{(1)}=e_{R}\qquad
Q_{L}^{(1)}=\left(\\!\\!\\!\begin{array}[]{c}u_{L}\\\
d_{L}\end{array}\\!\\!\\!\right)\quad u_{R}^{(1)}=u_{R}\qquad
d_{R}^{(1)}=d_{R}$ (99) $\displaystyle L_{L}^{(2)}$ $\displaystyle=$
$\displaystyle\left(\\!\\!\\!\begin{array}[]{c}\nu_{\mu L}\\\
\mu_{L}\end{array}\\!\\!\\!\right)\quad\ell_{R}^{(2)}=\mu_{R}\qquad
Q_{L}^{(2)}=\left(\\!\\!\\!\begin{array}[]{c}c_{L}\\\
s_{L}\end{array}\\!\\!\\!\right)\quad u_{R}^{(2)}=c_{R}\qquad
d_{R}^{(2)}=s_{R}$ (104)

with the chiral anomaly cancelling separately over each generation. This was
brilliantly confirmed by the 1974 discovery [65] of the $c$-quark — or rather
of a $c\bar{c}$ bound state, called the $J/\psi$.

The existence of two generations also provided an explanation for the mystery
of flavour-mixing, which had been postulated by Cabibbo [66] as early as 1963.
This was elaborated by GIM, using the Higgs doublet, as follows. The existence
of two generations permits the existence of mixed Yukawa couplings in the
quark sector of the form

${\cal S}_{\rm Yuk}^{(q)}=\int d^{4}x\
\sum_{a=1}^{2}\sum_{b=1}^{2}\left[y_{ab}^{(u)}\;\overline{Q_{L}^{(a)}}(x)\,\widetilde{\Phi}(x)\,u_{R}^{(b)}(x)+y_{ab}^{(d)}\;\overline{Q_{L}^{(a)}}(x)\,\Phi(x)\,d_{R}^{(b)}(x)+{\rm
H.c.}\right]$ (105)

using the notation of Eqn. (104), where, in order to keep the $U(1)_{Y}$
invariance, the first term uses the charge conjugate doublet

$\widetilde{\Phi}(x)\stackrel{{\scriptstyle\text{def}}}{{=}}i\sigma_{2}\Phi^{*}(x)=\left(\\!\\!\\!\begin{array}[]{c}\varphi^{0}\\\
-\varphi^{-}\end{array}\\!\\!\\!\right)$ (106)

Expanding and shifting $\varphi^{0}=v/\sqrt{2}+\eta^{0}(x)$, we obtain mass
terms of the form

$S_{m}=\int d^{4}x\
\sum_{a=1}^{2}\sum_{b=1}^{2}\left[\frac{y_{ab}^{(u)}v}{\sqrt{2}}\overline{u_{L}}u_{R}+\frac{y_{ab}^{(d)}v}{\sqrt{2}}\overline{d_{L}}d_{R}+{\rm
H.c.}\right]$ (107)

which can be written in matrix form as

$S_{m}=\int d^{4}x\
\left[\;\overline{U_{L}}\,\mathbb{M}^{(u)}\,U_{R}+\overline{D_{L}}\,\mathbb{M}^{(d)}\,D_{R}+{\rm
H.c.}\right]$ (108)

where
$\overline{U_{L,R}}=\left(\\!\\!\\!\begin{array}[]{cc}\overline{u_{L,R}}&\overline{c_{L,R}}\end{array}\\!\\!\\!\right)$
and
$\overline{D_{L,R}}=\left(\\!\\!\\!\begin{array}[]{cc}\overline{d_{L,R}}&\overline{s_{L,R}}\end{array}\\!\\!\\!\right)$,
while the mass matrices are

$\mathbb{M}^{(u)}_{ab}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{y_{ab}^{(u)}v}{\sqrt{2}}\qquad\qquad\mathbb{M}^{(d)}_{ab}\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{y_{ab}^{(d)}v}{\sqrt{2}}$
(109)

Each of these mass matrices $\mathbb{M}^{(u)}$ and $\mathbb{M}^{(d)}$ can be
diagonalised by bi-unitary transformations [67]

$U_{L}=\mathbb{V}_{L}^{(u)}U_{L}^{0}\qquad\qquad
U_{R}=\mathbb{V}_{R}^{(u)}U_{R}^{0}\qquad\qquad\mathbb{V}_{L}^{(u)}\mathbb{M}^{(u)}\mathbb{V}_{R}^{u\dagger}=\left(\\!\\!\\!\begin{array}[]{cc}m_{u}&0\\\
0&m_{c}\end{array}\\!\\!\\!\right)$ (110)

and

$D_{L}=\mathbb{V}_{L}^{(d)}D_{L}^{0}\qquad\qquad
D_{R}=\mathbb{V}_{R}^{(d)}D_{R}^{0}\qquad\qquad\mathbb{V}_{L}^{(d)}\mathbb{M}^{(d)}\mathbb{V}_{R}^{d\dagger}=\left(\\!\\!\\!\begin{array}[]{cc}m_{d}&0\\\
0&m_{s}\end{array}\\!\\!\\!\right)$ (111)

from which

$S_{m}=\int d^{4}x\
\left[m_{u}\,\overline{u_{L}^{0}}u_{R}^{0}+m_{c}\,\overline{c_{L}^{0}}c_{R}^{0}+m_{d}\,\overline{d_{L}^{0}}d_{R}^{0}+m_{s}\,\overline{s_{L}^{0}}s_{R}^{0}+{\rm
H.c.}\right]$ (112)

where the superscript $0$ denotes the physical quark state (often referred to
as the mass basis.)

When we come to the charged current interactions of Eqn. (89), however, we can
rewrite them as

$\displaystyle S_{\rm int}^{\rm cc}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\
\left[\frac{g}{2\sqrt{2}}\;\overline{U_{L}}\gamma^{\mu}(1-\gamma_{5})D_{L}W_{\mu}^{+}+{\rm
H.c.}\right]$ (113) $\displaystyle=$ $\displaystyle\int d^{4}x\
\left[\frac{g}{2\sqrt{2}}\;\overline{U_{L}^{0}}\mathbb{V}_{L}^{(u)\dagger}\gamma^{\mu}(1-\gamma_{5})\mathbb{V}_{L}^{(d)}D_{L}^{0}W_{\mu}^{+}+{\rm
H.c.}\right]$ $\displaystyle=$ $\displaystyle\int d^{4}x\
\left[\frac{g}{2\sqrt{2}}\;\overline{U_{L}^{0}}\gamma^{\mu}(1-\gamma_{5})\,\mathbb{K}_{2}D_{L}^{0}W_{\mu}^{+}+{\rm
H.c.}\right]$

where
$\mathbb{K}_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathbb{V}_{L}^{(u)\dagger}\mathbb{V}_{L}^{(d)}$
is known as the mixing matrix. The subscript $2$ indicates that it is a
$2\times 2$ matrix, and, of course, since all the $\mathbb{V}$’s are unitary,
$\mathbb{K}$ is also unitary. Such matrix can be parametrised in terms of one
angle parameter and three phase parameters; however, it can be shown [67] that
these three phase parameters can be absorbed in the phases of the four quark
fields. This makes $\mathbb{K}_{2}$ real, i.e. it has the form

$\mathbb{K}_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\\!\\!\\!\begin{array}[]{cc}\cos\theta_{C}&\sin\theta_{C}\\\
-\sin\theta_{C}&\cos\theta_{C}\end{array}\\!\\!\\!\right)$ (114)

where $\theta_{C}$ is the Cabibbo angle — postulated by Cabibbo in 1963. The
matrix $\mathbb{K}_{2}$ may, therefore, be referred to as the Cabibbo
matrix252525Strictly speaking, in 1963 Cabibbo [66] himself wrote down only
the mixing between the $d$ and the $s$ quarks, or rather, the corresponding
charged currents with a $\bar{u}$. The full matrix, which requires the $c$
quark, was written down only in 1970 by Glashow, Iliopoulos and Maiani [64]..

The electroweak model developed above was almost complete, but one important
feature was left unexplained. When parity ${\cal P}$ was shown not to be
conserved [56, 57], Landau had speculated [68] that what is conserved is the
combination262626This is really to say that while elementary particles may
have a parity-violating excess, the number of antiparticles in the Universe
will have an exactly complementary deficit, or vice versa, keeping the total
number unchanged. of parity ${\cal P}$ and charge conjugation ${\cal C}$,
i.e., the discrete quantum number ${\cal CP}$. This is precisely what happens
in the GIM model of the electroweak interactions, even if the Yukawa couplings
$y_{ab}^{(u,d)}$ are taken to be complex, since the phases of the mixing
matrix $\mathbb{K}_{2}$ will not appear. ${\cal CP}$, however, is not
conserved, as was proved in 1964 by the iconic Cronin-Fitch experiment [69].
In fact, unless ${\cal CP}$ is violated, there would be no baryon asymmetry in
the Universe, as pointed out by Sakharov [70], and hence the Universe of
matter which we live in and carry out experiments and observations could not
exist. It was in a perhaps desperate attempt272727Reminiscent of Pauli’s
’desperate attempt’ to save energy conservation by postulating the neutrino
[16]. to save the electroweak model that Kobayashi and Maskawa, in 1973,
suggested [71] that there may be a third generation of fermions, viz.

$L_{L}^{(3)}=\left(\\!\\!\\!\begin{array}[]{c}\nu_{L}^{(3)}\\\
\ell_{L}^{(3)}\end{array}\\!\\!\\!\right)\qquad\ell_{R}^{(3)}\qquad\qquad
Q_{L}^{(3)}=\left(\\!\\!\\!\begin{array}[]{c}u_{L}^{(3)}\\\
d_{L}^{(3)}\end{array}\\!\\!\\!\right)\qquad u_{R}^{(3)}\qquad d_{R}^{(3)}$
(115)

such that the Yukawa couplings and mass matrices would become $3\times 3$
matrices, and we would have to replace the $2\times 2$ mixing matrix
$\mathbb{K}_{2}$ by a $3\times 3$ version $\mathbb{K}_{3}$, which is now
called the Cabibbo-Kobayashi-Maskawa matrix, or CKM matrix for short. Now, a
$3\times 3$ unitary matrix has 3 angles and 6 phases, of which 5 phases can be
absorbed into the phases of the six quark fields. One phase, however, remains,
and this, according to Kobayashi and Maskawa [71], is the source of ${\cal
CP}$ violation.

Like Pauli’s bold prediction of the neutrino and GIM’s bold prediction of the
charm quark, the equally bold speculation of Kobayashi and Maskawa was proved
absolutely correct, when the fermions of the third generation began to be
discovered one by one. First came the tau lepton $\ell^{(3)}=\tau^{-}$ in 1975
[72], closely followed by the bottom quark $d^{(3)}=b$ in 1977 [73]. There
followed a 17-year hiatus till the 1994 discovery of the top quark $u^{(3)}=t$
[74], and another 6 years wait till the existence of the tau neutrino
$\nu^{(3)}=\nu_{\tau}$ was confirmed [75] in 2000.

By the turn of the century, it was clear that the electroweak model — dubbed
the ’Standard Model’, or SM for short, by Pais and Treiman [76] as early as
1975 — provides a logical and coherent structure for the weak and
electromagnetic interactions, and also intimately ties in with QCD through the
importance of three colours for anomaly cancellation. All that remained was
the cornerstone of the whole theory, viz., the Higgs boson, which had proved
elusive to experimental searches for several decades We may quickly recall
that it is by undergoing a phase transition in the early Universe, as it was
cooling down, that the Higgs field gave mass to all the other particles in the
SM (except photons and gluons with which it does not interact). And it is the
large masses of $W,Z$ bosons that makes the weak interaction weak, since they
are responsible for the smallness of the Fermi coupling constant. Therefore,
the Higgs boson lies at the very heart of weak interactions theory. However,
the electroweak action does not tell us anything about the mass of the Higgs
boson, which is a free parameter in the Glashow-Salam-Weinberg model.
Fortunately, there are elegant arguments why the mass of the Higs boson cannot
be very high. These ars outlined below.

It one considers the elastic scattering of $W$ bosons — a permitted process in
the GSW model — then it is easy to see that the longitudinal mode, i.e.

$W_{L}+W_{L}\longrightarrow W_{L}+W_{L}$ (116)

will have bad high energy behaviour because in that limit
$\varepsilon_{L}^{\mu}(p)\simeq p^{\mu}/M_{W}$. In fact, in the limit when the
$W$ and $Z$ masses can be neglected (but not the Higgs boson mass), the
amplitude for this process assumes the form

${\cal
A}_{L}(s,t)\approx-\sqrt{2}M_{H}^{2}\left(\frac{s}{s-M_{H}^{2}}+\frac{t}{t-M_{H}^{2}}\right)$
(117)

If we allow $M_{H}$ to grow without limit, we find that as $M_{H}\to\infty$
the amplitude reduces to

${\cal
A}_{L}(s,\theta^{*})\approx\frac{G_{F}}{\sqrt{2}}\,s\left(1+\cos\theta^{*}\right)$
(118)

where $\theta^{*}$ is the scattering angle. Obviously, as $s$ increases, this
will grow without limit, violating unitarity. It follows that we cannot let
$M_{H}$ grow without limit.

This argument can be sharpened [19] by invoking partial-wave unitarity of the
above amplitude, which can be expanded as

${\cal
A}_{L}(\theta^{*})=\sum_{\ell=0}^{\infty}(2\ell+1)A_{\ell}P_{\ell}(\cos\theta^{*})$
(119)

with the requirement $|A_{j}|^{2}\leq|{\rm Im}A_{j}|$, which can be rewritten
as $\mid\\!A_{j}\\!\\!\mid\leq\frac{1}{2}$. Applying this to the $j=0$ partial
wave (which happens to give the best bound among all the values of $j$)

$A_{0}=\frac{1}{16\pi s}\int_{-s}^{\infty}dt\ {\cal A}_{L}(s,t)$ (120)

one obtains an upper bound

$M_{H}^{2}\leq\frac{4\sqrt{2}\pi}{3G_{F}}\approx\left(700~{}{\rm
GeV}\right)^{2}$ (121)

Thus it was clear that the Higgs boson could not be very heavy, for in 1977,
thoughts of a 20 TeV LHC and a 40 TeV SSC were already in the air. In fact,
these machines (of which the SSC never materialised) were designed as machines
where the Higgs boson would certainly be discovered. However, successive
experiments came and went, but there was no sign282828It is well-known that
when Leon Lederman wrote a popular book with Dick Teresi on the hard-to-find
Higgs boson, he wanted to name his book The Goddamn Particle to express the
frustration of the high energy physics community. It is equally famous that
his scandalised publisher changed the title to The God Particle [77], thereby
creating an endless source of confusion in the minds of the public when the
elusive particle finally showed up. of the elusive particle. This explains the
jubilation when it was finally discovered at CERN [78] in 2012, and of course,
the prompt award of a much-belated Nobel Prize to François Englert and Peter
Higgs in the following year292929Robert Brout, Englert’s co-author, had died
in 2011.. The long road to this discovery makes, of itself, a fascinating
story, and it has been covered in several books published after 2012 [79].

With the discovery of the Higgs boson, the long development of the Standard
Model was complete. Till date, we do not have any better model for explaining
the experimental measurements made in all terrestrial experiments.

Part 2 : After the Higgs, What?

During the US Senate hearings on the SSC project, one Senator is reported to
have commented ”You have quite a few quarks already. What will you do with
another one?” While this story may be apocryphal, a similar question can be
asked in the context of extended Higgs sectors and, more generally, of all
physics beyond the SM. We need to briefly justify this before moving on to
more Higgs-specific discussions.

##  6 Impressionism: Beyond the Standard Model

The initial impetus to go beyond the SM (bSM) came from the fact that one
requires three colours — the basis of QCD — to get an anomaly-free theory of
electroweak interactions. The fact that QCD is also a gauge theory led to
speculations that a further level of unification, viz. strong and electroweak,
may occur if we embed both in a common gauge theory with a larger gauge group.
The earliest ideas in this direction came from Pati and Salam [80], who, in
1974, created a gauge theory based on the group $SO(4)$, where lepton number
is a fourth colour. A more comprehensive theory, based on the group $SU(5)$
was written down by Georgi and Glashow soon after [81]. This set off a vast
industry of Grand Unified Models or GUTs, which generically provided strong-
electroweak unification in addition to explaining one or more of the arbitrary
features of the SM.

A common feature of GUTs, however, is the prediction of proton decay through
the process

$p\longrightarrow\pi^{0}+e^{+}$ (122)

which would be mediated by a (massive) gauge boson $X$ of the higher (broken)
symmetry. The decay width will be suppressed by the mass of this heavy gauge
boson $X$, and the lifetime can be estimated as [82]

$\tau_{p}\sim\frac{M_{X}^{4}}{m_{p}^{5}}$ (123)

Though originally conceived by Sakharov [70], this prediction of proton decay
in the context of GUTs was first made by Pati and Salam [80] in 1974. Since
then, there have been intensive searches for proton decay303030Including
pioneering searches at the Kolar Gold Fields in India [83]., but none have
been successful in finding any evidence whatsoever. In the 1970s, the lower
bound on proton decay313131Since the age of the Universe is estimated to be
only about $1.38\times 10^{10}$ years [84], there is no imminent danger of the
Universe disappearing into a puff of pions and leptons. stood at around
$10^{29}$ years, which means that $M_{X}$ should be at least $10^{15}$ GeV.
This was known as the GUT scale. However, over the years, the lower bound on
the proton lifetime has crept up and up to around $1.67\times 10^{34}$ years
today [85], requiring $M_{X}$ to be greater than about $3\times 10^{16}$ GeV.
This is still about 400 times smaller then the Planck scale $M_{P}=\left(\hbar
c/G_{N}\right)^{1/2}$, where $G_{N}$ is the gravitational constant, and
therefore, it should be just about possible to construct a GUT without having
to seriously take into account the threshold effects of strong gravity.

Figure 4: Running coupling constants in the SM.

A more serious problem with GUTs arose after 1991 when the LEP-1 collider at
CERN produced its precision measurements of all the three gauge couplings
$\alpha_{1}$, $\alpha_{2}$ and $\alpha_{3}$ corresponding to $U(1)$, $SU(2)$
and $SU(3)$ respectively, at the mass scale $M_{Z}$. One could now determine
the beta functions of these couplings, and thereby predict their running till
the Planck scale. This exercise was done by several groups [86] with a
startling result. Instead of evolving logarithmically so that all three met at
a common scale where the expected GUT unification would occur, the
renormalisation group evolution of these couplings seemed to lead to pairwise
crossing at three different scales (the black dots in Figure 4) — just as if
there is no GUT unification. This was to prove a blow from which GUTs never
really recovered. Of course, it was almost immediately shown that the single-
scale unification could be restored by embedding the SM in a supersymmetric
model, and later it was established that one could also achieve it with a
bigger GUT group, with symmetry-breaking in multiple stages, where the beta
functions would change at each threshold [87]. But the pristine beauty of
$SU(5)$ grand unification was gone, and thenceforth, slowly but surely, GUT
theories moved from the mainstream to the fringes of high energy research.

Before fading into the background, however, GUT theories brought to the fore a
serious issue with the Higgs boson mass. The Higgs doublet, in order to
transform under the $SU(2)\times U(1)$ gauge group, must also transform under
the larger GUT gauge group $G\supset SU(2)\times U(1)$. The kinetic term for
Higgs bosons will, then, induce seagull terms of the form

$S_{\rm sg}=\int d^{4}x\ g_{X}X^{\mu}(x)X_{\mu}(x)\Phi^{\dagger}(x)\Phi(x0$
(124)

leading to the so-called ’sunset’ diagrams as shown in the figure, with a
heavy $X$ boson contributing to the Higgs boson self-energy at the one-loop
level. This contribution may be calculated, using the dimensional
regularisation scheme in $4-\varepsilon$ dimensions, as

$\delta
M_{H}^{2}(M_{X})=\frac{g_{X}}{32\pi^{2}}M_{X}^{2}\left(\frac{2}{\varepsilon}-\gamma+\ln\pi-1+\ln
M_{X}^{2}\right)$ (125)

where $\gamma$ is the Euler-Mascheroni constant [88]. Even apart from the
divergence as $\varepsilon\to 0$, this contribution is intolerably large,
given that $M_{X}\sim 10^{16}$ GeV and $M_{H}$ is 125.4 GeV. This is known as
the hierarchy problem and it can appear whenever there are two widely-
separated energy scales in a quantum field theory [89]. In this case, the
renormalisation technique cannot save the situation. One can, of course,
introduce a counterterm in the action to cancel this divergence in the
$\overline{\rm MS}$ scheme. However, if we recall that the $M_{X}$ is a
physical mass and will surely run with energy, this means that the divergence
will reappear at a different energy. One would then have to introduce a
running counterterm, which is hard to justify. This is known as the fine-
tuning problem.

It is important to point out at this stage that the hierarchy and fine-tuning
problems arise only when there is a GUT or other energy scale $M_{X}$ lying
between the electroweak scale $M_{H}$ and the Planck scale $M_{P}$. If there
is no such scale, and the SM holds all the way up to the Planck scale $M_{P}$,
there will be no hierarchy problem, since $\delta M_{H}^{2}$ will depend only
on $M_{P}$ — which does not run and can therefore be removed once and for all
in an $\overline{\rm MS}$ cancellation. We therefore need to look for a
stronger reason to have an intermediate scale $M_{X}$ – other than a naive
belief that there is a GUT somewhere in the background. This may be sought in
a consideration of the stability of the electroweak vacuum against quantum
corrections.

As is the case with all the coupling constants in the SM, the scalar quartic
coupling $\lambda$ in Eqn. (48) will also run. At the electroweak scale, we
can solve the two mass relations

$M_{W}=\frac{1}{2}gv\qquad\qquad M_{H}=\sqrt{2\lambda}v$ (126)

to obtain

$\lambda=\frac{\pi\alpha}{2\sin^{2}\theta_{W}}\left(\\!\frac{M_{H}}{M_{W}}\\!\right)^{2}\simeq
0.129$ (127)

plugging in the measured values. This value of $\lambda$ is already quite
small, and, as it happens, with increasing energy it decreases steadily, its
evolution being driven mostly by the large top quark Yukawa coupling. At a
scale somewhere between $10^{9}$ GeV and $10^{20}$ GeV, in fact, it will
become zero, and thenceforth run to negative values. The state of the art in
this analysis is reproduced un Figure 5 from Ref. [90], where the computation
has been carried to the three-loop level in the SM. The grey shaded region
shows the uncertainty in the extrapolated value of $\lambda$ as it runs with
increasing energy ($\mu$ in the figure). The two main sources of this error
are the values of the strong coupling constants $\alpha_{3}$ and the
uncertainty in the Yukawa coupling of the top quark, as currently measured.

Figure 5: Running of the Higgs quartic coupling in the SM. Reproduced from
Ref. [90]). The parameters have values as measured in 2013\. Values of
$\lambda$ is the light green-shaded region will correspond to an unstable
vacuum.

A glance at Figure 5 will make it obvious, that if $\lambda$ is non-positive,
it will be an absolute disaster for the SM, for then the potential will become
unbounded from below, and it will be possible for any particle to go down the
potential, emitting an infinite amount of energy, i.e. there will be a
catastrophic explosion of the Universe. Of course, a catastrophe is not
precluded just because it is a catastrophe. The argument here is that just
after the Big Bang, the newly-created Universe consisted of particles at
energies at the Planck scale, or just below it. If the electroweak vacuum had
been unstable at these energies, there would have been plenty of radiated
energy to keep the Universe in that hot condition, or make it even hotter.
This clearly did not happen, since the Universe has cooled down to as low as
2.73 K. Ergo, the electroweak vacuum has stayed stable throughout, and the
$\lambda$ coupling has never gone negative. This can only happen if, at some
higher scale, there is new physics beyond the SM, which changes the running of
the $\lambda$ coupling and keeps it from falling to zero.

Once can therefore make a strong claim that if $\lambda$ becomes negative at a
high energy scale below the Planck scale, we are bound to have new physics at
that scale, and this will immediately reinstate the hierarchy problem for the
Higgs boson mass. But can we be sure that $\lambda$ does, indeed become
negative at a value below the Planck scale? Figure 5 shows that because of the
uncertainty in parameters323232There are several uncertainties in the value of
the top quark mass which goes into the RGE evolution of $\lambda$. The
principal problem is that there is no measurable pole mass for the top quark,
since quark states are not asymptotically free. The experimentally measured
mass at the LHC is an invariant mass of final states where the interpretation
of hadronisation effects is model dependent. The best inference of the top
quark ’pole’ mass comes from the measurement of the $t\bar{t}$ production
cross section, but that is dependent on PDF uncertainties as well as the
renormalisation scheme. Most of these uncertainties do not affect common
processes, but even small changes affect the RGE scale, which has the top
quark-dependent parameters in the argument of the exponential function. we
cannot yet be sure about this, since the range of $\lambda$ values has one
edge that lies just around the limiting value, i.e. there is still a
possibility that the vacuum will remain stable all the way to the Planck scale
without invoking new physics.

There have been attempts to save the electroweak vacuum by postulating a more
complicated structure, equivalent to putting in a (non-renormalisable)
$\epsilon\varphi^{6}(x)$ term in the potential. In these models, there is
another, deeper vacuum, which is stable, and the current vacuum is a ’false
vacuum’, i.e., a metastable state [91]. The tunnelling time for this has been
calculated as $10^{10}$ years [90], which is reassuring, since it makes the
probability that it will happen in the next 10,000 years around $10^{-129}$.
However, it has been pointed out [92] that if the Universe underwent an
inflationary stage, as is commonly believed, then the tunnelling time would
have been much shorter — in fact short enough for the transition to the true
vacuum to have occurred. However,it was then pointed out [93] that this is
true only in a flat space calculation; if there is Higgs-curvature mixing with
a mixing parameter $\xi$, then it can be tuned to the so-called ’conformal
point’ $\xi=-1/6$ to keep the electroweak vacuum stable throughout the
inflation period. This was reinforced by a calculation of reheating effects
[94] which established that the fine tuning $\xi=-1/6$ is the only way to save
the electroweak vacuum. There the case rests.

However, long before the value of $\lambda$ was measured at any scale, the
influence of GUTs had inspired several bSM solutions for the hierarchy
problem. These are of two classes, as follows.

  1. 1.

Models where the cutoff scale is brought down to just above the electroweak
scale.

     * •

Technicolor: It was Wilson in 1971 who first pointed out [89] that gauge boson
and fermion self-energy corrections are protected from having quadratic
divergences by the gauge symmetry and the chiral symmetry, respectively. For a
scalar, however, there is no such symmetry, and hence, quadratic divergences,
such as Eqn. (125), cannot be avoided. Wilson went on to point out that this
was not a problem for scalars like the pions, since they were composites, and
the scale set by the binding energy would be a natural cutoff for self-energy
corrections. For pions, this is the QCD scale, which, at around 280 MeV, is
not so much larger than the pion mass.

By the later half of the decade, as the SM went from triumph to triumph, it
was clear that the hierarchy problem in the Higgs boson mass needed to be
addressed. The first solution to this was proposed by Weinberg [95] in 1976
and backed up by Susskind [96] in 1978. This was more-or-less a repetition of
Wilson’s argument for pions. It was assumed that the Higgs boson is actually a
composite of a new kind of confined fermions called techni-fermions, analogous
to qiarks, which were bound into a scalar by a $SU(N)$ gauge interaction
called technicolor. The cutoff scale could then be around a few TeV, the
analogy of the QCD scale, and there would be no hierarchy problem.

An obvious experimental check of the technicolour model would be the presence
of a heavy techni-rho $\rho_{T}^{0}$, being the spin-1 bound state partnering
the spin-0 Higgs boson, in the same way as the spin-1 $\rho^{0}$ partners the
$\pi^{0}$. This could be searched for through decays $\rho_{T}^{0}\to
W_{L}^{+}W_{L}^{-}$, where the $W_{L}^{\pm}$ are longitudinally-polarised $W$
bosons. Needless to say, no such signals have been found at the LHC or other
experiments. An even stronger argument against technicolor ideas came in 1990,
when Peskin and Takeuchi [97] showed that the prediction of the so-called
oblique parameter $S$ in technicolor models was much too large to be
compatible with the value obtained from experimental measurements at the CERN
LEP-1 collider. Since then, technicolor models have had few takers, except for
a few persistent souls who have crafted modifications of the minimal
technicolor model [98] to evade all experimental constraints. However, these
models have been acquiring an increasingly baroque nature, just as modern GUT
models have333333The partly-pejorative term ’epicycles’ is often applied to
these efforts, which is not without justification. However, a sobering thought
is that epicycles are just a two-dimensional version of the Fourier series,
discovered many centuries after Ptolemy..

     * •

Extra Dimensions: Extra dimensions had been introduced as a vehicle for
unification of gravity with electromagnetism in the early twentieth century
[99], but largely abandoned when it became clear that such ideas could not be
verified experimentally. In the 1970s, as gauge theories stated to become
popular, extra dimensions became a mainstay of string theories [100], which
visualised extra compact dimensions with a size around $M_{P}^{-1}$ and were
not unduly worried about experimental verification. In 1998, a sensation was
created when the trio of Arkani-Hamed, Dimopoulos and Dvali, or ADD for short,
proposed a model [101] where there would be extra dimensions of the size of a
millimetre, with the SM fields (which don’t seem to sense any extra
dimensions) confined to a wafer-thin slice of four-dimensional space inside
the higher-dimensional bulk343434The thickness of this slice must be smaller
that the smallest length scale to which the SM has curently been probed, i.e.
about $10^{-17}$ cm. This scenario was, in fact, nicely fulfilled by a
topological defect, called a $D_{3}$-brane, predicted in Type-II string
theories [102], and this led to all such subspaces being dubbed branes,
whether the model was embedded in a string theory or not. The ADD construction
would immediately bring the effective Planck scale down to around a TeV,
obviously removing the hierarchy problem. The model predicted novel spin-2
interactions from towers of Kaluza-Klein excitations of the graviton, which
would lead to observable consequences in high energy experiments.

Soon, however, it became apparent that the ADD model suffers from its own
hierarchy problem — not in the mass of the Higgs boson, but in the size of the
extra dimensions. Here too, quantum corrections were the problem, for they
would tend to shrink these dimensions down to the smallest length scale in the
theory, i.e. the Planck scale — thereby reinstating the hierarchy problem in
toto. To solve this problem, an ingenious construction was effected by Randall
and Sundrum in 1999 [103]. The Randall- Sundrum model, or RS model for short,
envisages two branes, separated by a small distance not much larger than the
Planck length, which communicate through gravitational interactions. By fine-
tuning cosmological constants on the two branes and in the intervening bulk
(which has to have the topology of a circle folded about one of its diameters
[104], i.e. a $\mathbb{S}^{1}/\mathbb{Z}_{2}$ orbifold) RS obtained a solution
for the gravitational field which falls off exponentially from one brane to
the other. It is thus possible for gravity to have electroweak strength on one
brane (dubbed the Planck brane) and its usual ultra-weak Newtonian strength on
the other (dubbed the TeV brane). The electroweak scale is the only scale in
this model and all large numbers are generated from this by exponentials with
small arguments. This theory predicts graviton Kaluza-Klein modes with masses
and coupling constants of electroweak strength, which can be easily looked for
as resonances in high energy collisions [105]. This model also has an issue
with the stability of the inter- brane distance, which was brilliantly solved
by Goldberger and Wise [106] through the introduction of a bulk scalar, which
appears on the TeV brane as a Higgs-like particle, called the radion.

It is worth mentioning, in this context, a third model with an
$\mathbb{S}^{1}/\mathbb{Z}_{2}$ orbifolded extra dimension — not because it
solves the hierarchy problem (it does not!), but because it is relevant for
Higgs boson physics. This model places all the SM fields in the bulk, with,
presumably some topological defects at the two ends of the extra dimension. In
this model, called the Universal Extra Dimensions, or UED, scheme [107], all
the SM fields have Kaluza-Klein excitations and the only new parameter in the
theory is the radius of the bulk.

All three extra-dimensional models are of decoupling nature, i.e. the
observable effects can be made arbitrarily small by tuning some parameter –
generally the size of the extra dimension. Indeed, this has been forced upon
us by the non-observation of any trace of extra dimensions. The negative
aspect of this is that the effective Planck mass keeps rising, and thereby we
have a little hierarchy problem, i.e. a ratio of order $10^{4}$ between the
Higgs boson mass and the minimum (bulk) Planck scale. This is a slightly
uncomfortable situation, but by no means as intolerable as having a cutoff at
the GUT scale of $10^{16}$ GeV.

  2. 2.

Models where the quadratic divergences in the Higgs self-energy cancel.

     * •

Supersymmetry: In Wilson’s pioneering work [89], it had been pointed out that
there are symmetries protecting the fermion and gauge boson masses from
quadratic divergences, but none protecting the scalar Higgs boson mass. If one
could find such a symmetry, clearly there would be no quadratic divergence in
$M_{H}^{2}$ and hence no hierarchy problem. As it happens, by the late 1970s,
such a symmetry lay ready at hand. This was supersymmetry. As is well known,
quantum mechanics assigns different roles to particles of spin-$\frac{1}{2}$
which are fermions and particles of integer spin which are bosons — this being
known as the spin-statistics theorem. Supersymmetry, however, transcends this
by allowing fermionic degrees of freedom to mix with bosonic degrees of
freedom and vice versa. This revolutionary idea was first used in string
theory, where the basic string excitations are all bosonic, and therefore one
needs supersymmetry to generate fermionic degrees of freedom [108]. It was
discovered by Wess and Zumino in 1974 [109] that in fact the effect of this
extra symmetry is to protect the Higgs boson mass from developing quadratic
divergences, thereby removing the hierarchy problem. The Wess-Zumino model is
the simplest supersymmetic model, being the supersymmetric version of a
$\lambda\varphi^{4}$ theory. In 1976, Pierre Fayet was the first to develop a
supersymmetric SM [110], but a phenomenologically viable model came only with
the work of Dimopoulos and Georgi in 1981[111]. In all of these models, the
Higgs boson mass remains free of quadratic divergences.

In the minimal supersymmetric SM, called the MSSM, and all of its different
variations, every boson in the SM has a fermionic partner and every fermion in
the SM has a bosonic partners. These additional fields are called
superpartners. In the limit of exact supersymmetry, the SM particles and their
respective superpartners have the same masses and coupling strengths. However,
since no such superpartners have been observed, it must be assumed that they
are too heavy to have been produced in experiments till date, i.e.
supersymmetry is explicitly broken by large masses of the
superpartners353535and by some other operators besides. Different patterns of
this symmetry-breaking create a landscape of supersymmetric models, almost all
of which have been diligently searched for at high energy experiments,
including the LHC. Unfortunately, nothing has been seen as of date, as a
result of which the lower bounds on superpartner masses are getting steadily
pushed up. Once again, there is nothing wrong in principle with such heavy
superpartners; however their failure to show up has led to a certain degree of
despondency in the high energy community.

One important feature of the MSSM and its variants is that there necessarily
has to be a second Higgs doublet. This has inspired a whole new field of
investigation, which is described in the next section.

     * •

Little Higgs Models: An ingenious construction, whose proximate inspiration
was no doubt extra-dimensional models, was proposed in 2008. However, this
idea really goes back to the times when the pion was thought to be an
elementary particle and the pion-nucleon Lagrangian had a spontaneously-broken
global symmetry, of which the pion was a Goldstone boson — or rather (since
the pion gets a small mass through explicit symmetry-breaking), a pseudo-
Goldstone boson. This was an elegant argument to keep the pion light, and the
idea of little Higgs models was to explain the lightness of the Higgs boson
using similar ideas. There are different versions of the model, but the
essential features are captured by a version where the scalar sector has a
local $\left[SU(2)\times U(1)\right]^{2}$ at high energies. This breaks
spontaneously around 8–10 TeV to the SM group $SU(2)_{L}\times U(1)_{Y}$, and
the Higgs boson is the Goldstone boson of this theory. This symmetry again
breaks spontaneously to $U(1)_{\rm em}$ at the electroweak scale, and it is at
that scale that the Higgs boson acquires a mass, becoming a pseudo-Goldstone
boson. However, quantum corrections to this cannot be too large, since
masslessness is guaranteed at the not-too-high scale 8–10 TeV. In practice,
the way this happens is that the one-loop corrections from the pair of gauge
bosons corresponding to the two $SU(2)\times U(1)$ symmetries, and these will
have opposite signs due to group theoretic factors which are induced by the
unbroken symmetry. Therefore the Higgs boson mass is generated at two loops
and is perturbatively small. In the so-called ’littlest Higgs model’, the
$\left[SU(2)\times U(1)\right]^{2}$ symmetry is obtained from the breaking
chain $SU(5)\to SO(5)\to SU(3)_{c}\times\left[SU(2)\times U(1)\right]^{2}$.

An immediate problem in such a model is that these are extra contributions to
the $S$ and $T$ precision oblique parameters at LEP, and these have a non-
decoupling component which makes them incompatible with the data, which are
very close to the Standard Model predictions. If there are two sets of gauge
bosons, however, we can assign a $\mathbb{Z}_{2}$ symmetry called $T$-parity,
such that the SM particles have $T=1$ and the new particles have $T=-1$. This
would ensure a cancellation between the corresponding contributions to the
oblique parameters, and indeed to any loop corrections, with any residual
effects being due to the mass differences. An immediate consequence is that
the lightest particle with $T=-1$ will not decay and can be a candidate for
dark matter, just as the LSP is.

A great deal of the research with bSM Higgs bosons has been in the context of
one or other of these models. In fact, there was a time when supersymmetry was
widely regarded as the ’standard’ bSM theory, but that alas! is no longer the
case.

We now come to the greatest failing of the SM, viz., its complete inability to
explain most of the composition of the Universe. These mysterious components
have been dubbed dark matter and dark energy, and the SM does not have any
candidate for either.

Dark matter was postulated by Zwicky in 1937, when he could not reconcile the
proper motion of galaxies in the Virgo super cluster with the mass inferred
from observation of the luminous matter [112]. Essentially the galaxies were
moving at speeds higher than the escape velocity corresponding to this mass,
and yet the system remains in a bound state, presumably under Newtonian
gravity. The solution, according to Zwicky, was that the supercluster contains
a great deal of matter which is not luminous, i.e. it does not radiate any
energy in the electromagnetic spectrum, and is, therefore,’dark’. Nearly four
decades later, Rubin [113] measured the transverse (rotational) velocity of
stars in the neighbouring Andromeda galaxy and a few others. The expectation
from Newtonian mechanics was that if the mass $M$ of the galaxy is
concentrated in the central region, the transverse velocity of stars in the
outer arms would fall off as $v(r)=\sqrt{G_{N}M/r}$. Rubin found, however,
that the measured velocities remain essentially constant as $r$ increases. She
correctly interpreted thus to mean that the galaxy is immersed in a ball of
invisible — dark — matter, such that the mass inside a sphere of radius $r$ is
$M(r)\propto r$. This immediately means that the dark matter distribution
inside a galaxy is $\rho(r)\propto r^{-2}$, with a cusp at the centre,
compatible with the existence of a black hole there. Since the 1970s, it has
been shown that practically every galaxy, and cluster of galaxies shows
similar behaviour. The existence of dark matter is also proved by a
consideration of gravitational lensing phenomena. Moreover, there is an
increasing body of evidence that the dark matter is particulate in nature and
does not consist of compact objects like mini black holes [114].

The problem with dark matter is that there is no particle in the SM which can
be described as dark matter. Since dark matter is believed to be non-baryonic
[115] and neutral, the only possible candidates are neutrinos and the Higgs
boson. The latter is ruled out because it is unstable, having a very short
lifetime of around $1.6\times 10^{-22}$ s. Neutrinos, on the other hand, may
be stable, but they are too light to satisfy the requirement that the dark
matter has cooled to the observed temperature (2.73 K) of the Universe. In
fact, the ideal dark matter candidate is a weakly-interacting massive particle
(WIMP) which is stable, or at least has a lifetime greater than the age of the
Universe [114]. This is simply not available in the SM.

The mystery that is dark matter pales in significance compared to what is
called ’dark energy’, which essentially means a positive cosmological
constant. The cosmological constant $\Lambda$ was originally introduced by
Einstein in 1917 to get a static Universe [116]. The best way to understand it
[117] is to write down the Robertson-Walker metric for a spherically-symmetric
Universe

$ds^{2}\stackrel{{\scriptstyle\text{def}}}{{=}}dt^{2}-R^{2}(t)\left(d\theta^{2}+\sin^{2}\theta\
d\varphi^{2}\right)$ (128)

and require the radial factor $R(t)$ to satisfy the Friedmann equation

$\frac{\ddot{R}}{R}=-\frac{4\pi
G_{N}}{3}\left(\rho+3p\right)+\frac{\Lambda}{3}$ (129)

where $p$ and $\rho$ stand for pressure and density respectively (assumed
uniform over cosmic scales) and $\Lambda$ is the cosmological constant.
Assuming $p$ and $\rho$ are positive, as would be the case for both matter and
energy, then $\Lambda=0$ would lead to $\ddot{R}<0$, i.e. a Universe which is
expanding but at a decelerated rate. The truth is exactly the opposite – it
has been shown by a study of very distant supernovae that the Universe is, in
fact expanding at a small accelerated rate [118]. This makes it imperative to
make the right side of the Friedmann equation positive, and the obvious way to
do this is to postulate a value $\Lambda>0$ which will lead to a small
positive residue on the right side.

Since the cosmological constant acts as a uniform matter density in the
Universe, it is commonly referred to as dark energy. Studies of the cosmic
microwave background indicate that the Universe is composed of $68\%$ of this
mysterious material, about $27\%$ of dark matter, and the remaining $5\%$ of
baryonic, i.e. SM matter. It would be easiest to identify the cosmological
constant or dark energy density with the vacuum energy associated with the
electroweak transition – unfortunately, that leads to an overestimate by a
factor of the order of $10^{60}$, leading to the so-called cosmological
constant problem. Many solutions of this have been proposed, from modification
of Newtonian dynamics (MOND) [120] to the use of effective field theories to
cancel the vacuum energy. There has also been a suggestion of a new kind of
particle called quintessence, which on cosmic length scales, has a repulsive
effect [119], thereby changing the sign of the pressure term and making a
nonzero cosmological constant unnecessary. These ideas are only peripheral to
this article, becoming relevant only when the Higgs sector acts as a portal to
these exotic particles through some kind of feeble coupling.

The signal failure of the first three runs of the LHC to find any evidence for
physics beyond the Standard Model has brought to the fore an alternative
paradigm. This is the development of bottom-up theories, i.e. those which
retain the gauge and global symmetries of the SM and make minimal (or just
beyond minimal) extensions to either the field content, or the specific
interactions. Among the first type are extensions of the gauge boson sector
(often associated with extra gauge symmetries of the simplest kind),
extensions of the Higgs sector, extensions of the fermion sector, or addition
of exotica such as leptoquarks and dileptons – almost all leading to new
interactions with the SM Higgs boson. There are also suggestions for
modification of the SM interaction vertices by introducing anomalous terms,
which can mostly be generated in effective field theories where there are new
heavy fields whose interactions at a high scale can be integrated out leaving
effective operators which then modify the SM vertices. In a combination of
these, models of dark matter often assume the existence of dark particles,
fermions or bosons, which interact with the SM fields through mediator fields,
and this can also modify the Higgs boson interactions.

The USP of these bottom-up approaches is claimed to be their model-independent
nature. However, while these models generally do not attempt to solve the
basic problems with the SM itself, they often make assumptions about new
fields and couplings which amount to a degree of model-building. A
comprehensive description of these ideas would require a different article
altogether. Hence, in the present work, we shall concentrate only on that part
of these extensions which is relevant specifically to the Higgs sector.

##  7 Expressionism: Extended Higgs Sectors

If we do not consider models which are derivable from some symmetry at a high
energy scale, then there is just one guideline to the kind of extra scalars
which can be added to the Standard Model. This comes from the so-called $\rho$
parameter (also called $T$ parameter), given by

$\rho\stackrel{{\scriptstyle\text{def}}}{{=}}\frac{M_{W}^{2}}{M_{Z}^{2}\cos^{2}\theta_{W}}$
(130)

In the Standard Model, $\rho=1$, and indeed its experimental value is [121]

$\rho_{\rm exp}=1.00038\pm 0.00020$ (131)

In a model with several extra Higgs multiplets $\Phi_{1},\Phi_{2},\dots$ with
different weak isospin and weak hypercharge assignments $(T_{i},Y_{i})$ where
$i=1,2,\dots$, the rho parameter (at tree-level) can be shown to have the form

$\rho=\frac{2v^{2}+\sum_{i}\left\\{4T_{i}(T_{i}+1)-Y_{i}^{2}\right\\}v_{i}^{2}}{2v^{2}+\sum_{i}2Y_{i}^{2}v_{i}^{2}}$
(132)

where $v_{i}$ is the vev of the neutral component in the $i$-th multiplet. It
is trivial to check that this remains unity if we add any number of scalar
doublets ($T=\frac{1}{2},Y=\pm 1$) or singlets ($T=0,Y=0$), irrespective of
the vacuum expectation values $v_{i}$. For other multiplets, the $v_{i}$ must
be fine-tuned suitably. It is not surprising, therefore, that the most popular
extensions of the Higgs sector are singlets and doublets. Some of the specific
ideas are now discussed in more detail.

###  7.1 Extra Real Singlet Scalar

The simplest extension of the Higgs sector is to add on a real scalar which is
a gauge singlet of $SU(2)_{L}\times U(1)$. An early exploration of this [122]
in the wake of the top quark discovery tried to satisfy the Veltman condition
of cancellation of quadratic divergences in the Higgs boson self-energy, but
this led to the prediction of a Higgs boson mass above 300 GeV, which is now
known to be unphysical. However, more recent analyses [123] take a less
ambitious approach, confining themselves to studying the parameter space and
discovery limits of such models. A singlet scalar does not, by definition,
have gauge interactions and hence its decays are solely governed by the
dynamics of the (extended) Higgs sector. This makes it, in many models,
practically stable, making it an attractive candidate for dark matter [124].

Denoting the real singlet by $\sigma$, the scalar potential of the theory in
such models becomes

$V(\Phi,\sigma)=-\mu^{2}\Phi^{\dagger}\Phi+\lambda\left(\Phi^{\dagger}\Phi\right)^{2}+\lambda_{1}\sigma+\frac{1}{2}\lambda_{2}\sigma^{2}+\lambda_{3}\sigma^{3}+\lambda_{4}\sigma^{4}+\lambda^{\prime}_{1}\Phi^{\dagger}\Phi\sigma+\lambda^{\prime}_{2}\Phi^{\dagger}\Phi\sigma^{2}$
(133)

where
$\Phi^{\dagger}\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\begin{array}[]{cc}\varphi^{-}&\varphi^{0\ast}+\frac{v}{\sqrt{2}}\end{array}\right)$.
Taking advantage of the singlet nature of the scalar $\sigma$, it is always
possible to make a redefinition such that its vacuum expectation value
vanishes, i.e. $\langle\sigma\rangle=0$. Minimising the potential now leads to
a mixing between the erstwhile Higgs boson and the singlet scalar, as follows

$\displaystyle h$ $\displaystyle=$ $\displaystyle\ \ {\rm Re}\,\varphi^{0}\
\cos\alpha+\sigma\ \sin\alpha$ $\displaystyle H$ $\displaystyle=$
$\displaystyle-{\rm Re}\,\varphi^{0}\ \sin\alpha+\sigma\ \cos\alpha$ (134)

where $M_{h}=125.4$ GeV is the mass of the elementary scalar discovered at the
LHC and $v=246.2$ GeV as in the SM. The free parameters of this theory can be
taken to be [123]

$M_{H}\qquad\qquad\sin\alpha\qquad\qquad\lambda_{3}\qquad\qquad\lambda_{4}\qquad\qquad\lambda^{\prime}_{2}$

in terms of which

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{1}{2v^{2}}\left(M_{h}^{2}\cos^{2}\alpha+M_{H}^{2}\sin^{2}\alpha\right)$
$\displaystyle\lambda_{2}$ $\displaystyle=$ $\displaystyle
M_{h}^{2}\sin^{2}\alpha+M_{H}^{2}\cos^{2}\alpha-\lambda^{\prime}_{2}v^{2}$
$\displaystyle\lambda^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2v}\left(M_{h}^{2}-M_{H}^{2}\right)\sin 2\alpha$ (135)

The mixing results will modify all the couplings of the $h$ and this will lead
to testable consequences at experimental facilities. For example, if
$M_{h}<2M_{H}$, it is easy to see that the Higgs production cross-section will
scale as $\cos^{2}\alpha$. Taking the experimental data from the LHC
experiments, it is shown in Ref. [123] that

$\mid\\!\sin\alpha\\!\mid\,<\,0.2$ (136)

In case $M_{h}>2M_{H}$, the decay channel $h\to HH$ will open up and the bound
will then be dependent on the other parameters as well. There will also be a
unitarity bound from the consideration of $h+h\to h+h$, which appears as

$M_{H}^{2}\leq\frac{16\pi v^{2}}{3}\csc^{2}\alpha-M_{h}^{2}\cot^{2}\alpha$
(137)

Taken in conjunction with Eqn. (136) this leads to an upper bound
$M_{H}\lesssim 5$ TeV, which gets relaxed as $\sin\alpha\to 0$. Only about
two-fifths of this range on $M_{H}$ would be accessible at the LHC.

In a variant of this idea, an extra $\mathbb{Z}_{2}$ symmetry
$\sigma\to-\sigma$ is included in the theory. In this case,
$\lambda_{1}=\lambda_{3}=\lambda^{\prime}_{1}$, but a glance at the third line
of Eqn. (135) will show that this is too simplistic. In fact, in such models,
we must set $\langle\sigma\rangle=v^{\prime}\neq 0$ in order to get nontrivial
results.

In a more exotic variant of this idea, a model with a $SU(2)_{L}\times
U(1)_{Y}\times U(1)_{Y^{\prime}}$ is considered, where there is an extra
$Z^{\prime}$ boson, which is massless even when the electroweak symmetry is
broken. Since there is no evidence for such a massless gauge boson, it must
acquire mass somehow. One way to do this is to break the extra
$U(1)_{Y^{\prime}}$ symmetry spontaneously by introducing some extra Higgs
multiplet which develops a vacuum expectation value of its own. Another
possibility, which has been investigated recently [125], is to add a
Stückelberg-type real scalar $\sigma$ (which transforms under the adjoint
representation of the gauge group, and is therefore not a singlet per se)
preserving the gauge symmetry, but giving a mass to the $Z^{\prime}$. This
model is highly predictive, since the $U(1)_{Y^{\prime}}$ symmetry forbids
many operators which would otherwise have been present.

###  7.2 Two Extra Real Singlets

If one can add on one real scalar singlet, why not two? This Two Real Singlet
Model (TRSM) [126] is equivalent to adding a single complex scalar doublet
(CxSM) [127]. The main motivation of such models is to obtain a viable dark
matter candidate in one of the physical scalars which result when these extra
scalars mix among themselves and with the SM scalar. The basic formalism is
not too different from a single real scalar. If we have two extra real scalars
$\sigma$ and $\sigma^{\prime}$, the most general scalar potential becomes very
messy. It is more usual, therefore, to assume an additional
$\mathbb{Z}_{2}\times\mathbb{Z}^{\prime}_{2}$ symmetry under transformations
of the form

$\displaystyle\mathbb{Z}_{2}$ $\displaystyle:$
$\displaystyle\sigma\to-\sigma\qquad\qquad\qquad\sigma^{\prime}\to\sigma^{\prime}$
$\displaystyle\mathbb{Z}^{\prime}_{2}$ $\displaystyle:$
$\displaystyle\sigma\to\sigma\qquad\qquad\qquad\ \
\;\sigma^{\prime}\to-\sigma^{\prime}$ (138)

where the SM fields are left invariant by both the transformations. The scalar
potential now becomes

$\displaystyle V(\Phi,\sigma,\sigma^{\prime})$ $\displaystyle=$
$\displaystyle-\mu^{2}\Phi^{\dagger}\Phi+\lambda\left(\Phi^{\dagger}\Phi\right)^{2}$
$\displaystyle+\frac{1}{2}\lambda_{2}\sigma^{2}+\lambda_{4}\sigma^{4}+\frac{1}{2}\lambda^{\prime}_{2}\sigma^{\prime
2}+\lambda^{\prime}_{4}\sigma^{\prime
4}+\eta_{1}\Phi^{\dagger}\Phi\sigma^{2}+\eta_{2}\Phi^{\dagger}\Phi\sigma^{\prime
2}+\eta_{3}\sigma^{2}\sigma^{\prime 2}$

where

$\Phi=\left(\begin{array}[]{c}\varphi^{+}\\\
\frac{\varphi_{0}+v}{\sqrt{2}}\end{array}\right)\qquad\qquad\sigma=\frac{s+w}{\sqrt{2}}\qquad\qquad\sigma^{\prime}=\frac{s^{\prime}+w^{\prime}}{\sqrt{2}}$
(140)

where the vacuum expectation values $\langle\sigma\rangle=w$ and
$\langle\sigma^{\prime}\rangle=w^{\prime}$ result in the spontaneous breaking
of the $\mathbb{Z}_{2}\times\mathbb{Z}^{\prime}_{2}$ symmetry.

It is convenient to take all the quartic couplings, viz.,
$\lambda,\lambda_{4},\lambda^{\prime}_{4},\eta_{1},\eta_{2},\eta_{3}$ as
independent parameters. The remaining parameters are then given by the
minimisation conditions as

$\displaystyle\mu^{2}$ $\displaystyle=$ $\displaystyle\lambda
v^{2}+\frac{1}{2}\eta_{1}w^{2}+\frac{1}{2}\eta_{2}w^{\prime 2}$
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda_{4}w^{2}+\frac{1}{4}\eta_{1}v^{2}+\frac{1}{4}\eta_{3}w^{\prime
2}$ $\displaystyle\lambda^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\lambda^{\prime}_{4}w^{\prime
2}+\frac{1}{4}\eta_{2}v^{2}+\frac{1}{4}\eta_{3}w^{2}$ (141)

where we can set $v=246.2$ GeV and choose $w,w^{\prime}$ appropriately as two
more free parameters of the theory. With these conditions, the potential can
be expanded to obtain mass terms resulting in a mixing between the neutral
scalar $\varphi_{0}$ of the SM and the two new scalars, of the form

$\left(\begin{array}[]{c}h_{1}\\\ h_{2}\\\
h_{3}\end{array}\right)=\mathbb{O}\left(\begin{array}[]{c}\varphi_{0}\\\
\sigma\\\ \sigma^{\prime}\end{array}\right)$ (142)

where the $h_{i}$ masses $M_{i}$ satisfy $M_{1}\leq M_{2}\leq M_{3}$ and the
orthogonal mixing matrix $\mathbb{O}$ satisfies the equations

$\displaystyle\sum_{i=1}^{3}M_{i}^{2}O_{i1}^{2}$ $\displaystyle=$
$\displaystyle 2\lambda
v^{2}\quad\sum_{i=1}^{3}M_{i}^{2}O_{i2}^{2}=2\lambda_{4}w^{2}\quad\sum_{i=1}^{3}M_{i}^{2}O_{i3}^{2}=2\lambda^{\prime}_{4}w^{\prime
2}$ (143) $\displaystyle\sum_{i=1}^{3}M_{i}^{2}O_{i1}O_{i2}$ $\displaystyle=$
$\displaystyle
2\eta_{1}vw\quad\\!\\!\\!\sum_{i=1}^{3}M_{i}^{2}O_{i1}O_{i3}=2\eta_{2}vw^{\prime}\quad\sum_{i=1}^{3}M_{i}^{2}O_{i2}O_{i3}=2\eta_{3}ww^{\prime}$

It is simple to now parametrise the mixing matrix $\mathbb{O}$ in terms of 3
mixing angles $\theta_{1},\theta_{2},\theta_{3}$, or rather their sines
$s_{i}=\sin\theta_{i}\ (i=12,3)$ and take these as well as as the 3 physical
masses $M_{1},M_{2},M3$ to replace the 6 free parameters in the potential
using Eqns. (7.2). We can also set one of the $M_{i}=125.4$ GeV and $v=246.2$
GeV. This leaves 7 free parameters — 3 mixing angkes, 2 scalar masses and 2
vev’s of the singlets. The potential and all the couplings of the model are
then determined in terms of these, and phenomenological studies can be made.

There are mild unitarity bounds on this model, but the phenomenological bounds
from electroweak precision tests and from high-energy collider searches can be
fairly complicated, depending on which of the different channels of $h_{i}\to
h_{j}+h_{k}$, where $ijk$ is some combination of $123$ is kinematically
permitted. Detailed studies may be found in Ref. [126]. What emerges is that
the current data permit a range of parameter space where one of the physical
scalars $h_{i}$ is coupled very lightly with matter. It can be produced by co-
annihilation of two of its sister scalars $h_{j}+h_{k}\to h_{i}\ (j,k\neq i)$,
but cannot decay back because of kinematic constraints, i.e.
$M_{i}<M_{j}+M_{k}$. This gives it the stability to become a candidate for
dark matter.

###  7.3 Two Higgs Doublet Models

While one can, in principle, go on adding more singlets scalars to the SM,
there is not much motivation for going beyond two singlets. Instead, the bulk
of efforts in this field has been in adding an extra doublet of scalars,
leading to the so-called two-Higgs doublet models (2HDM).

The basic formalism resembles the SM or its singlet extensions. If there are
two Higgs doublets $\Phi_{1}$ and $\Phi_{2}$ with identical gauge quantum
numbers, the most general potential has the form

$\displaystyle V(\Phi_{1},\Phi_{2})$ $\displaystyle=$
$\displaystyle-\mu_{1}^{2}\Phi_{1}^{\dagger}\Phi_{1}-\mu_{2}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\mu_{3}^{2}\left(\Phi_{1}^{\dagger}\Phi_{2}+{\rm
H.c.}\right)$ (144)
$\displaystyle+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\lambda_{4}\left\\{(\Phi_{1}^{\dagger}\Phi_{2})^{2}+(\Phi_{2}^{\dagger}\Phi_{1})^{2}\right\\}+\lambda_{5}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})$

As usual, we write

$\Phi_{1}\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\begin{array}[]{c}\varphi_{1}^{+}\\\
\frac{1}{\sqrt{2}}(\eta_{1}^{0}+ig_{1}^{0}+v_{1})\end{array}\right)\qquad\quad\Phi_{2}\stackrel{{\scriptstyle\text{def}}}{{=}}\left(\begin{array}[]{c}\varphi_{2}^{+}\\\
\frac{1}{\sqrt{2}}(\eta_{2}^{0}+ig_{2}^{0}+v_{2})\end{array}\right)$ (145)

The minimisation conditions for this potential are