The sensitivity to BSM in di-taon production at the LHC

Mariola Kłusek-Gawenda1$\star$, Antoni Szczurek1,2, Mateusz Dyndał3 and
Matthias Schott4

1 Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow,
Poland

2 Faculty of Mathematics and Natural Sciences, University of Rzeszow, Poland

3 AGH University of Science and Technology, Krakow, Poland

4 Johannes Gutenberg University, Mainz, Germany

*<EMAIL_ADDRESS>
Proceedings for the XXVIII International Workshop
on Deep-Inelastic Scattering and Related Subjects,
Stony Brook University, New York, USA, 12-16 April 2021
10.21468/SciPostPhysProc.?

## Abstract

We discuss the sensitivity of the $\gamma\gamma\to\tau^{+}\tau^{-}$ process in
ultraperipheral Pb+Pb collisions at LHC energies on the anomalous magnetic
moment of $\tau$ lepton ($a_{\tau}$). We derive the corresponding cross
sections considering semi-leptonic decays of both leptons in the fiducial
volume of ATLAS and CMS detectors. The expected limits on $a_{\tau}$ with the
existing Pb+Pb dataset are found to be better by a factor of two comparing to
current best experimental limits and can be further improved by another factor
of two at High Luminosity LHC. In addition, our results for tau lepton
electric dipole moment, $d_{\tau}$, can be competitive with the current best
limits that were obtained by the Belle experiment.

##  1 Theoretical framework

The physics of the ultraperipheral collisions (UPC) of heavy ions gives a good
opportunity to study several QED processes [1]. The Feynman diagram for the
Pb+Pb$\rightarrow$Pb+Pb+$\tau^{+}\tau^{-}$ process in Fig. 1 includes two
$\gamma\tau\tau$ vertices providing an enhanced sensitivity to the anomalous
magnetic ($a_{\tau}$) and electric ($d_{\tau}$) moments of the $\tau$ lepton.

Figure 1: Diagram for the di-taon production in ultraperipheral lead-lead
collisions. Main $\tau$ decay channels presented on the graphic,
$\tau^{\pm}\rightarrow\nu_{\tau}+\ell^{\pm}+\nu_{\ell}$
($~{}\ell=e,~{}\mu~{}$) and
$\tau^{\pm}\rightarrow\nu_{\tau}+\pi^{\pm}+n\pi^{0}~{}$, give approximately
$80\%$ of all $\tau$ decays.

Cross section for two-lepton production in heavy-ion collision is the
convolution of the elementary cross section for
$\gamma\gamma\to\tau^{+}\tau^{-}$ and photon fluxes. Due to large charge, ions
are surrounded by a strong electromagnetic field. In our approach, photon
fluxes depend not only on photon energy but also on the impact parameter [2].
The amplitude for the elementary cross section for the
$\gamma\gamma\to\ell^{+}\ell^{-}$ reaction in the $t$\- and $u$-channel was
derived in [3]:

$\displaystyle{\mathcal{M}}$ $\displaystyle=$
$\displaystyle(-i)\,\epsilon_{1\mu}\epsilon_{2\nu}\,\bar{u}(p_{3})\Big{(}i\Gamma^{(\gamma\ell\ell)\,\mu}(p_{3},p_{t})\frac{i(\not\\!p_{t}+m_{\ell})}{t-m_{\ell}^{2}+i\epsilon}i\Gamma^{(\gamma\ell\ell)\,\nu}(p_{t^{\prime}}-p_{4})$
(1)
$\displaystyle+i\Gamma^{(\gamma\ell\ell)\,\nu}(p_{3},p_{u})\frac{i(\not\\!p_{u}+m_{\ell})}{u-m_{\ell}^{2}+i\epsilon}i\Gamma^{(\gamma\ell\ell)\,\mu}(p_{u^{\prime}}-p_{4})\Big{)}v(p_{4})\,.$

Designating $p^{\prime}$ and $p$ as momenta of incoming and outgoing lepton
and defining $q=p^{\prime}-p$ as the momentum transfer, a photon-lepton vertex
function can be written as:

$i\Gamma^{(\gamma\ell\ell)}_{\mu}(p^{\prime},p)=-ie\left[\gamma_{\mu}F_{1}(q^{2})+\frac{i}{2m_{\ell}}\sigma_{\mu\nu}q^{\nu}F_{2}(q^{2})+\frac{1}{2m_{\ell}}\gamma^{5}\sigma_{\mu\nu}q^{\nu}F_{3}(q^{2})\right]\,,$
(2)

where $\sigma_{\mu\nu}=\dfrac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$,
$F_{1}(q^{2})$ and $F_{2}(q^{2})$ are the Dirac and Pauli form factors,
$F_{3}(q^{2})$ is the electric dipole form factor. The asymptotic values of
the form factors, in the $q^{2}\rightarrow 0$ limit, are the moments
describing the electromagnetic properties of the lepton: $F_{1}(0)=1$,
$F_{2}(0)=a_{\ell}$ and $F_{3}(0)=d_{\ell}\frac{2m_{\ell}}{e}$.

To study the experimental sensitivity on $a_{\tau}$ in the
$\gamma\gamma\rightarrow\tau^{+}\tau^{-}$ processes at the LHC, one has to
detect UPC events, which contain two reconstructed $\tau$ leptons and no
further activity in the detector. Since $\tau$ lepton is the heaviest lepton
with a lifetime of $3\times 10^{-13}$ s, it decays into lighter leptons
($\tau^{\pm}\rightarrow\nu_{\tau}+\ell^{\pm}+\nu_{\ell},~{}\ell=e,~{}\mu~{}$)
or hadrons ($\tau^{\pm}\rightarrow\nu_{\tau}+\pi^{\pm}+n\pi^{0}~{}$,
$\tau^{\pm}\rightarrow\nu_{\tau}+\pi^{\pm}+\pi^{\mp}+\pi^{\pm}+n\pi^{0}$)
happens before any direct interaction with the detector material. The
reconstruction of tau candidates depends, therefore, on the identification of
its unique decay signatures. Approximately 80% of all $\tau$ decays are one
charged particle type, and 20% of them are three-prong decays.

##  2 Calculation details

The nuclear cross section for the Pb+Pb$\rightarrow$Pb+Pb+$\tau^{+}\tau^{-}$
process is calculated in the equivalent photon approximation. Next, the
Pythia8.243 program is used to model $\tau$ decays. The QED effect of the
final state radiation from outgoing leptons is also simulated by Pythia8. The
$\gamma\gamma\rightarrow\tau^{+}\tau^{-}$ candidate events are selected by
requiring at least one $\tau$ lepton to decay leptonically, as this allows
that existing triggering algorithms of the ATLAS or CMS detector can be used
[4, 5]. We take into account the events with the limits for the leading
electron or muon: $p_{\mathrm{T}}>4$ GeV and $|\eta|<2.5$ . This operation
allows an efficient reconstruction and identification by the LHC detectors.

It is worth noting that most produced $\tau$ lepton pairs have relatively low
energy (equivalent to low transverse momentum). Therefore, the standard $\tau$
identification tools, developed by the ATLAS and CMS collaborations [6, 7] are
not expected to be applicable. We propose, therefore, to categorize the
$\gamma\gamma\rightarrow\tau^{+}\tau^{-}$ candidate events by their decay
mode. All charged-particle tracks from one- or three-prong decays are required
to have a transverse momentum of $p_{\mathrm{T}}>0.2$ GeV and a pseudo-
rapidity of $|\eta|<2.5$.

The number of events for Pb+Pb$\rightarrow$Pb+Pb+$\tau^{+}\tau^{-}$ process
[8] for different $a_{\tau}$ values can be translated into expected
sensitivity for limiting $a_{\tau}$. We treat SM results ($a_{\tau}=0$) as
background and the difference between $a_{\tau}=0$ and $a_{\tau}=X$
distributions as a signal. We use two values of expected systematic
uncertainty (5% and 1%) and two assumptions on Pb+Pb integrated luminosity (2
nb-1 for existing ATLAS/CMS dataset or 20 nb-1 for HL-LHC). The expected
significance can be directly transformed into expected 95% CL limits on
$a_{\tau}$. Smaller systematic uncertainty or larger value of luminosity
allows predicting a narrower limit on $a_{\tau}$ [8].

##  3 Main results

The DELPHI collaboration at LEP2 [9, 10] obtained the limit:
$-0.052<a_{\tau}<0.013~{}(95\%~{}CL)$. The experimental limits on $a_{\tau}$
were also derived by the L3 and OPAL collaborations in radiative
$Z\rightarrow\tau^{+}\tau^{-}\gamma$ events at LEP [11, 12], but they are
typically weaker by a factor of two comparing to the DELPHI limits. For
comparison, the theoretical Standard Model (SM) value of $a_{\tau}$ [13] is:
$a_{\tau}^{\textrm{th}}=0.00117721\pm 0.00000005$.

Table 1 contains a summary of the integrated fiducial cross sections at
$\sqrt{s_{NN}}=5.02$ TeV for different $a_{\tau}$ values. There is an
enumeration of the expected number of reconstructed events in ATLAS or CMS. We
assume 80% reconstruction efficiency within the fiducial region and two values
of integrated luminosity ($L_{int}$). The first one corresponds to the
existing LHC Pb+Pb dataset: $L_{int}=2$ nb-1, and the second one relates to
expected High Luminosity LHC dataset: $L_{int}=20$ nb-1. With the existing
Pb+Pb dataset, we expect each experiment to reconstruct about 5000
$\gamma\gamma\rightarrow\tau^{+}\tau^{-}$ events ($a_{\tau}=0$). The expected
number of reconstructed $\tau$ pairs grows to about 50 000 at the HL-LHC.

$a_{\tau}$ value |  $\sigma_{fid}$ [nb] | Expected events | Expected events
---|---|---|---
|  | ($L_{int}=2$ nb-1, $C=0.8$) | ($L_{int}=20$ nb-1, $C=0.8$)
$-0.1$ | 4770 | 7650 | 76 500
$-0.05$ | 3330 | 5350 | 53 500
$-0.02$ | 3060 | 4900 | 49 000
$0$ (SM) | 3145 | 5050 | 50 500
$+0.02$ | 3445 | 5500 | 55 000
$+0.05$ | 4350 | 6950 | 69 500
$+0.1$ | 7225 | 11550 | 115 500
Table 1: Integrated fiducial cross sections for
Pb+Pb$\rightarrow$Pb+Pb+$\tau^{+}\tau^{-}$ process for different values of
anomalous electromagnetic moments. The expected number of events assuming 80%
selection efficiency and $L_{int}=2$ nb-1 or $L_{int}=20$ nb-1 are also shown.

The number of events from Table 1 can be translated into expected sensitivity
for probing $a_{\tau}$. We use the RooFit toolkit for the statistical analysis
of the results. We perform fits to $R_{\ell}(p_{\mathrm{T}}^{lead~{}lepton})$
distribution by treating SM results ($a_{\tau}=0$) as background and the
difference between $a_{\tau}=0$ and $a_{\tau}=X$ distributions as a signal.
The procedure exploits both normalization and $p_{\mathrm{T}}^{lead~{}lepton}$
shape differences, providing extra sensitivity on $a_{\tau}$ measurement. We
use two values of expected systematic uncertainty (5% and 1%) and two
assumptions on Pb+Pb integrated luminosity (2 nb-1 for existing ATLAS/CMS
dataset or 20 nb-1 for HL-LHC).

(a) (b)

Figure 2: (a) Expected signal significance as a function of anomalous $\tau$
moment for different values of Pb+Pb integrated luminosity and total
systematic uncertainty.
(b) Expected 95% CL limits on $a_{\tau}$ measurement for different values of
the Pb+Pb integrated luminosity and total systematic uncertainty. Comparison
is also made to the existing limits from OPAL [12], L3 [11] and DELPHI [9]
experiments at LEP.

Figure 2(a) shows the expected signal significance as a function of
$a_{\tau}$. The observed asymmetry for positive and negative $a_{\tau}$ values
reflects the destructive interference between SM and the anomalous $\tau$
coupling. The expected significance can be directly transformed into expected
95% CL limits on $a_{\tau}$, shown in Fig. 2(b). Assuming 2 nb-1 of integrated
Pb+Pb luminosity and 5% systematic uncertainty, the expected limits are
$-0.021<a_{\tau}<0.017$, approximately two times better than the DELPHI limits
[9]. By collecting more data (20 nb-1) and with improved systematic
uncertainties, these limits can be further improved by another factor of two.
The expected results by studying ultraperipheral collisions at the LHC can
significantly improve the existing limits on $a_{\tau}$.

##  4 Conclusion

Here we presented a prediction on the cross section of the
$\gamma\gamma\to\tau^{+}\tau^{-}$ process and its dependence on anomalous
electromagnetic couplings of the tau lepton in ultraperipheral Pb+Pb
collisions at the LHC. We also investigated the expected sensitivity on
$a_{\tau}$ and $d_{\tau}$, assuming standard LHC detectors using the currently
available and future datasets. We propose to use cross section ratios of the
$\gamma\gamma\to\tau^{+}\tau^{-}$ and $\gamma\gamma\to
e^{+}e^{-}(\mu^{+}\mu^{-})$ processes to probe $a_{\tau}$, as several
systematic uncertainties cancel and the experimental knowledge of $a_{e}$ and
$a_{\mu}$ is several orders of magnitude more precise than $a_{\tau}$ itself.

Our studies suggest that the currently available datasets of the LHC
experiments are already sufficient to improve the sensitivity on $a_{\tau}$ by
a factor of two. Hence, we consider experimental analysis as highly
interesting and worthwhile to be done in the future. Future Belle-II
experiment should give much better constraints on $|a_{\tau}|<1.75\cdot
10^{-5}$ and $|d_{\tau}|<2.04\cdot 10^{-19}$ $e\cdot\textrm{cm}$ [14].

We have also studied the sensitivity on tau lepton electric dipole moment,
$d_{\tau}$. Our expected 95% CL sensitivity on $|d_{\tau}|$ assuming
$a_{\tau}=0$ is: $|d_{\tau}|<6.3~{}(4.4)\cdot 10^{-17}$ $e\cdot\textrm{cm}$ at
the LHC with 5% (1%) systematic uncertainty and $|d_{\tau}|<3.5\cdot 10^{-17}$
$e\cdot\textrm{cm}$ at HL-LHC (1% systematic uncertainty). For comparison, the
current best limits are measured by Belle experiment [15]:
$-2.2<Re(d_{\tau})<4.5~{}(10^{-17}~{}e\cdot\textrm{cm})$ and
$-2.5<Im(d_{\tau})<0.8~{}(10^{-17}~{}e\cdot\textrm{cm})$. Our projected
results on $d_{\tau}$ can be therefore competitive with the Belle limits.

## Acknowledgements

### Funding information

This study was partially supported by the Polish National Science Center grant
UMO-2018/31/B/ST2/03537 and by the Center for Innovation and Transfer of
Natural Sciences and Engineering Knowledge in Rzeszów. The project is co-
financed by the Polish National Agency for Academic Exchange within Polish
Returns Programme, Grant No. PPN/PPO/2020/1/ 00002/U/00001.

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# Catastrophe theoretic approach to the Higgs Mechanism

Samyak Jain  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400 076, India  Corresponding author<EMAIL_ADDRESS>A.
Bhagwat  Contributing author<EMAIL_ADDRESS>School of Physical Sciences,
UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Kalina
Campus, Mumbai 400 098, India

###### Abstract

A geometric perspective of the Higgs Mechanism is presented. Using Thom’s
Catastrophe Theory, we study the emergence of the Higgs Mechanism as a
discontinuous feature in a general family of Lagrangians. We show that the
Lagrangian that exhibits the Higgs Mechanism arises as a first-order phase
transition in this general family.

##  I Introduction

Catastrophe Theory is a geometrical framework developed to study sudden and
discontinuous changes in dynamical systems under smooth perturbations. Thom
showed (see [1]) that any smooth function of $n$ variables and $r$ parameters
$(r\leq 5)$ can be mapped to one and only one of 11 known families of
functions (catastrophes). These catastrophes have unique geometries, and have
already been studied in detail for sudden discontinuous changes under smooth
perturbations to their parameters, thus allowing us to study any $r$ parameter
smooth function by finding mappings that take us to one of these known
catastrophes (see [2] and [3] for details). Catastrophe theory deals with
systems that have these $r$ parameter functions as their potentials. We
emphasize that these catastrophes are unique and cannot related to each other
by smooth variable transformations.

Proposals to study quantum systems using Catastrophe theory have been
suggested (see [4]). Motivated by these, we hunt for discontinuous features
typical of catastrophes in the Lagrangian demonstrating the Higgs Mechanism.
In Section II, we give a brief introduction to the cusp catastrophe (following
the terminology of [4]), which is one of the simplest but most commonly
arising catastrophes. In Section III, following [5], we briefly outline how
the Higgs Mechanism arises from a combination of Spontaneous Symmetry Breaking
and local gauge invariance. In Section IV, we show how the Lagrangian (its
potential) used in Section III is related to the Cusp Catastrophe, which
describes a more general family of potentials. We show that attenuating the
parameters of this general family leads to a first-order phase transition, as
defined in [4]. We additionally observe a discontinuous change in an affine
property of the system (number of critical points of the Lagrangian’s
potential) as we arrive at the usual Lagrangian that shows the Higgs
Mechanism.

##  II The Cusp Catastrophe

As observed earlier, the cusp catastrophe is an extremely common catastrophe
in physical systems. The cusp potential reads as:

$\displaystyle V_{\text{cusp}}(a,b;x)=x^{4}+ax^{2}+bx,\quad x\in\mathbb{R}$

with $a,b$ as parameters. The critical points are defined as points where the
first derivative of this potential vanishes, and satisfy

$\displaystyle 4x_{c}^{3}+2ax_{c}+b=0$ (1)

This can be solved to see how the number of critical points $x_{c}$ varies
with $a,b$. We obtain a single critical point (a local minimum) for all
choices of parameters $a,b$ except for the region

$\displaystyle a<0,\quad|b|\leq\frac{4}{3\sqrt{6}}\sqrt{-a^{3}}$ (2)

where we have a local maxima sandwiched between 2 local minima. Fig.1 shows
how this looks in the $b-a$ plane. In Fig.(2) we show a clearer geometric
picture by plotting Eq.(1) in the $b-a-x$ space. This structure defines the
so-called catastrophe manifold (see [2]).

Figure 1: Variation in the number of critical points of the cusp potential in
the $b-a$ plane.  Figure 2:  The catastrophe manifold for the Cusp
Catastrophe. We can see how the critical points (number and values) vary
across the $b-a$ plane.

(a) $a=-1,b=-\frac{1}{4}$

(b) $a=-1,b=0$

(c) $a=-1,b=\frac{1}{4}$

Figure 3:  The first-order phase transition in the cusp potential is shown. We
fix $a=-1$, vary $b$ from $\frac{-1}{4}$ to $\frac{1}{4}$ and plot the cusp
potential. The ground states for the potentials are indicated by red dots.

a) For $b=-\frac{1}{4}$, we have two negative critical points, and the ground
state is at the positive critical point (a minimum). b) At $b=0$, the minima
become degenerate, and the ground state can be at either minimum. Till now,
the ground state energy has been decreasing with increasing $b$. c) As soon as
$b>0$, the ground state shifts abruptly to the negative minimum. The ground
state energy now suddenly starts decreasing with increasing $b$. Since the
ground state was never at $x=0$, the rate of change of energy given by Eq.(4)
changes from a strictly positive value to a strictly negative value. Thus the
rate of change of energy has shifted discontinuously, completing the first-
order phase transition.

As in [4], we assume that for any fixed choice of parameters, the energy of
the system described by this potential dwells at the lowest minima of the
potential. Consider $a<0$ and $b$ varying from negative to positive values. We
see that at $b=0$, the minima become degenerate, and the lowest potential
minimum shifts abruptly (see Fig. 3). We can show that while the ground state
energy $E_{g}$ is continuous (because the shift happens when the minima are
degenerate), the rate of change of the ground state energy $\frac{dE_{g}}{db}$
is discontinuous:

$\displaystyle\frac{dE_{g}}{db}=\frac{dE(x_{c},a,b)}{db}=\frac{\partial{E(x_{c},a,b)}}{\partial
b}$ $\displaystyle+\frac{dE(x_{c},a,b)}{dx_{c}}\frac{dx_{c}}{db}$ (3)

Note that $x_{c}$ here is the particular critical point at which the ground
state dwells. Now, the second term vanishes because
$\frac{dE(x_{c},a,b)}{dx_{c}}=0$ by the definition of $x_{c}$. Thus

$\displaystyle\frac{dE_{g}}{db}=x_{c}$ (4)

Since $x_{c}$ is discontinuous at $b=0$, the first derivative of the ground-
state energy is discontinuous as well. This is called a first-order phase
transition ([4]).
As shown in [4], we obtain a second-order phase transition (the second
derivative being discontinuous) at $a=b=0$. We do not describe this in detail
because, as we shall see in Section IV, we are concerned with only the first-
order phase transition in this cusp potential.

##  III The Higgs Mechanism

We begin with the Mexican Hat Lagrangian that demonstrates spontaneous
symmetry breaking [5]

$\displaystyle\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi_{1}^{*})(\partial^{\mu}\phi_{1})+\frac{1}{2}(\partial_{\mu}\phi_{2}^{*})(\partial^{\mu}\phi_{2})$
$\displaystyle+\frac{\mu}{2}(\phi_{1}^{2}+\phi_{2}^{2})-\frac{\lambda}{4}(\phi_{1}^{2}+\phi_{2}^{2})^{2}$
(5)

We define

$\phi=\phi_{1}+i\phi_{2},\quad\phi^{*}=\phi_{1}-i\phi_{2}$ (6)

The Lagrangian thus becomes

$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{*}(\partial^{\mu}\phi)+\frac{\mu}{2}(\phi^{*}\phi)-\frac{\lambda}{4}(\phi^{*}\phi)^{2}$
(7)

Enforcing local gauge invariance, we introduce a field $A^{\mu}$ that
transforms like

$A^{\mu}\rightarrow A^{\mu}+\partial^{\mu}\lambda$ (8)

under a transformation

$\phi\rightarrow e^{i\theta(x)}\phi$ (9)

We further complete the Proca Lagrangian by introducing the term
$\frac{-1}{16\pi}F^{\mu\nu}F_{\mu\nu}$. Finally, we replace partial
derivatives with covariant derivatives to obtain the locally gauge-invariant
Lagrangian:

$\displaystyle\mathcal{L}=\frac{1}{2}\left(\left[(\partial_{\mu}-iqA_{\mu}\right]\phi^{*}\right)\left(\left[\partial^{\mu}+iqA^{\mu})\right]\phi\right)$
$\displaystyle+\frac{1}{2}\mu^{2}(\phi^{*}\phi)-\frac{1}{4}\lambda^{2}(\phi^{*}\phi)^{2}-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}$
(10)

The Higgs Mechanism is seen by expanding this Lagrangian density about the
minima (ground state) of the potential of the initial Lagrangian :

$V=\frac{\lambda}{4}(\phi^{*}\phi)^{2}-\frac{\mu}{2}(\phi^{*}\phi)$ (11)

The ground state is obtained at

$\displaystyle\phi^{*}\phi=\frac{\mu^{2}}{\lambda^{2}}$ (12)

We conveniently define

$\displaystyle\phi_{1}=\eta+\frac{\mu}{\lambda},\phi_{2}=\xi$ (13)

As shown in [5], upon rewriting the Lagrangian in Eq. (10) in terms of these
new fields we obtain a new particle defined by the field $\eta$ (the Higgs)
and other interaction terms. A Goldstone Boson ($\xi$) is obtained as well,
along with a bilinear term

$\displaystyle-2i\left(\frac{\mu}{\lambda}\frac{q}{\hbar
c}\right)\left(\partial_{\mu}\xi\right)(A^{\mu})$ (14)

This term cannot be interpreted as an interaction term, because it implies
that the particles defined by $\xi$ and $A^{\mu}$ cannot exist independently.
To get rid of the Goldstone Boson and this bilinear term, [5] exploits the
local gauge invariance of the Lagrangian defined by Eq.(10), and chooses

$\displaystyle\theta=-\tan^{-1}\left(\frac{\phi_{2}}{\phi_{1}}\right)$ (15)

In this gauge, $\phi_{2}$ trivially vanishes, thus eliminating both the
Goldstone Boson and the bilinear term. This completes the Higgs Mechanism.
We emphasize that at no step was the initial Lagrangian given by Eq.(10)
altered, it was only redefined by simple re-parameterization of the initial
fields $\phi_{1}$ and $\phi_{2}$, and a suitable $\theta$ was chosen at the
end. At any point, we can revert to the initial choice of fields and arrive at
the initial Lagrangian. The key features of the Mechanism are the emergence of
the Higgs ($\eta$), elimination of the Goldstone Boson ($\xi$), and the non-
physical interaction term (Eq.(14)) when the Lagrangian is expanded about its
potential’s minimum and a suitable gauge is chosen.

##  IV Mapping to the Cusp Potential

In this section, we study the rather unique choice of Lagrangian defined by
Eq.(7). The Higgs Mechanism as detailed above is a very specific phenomenon,
and need not arise in a more general Lagrangian. We thus now apply a
catastrophe theoretic approach to the potential defined by Eq.(11).

We see that the potential only depends on $\phi^{*}\phi$, and effectively is a
one-variable function. We define $r=\sqrt{\phi^{*}\phi}$ and obtain:

$V=\frac{\lambda^{2}}{4}r^{4}-\frac{\mu}{2}r^{2}$ (16)

This potential can be mapped to the cusp potential defined by Eq.(II) via

$\displaystyle x=\sqrt{\frac{\lambda}{2}}\,r,\quad
a=-\frac{\mu}{\lambda},\quad b=0$ (17)

We note that this automatically assigns a unique geometry (that of the Cusp
Catastrophe) to this family. The family cannot be mapped to any other
catastrophe. As emphasized in Section I, each Catastrophe has a unique
geometry, and thus a family of functions can never be mapped to more than one
catastrophe. All the re-parameterizations of the initial Lagrangian in Section
III can be mapped to (only) the Cusp Catastrophe, simply because the initial
Lagrangian maps to the Cusp Catastrophe.
We immediately see that we indeed have something special happening. We have
already shown in Section II that we have a first order-phase transition at
$b=0$ for a fixed $a<0$, which is clearly true for our potential. This means
that the potential that gave rise to the Higgs Mechanism in Section III can be
generalized by adding a term $br$ with $b\neq 0$. Thus the Higgs Mechanism
arises with a first-order phase transition in the family of Lagrangians with
potentials given by

$\displaystyle
V_{f}=\frac{\lambda}{4}r^{4}-\frac{\mu}{2}r^{2}+br,b\in\mathbb{R}$ (18)

We will later show that the Higgs Mechanism does not arise in a general
Lagrangian of this family.
We further note that $r=\sqrt{\phi^{*}\phi}$ is strictly non-negative. It is
easy to check that if we fix $a<0$, we have our critical points as

$\displaystyle 0<b<\frac{4}{3\sqrt{6}}\sqrt{-a^{3}}\longrightarrow\text{two
positive, one negative}$ $\displaystyle b=0\longrightarrow\text{one positive,
one negative and one as 0}$
$\displaystyle-\frac{4}{3\sqrt{6}}\sqrt{-a^{3}}<b<0\longrightarrow\text{two
negative, one positive}$

Enforcing $r\geq 0$, we are left with 2 critical points for $b\geq 0$ and 1
critical point for $b<0$. Thus, in addition to the first-order phase
transition, the number of critical points changes as well when the Higgs
Mechanism arises.
It is easy to verify that the Higgs Mechanism cannot show up with the new term

$\displaystyle
br=b\sqrt{\phi^{*}\phi}=\sqrt{\left(\frac{\mu}{\lambda}+\eta\right)^{2}+\xi^{2}}$
(19)

There is no way to interpret this term as an interaction. The fields cannot be
promoted to operators under canonical quantization. We cannot exploit the
local gauge invariance of $\mathcal{L}$ to get rid of this term,
$\sqrt{\phi^{*}\phi}$ is trivially invariant under the transformation defined
by Eq.(9), and vanishes only for $\phi_{1}=\phi_{2}=0$. Thus the Higgs
mechanism does not show up in this general Lagrangian family, it arises
suddenly only when $b=0$.

##  V Summary and Conclusion

We present a catastrophe theoretic interpretation of the Higgs Mechanism. We
showed that the Lagrangian demonstrating the Higgs Mechanism (Eq.(7)) is a
special member of a general family of Lagrangians. The potential of this
family of Lagrangians can be easily mapped to the cusp catastrophe, thus
assigning a unique geometry to the family. We find that the family
demonstrates a first-order phase transition (as defined in Section II) when
its parameters are varied continuously so as to arrive at the usual Lagrangian
exhibiting the Higgs Mechanism.

##  VI Acknowledgements

We would like to thank Urjit A. Yajnik for his valuable insight into our work.
We thank Ajay Patvardhan for his suggestions.

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