# Unfolding Particle Physics Hierarchies with Supersymmetry and Extra
Dimensions

Raman Sundrum

(May 31, 2023)

###### Abstract

This is a written version of lectures delivered at TASI 2022 “Ten Years After
the Higgs Discovery: Particle Physics Now and Future”. Mechanisms and
symmetries beyond the Standard Model (BSM) are presented capable of elegantly
and robustly generating the striking hierarchies we observe in particle
physics. They are shown to be among the central archetypes of quantum
effective field theory and to strongly resonate with the tight structure and
phenomenology of the Standard Model itself, allowing one to motivate, develop
and test a worthy successor. The (Little) Hiearchy Problem is discussed within
this context. The lectures culminate in specific BSM case-studies, gaugino-
mediated (dynamical) supersymmetry breaking to generate the weak/Planck
hierarchy, and (in less detail) extra-dimensional wavefunction overlaps to
generate flavor hierarchies.

##  1 Introduction

The Standard Model (SM) describes an orchestra of elementary particles playing
tightly interwoven melodies in the symphony of Nature [1]. And yet, it is an
unfinished symphony. The SM gauge and Yukawa couplings display intriguing
patterns that require new mechanisms to explain. The enigmas of Dark Matter
and the origins of the matter-antimatter asymmetry also point beyond the SM.
The incorporation of a fully realistic quantum gravity represents a
significant challenge. Against the backdrop of plausible BSM physics at far-UV
scales, electroweak symmetry breaking (EWSB) is very fragile, posing another
thorny mystery, the Hierarchy Problem.

In these lectures, I want to survey the direct and indirect evidence for the
very hierarchical structure of fundamental physics and to provide powerful and
overarching quantum field theory (QFT) mechanisms beyond the Standard Model
(BSM) capable of elegantly generating this structure. Part of the job is to
fully appreciate the beautiful themes already at work within the SM, so as to
guide us in how the symphony might extend further. New themes should harmonize
with the old. Of course, we will need to carefully account for the state of
play along different experimental frontiers and the prospects for their
improvement. It is in the context of this ambitious BSM undertaking that the
(in)famous Hierarchy Problem will be discussed intuitively, rather than as a
philosophically dubious concern of the SM in isolation.

The lectures will culminate in a specific BSM structure, not because I think
it is the inevitable successor to the SM but because it makes a good “case
study”, illustrating robust QFT principles, methodology and phenomenological
detective work along the way. The key new ingredients are extensions of
relativistic spacetime, supersymmetry (SUSY) and extra dimensions, which will
be strongly motivated from both top-down and bottom-up considerations. The key
old ingredient to be recycled is dimensional transmutation as seen in QCD,
capable of generating exponential hierarchies. The Hierarchy Problem will be
solved (modulo the Little Hierarchy Problem) within the framework of “Gaugino-
Mediated SUSY Breaking” [2], where SUSY is ultimately broken “dynamically”
(via dimensional transmutation). Along the way, I will give a low-resolution
introduction to the extra-dimensional wavefunction overlap mechanism for
generating flavor (Yukawa-coupling) hierarchies [3].

Apology: The goal here is to present a coherent conceptual framework for
particle physics, but of course to do that concretely requires equations,
which necessarily involve factors of $2,\pi,i$ and minus signs. I have done a
modest job of trying to self-consistently get the right factors of $2$ and
minus signs in the time I had, but I am sure that there are still several
errors. I have done a better job with factors of $\pi$ and $i$. I hope this
still allows the lectures to be readily comprehensible, and the reader can go
through derivations more carefully for themselves or consult the more careful
references provided (accounting for slight differences of convention). Since
the lectures are founded on the profound mathematical identity,

$e^{\rm moderate}={\rm Large},$ (1)

I have been especially careful to ensure that I have no mistakes in the
exponents that appear.

###  1.1 The End of Particle Physics

Let me begin with a few things you all know, but I want to look again with
fresh eyes and marvel at the enigmas of fundamental physics. Elementary
particles are categorized in terms of two spacetime quantum numbers, Mass and
Spin, as well as some internal quantum numbers. Both the mass and spin have
maximum allowed values, in each case involving General Relativity in
interesting but different ways. These are the “ends” of particle physics in
the mass and spin directions.

Since $E=mc^{2}=\$$ is the central consideration in particle physics, we begin
with mass. A point particle has a classical Schwarzchild radius $\sim G_{N}m$
as well as effectively a quantum mechanical “size” given by its Compton
wavelength $1/m$. When the former is larger, the particle effectively is
within its Schwarzchild radius and is predominantly a classical black hole
rather than an elementary quantum particle. This happens when $m\gg
1/\sqrt{G_{N}}$, so that the Planck scale $M_{Planck}\equiv 1/\sqrt{8\pi
G_{N}}=2\times 10^{18}$ GeV marks the high end of particle physics in the mass
direction, and the onset of black hole physics.

Relatedly, particle physics is the exploration of the smallest distances
$\ell$, which by the uncertainty principle and relativity requires
concentrating energy $E\geq|\vec{p}\,|>1/\ell$. And yet if this concentration
of energy is within its Schwarzchild radius $\ell\ll G_{N}E$, it will again
gravitationally form a black hole. We are therefore unable to probe distances
shorter than the Planck length $\ell_{Planck}=\sqrt{G_{N}}\sim 1/M_{Pl}$.

Given these considerations, we are led to the following paradigm. A full
dynamics of quantum gravity operating at the highest energy/mass scales
matches onto ( reduces to) a quantum effective field theory (EFT) below
$M_{Pl}$, describing matter, radiation and general relativity with pointlike
quanta. This EFT unfolds as we follow its renormalization group (RG) flow to
lower energies and larger distances. A roughly parallel unfolding takes place
in cosmic history as the universe expands and cools from high temperatures at
the Big Bang to the cold temperatures of outer space today.

###  1.2 The View from the Top

Superstring theory offers a UV-complete formulation of quantum gravity [4]. It
has a tight internal consistency, great beauty and unity, and the virtue of
concreteness of its perturbative structure. String constructions can have
quasi-realistic features. And who knows, maybe some incarnation of string
theory is even true. But it at least gives us a concrete means to envision the
quantum gravity heights and what might descend from there into EFT.

The fundamental objects are one-dimensional strings which approximate point-
particles on distances longer than the string length parameter $\ell_{\rm
string}>\ell_{\rm Planck}$. The lightest vibrational modes of the superstring
appear as pointlike gravitons, gauge bosons and charged particles at low
energies, but excited vibrational modes of the string appear as high mass
$\sim m_{\rm string}\equiv 1/\ell_{\rm string}<M_{Pl}$ and higher-spin
resonances. The non-renormalizable perturbative expansion of quantum general
relativity, in terms of the dimensionless coupling $G_{N}.Energy^{2}$,
threatens to exit perturbative control as energies grow towards $M_{Pl}$. But
in string theory this UV growth of the effective coupling is cut off while it
is still weak, by energies $\sim m_{\rm string}$, retaining perturbative
control.

Remarkably, the simplest string theory constructions require higher-
dimensional spacetime and SUSY for their self-consistency. There is a stringy
analog of the kind of gauge-anomaly cancellation requirement familiar in
chiral gauge theories such as the SM itself, which restricts the possible
field content. But in string theory, spacetime dimensions are themselves
fields (on the string worldsheet), and anomaly cancellation determines their
number to be $10$! Furthermore, for the stringy vibrational modes to include
fermionic effective particles and to ensure vacuum stability (absence of
tachyons), SUSY is required. In this way, extra spacetime dimensions and SUSY
are strongly motivated from a top-down quantum gravity perspective. The only
question is how low in energies these exotic ingredients and their associated
higher symmetries are manifest, and then ultimately hidden at current
experimental energies.

###  1.3 Fundamental Physics is Hierarchical!

Figure 1: Cartoon of the hierarchical structure of fundamental physics

Consider the cartoon of fundamental physics in figure 1, laid out in terms of
energy/mass scales. It is meant to be like an ancient map, starting with
actual experimental data but trailing off into broad theoretical prejudices
and guesswork, and far from complete. It is an explorers map for those who
hope to sail to its extreme reaches by every means possible. This kind of
synthesis of precision data and plausible theory is illustrated in figure 2.

Figure 2: Running of SM gauge couplings at one loop, hinting at a grand
unification at extremely high energies

Starting with the disparate measured values of Standard Model (SM) gauge
couplings at the weak scale, their SM RG evolution shows a striking “near”
coincidence at extremely high energies, suggesting a common origin. Indeed, on
closer inspection, the different SM fields and their quantum numbers fit
neatly like puzzle pieces into “grand unified theories” (GUTs), where some
missing pieces have gotten very large masses $\sim M_{GUT}$ by a grand version
of the Higgs mechanism. See [5] for a review. GUTs would explain why the gauge
couplings run up to a nearly unified value at $\sim M_{GUT}$. As you probably
know, there are attractive realizations of GUTs involving Supersymmetry (SUSY)
and the extra puzzle pieces it necessitates, but here I wanted to show that
just the data and SM extrapolation already suggests something interesting is
going on orders of magnitude above collider energies.

From the gargantuan size of our universe to the Planck scale, and everything
in between, fundamental physics seems remarkably hierarchical. What powerful
and economical mechanisms might underlie such hierarchical structure? One goal
of these lectures is to show how supersymmetric and higher-dimensional
dynamics can robustly and elegantly generate the observed hierarchical
structure in particle physics, at least in its non-gravitational aspects.

###  1.4 Exponential Hierarchy from Non-Perturbative Physics

Rather than giving some sort of formal definition of what it means to
successfully “explain” hierarchical structure, I want to just remind you of a
beautiful mechanism that captures its spirit, namely Dimensional
Transmutation. Our goal will then be to generalize this in some way that
applies it to the broader set of particle physics hierarchies. Let us
specialize to the case of QCD, for simplicity approximating the light “up” and
“down” quarks as massless, and neglecting the other quark flavors altogether.
We can imagine this arising as an EFT just below the Planck Scale and ask how
robust it is that the observed proton mass is so many orders of magnitude
lighter.

Massless QCD is parametrized by $\alpha_{QCD}(\mu)$ where $\mu$ is the RG
scale. The physical proton mass must be an RG-invariant function of
$\alpha_{QCD}(\mu)$ and $\mu$, which uniquely determines its form,

$\displaystyle
m_{proton}=\mu\,e^{-\int_{x_{0}}^{\alpha(\mu)}dx\,\,\left(\nicefrac{{1}}{{\beta(x)}}\right)}.$
(2)

The RG flow is given by $\frac{d\alpha}{d\ln\mu}\equiv\beta(\alpha)$111It is
straightforward to check that Eq. (2) is indeed RG-invariant by taking
$d/d\ln\mu$ of it.. The parameter $x_{0}$ is just an order one integration
constant of the RG solution. From the perspective of the Planck scale,

$\displaystyle
m_{proton}=M_{Pl}\,e^{-\int_{x_{0}}^{\alpha(M_{Pl})}dx\,\,\left(\nicefrac{{1}}{{-bx^{2}+\cdots}}\right)}\sim{\cal
O}(M_{Pl})\,\,e^{\nicefrac{{-1}}{{b\alpha(M_{Pl})}}}\,,$ (3)

where we have chosen $\mu\sim M_{Pl}$ and used asymptotic freedom of QCD at
such high scales to $1$-loop approximate $\beta(\alpha)\approx-b\alpha^{2}$.
Here $b=29/6\pi$ is the gauge-algebraically determined $1$-loop coeffcient for
$2$-flavor QCD.

We see that $m_{proton}\approx$ GeV, $m_{proton}/M_{Pl}\sim 10^{-18}$ emerges
if $\alpha_{QCD}(M_{Pl})\sim{\rm few}\times 10^{-2}$. If we consider any value
of $\alpha_{QCD}(M_{Pl})$ between say $0$ and $\sim 1$ to have been equally
likely to have emerged upon matching to some unspecified Planckia quantum
gravity a priori, then we only need to be “lucky” enough that $\alpha_{QCD}$
is modestly small in order to understand why the proton is orders of magnitude
lighter than $M_{Pl}$.222We could more generally consider any other smooth
likelihood for $\alpha_{QCD}$ in this range. Note that this powerful mechanism
is fundamentally non-perturbative in nature, $e^{-{\rm constant}/\alpha}$ is
the classic function whose perturbative Taylor expansion in $\alpha^{n}$
vanishes to all orders.

The inspiration and goal that follows from this example is to discover BSM QFT
mechanisms so that modest hierarchies ${\cal O}(10^{-1}-10^{-2})$ in far-UV
couplings and mass-parameters/$M_{Pl}$ unfold in the IR in some roughly
analogous manner, being exponentially stretched out to the very hierarchical
structure we observe or anticipate in particle physics. I hope you agree that
this should be one of our ends for particle theory. Let us proceed to uncover
the means to this end.

##  2 Spinning Tales

Thus far, particle experiments operate at $E\ll M_{Pl}$ and therefore can only
detect particles with $m\ll M_{Pl}$. That is, from a Planckian view of
particle physics the particles we have seen crudely satisfy $m\approx 0$.
Symmetries (and approximate symmetries) provide plausible, economic mechanisms
for understanding the robustness of $m=0$ (and $m\approx 0$). These
“protective” symmetries vary according to the spins of the (nearly) massless
particles under consideration, and together provide a powerful grammar
underlying the story of particle physics, what we have seen and have not seen,
and what we can hope to see.

###  2.1 Spin-$0$

The classic mechanism behind having massless spin-$0$ particles is their
realization as Nambu-Goldstone (NG) bosons of the spontaneous symmetry
breaking (SSB) of an internal global symmetry. The symmetry’s Noether current
can be approximated in terms of the NG field $\phi$:

$J_{\mu}\sim\partial_{\mu}\phi+\text{non-linear~{}in~{}fields}.$ (4)

Conservation of this current,

$0=\partial^{\mu}J_{\mu}\sim\Box\phi+\text{non-linear~{}in~{}fields},$ (5)

then implies $\phi$ is massless. If there is a small explicit symmetry
breaking as well, current conservation is imperfect and $m_{\phi}$ is small
but non-vanishing, making $\phi$ a “pseudo-NG boson” (PNGB).

While (P)NGBs can readily be kinematically light enough to discover, there is
a downside. Under the associated internal symmetry transformation,
$\phi\rightarrow\phi+{\rm constant}+...$, so that symmetric couplings are
necessarily derivatively-coupled, that is constructed from the invariant
$\sim\partial_{\mu}\phi$. This implies that $\phi$ couplings rapidly drop (at
least linearly) as we move into the IR, making $\phi$ difficult to detect even
if it is light enough to be produced kinematically. We can compare this with
the massless photon, whose coupling $\alpha_{em}$ also runs to weak coupling
in the IR, but only logarithmically.

It is therefore not surprising that we easily detect photons, but are still
struggling experimentally to discover very light axions, even if these
generically motivated $U(1)$ PNGBs exist. We have done better with composite
PNGBs, such as QCD pions. Their couplings also get weaker in the IR but only
starting from their compositeness or strong-coupling scale $\sim$ GeV down to
their mass $\sim{\cal O}(100)$ MeV. The spin-$0$ Higgs boson is an enigmatic
case. Certainly $m_{higgs}=125{\rm GeV}\ll M_{Pl}$ but in the SM theory there
is no protective mechanism for why “$m_{higgs}\approx 0$”. But in the BSM
Compositeness paradigm, the Higgs boson may in fact be a kind of PNGB
composite of some new strong force [6], analogous to the pion, although we
have still not detected any corroborating compositeness effects.

We have not seen any spin-$0$ particle with sizeable couplings which we can
confidently say is an elementary particle, the jury still being out on the
Higgs boson. We have seen lots of interacting spin-$1/2$ and spin-$1$
elementary particles, but nature does seem to be very stingy with interacting
elementary spin-$0$. If the only protective mechanism for light spin-$0$ is
based on PNGBs, then we can roughly understand why this is. Needless to say,
it would be very interesting to uncover any other protective mechanism for
light spin-$0$ particles which are not necessarily derivatively-coupled and
therefore not very weakly coupled in the IR. And we will.

###  2.2 Spin-$1/2$

For spin-$1/2$ the protective symmetry of $m=0$ is famously chiral symmetry,
either gauged or global. For example, for a (single) standard Dirac fermion,
$m\bar{\psi}\psi=m\bar{\psi}_{L}\psi_{R}+m\bar{\psi}_{R}\psi_{L}$ is forbidden
if we require chiral invariance under $\psi_{L}\rightarrow
e^{i\theta_{L}}\psi_{L},~{}\psi_{R}\rightarrow e^{i\theta_{R}}\psi_{R}$, where
$\theta_{L}\neq\theta_{R}$. The same is true of any other Lorentz-invariant
(Majorana) mass terms.

If chiral symmetry is broken at low scales or by small couplings, it explains
$m\approx 0$ on the big stage of particle physics. Indeed this is just what
happens in the SM, where the chiral electroweak gauge symmetries are Higgsed
at the weak scale $\ll M_{Pl}$ and communicated to spin-$1/2$ fermions by
(mostly small) Yukawa couplings.

###  2.3 Spin-$1$

We begin with the Proca equation for massive spin-$1$ fields:

$\displaystyle\partial^{\mu}F_{\mu\nu}+m^{2}A_{\nu}=0\,,$ (6)

where $F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ as in
standard Maxwell theory. It is best if you pretend you know nothing about
electromagnetism and Maxwell’s equations. Rather, you can check that the Proca
equation is effectively the unique relativistically covariant field equation
which is local (that is, a differential equation) associated to a free massive
spin-$1$ particle. It is particularly straightforward to interpret in
$4$-momentum space in the rest frame $\vec{p}=0$:

$\displaystyle(E^{2}-m^{2})A_{i}$ $\displaystyle=0$ $\displaystyle m^{2}A_{0}$
$\displaystyle=0\,.$ (7)

We see that in this rest frame it matches our non-relativistic expectation
that the three polarization form a spatial vector $\vec{A}$, whose
(anti-)particles satisfy $E=m(c^{2})$ at rest. The $A_{0}$ “polarization’
formally appearing by relativistic covariance of the equations of motion also
vanishes by these same equations.

Now we take the $m\rightarrow 0$ limit of Proca. Furthermore, if the spin-$1$
field is not free, we can add a non-trivial right-hand side made out of the
other fields that couple to it. In this way we arrive at Maxwell equations as
effectively the only option for a massless spin-$1$ field:

$\displaystyle\partial^{\mu}F_{\mu\nu}=J_{\nu}\,.$ (8)

Of course, this is not the historical empirically-based path to Maxwell, but
rather the only logical option for interacting massless spin-1 in a
relativistic theory.

The massless limit has resulted in two remarkable properties. The first
follows by noting that taking the $4$-divergence ($\partial^{\nu}$) of
Maxwell’s equations implies that $J_{\nu}$ made out of other fields is a
conserved “current”:

$\displaystyle
0=\partial^{\nu}\partial^{\mu}F_{\mu\nu}=\partial^{\nu}J_{\nu}=0\,.$ (9)

Globally, the total charge $Q=\int d^{3}xJ_{0}$ is therefore conserved in
time. The other property is that Maxwell’s equations are gauge-invariant under

$\displaystyle A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\xi\,.$ (10)

This is related to the fact that massless spin-$1$ has only two propagating
polarizations compared to the three of massive spin-$1$.

When there are multiple massless spin-$1$ fields, $A_{\mu}^{a}$, possibly
interacting among themselves so that they appear non-linearly in each other’s
currents, the general self-consistent structure is precisely non-abelian gauge
theory.333It is interesting to check that even the standard non-abelian gauge
theory field equations can be written in the “abelian-like” forms of Eq. (8)
with abelian-like field strengths and conserved currents,
$F_{\mu\nu}^{a}\equiv\partial_{\mu}W^{a}_{\nu}-\partial_{\nu}W^{a}_{\mu},~{}\partial^{\mu}J_{\mu}^{a}=0$
(that is, with just ordinary partial derivatives as opposed to covariant
derivatives). In this way, (non-abelian or abelian) gauge-invariance can be
thought of as the protective symmetry of $m=0$ for spin-$1$ particles.

The next question is how to break gauge symmetry by a “small amount” so as to
realize $m\approx 0$. This is subtle because the $m=0$ limit of spin-$1$ is
not smooth, in that it has two polarization for spin-$1$ while all non-zero
masses have three polarizations. But we know the nuanced answer, it is
precisely the Higgs mechanism which maintains the total number of physical
polarizations in the massless spin-$1$ limit by including spin-$0$ fields.
Clearly Nature gives us several examples of massless and Higgsed gauge
particles/fields.

Let us jump over spin-$3/2$ for the moment and survey even higher spins.

###  2.4 Spin-$2$

We again begin with $m\neq 0$, which for relativistic spin-$2$ satisfies the
Pauli-Fierz equation for a symmetric tensor field $h_{\mu\nu}$,

$\displaystyle\Box
h_{\mu\nu}-\partial_{\sigma}\partial_{\mu}h^{\sigma}_{\nu}-\partial_{\sigma}\partial_{\nu}h^{\sigma}_{\mu}+\partial_{\mu}\partial_{\nu}h+\eta_{\mu\nu}\left(\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}-\Box
h\right)-m^{2}(h_{\mu\nu}-h)=0$ (11)

where $h\equiv\eta^{\mu\nu}h_{\mu\nu}$. In $4$-momentum space, its equations
of motion in the rest frame $\vec{p}=0$ reduce to $h_{jj}=h_{00}=h_{0j}=0$,
while the remaining traceless symmetric spatial tensor components satisfy
$(E^{2}-m^{2})h_{ij}=0$. This again matches our non-relativistic expectation,
in this case that the five spin-$2$ polarizations form a symmetric traceless
spatial tensor with rest energy $m$.

Now we take the $m\rightarrow 0$ limit. You can check that the Pauli-Fierz
equation then reduces to the linearized Einstein Equations

$\displaystyle\Box
h_{\mu\nu}-\partial_{\sigma}\partial_{\mu}h^{\sigma}_{\nu}-\partial_{\sigma}\partial_{\nu}h^{\sigma}_{\mu}+\partial_{\mu}\partial_{\nu}h+\eta_{\mu\nu}\left(\partial_{\alpha}\partial_{\beta}h^{\alpha\beta}-\Box
h\right)\propto T_{\mu\nu}.$ (12)

These happen to be the usual Einstein Equations in terms of a dynamical
spacetime metric $g_{\mu\nu}\equiv\eta_{\mu\nu}+h_{\mu\nu}$ but keeping only
first-order terms in $h_{\mu\nu}$, that is the linearized Einstein Equations.
I have again put a non-vanishing right-hand side, $T_{\mu\nu}$, to represent
any non-linear terms in the spin-$2$ or other fields that might arise beyond
free field theory. Again, this is not the historical emperically-based path
with Newton’s Law and the Equivalence Principle as guides, but rather the only
logical option for interacting massless spin-2 in a relativistic theory.

And again, there are two remarkable properties in this limit. There is a gauge
invariance,

$h_{\mu\nu}\rightarrow
h_{\mu\nu}+\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu},$ (13)

with a $4$-vector gauge transformation $\xi_{\mu}(x)$. Secondly, by taking the
$4$-divergence $\partial_{\mu}...$ of Eq. (12), we see that $T_{\mu\nu}$ must
be a conserved tensor current, $\partial^{\mu}T_{\mu\nu}=0$, with globally
conserved charges $P_{\mu}=\int d^{3}\vec{x}~{}T_{\mu 0}(x)$. The only such
$4$-vector charge consistent with relativistic interactions is the familiar
$4$-momentum itself, one consequence of the Coleman-Mandula Theorem. Thus we
know that $T_{\mu\nu}$ can be nothing else but the energy-momentum or stress
tensor of all the fields.

Exercise: Consider the example of $2\rightarrow 2$ scattering, involving four
different species of spin-$0$ particles. Of course, $4$-momentum $P_{\mu}$ is
conserved by translation invariance, but let us ask if there could be another
$4$-vector conserved charge $Q_{\mu}=\int d^{3}\vec{x}$ (local charge
density). If so, in the far past and far future when all particles are well-
separated, total $Q_{\mu}$ would be the sum of $Q_{\mu}$ contributions of each
individual particle in isolation. Since the only $4$-vector attached to each
isolated spinless particle is its $4$-momentum, we must have
$Q_{\mu}^{particle}\propto p_{\mu}^{particle}$. But it is a priori possible
that the proportionality constant is different for each particle species. Show
that given standard $4$-momentum conservation for a generic nontrivial
scattering angle (in center-of-momentum frame) that the proportionality
constant must be universal for all the particles involved. Therefore $Q_{\mu}$
must be nothing but $P_{\mu}$ up to this overall constant.

Because the spin-$2$ particles must exchange $4$-momentum among themselves and
with other particles in an interacting theory, in order to be conserved
$T_{\mu\nu}$ must contain non-linear terms in $h_{\mu\nu}$ to represent their
energy-momentum. That is, the spin-$2$ particles must be self-coupled. This
situation is therefore analogous to the case of several self-interacting
massless spin-$1$ particles, the self-consistent form resulting in non-abelian
gauge theory with non-abelian gauge invariance, despite being expressable in
abelian-like form. For spin-$2$, we also have an apparent abelian-like gauge
transformation, but the self-coupling means that including the explicit form
of $T_{\mu\nu}$ requires finding the non-abelian extension of the gauge
invariance. This is precisely the gauge symmetry of general coordinate
invariance, and the explicit form of Eq. (12) is then the generally
coordinate-invariant non-linear Einstein Equations.

Exercise: Consider a general coordinate transformation,
$y^{\mu}=x^{\mu}-\xi^{\mu}(x)$. Given a proper distance function on spacetime
in terms of a general metric, $ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}$, re-
express it in terms of $y$, $ds^{2}=g^{\prime}_{\mu\nu}(y)dy^{\mu}dy^{\nu}$.
Writing
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},g^{\prime}_{\mu\nu}=\eta_{\mu\nu}+h^{\prime}_{\mu\nu}$,
and restricting $h,h^{\prime}$ and $\xi$ to being infinitesimally small (that
is, working strictly to first order in these) show that

$h_{\mu\nu}^{\prime}(x)=h_{\mu\nu}(x)+\partial_{\mu}\xi_{\nu}(x)+\partial_{\nu}\xi_{\mu}(x),$
(14)

the abelian-like gauge transformation we derived for spin-$2$ above. Removing
the restriction of being infinitesimal gives a fully non-abelian
generalization of the gauge symmetry, namely that of general coordinate
invariance.

The uniqueness of energy-momentum conservation as a conserved $4$-vector
charge implies there could only be one massless spin-$2$ field, and it must
incarnate as General Relativity. Indeed, Nature has given this to us.

###  2.5 Spin $>2$ (the other end of particle physics)

In brief, in analogy to spins $1$ and $2$, in order to be massless, higher
spin fields would have to couple to conserved currents of the appropriate
higher Lorentz representation. For example, a spin-$3$ field would have
require a conserved symmetric $3$-tensor current. But this would imply a
globally conserved $2$-tensor charge $Q_{\mu\nu}$. Such a conserved charge is
inconsistent with interactions, again as a consequence of the Coleman-Mandula
Theorem.

Exercise Again consider $2\rightarrow 2$ scattering, for simplicity involving
a single species of spin-$0$ particle. Suppose there were a conserved
traceless symmetric tensor conserved charge,

$Q_{\mu\nu}=\int d^{3}\vec{x}~{}(\rm local~{}charge~{}density),$ (15)

so that in the far past and far future it is the sum of individual particle
charges. Again for an isolated particle, its charge must be constructed out of
its $4$-momentum, in this case $Q_{\mu\nu}\propto
p_{\mu}p_{\nu}-\eta_{\mu\nu}p^{2}/4$. Show that conservation of total
$Q_{\mu\nu}$ is inconsistent with a generic scattering angle (in center-of-
momentum frame).

Therefore for spin $>2$ we cannot have $m=0$ or even $m\approx 0$, but
interacting spin $>2$ “particles” can exist if their masses are comparable to
the scale at which the EFT describing them breaks down in the UV,
$m\sim\Lambda_{UV}$. Operationally, such particles are “composite”, in that
the energies high enough to create them are close to the energies at which
they cease to be described by point-particle EFT. Nature gives us many
examples, such as the high-spin hadron composites of QCD, $m\gtrsim$ GeV.
String Theory as a perturbative theory of quantum gravity also contains many
higher-spin excitations “composed” of the fundamental string, with masses
$\sim m_{string}\lesssim M_{Pl}$. Their stringy structure becomes apparent
when probed in their relativistic regime, $E\sim m_{string}$.

In this sense, we see that spin-$2$ (and hence, General Relativity) is the
“end” of point-particle physics in the spin direction. Finally, we return to
the exciting case of …

###  2.6 Spin-$3/2$

For $m\neq 0$, the Rarita-Schwinger equation for a vector-spinor field
$\psi_{\mu\alpha}$, where I am choosing $4$-vector indices from the middle or
later parts of the Greek alphabet (in this case $\mu$) and spinor indices from
the early part of the Greek alphabet (in this case $\alpha$), reads

$\displaystyle\left(\epsilon^{\mu\nu\rho\kappa}(\gamma_{5}\gamma_{\nu})_{\alpha\beta}\partial_{\rho}+\frac{m}{2}\left[\gamma^{\mu},\gamma^{\kappa}\right]_{\alpha\beta}\right)\psi_{\kappa\beta}=0.$
(16)

The $\gamma$’s are the familiar Dirac matrices. As for Dirac spin-$1/2$, in
the rest-frame of $4$-momentum space this equation describes a particle
($E=m$) and its antiparticle ($E=-m$), each with effectively two independent
spinor components. Focusing on just the particle and its independent two-
component spinor, the above is essentially the unique relativistically
covariant equation which reduces in the rest frame to
$\psi_{\mu=0,\alpha}=0,~{}(\sigma_{i})_{\alpha\beta}\psi_{\mu=i,\beta}=0,(E-m)\psi_{\mu=i,\alpha}=0$,
where $\sigma_{i}$ are the Pauli matrices. Non-relativistically, the desired
spin-$3/2$ representation is contained in the product representation
$\psi_{\mu=i,\beta}$ of spin-$1$ and spin-$1/2$, while the second of these
equations projects out the unwanted spin-$1/2$ representation in this product.

Taking the $m\rightarrow 0$ limit of the Rarita-Schwinger equation and
introducing a non-zero right-hand side to represent non-
linearities/interactions as previously,

$\displaystyle\epsilon^{\mu\nu\rho\kappa}\sigma_{\nu}^{\alpha\beta}\partial_{\rho}\psi_{\kappa\beta}={\cal
J}^{\mu\alpha}(x)\,.$ (17)

Here, again in analogy to the Dirac equation, we have taken advantage of the
fact that in the massless limit the four-component spinor equations split into
decoupled equations for two-component or Weyl spinors, of which we keep just
the left-handed one. In this left-handed two-component spinor space, the
$\sigma_{\nu}$ are again the Pauli matrices $\sigma_{i}$ along with
$\sigma_{0}$ being the identity.

One might wonder whether massless spin-$3/2$ is more like massless spin-$1$
which allows an arbitrary number of distinct but interacting fields with that
spin, or whether it is like massless spin-$2$ which allows only one field with
that spin, namely the graviton field of GR. It turns out that the answer is
intermediate, in that the maximum number of species of distinct $\psi$ fields
is $8$, but we will not do the detectivework here to show this. Instead, we
will just consider a single species of $\psi_{\mu\alpha}$ as above, this being
the phenomenologically most interesting case.

Once again we get a new gauge symmetry, under
$\psi_{\mu\alpha}\rightarrow\psi_{\mu\alpha}+\partial_{\mu}\xi_{\alpha}$,
where the gauge transformation $\xi_{\alpha}(x)$ is a spinor field. And by
taking the $4$-divergence of Eq. (17) we see that ${\cal J}_{\mu\alpha}$ must
be a conserved vector-spinor current, $\partial^{\mu}{\cal J}_{\mu\alpha}=0$.
This implies a globally conserved spinor charge, $Q_{\alpha}\equiv\int
d^{3}\vec{x}{\cal J}_{\mu=0,\alpha}$. Since $2$-component spinors are
necessarily complex, so are $\psi,{\cal J}$, and therefore the $Q_{\alpha}$
have distinct conjugate conserved charges $\bar{Q}_{\alpha}$.

What are these unfamiliar spinor charges, “halfway” between standard scalar
internal charges and the spacetime energy-momentum charges?

##  3 The Supersymmetry Charge Algebra

Before trying to answer this, let us see how we can relate charges in the most
familiar case of some scalar conserved charges of internal symmetries,
$Q_{a}=\int d^{3}{\vec{x}}J_{0}^{a}$. Clearly if $Q_{a}$ and $Q_{b}$ are
conserved in time, then so is their product $Q_{a}Q_{b}=\int
d^{3}{\vec{x}}J_{0}^{a}(x)\int d^{3}{\vec{x}}^{\prime}J_{0}^{b}(x^{\prime})$.
But the great practical utility of charge conservation comes from the charge
being the sum of local contributions, whereas this product charge is clearly
not, with the ${\vec{x}}$ and ${\vec{x}}^{\prime}$ contributions being
arbitrarily far apart. Instead, let us consider the commutator,
$[Q_{a},Q_{b}]$. This is indeed a new charge with local contributions, if it
is non-vanishing. To see this, first note that $J_{0}^{a}(x)$ will be made of
some products of boson fields (or field momenta) and bilinears in fermion
fields at $x$, and their derivatives. (Since fermions are half-odd-integer
spinors by the spin-statistics connection they must appear in even numbers in
any $4$-vector $J_{\mu}$.) Let us pretend for just a moment that in the QFT
all bosonic fields and their derivatives commute, and all fermions and their
derivatives anticommute, and all fermions commute with bosons. (Only the last
of these statements is true, but let us just pretend.) Therefore, given this
fiction, all bosons and fermion bilinears and their derivatives would commute,
in which case clearly $[Q_{a},Q_{b}]=0$. Therefore in truth, the only way to
get a non-zero commutator is because of retaining at least one non-trivial
commutator between boson fields (including bosonic field momenta or time-
derivatives of boson fields) or fermion bilinears at $x$ and bosons or fermion
bilinears at $x^{\prime}$. Such commutators always contain
$\delta^{3}({\vec{x}}-{\vec{x}}^{\prime})$ factors, so $[Q_{a},Q_{b}]$ must be
another charge with local contributions! In this way, the “useful” charges
form a standard Lie algebra.

We can apply this kind of detective work to the more enigmatic spinor charges
$Q_{\alpha},\bar{Q}_{\beta}$. To be precise, let us remind ourselves of the
$2$-component spinor representation of the Lorentz group and the relevant
notation. It is based on the relativistic analog of the familiar isomorphism
of the rotation group, $SO(3)\equiv SU(2)$444We will not be careful about the
fact that globally $SU(2)$ is the double-cover of $SO(3)$, it is isomorphic in
the neighbourhood of the identity element., namely $SO(3,1)\equiv SL(2,C)$.
That is, the Lorentz group can be represented by complex $2\times 2$ matrices
$\Lambda$ with unit determinant. We can define a left-handed representation
$\psi_{L}$ to be a $2$-component spinor transforming as
$\psi_{L}\rightarrow\Lambda\psi_{L}$, where matrix-multiplication is implied
on the right side.

We will use the “bar” notation to be the same as conjugation
$\bar{\psi}_{\alpha}\equiv\psi_{\alpha}^{\dagger}$, and without explicitly
writing indices it will represent a row vector
$\bar{\psi}\equiv\psi^{\dagger}$ obtained from the hermitian conjugate of the
column vector $\psi$. Therefore under Lorentz transformation,
$\bar{\psi}\rightarrow\bar{\psi}\Lambda^{\dagger}$ with matrix multiplication
implied. One can then form Lorentz $4$-vectors from the spinor products
$\bar{\psi}\sigma^{\mu}\chi$, where again $\sigma^{0}$ is the identity matrix
and $\vec{\sigma}$ are Pauli matrices. Together the $\sigma^{\mu}$ are a
Hermitian basis for $2\times 2$ complex matrices.

As with internal charges we can start with the product charges
$Q_{\alpha}\bar{Q}_{\beta}$. Once again, they are conserved but not in a
useful way since they are not the sum of local contributions. But now we can
focus on the anticommutator $\\{Q_{\alpha},\bar{Q}_{\beta}\\}$, which is a sum
of local contributions. The reason that it is the anticommutator which has
this property is because any local spinor-vector current such as ${\cal
J}_{\mu\alpha}(x)$ must necessarily be the product of some bosons and an odd
number of fermionic fields and derivatives at $x$, by the spin-statistics
connection. By the analogous argument to the case of scalar charges, it is
$\\{Q_{\alpha},\bar{Q}_{\beta}\\}$ which necessarily contains a factor of
$\delta^{3}({\vec{x}}-{\vec{x}}^{\prime})$.

What is interesting is that by the spinor algebra reviewed above, and the fact
that the $\sigma^{\mu}$ are a hermitian basis for all $2\times 2$ matrices, it
follows that $\\{Q_{\alpha},\bar{Q}_{\beta}\\}$ necessarily transforms as a
Lorentz $4$-vector of conserved charges, and the only such conserved
$4$-vector that is the sum of local contributions allowed in local QFT is
$4$-momentum:

$\displaystyle\\{Q_{\alpha},\overline{Q}_{\beta}\\}=2P_{\mu}\sigma^{\mu}_{\alpha\beta}.$
(18)

The factor $2$ is chosen by convention in the normalization of $Q_{\alpha}$.

By analogous detective work, the anticommutators $\\{Q_{\alpha},Q_{\beta}\\}$
and $\\{\bar{Q}_{\alpha},\bar{Q}_{\beta}\\}$ can also be conserved charges
with local contributions. But these transform as Lorentz anti-symmetric
tensors, and no such conserved tensor charges are possible in interacting
QFT.555The famous antisymmetric tensor charges of QFT are the angular momentum
tensor $J_{\mu\nu}$, but they are not all conserved in the usual sense of
commuting with the Hamiltonian $P_{0}$, in particular the boost generators
$J_{0i}$ do not. Therefore, we can only have

$\displaystyle\\{Q_{\alpha},Q_{\beta}\\}=0,~{}\\{\bar{Q}_{\alpha},\bar{Q}_{\beta}\\}=0.$
(19)

The fact that the spacetime $P_{\mu}$ charges appear in our anticommutation
relations, mean that the new symmetry charges correspond to an extension of
spacetime symmetry beyond the usual algebra of Poincare generators
$P_{\mu},J_{\mu\nu}$ and their commutation relations. In addition to all of
these, there are also commutation relations between the
$Q_{\alpha},\bar{Q}_{\beta}$ and the $P_{\mu},J_{\mu\nu}$. The commutation
relations with the angular momentum generators $J_{\mu\nu}$ merely express the
spinor properties of the $Q_{\alpha},\bar{Q}_{\beta}$, so I will not bother to
write these out. The commutation relations with $P_{\mu}$ express charge
conservation,

$[Q_{\alpha},P_{\mu}]=0,~{}[\bar{Q}_{\beta},P_{\mu}]=0.$ (20)

(The $P_{0}=H$ commutator is literally conservation of charge, while the
$\vec{P}$ commutator vanishes because the charge is the total across space, so
a spatial translation does not change this total.)

If we put together the anticommutation algebra of conserved spinor charges
eqs. (18) and (19), the usual commutator algebra of Poincare generators, and
the conservation properties, Eq. (20), and (spinor) Lorentz transformation
properties of $Q_{\alpha},\bar{Q}_{\beta}$, we arrive at the minimal or ${\cal
N}=1$ superalgebra. The $Q_{\alpha},\bar{Q}_{\beta}$ are “supercharges”.666
${\cal N}$ counts the number of different “flavors” of $Q_{\alpha}$, but only
${\cal N}=1$ SUSY QFT is compatible with having chiral fermions, such as
Nature exhibits.

Does supersymmetry (SUSY) exist in the space of QFTs? That is, are there QFTs
that contain the superalgebra of charges, with charges realised as integrals
over local charge densities? In particular, is there a supersymmetric
extension of the SM? What is the structure of such theories? How can they be
realistic? Let us continue the qualitative deductive reasoning a little
further. See Ref. [7] for a canonical text on general SUSY field theory
construction, and Ref. [8] for construction and phenomenology of SUSY
extension of the Standard Model. See also Refs. [9] and [10].

##  4 Superpartners

Broadly, SUSY charges relate fermions and bosons, remarkable given their very
different behavioral properties. To see this, start with the conservation of
$Q_{\alpha}$, expressed canonically as $[H,Q_{\alpha}]=0$. Consider the energy
eigenstate of any bosonic particle, $|B\rangle$, with energy $E_{B}$. Then
$|F\rangle\equiv Q_{\alpha}|B\rangle$ must be a fermion by the spin-statistics
connection because $Q_{\alpha}$ changes the angular momentum of the state it
is acting on by a $1/2$ unit. Thus, $[H,Q_{\alpha}]|B\rangle=0$ implies that
$|F\rangle$ must also have the same energy, $E_{F}=E_{B}$. Working within each
momentum subspace, this means that their masses are also the same,
$m_{F}=m_{B}$. In this way, we see that in SUSY particles come in boson-
fermion degenerate pairs, or “superpartners”, which differ in their spin by a
half unit.

The superpartner of the massless spin-$2$ graviton $h_{\mu\nu}$ must be
therefore be a massless spin-$3/2$ particle/field $\psi_{\mu\alpha}$, which we
can call the “gravitino”. Note, the fermion had to have spin $1/2$ unit
different from $2$, but it could not have been spin-$5/2$ because massless
spins $>2$ are forbidden as discussed earlier.

The superpartner of a massless spin-$1$ gauge boson $A_{\mu}^{a}$ must be a
massless spin-$1/2$ fermion $\lambda_{\alpha}^{a}$, a “gaugino”. The ‘a’ index
is the internal (non-spacetime) index labelling gauge generators, so that the
gaugino is a fermion in the adjoint representation of the gauge group (the
representation furnished by the gauge group generators themselves). Again, one
might have thought the fermion could have been massless spin-$3/2$, but that
would make it a second spin-$3/2$ field, the first being the gravitino. But
this would correspondingly require two species of supercharges $Q_{\alpha}$,
incompatible with the minimal ${\cal N}=1$ SUSY we are studying.

Turning to standard (non-gaugino) spin-$1/2$ fermions, it will be convenient
from here on to exclusively use left-handed representation. We can convert
conventional right-handed fields into left-handed fields by charge
conjugation. That is, given a right-handed $2$-component spinor field
$\psi_{R}$, its charge conjugate $(\psi_{R})^{c}\equiv\sigma_{2}\psi_{R}^{*}$
is a left-handed spinor field.777 Often in the literature, one distinguishes
“dotted” and “undotted” indices, where the dotted indices label the conjugates
of the left-handed spinors. I will reduce “clutter” by not making this
distinction in these lectures, hoping that context will give away what is
intended.  Here $\sigma_{2}$ is the second Pauli matrix, and matrix-
multiplication is implied. The Lorentz-invariant one can then construct from
two left-handed fields is
$(\psi_{L}\chi_{L})\equiv\epsilon^{\alpha\beta}\psi_{L\alpha}\chi_{L\beta}=\psi_{L\alpha}\chi_{L}^{\alpha}$.
Here, $\epsilon$ is the completely antisymmetric tensor, and one can think of
it as a “metric” on spinors allowing one to “raise” the indices of a standard
left-handed spinor. In this way the SM spin-$1/2$ chiral fermions can be
listed as

$\displaystyle\psi_{\alpha}=q_{\alpha},\ell_{\alpha},u^{c}_{\alpha},d^{c}_{\alpha},e^{c}_{\alpha}.$
(21)

The superpartners of these standard fermions must be spin-$0$ “sfermion”
scalars,

$\displaystyle\phi=\tilde{q},\tilde{\ell},\tilde{u}^{c},\tilde{d}^{c},\tilde{e}^{c},$
(22)

where the tilde just labels the superpartner of the related SM field. (They
could not be spin-$1$ because then they would have to be gauge fields in the
adjoint representation of gauge groups and the fermions would be gauginos,
which does not fit SM quantum numbers.)

The only other field in the SM is the Higgs doublet scalar field $H(x)$, and
this must have a spin-$1/2$ “Higgsino” superpartner $\tilde{H}_{\alpha}$.
There is an important subtlety regarding the SUSY Higgs content, but we will
get to that later.

If we can realize SUSY in QFT it predicts a remarkable feature. Earlier, we
could only identify Nambu-Goldstone bosons as the means for having robustly
massless spin-$0$ particles, but they have the limitation that their
(derivative) couplings rapidly become weak in the IR. In particular, it was
not clear if the Higgs boson could be considered as at least approximately a
NG boson in some BSM scenario given its substantial couplings. But in SUSY
because of superpartner mass-degeneracy, a massless scalar is a robust
possibility if it is the superpartner of a spin-$1/2$ fermion which is
robustly massless because of a chiral symmetry. If the fermion has standard
gauge and Yukawa couplings, so will the scalar, such couplings falling at most
logarithmically in the IR. SUSY then offers an attractive mechanism for
robustly having light interacting scalars in QFT, such as the Higgs boson
(seen from our Planckian perspective).

##  5 Supergravity, SUSY Breaking and the $G_{N}\rightarrow 0$ Limit

Figure 3: A plausible cartoon spectrum of a SUSY extension of the standard
model. SUSY (degeneracy) is sufficiently broken to be consistent with our not
having discovered superpartners yet, while being approximately valid far above
the weak scale.

As we have seen, the graviton $h_{\mu\nu}$ is a “gauge field” whose associated
charge is energy-momentum $P_{\mu}$, while the gravitino $\psi_{\mu\alpha}$ is
a “gauge field” with associated charges $Q_{\alpha},\bar{Q}_{\beta}$. But
these charges obey a non-abelian type of superalgebra, Eq. (18), so together
the graviton and gravitino superpartners are “gauge fields” of SUSY. This
gauging of the global SUSY superalgebra is sometimes called “local SUSY”. But
since it also gives a supersymmetric theory including gravity, it is also
called “supergravity”, or SUGRA for short. Like standard simple non-abelian
gauge theories, local SUSY has a single “gauge” coupling, $G_{\rm Newton}$.

If SUSY were an exact symmetry then there would be a charged selectron with
the electron’s mass, and this is not seen in Nature. More generally, the
absence of superpartners in experiments to date implies that SUSY must be an
approximate symmetry at best, part of the robust QFT grammar of options for
$m\approx 0$ from a UV perspective.

We can guess a cartoon of a realistic particle spectrum with broken SUSY as
given in figure 3. The superpartners are thereby heavy enough to have evaded
LHC and other searches, but close enough to the weak scale that it might be
playing a strong role in making a weak-scale interacting spin-$0$ Higgs a
robust QFT feature. While the broken SUSY is clearly a big effect for human
particle experimentalists, it represents a “small” breaking compared to the
fundamental scale $\lesssim M_{Pl}$ at which QFT is born from quantum gravity.

Since fundamentally, SUSY is a gauged symmetry of SUGRA, its breaking must be
due to a “super-Higgs” effect [11]. But in the global limit of this gauge
theory, that is when we work in the approximation of vanishing gauge coupling
$G_{N}\approx 0$, this Higgs-like breaking must become spontaneous SUSY
breaking, analogous to what happens in the standard Higgs mechanism in the
global limit.

I have been trying to make the case to you that Nature may well be playing
every trick in the book as far as $m\approx 0$ is concerned. But does it
contain $m\approx 0$ for spin-$3/2$? The reason we do not know yet may not be
that such a particle lies above our puny energy reach (although maybe it does)
but because, like PNGBs, it has a reason to be extremely weakly coupled. And
unlike the graviton, which shares its $G_{N}$ coupling, it cannot take
advantage of Bose statistics to at least appear to us in observable classical
fields. Nevertheless, its existence necessitates all of SUSY structure, and
clearly searching for that experimentally is strongly motivated.

How can we construct SUSY QFT and spontaneous SUSY breaking? To motivate the
strategy, I want to digress into another important ingredient which is
somewhat more intuitive (at least for theorists), namely higher-dimensional
spacetime.

##  6 Higher Powers and Hierarchies from Higher Dimensions

Let us consider the simplest higher-dimensional extension of 4D Minkowski
spacetime to 5D Minkowski spacetime:

$\displaystyle 4D\,\,x^{\mu}\quad$ $\displaystyle\longrightarrow\quad
5D\,\,X^{M}=x^{\mu},x_{5}$ $\displaystyle 4D\,\,\text{Poincare symmetry}\quad$
$\displaystyle\longrightarrow\quad 5D\,\,\text{Poincare symmetry}$
$\displaystyle x^{\mu}\rightarrow\Lambda^{\mu}_{\nu}x^{\nu}+a^{\mu}\quad$
$\displaystyle\longrightarrow\quad X^{M}\rightarrow\Lambda^{M}_{N}X^{N}+A^{M}$
$\displaystyle\,\,\Lambda\in SO(3,1)\qquad$
$\displaystyle\qquad\quad\Lambda\in SO(4,1)\supset SO(3,1)$ $\displaystyle
4D\text{ Fields}\,\,\phi(x)\quad$ $\displaystyle\longrightarrow\quad 5D\text{
Fields}\,\,\phi(X)$ (23)

As presented, we posited a higher-dimensional spacetime and then noted that it
could enjoy a larger 5D Poincare symmetry. But to pave the way for our later
SUSY development, it is useful to think of it the other way around: if I know
I have a QFT with an extension of the usual 4D Poincare spacetime symmetry,
the simplest way of incorporating that is by extending the spacetime so that
it is symmetric under the extended symmetry. In this way, demanding 5D
Poincare symmetry leads to fields living on 5D Minkowski spacetime.

###  6.1 Compactification of the Extra Dimension

Figure 4: A tiny extra “5th” dimensional interval of size $L$, deep within
which 5D Poincare invariance is approximately valid. At much larger
wavelengths, the extra dimension is invisible, 5D Poincare invariance is badly
broken, and only 4D Poincare invariance survives.

The Poincare-invariant field equation, say Klein-Gordon, also extends
straightforwardly:

$\displaystyle(\partial_{\mu}\partial^{\mu}+m_{4}^{2})\phi(x)=0\longrightarrow(\partial_{\mu}\partial^{\mu}-\partial_{5}^{2}+m_{5}^{2})\phi(X)=0.$
(24)

Now, of course we do not live in a perfect 5D Minkowski spacetime with perfect
5D Poincare symmetry, so at best such a symmetry is badly broken at the “low”
energies we currently probe. This allows me to introduce the notion of soft
symmetry breaking, that is breaking by a dimensionful energy/mass scale such
that at much higher energies (short distances) there is a very good
approximate symmetry and at lower energies (longer distances) the symmetry is
not apparent. In the case of 5D, the simplest such soft breaking is depicted
in figure 4. The 5th dimension is a finite interval of microscopic length $L$,
so that spacetime is a 5D slab-like “bulk”, sandwiched between two 4D
boundaries or “branes”. For short distance scattering and short wavelengths
$\ll L$ in the bulk interior, as illustrated, 5D Poincare symmetry
approximately holds. For long wavelengths $\gg L$ only the 4D Poincare sub-
symmetry is apparent. As depicted in figure 4, at these long wavelengths the
extra dimension itself cannot be resolved. We can say that the 5D Poincare
symmetry is softly broken at the “Kaluza-Klein” (KK) scale $\mu_{KK}=1/L$ down
to 4D Poincare symmetry.

Let us solve the 5D field equation on the right side of Eq. (24) by separation
of variables:

$\displaystyle\phi(X)=f(x_{5})\phi_{4}(x),$ (25)

where the $\phi_{4}(x)$ factor satisfies the 4D Klein-Gordon equation

$\displaystyle(\partial_{\mu}\partial^{\mu}+m_{4}^{2})\phi_{4}(x)=0.$ (26)

Furthermore, let us assume that the fifth dimension is hidden far in the UV
and that any fields the effective 4D experimentalist can probe have
$m_{4}\approx 0$. Then clearly the general 5D solution is given by

$\displaystyle\phi(X)$
$\displaystyle=\left(Ae^{m_{5}x_{5}}+Be^{-m_{5}x_{5}}\right)\phi_{4}(x)$
$\displaystyle\underset{\text{typical
b.c}}{\sim}e^{-m_{5}x_{5}}\phi_{4}(x)\quad\text{or}\quad
e^{+m_{5}(x_{5}-L)}\phi_{4}(x).$ (27)

The integration constants $A,B$ have to be determined by the boundary
conditions at $x_{5}=0,L$. We will not detail those, but it is easy to see
that for $m_{5}>L$ (the exponentials are varying signficantly across the fifth
dimension) typical boundary conditions will result in the solution leaning
strongly towards one boundary or the other, as indicated in the second line.
These $m_{4}\approx 0$ solutions, which determine (with boundary conditions)
the 5D solutions, including 5D profiles, are called “zero modes” (or near-zero
modes more accurately). In essence, at experimental “low” energies, the theory
reduces to a 4D EFT of these zero-modes.

See [12] for reviews of extra-dimensional field theory and phenomenology.

###  6.2 Emergence of (Yukawa) Coupling Hierarchies

Figure 5: Some typical “zero”-modes for three 5D fields, exhibiting
exponential profiles depending on their 5D masses.

Now consider three species of 5D fields, $\chi(X),\psi(X),\phi(X)$, where the
first two have zero-modes leaning towards $x_{5}=0$ and the last towards
$x_{5}=L$, as depicted in figure 5. We will consider a simple trilinear
coupling between them,

$\displaystyle S_{5D}$ $\displaystyle\supset\lambda_{5}\int
d^{4}x\int_{0}^{L}dx_{5}\,\,\psi(X)\chi(X)\phi(X)$
$\displaystyle\underset{\text{zero-modes}}{=}\lambda_{5}\int
d^{4}x\int_{0}^{L}dx_{5}\,\,e^{-m_{5\chi}x_{5}}e^{-m_{5\psi}x_{5}}e^{m_{5\phi}(x_{5}-L)}\chi_{4}(x)\psi_{4}(x)\phi_{4}(x)$
$\displaystyle\underset{m_{5\phi}>m_{5\psi}+m_{5\chi}}{\sim}\lambda_{5}e^{-(m_{5\chi}+m_{5\psi})L}\int
d^{4}x\,\,\chi_{4}(x)\psi_{4}(x)\phi_{4}(x).$ (28)

At low energies, this reduces to plugging in the zero-modes to get the second
line. Note that now the $x_{5}$ integral can explicitly be done, resulting in
an effective coupling in the 4D EFT of the zero-modes. Our philosphy will be
that the fundamental 5D theory has only modest hierarchies, that is that all
mass parameters and couplings are $\sim{\cal O}(1/10)-{\cal O}(10)$ in units
of the KK scale $1/L$ say. But at low energies ($m_{4}\approx 0$) the 4D EFT
can readily generate exponential hierarchies. I illustrate this on the third
line with the robust possibility that $m_{5\phi}>m_{5\psi}+m_{5\chi}$. We see
that in this case, the effective 4D trilinear coupling can be exponentially
small given that the $m_{5}L$ are just modestly large,

$\displaystyle\lambda_{4,\text{eff}}\sim\lambda_{5}e^{-(m_{5\chi}+m_{5\psi})L}.$
(29)

Let us apply this result to a toy model of SM quark Yukawa couplings, where
ignoring spin and chirality details, $\chi$ represents the $i$th generation of
a quark electroweak doublet, $\psi$ represents the $j$th generation of a quark
electroweak singlet, and $\phi$ represents the Higgs doublet field. These SM
fields start as fundamentally 5D fields, but it is only their 4D zero-modes
that have been discovered. Then the trilinear coupling $\lambda$ we just
studied represents the Yukawa coupling $Y_{ij}$. We therefore see that
starting from a fundamental 5D Yukawa matrix of couplings $Y_{5ij}$ which are
more or less randomly distributed without large hierarchies, one predicts an
exponentially hierarchical effective 4D Yukawa matrix,

$\displaystyle Y_{4,\text{eff}}^{ij}\sim Y_{5}^{ij}e^{-(m_{5i}+m_{5j})L}.$
(30)

This is a very interesting structure qualitatively. If the 5D masses are not
particularly degenerate, one can approximately diagonalize this Yukawa matrix
to give quark mass eigenvalues and CKM mixing angles

$\displaystyle m_{4,i}$ $\displaystyle\sim e^{-2m_{5i}L}v_{\rm electroweak}$
$\displaystyle V_{ij}^{\text{CKM}}$
$\displaystyle\underset{m_{5i}>m_{5j}}{\sim}\frac{e^{-m_{5i}L}}{e^{-m_{5j}L}}\sim\sqrt{\frac{m_{4i}}{m_{4j}}}.$
(31)

This is not a bad qualitative fit to the hierarchical quark mass and CKM
structure and the correlations between them that we observe!

To conclude, we have just found an attractive mechanism for understanding the
hierarchical form of the Yukawa structure that would otherwise remain
mysterious in the SM. See [3] for a rapid review of obtaining realistic flavor
hierarchies from extra-dimensional wavefunction overlaps, plus original
references.

###  6.3 Boundary-localized fields and Sequestering

Figure 6: Two effectively boundary-localized fields, $\psi_{4}$ and
$\phi_{4}$, interacting via exchange of a bulk $\chi$ field. This exchange is
Yukawa-suppressed if the bulk mass is large $m_{5}L\gg 1$.

There are a couple of more concepts to introduce in 5D, which will be
important later. Let us return to $\chi,\psi,\phi$, keeping them general
(without identifying them with quarks and Higgs fields). The first concept is
the approximation of boundary-localization (or “brane-localization”), which
applies when $m_{5}L\gg 1$. In this limit clearly the zero-modes are closely
stuck close to one or other of the boundaries, and one can approximate them as
propagating exclusively in 4D, either restricted to $x_{5}=0$ or $x_{5}=L$.
Given the exponential profiles of zero modes, this approximation kicks in
quickly, so $m_{5}L$ need not be too big. Having seen how we can approach this
boundary-localization limit, we can simply impose boundary-localization as
fundamental. For example, we can take $\psi_{4}$ and $\phi_{4}$ to be
perfectly localized at $x_{5}=0$ and $x_{5}=L$ respectively. Consequently,
they can each self-interact with themselves without suppression, but they
cannot directly interact with each other by locality, since they live on the
two separated boundaries. But we first take $m_{5\chi}L>1$ to not be too
large. This allows $\psi$ and $\phi$ to interact by exchange of $\chi$ if they
have suitable trilinear couplings,

$\displaystyle\lambda^{\prime}_{5}\int
d^{4}x\int_{0}^{L}dx_{5}\,\psi^{2}(X)\chi(X)$
$\displaystyle\sim\lambda^{\prime}_{5}\int
d^{4}x\,\psi^{2}_{4}(x)\,\chi(x,x_{5}=0)$
$\displaystyle\lambda^{\prime\prime}_{5}\int
d^{4}x\int_{0}^{L}dx_{5}\,\phi^{2}(X)\chi(X)$
$\displaystyle\sim\lambda^{\prime\prime}_{5}\int
d^{4}x\,\phi^{2}_{4}(x)\,\chi(x,x_{5}=L).$ (32)

Therefore the $\psi-\phi$ interaction is given in terms of the 5D $\chi$
exchange,

$\propto\langle
0|T\\{\chi(x,x_{5}=0)~{}\chi(x^{\prime},x_{5}^{\prime}=L)\\}|0\rangle\propto
e^{-m_{5\chi}L}.$ (33)

Without needing the detailed form of the 5D propagator, this exponential is
just the Yukawa-suppression one expects whenever a massive mediator field has
to virtually traverse a distance $L$ beyond its Compton wavelength at low
energies ($<m_{5\chi}$). The full exchange is depicted in figure 6. Obviously,
for $m_{5\chi}L\gg 1$, the $\psi-\phi$ interactions effectively shut off.
Later in the SUSY context, we will use this natural mechanism for suppressing
interactions between sets of boundary-localized 4D fields. It is known as
“sequestering” [13].

###  6.4 Non-renormalizability of Higher-dimensional EFT

Higher-dimensional field theories are non-renormalizable.888Scalar field
theory with only trilinear couplings is renormalizable but such a cubic
potential is unbounded from below and therefore unphysical. Consider Yang-
Mills theory, which in 4D is the classic renormalizable QFT. In 5D,

$\displaystyle S_{5D}=-\frac{1}{4g_{5}^{2}}\int
d^{4}x\int_{0}^{L}dx_{5}G_{MN}^{a}G_{a}^{MN},$ (34)

where we have chosen for later convenience the non-canonical normalization of
the gauge fields $A^{a}$ such that the coupling appears out the front of the
action rather than in the interaction terms,

$\displaystyle D_{M}$ $\displaystyle=\partial_{M}+it^{a}A_{M}({\rm x})$
$\displaystyle G_{MN}^{a}$
$\displaystyle=\partial_{M}A_{N}^{a}-\partial_{N}A_{M}^{a}+f^{abc}A_{M}^{b}A_{N}^{c}.$
(35)

(The field redefinition $A_{M}\rightarrow g_{5}A_{M}$ restores canonical
normalization.) We see that the 5D gauge coupling $g_{5}$ has mass dimension
$-1/2$ and therefore the theory is non-renormalizable in the 5D regime.

But this behavior changes below $\mu_{KK}$ for the 4D EFT. Since $m_{5}=0$ by
5D gauge invariance, the zero-modes have a $x_{5}$-independent profile, so
that doing the $x_{5}$ integral for these modes yields

$\displaystyle S_{4,\text{eff}}=-\frac{L}{4g_{5}^{2}}\int
d^{4}x\,G_{\mu\nu}^{a}G^{\mu\nu}_{a}+A_{5}\text{-terms}\,.$ (36)

We see that the effective 4D gauge coupling is now dimensionless and given by

$\displaystyle\frac{1}{g_{4,\text{eff}}^{2}}=\frac{L}{g_{5}^{2}}\,.$ (37)

This 4D Yang-Mills is asymptotically free, $g_{4,\text{eff}}$ running to
become stronger in the IR.

Non-renormalizability of 5D field theories is not a disaster, it however does
mean that they have to be treated by the methods of non-renormalizable EFT
[14], with a finite energy range of validity above which some more UV-complete
description must take over.

###  6.5 The Ultimate $m\rightarrow 0$ Limit

SUSY has been motivated from the top down, as a possible remnant of a
superstring realization of quantum gravity, as well as from the bottom up if
Nature gives us every spin of elementary particle possible with $m\approx 0$,
including spin $3/2$. But so far, extra spacetime dimensions have only been
motivated from the top down. Yet they too have a robust bottom-up motivation.
I have surveyed the different possible particle spins one at a time in terms
of what mechanisms and symmetries robustly yield $m=0$, and then the sense in
which these structures might only be approximate, resulting in $m\approx 0$. I
could do this at weak coupling because then each particle approximately
corresponds to a free field. But what about strong coupling and $m\rightarrow
0$? We have already seen that non-perturbatively, there can be emergent mass
scales via dimensional transmutation. If we really wish to go all the way to a
completely massless theory non-perturbatively, both explicit mass scales and
emergent mass scales must be absent. In particular, this requires a theory
where even the dimensionless couplings do not run, $\beta(\alpha)=0$. Lacking
characteristic mass scales such a theory also lacks characteristic length
scales, and therefore has scale-invariance as a new symmetry. There is a
strong conjecture that scale-symmetry combined with Poincare symmetry and the
locality of QFT “accidentally” implies an even larger symmetry, namely
conformal symmetry. See for example [15]. A 4D QFT with conformal symmetry is
called a conformal field theory (CFT).

Remarkably, the famous AdS5/CFT4 correspondence re-expresses 4D CFT
“holographically” as an equivalent (or “dual”) 5D quantum gravity theory [16]!
The 5D theory realizes the conformal symmetry as the symmetry (isometry) of
the background curved spacetime geometry, 5D Anti-de Sitter (AdS):

$\displaystyle
ds^{2}_{\text{AdS}_{5}}=R^{2}\,\,\frac{\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dx_{5}^{2}}{x_{5}^{2}}\,,\quad
x_{5}>0$ $\displaystyle\text{where }R=\text{constant AdS radius of
curvature}\,.$ (38)

This is analogous to how the 5D Poincare group is the symmetry of 5D Minkowski
spacetime geometry. Of course, such a CFT can only be a subsector of the real
world, since we know we have 4D GR with its characteristic mass scale
$M_{Pl}$, and the SM at least. If one includes these elements of realism the
holographically dual description is given by the Randall-Sundrum II (RS2)
scenario [17].

It is also possible not to have an exact CFT (exactly massless strongly
interacting QFT in the IR) but one in which conformal symmetry is approximate
and broken spontaneously or softly in the IR ($\ll M_{Pl}$) as well as in the
UV ($\sim M_{Pl}$). These have holographic duals called Randall-Sundrum I
(RS1) models or “warped extra dimensions” [18]. They are similar to the extra-
dimensional framework we have sketched in the earlier subsections, the major
difference being that the bulk 5D spacetime is highly curved in a manner that
however cannot be detected if one does not have the energy to resolve the
extra dimension. This is the meaning of “warped” in this context. Much of the
modeling of the Composite Higgs paradigm currently takes place in this 5D EFT
framework [12]. The warped variant of generating Yukawa hierarchies from
extra-dimensional wavefunction overlaps is CFT/AdS dual to the mechanism of
“Partial Compositeness” in purely 4D [19] [20], where the Yukawa hierarchies
are created by strong-coupling non-perturbative RG effects, closely related to
the physics of dimensional transmutation we have discussed.

The advantage of the 5D dual RS-level descriptions is that one does not need
to completely specify the 4D CFT in detail, which would be equivalent to
specifying the 5D quantum gravity in detail. Rather, one can describe the 5D
theory with non-renormalizable EFT. It is easier (but still non-trivial) to
check self-consistency of a 5D EFT than to check that one has a viable and UV-
complete 5D quantum gravity! In this way, one can explore interesting physics
that strongly coupled 4D theories might produce. For example, the possibility
of traversable wormholes is explored within an RS2-like framework in [21]. The
Composite Higgs scenario is conveniently explored within the RS1 framework.

##  7 Superspace and Superfields

With extended spacetime now being motivated as a convenient and simple way of
representing theories with extended spacetime symmetries, we return to the
extended spacetime symmetry of ${\cal N}=1$ SUSY. We will try to write SUSY
QFTs in terms of “superfields” $\Phi(X)$, fields living on an extended
spacetime, “superspace” [7, 8], which has SUSY as its geometric symmetry
algebra:

$\displaystyle X$
$\displaystyle\equiv(x^{\mu},\theta_{\alpha},\overline{\theta}_{\beta}).$ (39)

The extra coordinates allow us to represent the action of the supercharges in
the simplest possible way, as “supertranslations”, akin to the action of
translations on Minkowski coordinates:

$\displaystyle\theta_{\alpha}$
$\displaystyle\rightarrow\theta_{\alpha}+\xi_{\alpha}\,,\,\,\overline{\theta}_{\beta}\rightarrow\overline{\theta}_{\beta}+\overline{\xi}_{\beta}.$
(40)

The $\xi_{\alpha}$ are the transformation parameters we associate with the
action of $Q_{\alpha}$ (analogous to a translation vector $a^{\mu}$ associated
to the translation generator $P_{\mu}$), and therefore must be spinorial. They
are independent of $x$ because we are studying the global limit of SUSY here,
not local SUSY. Furthermore, since the $Q$ are anticommuting charges
(ulimately coming from the fact that they come from a current that couples to
the fermionic spin-$3/2$ gravitino) the $\xi_{\alpha}$ must be Grassmann
numbers. For compatibility, the $\theta_{\alpha}$ must also be spinor
Grassmann coordinates, unlike the usual c-number $x^{\mu}$ coordinates. This
is what will separate SUSY from standard extra dimensions. Matching the
conjugate relationship of $Q$ and $\bar{Q}$, $\bar{\theta}_{\alpha}$ is just
the conjugate of $\theta_{\alpha}$.

In order for superspace to exhibit the “non-abelian” aspect, Eq. (18), of
SUSY, it is crucial that the $x^{\mu}$ also transform under the supercharges:

$\displaystyle x^{\mu}$ $\displaystyle\rightarrow
x^{\mu}-i\theta_{\alpha}\overline{\xi}_{\beta}\sigma_{\alpha\beta}^{\mu}+i\xi_{\alpha}\overline{\theta}_{\beta}\sigma_{\alpha\beta}^{\mu}$
(41)

Note that the two transformation terms are hermitian conjugates of each other,
keeping $x$ “real”, or, more correctly hermitian as an operator. To check
these symmetry transformations, we first recall how ordinary (infinitesimal)
translations work on fields on ordinary spacetime,
$\phi(x)\rightarrow\phi(x+a)=\phi(x)+a^{\mu}\partial_{\mu}\phi(x)$. We thereby
identify the associated $4$-momentum charge as the translation generator
$P_{\mu}=i\partial_{\mu}$ (the “$i$” for hermiticity). Similarly, supercharges
encoding infinitesimal supertranslations on superfields on superspace are
represented as differential operators on superspace:

$\displaystyle Q_{\alpha}$
$\displaystyle=\frac{\partial}{\partial\theta_{\alpha}}+i\overline{\theta}_{\gamma}\sigma_{\alpha\gamma}^{\mu}\partial_{\mu}$
$\displaystyle\overline{Q}_{\beta}$
$\displaystyle=\frac{\partial}{\partial\overline{\theta}_{\beta}}+i\theta_{\gamma}\sigma_{\gamma\beta}^{\mu}\partial_{\mu}\,.$
(42)

The first of these encodes the transformation of superspace by $\xi$, and the
second by $\bar{\xi}$. In each case, the first term on the right encodes
infinitesimal translation of the $\theta$ coordinates. The second term encodes
the fact that $x^{\mu}$ also receives a $\bar{\theta}$ (or $\theta$) dependent
infinitesimal translation. You can check that these supercharges indeed obey
the superalgebra of eqs. (18), (19), and (20).

Fortunately, we will only have explicit need of the simplest type of
superfield, namely a scalar field on superspace, $\Phi(X)$, where by “scalar”
I mean that only its spacetime argument transforms under SUSY as described
above.

##  8 Chiral Superspace and Chiral Superfields

We have seen how SUSY acts on the three types of superspace coordinates
$x^{\mu},\theta_{\alpha},\bar{\theta}_{\beta}$. Remarkably, there is a kind of
“projection” of full superspace down to “chiral superspace” which is closed
under SUSY transformations. It has just two types of coordinates
$Y\equiv(y^{\mu},\theta_{\gamma})$, where

$\displaystyle y^{\mu}\equiv
x^{\mu}-i\theta_{\alpha}\sigma_{\alpha\beta}^{\mu}\overline{\theta}_{\beta},$
(43)

in terms of the original supercoordinates. Note that because of the “$i$”, the
$y^{\mu}$ coordinate is not real (hermitian). Given the original superspace
transformations, it is straightforward to check that under SUSY,

$\displaystyle(y^{\mu},\theta_{\gamma})\rightarrow(y^{\mu}-2i\theta_{\alpha}\sigma_{\alpha\beta}^{\mu}\overline{\xi}_{\beta},\theta_{\gamma}+\xi_{\gamma}),$
(44)

independently of $\bar{\theta}$.

We can therefore define a special kind of scalar superfield which only depends
on chiral superspace, $\Phi(y^{\mu},\theta_{\gamma})$. Clearly, under SUSY
transformations such a “chiral superfield” retains its form, that is
independent of $\bar{\theta}$ except implicitly within $y$.

In this simpler context, it is time to make a central point, that superfields
are really just a finite collection of ordinary component fields which are in
the same supermultiplet, that is they transform into each other under SUSY. To
see this we simply Taylor expand the $\theta$ dependence of the chiral
supefield,

$\displaystyle\Phi(y,\theta)=\phi(y)+\sqrt{2}\psi_{\alpha}(y)\theta^{\alpha}+F(y)\theta^{2}$
(45)

Because there are only two Grassmann coordinates here, $\theta_{\alpha}$, and
each of their squares vanishes because of their anticommuting with themselves,
the Taylor expansion can not contain more than one power of each Grassmann
coordinate. In particular the highest power of $\theta$s in the Taylor
expansion must be $\theta_{1}\theta_{2}=-\theta_{2}\theta_{1}$, or
equivalently in the manifestly Lorentz invariant form
$\theta^{2}\equiv\epsilon^{\alpha\beta}\theta_{\alpha}\theta_{\beta}$. With a
finite Taylor expansion, there are a finite number of Taylor coefficients
which are fields of $y^{\mu}$ alone. To make $\Phi$ Lorentz-scalar, $\phi$ and
$F$ must be complex Lorentz scalars, while $\psi_{\alpha}$ must be a chiral
two-component spinor.

For some purposes, this $y$ notation is compact and makes SUSY transformations
look simpler, but in general, we will want to write the fields in terms of
$x$, corresponding to real spacetime. That means each component field has to
be further Taylor expanded in the $-i\theta\sigma^{\mu}\bar{\theta}$ deviation
of $y$ from $x$:

$\displaystyle\Phi(y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\overline{\theta},\theta)$
$\displaystyle=\phi(x)-i\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\phi(x)-\frac{1}{4}\theta^{2}\overline{\theta}^{2}\square\phi(x)$
$\displaystyle\quad+\sqrt{2}\psi_{\alpha}(x)\theta^{\alpha}+\frac{i}{\sqrt{2}}\theta^{2}\overline{\theta}_{\alpha}\sigma^{\mu}_{\alpha\beta}\partial_{\mu}\psi_{\beta}(x)+\sqrt{2}F(x)\theta^{2}.$
(46)

Obviously there is a totally analogous conjugate notion of anti-chiral
superspace,
$\\{\bar{Y}\equiv(\bar{y}^{\mu}=x^{\mu}+i\theta_{\alpha}\sigma^{\mu}_{\alpha\beta}\bar{\theta}_{\beta},\bar{\theta}_{\gamma})$,
and the analogous anti-chiral superfield $\bar{\Phi}(\bar{y},\bar{\theta})$.
We need this because it houses the conjugates of the component fields of the
chiral superfield.

##  9 The Wess-Zumino Model

Finally, we are in position to write an actual simple renormalizable SUSY QFT,
with remarkable properties. It is built from a single chiral superfield
$\Phi(Y)$ (and of course its anti-chiral conjugate $\bar{\Phi}(\bar{Y}$).

The question is how to build a SUSY invariant action. First recall how we
build Poincare invariant actions,

$\displaystyle S=\int d^{4}x\,\,{\rm Lorentz-scalar}(x).$ (47)

The integration measure $d^{4}x$ is Poincare invariant and the Lagrangian
integrand is a (composite) Lorentz-scalar field, which upon integration over
all its translations (all values of $x$) becomes Poincare-invariant. We build
SUSY-invariant actions the same way, find a SUSY-invariant integration measure
and have the integrand be a (composite) scalar superfield:

$\displaystyle S=\int d^{4}x\int d^{2}\theta\int
d^{2}\overline{\theta}\,\,K(\Phi(Y),\overline{\Phi}(\overline{Y}))+\int
d^{4}y\int d^{2}\theta\,\,W(\Phi(Y))+\int d^{4}\overline{y}\int
d^{2}\overline{\theta}\,\,\overline{W}(\overline{\Phi}(\overline{Y})).$ (48)

Let us begin with the first term. Here, the Grassmann integral measure,
$d^{2}\theta\equiv\theta_{1}d\theta_{2}=1/2\epsilon^{\alpha\beta}d\theta_{\alpha}d\theta_{\beta}$
is Lorentz invariant just like $d^{4}x$. This and the conjugate integral
measure $\int d^{2}\theta\int d^{2}\bar{\theta}$ are usually abbreviated to
“$\int d^{4}\theta$”. You can check easily that the full measure $\int
d^{4}xd^{4}\theta$ is invariant under supertranslations, since these look like
$x$-independent translations of $x$ and simple translations of
$\theta,\bar{\theta}$. Since products of scalar fields are scalar fields,
including scalar superfields on superspace, we can make the integrand by (sums
of) products of the superfields $\Phi,\bar{\Phi}$. This integrand is called
the Kahler potential, $K$. We require that $K$ is hermitian so that the
Lagrangian and Hamiltonian are.

The action made from the Kahler potential would be SUSY invariant if $\Phi$
were any scalar superfield. But the fact that $\Phi$ is a chiral superfield
offers another possibility in the second term of the action. Here the
integrand is only made of sums and products of the chiral superfield, so that
$W$, the “superpotential”, is also a composite chiral superfield on chiral
superspace. Therefore the integration measure is only over chiral superspace
coordinates. Again, it is straightforward to check the measure’s SUSY
invariance given Eq. (44). The third term is just the conjugate of the second
to keep the sum hermitian.

The above action is the most general SUSY action with the fewest explicit
derivatives, namely none, which is why both $K$ and $W$ are called
“potentials”. But derivatives will indeed arise from the Taylor expansion of
the superfield into component fields, Eq. (8).

The fact that the superpotential is a function of $y$ but not $\bar{y}$ means
we can shift the integration variable from $y$ to just $x$,

$\displaystyle\int d^{4}y\int d^{2}\theta\,\,W(\Phi(y,\theta))=\int d^{4}x\int
d^{2}\theta\,\,W(\Phi(x,\theta)).$ (49)

As can be seen in Eq. (45), there are no derivatives hidden in the Taylor
expansion of $\Phi(x,\theta)$, so the superpotential part of the action does
not give rise to derivatives, these only arise from the Kahler potential which
depends on both $y$ and $\bar{y}$ which cannot both be shifted to $x$
simultaneously.

It is notable that $W$ is a complex analytic, or “holomorphic”, function of
the complex chiral superfield because it does not depend on the conjugate
anti-chiral superfield. This yields deep insights peculiar to SUSY, which we
only touch on later.

Let us do some quick dimensional analysis to write a renormalizable theory.
Given the SUSY algebra we see that the Grassmann coordinates
$\theta,\bar{\theta}$ have mass dimension $-1/2$. Given that the component
spinor fermion will turn out to be a canonical fermion with mass dimension
$3/2$, and component scalar field $\phi(x)$ will be a canonical scalar field,
the superfield $\Phi$ has mass dimension $1$. The component scalar field $F$
has the unusual dimension $2$, and we will see why soon. The Grassmann measure
$d\theta$ has the opposite dimension to the Grassmann coordinate, namely
$+1/2$, to satisfy the axiomatic $\int d\theta~{}\theta=1$. This means $K$ has
dimension $2$, and $W$ has dimension $3$. In order to not use couplings with
negative mass dimensions, the standard indicator of non-renormaliability, we
are restricted to:

$\displaystyle K=\overline{\Phi}\Phi,\quad
W=\frac{\lambda}{3}\Phi^{3}+\frac{m}{2}\Phi^{2}.$ (50)

Note, $K$ needs both chiral and antichiral fields to be non-trivial, so
dimension $2$ restricts it to the above, and it has coefficient $1$ by choice
of the normalization of $\Phi$. $W$ could clearly be any cubic polynomial in
$\Phi$, but a constant term would not survive the $\int d^{2}\theta$
integration, and a linear term could be redefined away by a
$\Phi\rightarrow\Phi+$ constant shift.999This is as long as there are
$\Phi^{2}$ or higher order terms. We will study a case where this is not true
soon. This renormalizable structure is the Wess-Zumino model.

We can now write this out in terms of the component fields using Eq. (8), and
then do the Grassmann integrations:

$\displaystyle S$ $\displaystyle=\int
d^{4}x\\{\partial_{\mu}\overline{\phi}(x)\partial^{\mu}\phi(x)+\overline{\psi}i\sigma^{\mu}\partial_{\mu}\psi(x)+\overline{F}F(x)$
$\displaystyle\quad-\lambda\phi(x)\psi^{2}(x)-\frac{m}{2}\psi^{2}(x)+(\lambda\phi^{2}+m\phi)F(x)+h.c\\}.$
(51)

We have done an integration by parts to put it in this form. Here the first
line comes from $K$ and the second from $W$. We see that the dimension $2$
scalar fields $F,\bar{F}$ appear quadratically and without derivatives. If you
think of this action in a path integral, this means that you can easily do the
Gaussian integrals over $F,\bar{F}$ by “completing the square”. It is entirely
equivalent to just solving the $F,\bar{F}$ equations of motion for these
“auxiliary” fields in terms of the other fields, and plugging the result back
into the action. The result is

$\displaystyle S=\int
d^{4}x\\{\partial_{\mu}\overline{\phi}\partial^{\mu}\phi+\overline{\psi}i\sigma^{\mu}\partial_{\mu}\psi-|m\phi+\lambda\phi^{2}|^{2}-(\frac{m}{2}+\lambda\phi)\psi^{2}(x)+h.c.\\}.$
(52)

We have arrived at a QFT with renormalizable interactions consisting of Yukawa
interactions between fermion and scalar, and scalar trilinear and quartic
self-interactions. But the various couplings and masses are strongly
correlated, as dictated by its manifestly SUSY construction. For example the
quartic scalar coupling is the square of the Yukawa coupling. But in
particular, we see the anticipated SUSY degeneracy of fermion and boson
superpartners, $m_{\phi}=m_{\psi}=m$. SUSY guarantees this structure is
maintained radiatively, upon renormalization the counterterms must also have
this special form. The Wess-Zumino model finally shows us that the abstract
SUSY algebra, which we deduced from general considerations, is actually
realizable within an interacting QFT.

###  9.1 Robust interacting massless scalar

We can further consider the massless limit $m\rightarrow 0$:

$\displaystyle S\underset{m\rightarrow 0}{\rightarrow}\int
d^{4}x\\{\partial_{\mu}\overline{\phi}\partial^{\mu}\phi+\overline{\psi}i\sigma^{\mu}\partial_{\mu}\psi-\lambda|\phi|^{4}-\lambda\phi\psi^{2}(x)+h.c.\\}.$
(53)

Note that this limit is robust, because there is a new chiral symmetry when
$m=0$,

$\displaystyle\psi\rightarrow e^{-i\alpha/3}\psi,\quad\phi\rightarrow
e^{+2i\alpha/3}\phi.$ (54)

Such chiral symmetries can make fermions robustly massless in Yukawa QFTs, but
what is special is that the robust masslessness extends to the scalar because
of SUSY degeneracy. In this way, we have an interacting massless (or light)
scalar with order-one coupling (with only logarithmic running). SUSY is the
only known robust mechanism for such scalars, at least in the perturbative
regime.

###  9.2 R-symmetries

The chiral symmetry is at first sight a bit strange in that it acts
differently on the two superpartners, in particular it appears we cannot
assign the entire superfield $\Phi(X)$ a charge under this symmetry. Indeed,
this cannot be done if we think of it as an “internal” symmetry which does not
act on the superspace $X$ argument of the superfield. But if we also rotate
the complex Grassmann coordinates, $\theta\rightarrow
e^{i\alpha}\theta,\bar{\theta}\rightarrow e^{-i\alpha}\bar{\theta}$, then we
see that by Eq. (45), we can assign $\Phi$ intrinsic charge $2/3$. Such
symmetries which rephase $\theta$ are known as R-symmetries. They are not
required by SUSY, for example the massive Wess-Zumino model is supersymmetric
but has no R-symmetries. (But even when absent the nature of their violation
can be useful to track.)

###  9.3 Non-renormalizable SUSY EFT

We can also take $K$ and $W$ to be more general, in which case they describe a
non-renormalizable but supersymmetric EFT. We can again Taylor expand $\Phi$
in Grassmann coordinates, Eq. (8), and use that to then Taylor expand $K$ and
$W$. Doing the various straightforward Grassmann coordinate integrations (and
some integrations by parts) yields the action in terms of the component
fields,

$\displaystyle{\cal L}$
$\displaystyle=\frac{\partial^{2}K}{\partial\Phi\partial\overline{\Phi}}(\phi(x))\\{|\partial_{\mu}\phi|^{2}+\overline{\psi}i\sigma\cdot\partial\psi+|F|^{2}\\}+\frac{1}{4}\frac{\partial^{4}K}{\partial^{2}\Phi\partial^{2}\overline{\Phi}}\psi^{2}(x)\psi^{2}(x)$
$\displaystyle\quad+\left(\frac{\partial^{3}K}{\partial^{2}\Phi\partial\overline{\Phi}}(\frac{i}{2}\bar{\psi}\sigma^{\mu}\psi\partial_{\mu}\phi+\psi^{2}\overline{F})+\frac{\partial^{2}W}{\partial\Phi^{2}}(\phi(x))\psi^{2}+\frac{\partial
W(\phi)}{\partial\Phi}F+h.c.\right)$
$\displaystyle\overset{\text{solve}}{\underset{F,\overline{F}\text{
eqs}}{\longrightarrow}}\frac{\partial^{2}K}{\partial\Phi\partial\overline{\Phi}}(\phi(x))\\{|\partial_{\mu}\phi|^{2}+\psi
i\sigma\cdot\partial\psi\\}+\left(\frac{\partial^{2}W}{\partial\Phi^{2}}(\phi)\psi^{2}(x)+h.c.\right)$
$\displaystyle\quad+\left(\frac{\partial^{3}K}{\partial^{2}\Phi\partial\overline{\Phi}}(\frac{i}{2}\bar{\psi}\sigma^{\mu}\psi\partial_{\mu}\phi+h.c.\right)$
$\displaystyle+\frac{1}{4}\frac{\partial^{4}K}{\partial^{2}\Phi\partial^{2}\overline{\Phi}}\psi^{2}(x)\psi^{2}(x)\quad-\left(\left|\frac{\partial
W}{\partial\Phi}(\phi)+\frac{\partial^{3}K}{\partial^{2}\overline{\Phi}\partial\Phi}\overline{\psi}^{2}\right|^{2}\right)\Bigg{/}\left(\frac{\partial^{2}K}{\partial\Phi\partial\overline{\Phi}(\phi)}\right).$
(55)

In the second step we have again integrated out the auxiliary fields
$F,\bar{F}$.

This is the most general SUSY EFT of the chiral superfield at $2$-derivative
order ($1$-derivative in fermions). We will give a simple application of this
structure in the next section on SUSY breaking. Terms with more derivatives
would require generalizing the SUSY Lagrangian beyond $K,W$ to integrands with
explicit superspace derivatives. But in EFT the higher derivatives would be
less important in the IR and therefore can be consistently dropped.

##  10 An EFT of Spontaneous SUSY Breaking

We can now give the simplest example of spontaneous breaking of SUSY, with a
non-renormalizable EFT. This is somewhat analogous to the well-known non-
linear $\sigma$-models that describe spontaneous breaking of ordinary (non-
abelian) internal symmetries in non-renormalizable EFT, which may then be
further UV-completed at higher energies.

Before specifying the theory, it is worth seeing one general implication of
spontaneous SUSY breaking. By taking the spinor trace of Eq. (18), we see that
the Hamiltonian in SUSY is a sum of squares of supercharges and therefore
positive semi-definite101010It is important for this that we are restricting
ourselves to the $G_{N}\rightarrow 0$ limit in which SUGRA decouples and SUSY
is a global symmetry.:

$\displaystyle H=|Q_{1}|^{2}+|Q_{2}|^{2}\geq 0.$ (56)

If the vacuum breaks SUSY, then at least one supercharge must not annihilate
it,

$\displaystyle Q_{1}|0\rangle\text{ or }Q_{2}|0\rangle\neq 0,$ (57)

which then implies that

$\displaystyle\langle
0|H|0\rangle\underset{\text{pert.}}{=}V_{\text{effective}}>0.$ (58)

In a perturbative theory, this means that the minimum of the effective
potential energy will be positive.

In honor of non-supersymmetric $\sigma$ models, we will call our chiral
superfield
$\Sigma(y,\theta)=\sigma(y)+\psi_{\Sigma}\theta+F_{\Sigma}\theta^{2}$, rather
than the generic “$\Phi$”. Specifically, we take

$\displaystyle K$
$\displaystyle=\overline{\Sigma}\Sigma-\frac{\overline{\Sigma}^{2}\Sigma^{2}}{4M^{2}},\quad
M\sim{\cal O}(M_{Pl})$ $\displaystyle W$ $\displaystyle=\Lambda^{2}\Sigma.$
(59)

We note that $K$ has a non-renormalizable term, with a negative-dimension
coupling. We will take the scale of nonrenormalizability to be Planckian,
$M\sim{\cal O}(M_{Pl})$, so that it only requires UV-completion in the
ultimate quantum-gravity/string-theory regime. Also note that we have a linear
superpotential, and it cannot be redefined away as we did in the Wess-Zumino
model because that required quadratic or cubic terms to do.

Figure 7: The scalar potential for our SUSY EFT has positive minimum,
indicating spontaneous SUSY breaking.

From Eq. (9.3), we read off the scalar potential,

$\displaystyle V(\sigma)=\left|\frac{\partial
W}{\partial\Sigma}\right|^{2}(\sigma)\Bigg{/}\frac{\partial^{2}K}{\partial\Sigma\partial\overline{\Sigma}}(\sigma)=\frac{|\Lambda|^{4}}{1-\frac{|\sigma|^{2}}{M^{2}}},$
(60)

illustrated in figure 7. Clearly, it has a local minimum at the origin of
field space, $\langle\sigma\rangle=0$, and indeed it is positive,

$\displaystyle\langle\sigma\rangle=0,\quad V_{\text{vacuum}}=|\Lambda|^{4}>0.$
(61)

You may also notice that, taken literally, the potential can become negative
for $|\sigma|>M$, suggesting that $\sigma=0$ is not the true ground state and
that in fact the potential is unbounded from below, which is unphysical.
However, we cannot trust such a conclusion because $|\sigma|>M$ lies outside
strict EFT control since the EFT is breaking down at scales of order $M$. To
know what really happens at large $\sigma$ would require the UV completion of
this EFT to all scales. Such renormalizable SUSY breaking models do indeed
exist, but we will not need them here. In this non-renormalizable EFT, we can
still conclude that $\sigma=0$ is potentially a metastable (on cosmological
timescales) or absolutely stable vacuuum, depending on the details of a full
UV complete extension of the EFT. Either way, $\sigma=0$ has positive vacuum
energy and therefore represents spontaneous SUSY breaking (at least for
cosmological timescales).

We can go back and calculate $\langle F_{\Sigma}\rangle$ from the auxiliary
field equation of motion,

$\displaystyle
F_{\Sigma}=\frac{\Lambda^{*2}}{1-\frac{|\sigma|^{2}}{M^{2}}},\quad\langle
F_{\Sigma}\rangle=\Lambda^{*2}\neq 0.$ (62)

Non-vanishing $\langle F\rangle$ VEVs are order-parameters for spontaneous
SUSY breaking. This is because when we solve for the auxiliary field $F$, its
square (or sum of squares when there are several chiral superfields)
contributes to the effective potential energy.

Phenomenologically, the thing that stands out is that superpartners are no
longer degenerate after spontaneous SUSY breaking,

$\displaystyle m_{\sigma}^{2}$ $\displaystyle=\frac{|\Lambda|^{4}}{M^{2}}$
$\displaystyle m_{\psi_{\Sigma}}$ $\displaystyle=0.$ (63)

###  10.1 The Goldstino and the Gravitino

The vanishing of the fermion mass is not specific to this particular model of
spontaneous SUSY breaking. Rather it is a parallel of Goldstone’s Theorem for
ordinary internal symmetries. There, there is a massless scalar robustly
arising, but for spontaneous SUSY breaking it is a massless fermion that is
robustly predicted [7][8][10]. In honor of the parallel this massless fermion
is called a Goldstone fermion, or more often, “Goldstino”.

Again paralleling internal symmetries, where when the symmetry is gauged the
gauge field “eats” the Goldstone particle to become massive, here too when we
do include SUGRA, the gravitino “eats” the Goldstino to become massive [11],

$\displaystyle m_{\text{gravitino}}=``m_{3/2}"\sim\frac{\Lambda^{2}}{M_{Pl}}.$
(64)

As a result, the massless Goldstino is ultimately not part of the physical
spectrum once SUGRA is included, but roughly describes the longitudinal
polarizations of the massive gravitino. This is the “super-Higgs” mechanism.

We will refer to a sector of particle physics which spontaneously breaks SUSY
as a “hidden sector” for phenomenological purposes, hidden in the sense that
it is taken not to have SM-charged fields within it. The simplest hidden
sector, which is mostly what we consider, is the non-renormalizable model we
have just discussed with just the chiral gauge-singlet superfield $\Sigma$.

##  11 The Renormalizable Minimal Supersymmetric SM (MSSM) [8]

Before we consider the gauge superfields, the SM matter fields can be elevated
to chiral superfields,

$\displaystyle Q_{i}$
$\displaystyle=\tilde{q}_{i}(y)+\sqrt{2}q_{i}(y)\theta+F_{q_{i}}\theta^{2}$
$\displaystyle L_{i}$
$\displaystyle=\tilde{l}_{i}(y)+\sqrt{2}l_{i}(y)\theta+F_{l_{i}}\theta^{2}$
$\displaystyle U_{i}^{c}$
$\displaystyle=\tilde{u}_{i}^{c}(y)+\sqrt{2}u_{i}(y)\theta+F_{u_{i}}\theta^{2}$
$\displaystyle D_{i}^{c}$
$\displaystyle=\tilde{d}_{i}^{c}(y)+\sqrt{2}d_{i}(y)\theta+F_{d_{i}}\theta^{2}$
$\displaystyle E_{i}^{c}$
$\displaystyle=\tilde{e}_{i}^{c}(y)+\sqrt{2}e_{i}^{c}(y)\theta+F_{e^{c}_{i}}\theta^{2}$
$\displaystyle{\cal H}_{u}$
$\displaystyle=H_{u}(y)+\sqrt{2}\tilde{H}_{u}(y)\theta+F_{H_{u}}\theta^{2}$
$\displaystyle{\cal H}_{d}$
$\displaystyle=H_{d}(y)+\sqrt{2}\tilde{H}_{d}(y)\theta+F_{H_{d}}\theta^{2},$

(almost) as anticipated in section 4, and given a renormalizable SUSY action
which is a straightforward generalization of the Wess-Zumino model. The $i$
indices refer to SM generations. Anticipating gauging, we restrict the theory
to respect global internal (non-R) symmetries $SU(3)_{QCD}\times
SU(2)_{EW}\times U(1)_{Y}$ prior to their gauging. The minimal superpotential
with these properties and which contains all the SM Yukawa couplings in its
component expansion is111111In these lectures, for simplicity I will
completely neglect neutrino masses.

$\displaystyle W_{\text{Yukawa}}=Y_{ij}^{u}U_{i}^{c}{\cal
H}_{u}Q_{j}+Y_{ij}^{d}D_{i}^{c}{\cal H}_{d}Q_{j}+Y_{ij}^{e}E_{i}^{c}{\cal
H}_{d}L_{j}\supset Y_{ij}^{e}e_{i}^{c}H_{d}l_{j}+\cdots$ (66)

The subtlety arises because of the holomorphy of $W$ in chiral superfields. In
the SM one uses the Higgs doublet to Yukawa couple to up-type fermions and its
conjugate to couple to down-type fermions, but we cannot use the conjugate
anti-chiral superfield in $W$. We are therefore forced to introduce separate
Higgs chiral superfields (with conjugate electroweak quantum numbers) for up-
type and down-type Yukawa couplings, as indicated. Therefore there is another
electroweak-invariant renormablizable superpotential term just involving these
two Higgs doublet superfields, called the “$\mu$” term:

$\displaystyle W_{\mu\text{-term}}=\mu{\cal H}_{u}{\cal H}_{d}.$ (67)

There are other possible $SU(3)_{QCD}\times SU(2)_{EW}\times
U(1)_{Y}$-symmetric renormalizable superpotential terms possible, such as
$D^{c}LQ$, but all of these can be forbidden if we impose another symmetry,
“R-parity”, in which every SM field is parity-even and every superpartner of a
SM field is parity-odd.121212Both Higgs scalar fields are even, and their
Higgsino superpartners are odd. R-parity is not strictly necessary, but it is
definitely simplifying (my main reason here) and also makes the lightest
superpartner stable and therefore potentially a dark matter candidate. The
Kahler potential is so tightly constrained by renormalizablility and the SM
symmetries, that (after familiar wavefunction diagonalization and
normalization) it has the canonical form

$\displaystyle
K=\overline{Q}_{i}Q_{i}+\overline{L_{i}}L_{i}+\overline{U}_{i}^{c}U_{i}^{c}+\overline{D}_{i}^{c}D_{i}^{c}+\overline{E}_{i}^{c}E_{i}^{c}+\overline{{\cal
H}_{u}}{\cal H}_{u}+\overline{{\cal H}_{d}}{\cal H}_{d}.$ (68)

###  11.1 Gauge Superfields

Given that ordinary gauge fields $A_{\mu}$ are real-valued, their superfields
including gauginos are found in what are called “real superfields”, but more
precisely they are scalar superfields which are hermitian (and not chiral):

$\displaystyle
V(x,\theta,\overline{\theta})=\overline{V}(x,\theta,\overline{\theta}).$ (69)

They are written “$V$” rather than “$\Phi$” because they are also sometimes
called “vector superfields”, presumably because they contain the $4$-vector
gauge fields, but they are scalar fields on superspace in that only their
superspace arguments transform under SUSY. They are also called “gauge
superfields”.

Now turn to charged matter fields. Ordinarily a gauge transformation of a
matter field transforms as

$\displaystyle\psi(x)\longrightarrow e^{i\alpha^{a}(x)t^{a}}\psi(x),$ (70)

where the $t^{a}$ are relevant matrix-valued generators of the gauge group
under which $\psi$ is a charged multiplet. But in SUSY these fields, including
the gauge transformation itself, must be elevated to superfields. Since our
matter fields are chiral superfields, this must also be true of the gauge
transformation. We therefore have

$\displaystyle\Phi(y,\theta)$ $\displaystyle\longrightarrow
e^{\Lambda^{a}(y,\theta)\,t^{a}}\,\,\Phi(y,\theta)$
$\displaystyle\overline{\Phi}(\overline{y},\overline{\theta})$
$\displaystyle\longrightarrow\overline{\Phi}(\overline{y},\overline{\theta})\,\,e^{\overline{\Lambda}^{a}(\overline{y},\overline{\theta})\,t^{a}}\,.$
(71)

$\bar{\Phi}$ is taken to be a conjugate row vector of the charged multiplet,
while $\Phi$ is a column. Note that the gauge transformation $\Lambda^{a}$ can
no longer be real (hermitian), so we can absorb the usual “i” in Eq. (70) into
its definition.

The need for gauge superfields to ensure gauge invariance of the kinetic terms
($K$) of the matter fields is easy to see. For simplicity consider the
renormalizable abelian case without gauge superfields:

$\displaystyle
K=\overline{\Phi}\Phi\longrightarrow\overline{\Phi}\,\,e^{\overline{\Lambda}}\,e^{\Lambda}\,\,\Phi(y,\theta).$
(72)

It is not invariant because $\Lambda\neq-\bar{\Lambda}$, they are not even
fields of the same type. This is fixed by introducing the abelian gauge
superfield $V$, transforming as

$\displaystyle V(x,\theta,\overline{\theta})\longrightarrow
V(x,\theta,\overline{\theta})-\Lambda(y,\theta)-\overline{\Lambda}(\overline{y},\overline{{\theta}}).$
(73)

Straightforwardly then,

$\displaystyle S$ $\displaystyle\supset\int d^{4}x\int
d^{4}\theta\,\,\overline{\Phi}e^{V}\Phi$ (74)

is both gauge-invariant and SUSY-invariant. Even though we will continue to
write $K=\bar{\Phi}\Phi$, think of it as short-hand for this gauge-invariant
inclusion of $V$ to make the Lagrangian gauge-invariant.

For non-abelian gauge theory, we still have

$\displaystyle S$ $\displaystyle\supset\int d^{4}x\int
d^{4}\theta\,\,\overline{\Phi}e^{V^{a}\,t^{a}}\Phi,$ (75)

with gauge-invariance following from

$\displaystyle e^{V^{a}\,t^{a}}\longrightarrow
e^{-\overline{\Lambda}^{a}\,t^{a}}\,e^{V^{b}\,t^{b}}\,e^{-\Lambda^{c}\,t^{c}}.$
(76)

###  11.2 Wess-Zumino Gauge

Ordinary axial gauge $A_{3}(x)=0$ is a Lorentz-violating but sometimes useful
partial gauge-fixing condition, which effectively reduces the number of gauge-
field components. A general gauge field can be put in this form by a suitable
gauge transformation. In analogy, Wess-Zumino gauge is a SUSY-violating but
(Lorentz-preserving) partial gauge-fixing which reduces the number of
component fields of $V$:

$\displaystyle
V^{a}(x,\theta,\overline{\theta})=A^{a}_{\mu}(x)\theta_{\alpha}\bar{\sigma}_{\alpha\beta}^{\mu}\overline{\theta}_{\beta}+(\lambda^{a}(x)\theta)\overline{\theta}^{2}+(\overline{\lambda}^{a}(x)\overline{\theta})\theta^{2}+\frac{1}{2}D^{a}(x)\theta^{2}\overline{\theta}^{2},$
(77)

where $\bar{\sigma}^{i}=-\sigma^{i},\bar{\sigma}^{0}=\sigma^{0}$.

###  11.3 Gauge field strength and gauge field action

It is possible to construct a gauge-invariant chiral superfield as a composite
of the gauge superfield $V$,

$\displaystyle{\cal W}_{\alpha}^{a}{\cal W}^{a\,\alpha}(y,\theta)$
$\displaystyle=\frac{1}{2}\lambda_{\alpha}^{a}\lambda^{a\,\alpha}+\cdots+$
$\displaystyle\quad+(\frac{1}{4}G_{\mu\nu}^{a\,2}+\frac{i}{4}\tilde{G}_{\mu\nu}^{a}G^{a\,\mu\nu}+\overline{\lambda}^{a}i\sigma.D\lambda^{a}+\frac{1}{2}D^{a}D^{a})\theta^{2},$
(78)

where $\tilde{G}_{\mu\nu}\equiv
1/2~{}\epsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}$. We will not need the terms
linear in $\theta$. Each factor ${\cal W}^{a}_{\alpha}$ is a spinor chiral
superfield generalizing the covariant gauge field strength, which I have not
defined, but we will only need its gauge-invariant “square” ${\cal
W}_{\alpha}{\cal W}^{\alpha}$ which transforms under SUSY like the elementary
chiral superfields we have discussed. Clearly it has mass dimension $3$. Since
it is chiral, we can write a renormalizable action,

$\displaystyle S_{\rm gauge}$ $\displaystyle=\frac{1}{4g^{2}}\int
d^{2}\theta\,{\cal W}_{\alpha}^{a}{\cal W}^{a\,\alpha}+h.c.$
$\displaystyle=-\frac{1}{4g^{2}}G_{\mu\nu}^{a\,2}+\frac{1}{g^{2}}\overline{\lambda}^{a}i\sigma.D\lambda^{a}+\frac{D^{a}D^{a}}{g^{2}}.$
(79)

We see that there is a new set of auxiliary fields, $D^{a}$. I am again using
the normalization where the gauge coupling is out the front of the action and
not in interaction terms, but one can return to the canonical normalization by
field redefinition.

It will be useful later to also allow non-renormalizable interactions at the
two-derivative level (one-derivative for fermions) of EFT, between chiral
matter superfields and the gauge superfields,

$\displaystyle\delta S=\int d^{2}\theta\,\,f(\Phi){\cal W}_{\alpha}^{a}{\cal
W}^{a\,\alpha}+h.c.$ (80)

The function of chiral superfields $\Phi$ is holomorphic for the same reason
the superpotential is, and is known as the “gauge coupling function” because
it is clearly a generalization of the renormalizable coupling $1/g^{2}$. That
is, it is a field-dependent gauge coupling. We will allow an independent gauge
coupling function for each gauge group. We take it to be a gauge-invariant
function of fields, as is ${\cal W}^{a}_{\alpha}{\cal
W}^{a\alpha}$.131313There is a more general option $f_{ab}(\Phi){\cal
W}^{a}_{\alpha}{\cal W}^{b\alpha}$ where $f_{ab}$ is not gauge-invariant but
we will not need this, and discard it for simplicity.

###  11.4 Component form of charged field gauged kinetic terms

Unpacking the $\int d^{4}\theta\,\bar{\Phi}e^{V}\Phi$, the couplings of $V$
introduce the expected
$\partial_{\mu}\rightarrow\partial_{\mu}-iA_{\mu}^{a}t^{a}$ in all the
derivatives of the charged fields. In addition, they give rise to a set of
Yukawa-type interactions of the gaugino,

$\displaystyle{\cal
L}_{\lambda}=\frac{1}{g^{2}}\overline{\lambda}^{a}i\sigma.D\lambda^{a}+\overline{\phi}\lambda_{\alpha}^{a}t^{a}\psi_{\beta}\epsilon^{\alpha\beta}+h.c.$
(81)

where the first term comes from the the gauge superfield action, Eq. (11.3).
Finally, there are terms linear in the auxiliary fields $D^{a}$. Taking into
account the quadratic terms in these fields from the gauge action, Eq. (11.3),
and eliminating these fields using their equations of motion, gives rise to a
scalar potential called the “D-term potential”,

$\displaystyle
V_{D}=\frac{1}{2}\sum_{a}g_{a}^{2}\left(\sum_{i}\overline{\phi}_{i}t^{a}_{(i)}\phi_{i}\right)^{2}.$
(82)

The index $a$ labels all gauge generators of all gauge groups, and $g_{a}$
denotes the gauge coupling appropriate to that generator (so a simple gauge
group has the same $g_{a}$ for all its generators, $a$). The index $i$ labels
the different charged species and $t^{a}_{(i)}$ are the gauge generator
matrices in the representation $i$. The full scalar potential is then given by
the sum of D- and F-term potentials,

$\displaystyle V_{\rm scalar}(\phi,\overline{\phi})=V_{D}+V_{F}.$ (83)

###  11.5 Renormalizable Feynman rules

It is useful to picture the structure of the renormalizable MSSM in general
terms. This is shown schematically in figure 8. But I have reverted to
canonical normalization for the gauge fields where gauge couplings $g$ appear
only in interactions not propagators, as is useful in practical work, rather
than the normalization with $1/g^{2}$ multiplying the entire gauge kinetic
terms, which is useful in some of our more theoretical and conceptual
discussion. Detailed forms of these rules with all the relevant factors and
representation matrices can be found in [8]. (One passes from the latter to
canonical normalization by the field redefinition $A\rightarrow gA$.)

The first interaction shown is the Yukawa-type interaction of the gaugino
mentioned above. The $g^{2}$ scalar quartic interaction comes from $V_{D}$.
All the other couplings of strength $g$ or $g^{2}$ just follow from gauge
invariance without any special SUSY considerations, just the gauge charge of
the ordinary component fields. The various Yukawa couplings of strength $Y$
arise from expanding the superpotential action with two fermions and one
scalar field. The last two interactions arise from $V_{F}$. In particular, the
last interaction arises from the cross-term in $V_{F}$ depending on both the
$\mu$-term and the Yukawa coupling.

Figure 8: The schematic structure of the Feynman rules of the MSSM component
fields, or indeed a generic renormalizable SUSY gauge-Higgs theory.

##  12 Soft SUSY breaking from Spontaneous SUSY breaking

Consider the following non-renormalizable couplings between the MSSM matter
and the simple hidden sector of SUSY breaking discussed earlier:

$\displaystyle{\cal L}_{\rm matter-hid}$ $\displaystyle=\int
d^{4}\theta\,\frac{\overline{\Sigma}\Sigma}{M_{Pl}^{2}}\left(c_{ij}^{Q}\overline{Q}_{i}Q_{j}+c_{ij}^{U}\overline{U}_{i}^{c}U_{j}^{c}\right.$
$\displaystyle\quad+c_{ij}^{D}\overline{D}_{i}^{c}D^{c}_{j}+c_{ij}^{L}\overline{L}_{i}L_{j}+c_{ij}^{E}\overline{E}_{i}^{c}E_{j}^{c}$
$\displaystyle\left.\quad+c_{u}\overline{{\cal H}}_{u}{\cal
H}_{u}+c_{d}\overline{{\cal H}}_{d}{\cal H}_{d}\right),$ (84)

where all the $c$ coefficients are taken to be roughly ${\cal O}(1)$ in size
and where $i,j=1,2,3$ label the three SM generations.

Now, upon spontaneous SUSY breaking in the hidden sector, we have
$\langle\Sigma\rangle=\Lambda^{2}\theta^{2}$. If we plug this VEV into the
coupling to the MSSM matter, the $\int d^{4}\theta$ is forced to integrate the
Grassmann coordinates in $\langle\bar{\Sigma}\Sigma\rangle$ and not the MSSM
superfields. Therefore the result can only be a potential for the MSSM
scalars,

$\displaystyle{\cal L}_{\rm matter-
hid}\underset{\Sigma\rightarrow\langle\Sigma\rangle}{\longrightarrow}\frac{\Lambda^{4}}{M_{Pl}^{2}}(c_{ij}^{Q}\overline{\tilde{q}}_{i}\tilde{q}_{j}+c_{ij}^{U}\overline{\tilde{u}}_{i}^{c}\tilde{u}_{j}^{c}+c_{ij}^{D}\overline{\tilde{d}}_{i}^{c}\tilde{d}_{j}^{c}+c_{ij}^{L}\overline{\tilde{l}}_{i}\tilde{l}_{j}+c_{ij}^{E}\overline{\tilde{e}}_{i}^{c}\tilde{e}_{j}^{c}+c_{u}\overline{H}_{u}H_{u}+c_{d}\overline{H}_{d}H_{d}).$
(85)

That is, we get a set of mass terms for all the scalars, roughly

$\displaystyle m^{2}_{\rm
scalar}\sim\left(\frac{\Lambda^{2}}{M_{Pl}}\right)^{2}\gtrsim v_{weak}^{2},$
(86)

but not for their superpartner fermions. From the MSSM viewpoint this is
effectively soft SUSY breaking roughly at the scale $\sim\Lambda^{2}/M_{Pl}$.
As we anticipated earlier by qualitative reasoning, we will take this scale of
soft SUSY breaking to be (very roughly) comparable to the weak scale
$v_{weak}$. Yes, order $1-10$ factors matter hugely to experimentalists
searching for such squarks and sleptons (and extra Higgs scalars) but we are
not yet ready to confront experiment at this point, so let us go with this
thumbnail sketch for now.

Solving for the scale of fundamental spontaneous SUSY breaking in the hidden
sector,

$\displaystyle\Lambda\gtrsim\sqrt{M_{Pl}v_{weak}}\sim 10^{11}\,{\rm GeV}.$
(87)

Because it is roughly the geometric mean of the Planck and weak scales it is
known as the “intermediate scale”. It is consistent to keep this big VEV of
$\Sigma$ while neglecting its fluctuations in their impact on the MSSM because
the fluctuation fields $\sigma,\psi_{\Sigma}$ have Planck-suppressed couplings
which are tiny at accessible energies. By comparison, the intermediate scale
VEV is so large that even after Planck-suppression its impact on the MSSM is
comparable to the weak scale.

Similarly, we can also consider a non-renormalizable hidden sector coupling to
the gauge superfields,

$\displaystyle{\cal L}_{\rm gauge-hid}$ $\displaystyle\,\,=\int
d^{2}\theta\,\frac{\Sigma}{M_{Pl}}c{\cal W}_{\alpha}^{a}{\cal W}^{a\,\alpha}$
$\displaystyle\underset{\Sigma\rightarrow\langle\Sigma\rangle}{\longrightarrow}\,\,\frac{c\Lambda^{2}}{M_{Pl}}\lambda_{\alpha}^{a}\lambda^{a\,\alpha},$
(88)

where there is an (implicit) independent $c$ for each MSSM gauge group. The
MSSM sees this coupling as effectively a soft SUSY-breaking gaugino mass of
comparable size to the scalar soft masses,

$\displaystyle m_{\lambda}\sim\frac{\Lambda^{2}}{M_{Pl}}.$ (89)

Putting it all together, the MSSM is effectively softly broken by the
couplings to the hidden sector, at the rough scale

$\displaystyle m_{\rm superpartners}\sim\frac{\Lambda^{2}}{M_{Pl}}\gtrsim
v_{weak}.$ (90)

With this simple form of soft SUSY breaking only affecting superpartner
masses, the Feynman diagrammatics above will look schematically the same as
the SUSY limit, but in detail the propagators for the different component
particles will reflect the SUSY-breaking masses.

##  13 The “$\mu$-Problem” and the Giudice-Masiero Mechanism

There is only one explicit mass parameter (not arising from dimensional
transmutation) in the MSSM prior to soft SUSY breaking, namely the $\mu$
parameter:

$\displaystyle\int d^{2}\theta\,\mu{\cal H}_{u}{\cal
H}_{d}=\mu\tilde{H}_{u}\tilde{H}_{d}+\cdots$ (91)

One can think of it as primarily giving rise to a Dirac mass, $\mu$, for
Higgsinos. In the SUSY limit it also gives the same mass to the Higgs scalars
of course, but after SUSY breaking we have seen that there are other
significant corrections to the latter. There are then two parametrically
distinct mass scales, $\mu$ and $\Lambda^{2}/M_{Pl}$, the first supersymmetric
in form and the second the result of SUSY breaking. If
$\mu\ll\Lambda^{2}/M_{Pl}\sim v_{weak}$ then the Higgsinos would be much
lighter than the weak scale and would have already been discovered given their
electric and weak charges. And if $\mu\gg\Lambda^{2}/M_{Pl}$ then the entire
${\cal H}_{u,d}$ multiplets would have supersymmetric mass $\mu$ far above the
weak scale so that they could not execute EWSB. Therefore, we need
$\mu\sim\Lambda^{2}/M_{Pl}$. However, given the orders of magnitude hierarchy
between the Planck and weak scales, this is a striking and unexplained
coincidence, an unsatisfactory feature in our push to understand all large
hierarchies where seen and absences of large hierarchies where not seen. It is
called the “$\mu$ problem”. We turn now to its simplest and ultimately quite
satisfying solution.

First notice that $\mu=0$ is robust in that it is protected by a new global
symmetry in that limit, “Peccei-Quinn” (PQ) symmetry,

$\displaystyle U(1)_{\rm PQ}:\quad$ $\displaystyle{\cal H}_{u}\longrightarrow
e^{i\alpha}{\cal H}_{u}$ $\displaystyle{\cal H}_{d}\longrightarrow
e^{i\alpha}{\cal H}_{d}.$ (92)

Instead, we introduce new couplings to the hidden sector that are peculiar to
the two Higgs supermultiplets because they are in conjugate gauge
representations:

$\displaystyle{\cal L}_{\rm Higgs-hid}$ $\displaystyle\,\,=\int
d^{4}\theta\,\left(c^{\prime}\frac{\overline{\Sigma}}{M_{Pl}}{\cal H}_{u}{\cal
H}_{d}+c^{\prime\prime}\frac{\overline{\Sigma}\Sigma}{M_{Pl}^{2}}{\cal
H}_{u}{\cal H}_{d}+h.c.\right)$
$\displaystyle\overset{\Sigma\rightarrow\langle\Sigma\rangle}{\underset{\overline{\Sigma}\rightarrow\langle\overline{\Sigma}\rangle}{\longrightarrow}}\left(\frac{c^{\prime}\Lambda^{2}}{M_{Pl}}\int
d^{2}\theta\,{\cal H}_{u}{\cal
H}_{d}+h.c.\right)+\left(\frac{c^{\prime\prime}\Lambda^{4}}{M_{Pl}^{2}}\,H_{u}H_{d}+h.c.\right).$
(93)

We see that the $c^{\prime}\sim{\cal O}(1)$ term gives us an effective
$\mu_{eff}=c^{\prime}\Lambda^{2}/M_{Pl}$ term after SUSY breaking in the
hidden sector, and solves the $\mu$ problem by realizing this apparent SUSY-
preserving mass as fundamentally a SUSY breaking effect. This is the Giudice-
Masiero mechanism [22]. It can be viewed as preserving Peccei-Quinn symmetry
if we assign $\Sigma$ charge $2$ under it. The second term is called an
effective soft-SUSY-breaking “$B\mu$ term”
$=c^{\prime\prime}\Lambda^{4}/M_{Pl}^{2}$, and can be useful in model-building
EWSB as we will see later. But clearly this would violate Peccei-Quinn
symmetry. The easiest way to allow both terms is to simply assume that the
hidden sector and its couplings violate Peccei-Quinn explicitly, but the MSSM
in isolation does not. We will make this more plausible, and radiatively
stable, from an extra-dimensional perspective later.

##  14 SUSY Phenomenology

###  14.1 Importance of the $125$ GeV Higgs Sector

Figure 9: Top/stop loops can have a significant impact on Higgs quartic self-
coupling, and hence physical Higgs scalar mass, if the stop-top mass splitting
is significantly larger than the weak scale.

Here, I want to give a schematic sense of the importance of the Higgs mass and
SUSY-breaking radiative corrections, ignoring some of the subtleties related
to the $2$-Higgs-doublet nature of the MSSM. We will do a more refined job
once we have sharpened a model of SUSY-breaking $c$ parameters.

In the non-supersymmetric SM, the physical Higgs boson mass is given at tree-
level by

$\displaystyle m_{h}^{2}=\lambda_{h}v_{weak}^{2},$ (94)

where $\lambda_{h}$ is the Higgs-doublet quartic self-coupling. In the
(softly-broken) MSSM, you can see from the scalar potential contributions, the
only self-couplings of the (two) Higgs doublets arise in the D-term potential,
and therefore $\sim g_{EW}^{2}$. In detail this implies a tree-level bound,
$m_{h}\leq m_{Z}$, regardless of how EWSB is shared between the two Higgs
doublet VEVs. In terms of the mass-squareds (more relevant for bosons) we see
that the observed Higgs strongly violates this bound:

$\displaystyle m^{2}_{h}=(125\,{\rm GeV})^{2}\sim 2m^{2}_{Z}!$ (95)

This implies that if SUSY is playing a role near the weak scale, Higgs
radiative corrections are important. The largest such radiative corrections
are expected from the largest Higgs coupling, the Yukawa coupling to the top
quark, as illustrated in figure 9.

We see that the simplest fit to $m_{h}=125$ GeV is to have a rather heavy
stop, $m_{\tilde{t}}\sim 10$ TeV, “just” out of LHC reach!

###  14.2 Direct LHC Searches

Figure 10: Gluino pair production at a hadron collider, initiated by gluons,
decaying into quarks and neutralinos. If the neutralinos are stable they would
be a good candidate for dark matter particles.

As you know, all searches for superpartners to date have come up empty, so
these translate into bounds on superpartner masses/couplings [23]. Very
roughly, a variety of LHC searches have resulted in bounds,

$\displaystyle m_{\rm gluino}\gtrsim 2\,{\rm TeV},\,\,m_{\rm Wino}\gtrsim{\rm
TeV}$ $\displaystyle m_{\rm squarks}\gtrsim{\rm TeV},\,\,m_{\rm
sleptons}\gtrsim 100(s)\,{\rm GeV}.$ (96)

A typical process that the LHC searches for is given in figure 10. Here, two
gluonic partons from within the colliding protons are pair-creating a pair of
gluinos. Each gluino then promptly decays into a (anti-)quark (jet) and
squark, and the squark in turn decays into a quark (jet) and “neutralino”. We
are assuming that the lightest superpartner, stable by $R$-parity
conservation, is electrically and color-neutral, hence “neutralino”. I have
drawn it as one of the neutral electroweak gauginos (though it may be a more
complicated superposition of superpartner gauge-eigenstates). Such a stable
neutralino is an example of a WIMP (Weakly Interacting Massive Particle,
implicitly stable). Because of its neutrality (and color-neutrality), a WIMP
behaves roughly like a heavy neutrino, escaping the LHC detectors, only
“detected” as missing (or imbalance in the) energy-momentum of the event.

###  14.3 WIMP Dark Matter Direct Detection

A stable WIMP is, broadly speaking, a good candidate for the particles making
up the Dark Matter of the universe [24]. If that is the case, then beyond
trying to pair-produce WIMPs at colliders as sketched above, we can also try
to detect WIMPs in our galaxy sweeping through the Earth. For example, the
WIMPs can scatter off nuclei in underground detectors, the details depending
on the specifics of their weak interactions. The likelihood of detection is
amplified by the potentially large flux of WIMPs and the large volume of the
detector target, as compared to pair-production and detection at particle
colliders. Again, as you know, despite heroic efforts WIMP dark matter
particles have not been directly detected to date. But there still remains
room for discovery.

##  15 Flavor and CP Problems of SUSY (and BSM more generally)

In the experimental search for new BSM physics there are two modes. There is
the mode of simply producing and detecting the new particles “on-shell” (or
having them be created by the early universe and hit our detectors). But in
this section we consider the other mode of processes involving SM particles
mediated by virtual “off-shell” exchanges of the new particles, typically
because the energy of the process is well below the BSM masses as the price
for higher statistics and hence precision. Our greatest ability to
discriminate BSM effects is within processes in which the competing SM effects
are relatively small for SM-structural reasons. Flavor-violating and CP-
violating processes at low energies (by high-energy collider standards) are
two of the most powerful categories in this regard, because in the SM they are
strongly constrained by the CKM flavor/weak-interaction structure and the
resulting GIM mechanism.

###  15.1 SM FCNCs and CP-Violation

Figure 11:  Figure 12: Tree-level neutral current processes in the standard
model are flavor-preserving, a striking consequence of its minimal structure.

The SM GIM mechanism strongly constrains Flavor-Changing Neutral Currents
(FCNCs). The classic “neutral current” is virtual $Z^{0}$-exchange,
illustrated in figure 11 between quarks, the neutral version of the “charged
current” exchange of $W^{\pm}$ in the standard Fermi approximation to weak
interactions. For charged currents the $W$ has coupling $gV^{CKM}_{ij}$ to any
pair of quark and antiquark flavors $i,j$ with net charge $1$, where $V^{CKM}$
is the CKM matrix emerging from the Yukawa couplings. By contrast, the $Z^{0}$
coupling is necessarily diagonal in the species of the pair of quarks it
couples to. This is a consequence of the minimal gauge coupling from covariant
derivatives in the quark kinetic terms. Therefore flavor $i$ is both entering
and leaving the effective neutral current exchange, and the same is true of
flavor $j$. (Or equivalently, quark flavors $j$ and antiquark flavor $j$ are
being produced.) So this is a flavor-conserving neutral current, not a flavor-
violating one. A less trivial “neutral current” arises from Higgs $h^{0}$
exchange, illustrated between quarks in figure 12. Even though the Yukawa
coupling matrix is not diagonal in the gauge basis, the quark mass matrix is
proportional to it, $m_{ij}=Y_{ij}v_{weak}$, so that the Yukawa couplings are
flavor-diagonal in the mass eigenbasis. So once again, this neutral current is
flavor-conserving.

Figure 13: FCNCs appear at one loop within the standard model by sewing
together a pair of tree-level flavor-changing charged currents. Even here,
there are significant suppressions due to the unitarity of the CKM matrix.

We have to go to $1$-loop to produce FCNCs in the SM, where we can create an
effective neutral current by exchanging a $W^{+}W^{-}$ pair, and taking
advantage of the flavor-changing $W$ couplings. This is illustrated in the
box-diagram of figure 13 for (virtual) $W$ exchanges between down-type quarks
of various generations, at low energies (well below $m_{W}$). I have just
given the dependence on the CKM matrix elements from the four vertices, while
the loop integral itself is given by $f$ at low energies. I will make my point
by pretending that all the up-type quarks are light, $m_{j},m_{r}\ll m_{W}$.
Then the loop-level FCNC in the figure simplifies to $\approx
f(0,0)\delta_{ik}\delta{\ell s}+$ small, by the unitarity of the CKM matrix in
the SM. That is, even at loop level, the individual contributions to FCNCs
cancel to a large extent!141414Obviously the top quark violates the assumption
of small quark mass and requires special treatment, which I will not go into.
But suffice it to say, it enters with rather small mixing angles
$V^{CKM}_{ts},V^{CKM}_{td}$, so that its contributions to FCNCs are also
relatively small. The small deviations from this limit predict small but
observable FCNCs which closely match observation. Any contributions from BSM
must be small enough to hide in the current experimental and calculational
error bars!

Now let us turn to CP violation, which can appear in conjunction with flavor-
changing processes or within purely flavor-conserving processes. In QFT, CP-
violation arises from complex phases in couplings that cannot be eliminated by
field redefinitions. In the renormalizable SM, with its three generations (and
neglecting the tiny neutrino masses for simplicity), at the perturbative level
there is just one such phase in $V^{CKM}$. This gives rise to a highly
restrictive form for CP-violating effects in weak-interaction processes within
hadronic flavor physics, and there is a beautiful body of experimental
evidence corroborating this single-source of CP-violation. Non-perturbatively
and subtly, there is another complex phase that is allowed, the “$\Theta$
angle”, which would give rise to CP-violation in strong-interaction hadron
physics (without involving weak-interactions at all), most famously predicting
a neutron electric dipole moment. But no such strong-interaction CP-violation
has been observed, giving rise to the stringent limit $\Theta<10^{-10}$. Other
than $\Theta$, Nature seems to have turned on every coupling allowed by the
renormalizable structure of the SM at an observable level, so the nearly-
vanishing $\Theta$ poses a puzzle “The Strong CP Problem”. Famously, there is
a plausible BSM resolution to this puzzle by adding an axion to the SM,
subject to the brilliant Peccei-Quinn mechanism. For a review see [25]. We
will assume that this addition to the SM has been made, so that only the CKM
phase remains as the source of CP violation in the SM$+$axion theory.

Given the SM ($+$ axion) structure that so strongly suppresses FCNCs and CP-
violation, in agreement with current precision low-energy flavor and CP tests,
makes these very sensitive channels for searching for BSM. Let us see how
these channels are sensitive to SUSY.

###  15.2 Superpartner-mediated FCNCs

Figure 14: The difference between the mass eigenbases of quarks and squarks
after generic SUSY breaking leads to FCNCs mediated by gluinos.

To exemplify how SUSY can introduce new FCNC contributions we consider new box
diagrams, analogous to those of the SM but with the $W$s replaced by gauginos.
Gluino exchange comes with the strongest coupling, so we focus on this in
figure 14.151515Note that gluon exchange in a box diagram itself would not
give FCNCs because gluons always couple the same quark and antiquark flavor,
unlike $W$s. The same is true for gluino couplings before SUSY breaking, but
not after SUSY breaking, as figure 14 schematically illustrates. In the figure
the quark flavors are in their mass basis, and I am labelling the squark
flavors in this same basis, so that gluinos connect quarks and squarks of the
same flavor. However, I have indicated the soft SUSY-breaking squark mass-
squareds which are not necessarily diagonal in this flavor basis because the
$c_{ij}$ are a priori general matrices in flavor-space, as introduced in
section 12. If we were to diagonalize these soft SUSY-breaking squark mass2
matrices, we would find that the quark mass basis was not the same as the
squark mass basis. This is in analogy to how the up-type quark mass basis is
different from the down-type mass basis in the SM, so there is the nontrivial
CKM generation-changing matrix for up-down couplings to $W$ after EWSB.
Similarly the gluino will have generation-changing couplings between quark and
squark mass eigenstates after SUSY breaking. Unlike (most of) the up-type
quarks in figure 13, all the squarks in figure 14 are comparably as heavy as
the gluino, and therefore we expect significant FCNCs for generic
$c_{ij}\sim{\cal O}(1)$.

The effective flavor-violating four-quark interaction arising from integrating
out the superpartners is however suppressed by $\sim 1/m_{superpartner}^{2}$,
by dimensional analysis. Therefore for generic $c_{ij}$ we can suppress BSM
FCNCs by taking large enough superpartner masses. Given generic $c_{ij}$ and
current constraints, this leads to bounds of roughly $m_{squarks}>1000$(s) of
TeV! [26]

###  15.3 Superpartner-mediated CP-violation

Figure 15: The classic diagram generating the anomalous magnetic moment of the
electron in QED. Figure 16: A supersymmetric-loop analog of the previous
diagram which can generate a distinctive CP-violating electric-dipole moment
if there are CP-violating phases in the superpartner masses/couplings.

We can illustrate non-flavor-violating CP-violation by the CP-violating
electron electric dipole moment (EDM). We can orient ourselves by first
considering the classic $1$-loop QED contribution to the electron magnetic
dipole moment of figure 15. QED does not give rise to an electron EDM because
it does not contain CP-violating phases in the renormalizable theory. But we
can now consider a superpartner analog of this loop, figure 16. It will give a
BSM contribution to the electron magnetic dipole moment. Because it is
suppressed by one power of the superpartner mass, it can easily be much
smaller than the QED contribution and therefore hard to detect. But if there
are irreducible CP-violating phases in the gaugino or slepton masses/couplings
then the diagram will also contribute to an electron EDM. For ${\cal O}(1)$
phases, the non-observation of such an EDM then requires $m_{superpartner}\gg$
TeV [27].

###  15.4 Moral for BSM from flavor and CP considerations

We have reviewed the key to how the SM ($+$ axion) strongly suppresses FCNCs
and CP-violation, namely that the only sources of flavor and CP violation
appear in the Yukawa coupling matrices, $Y^{u}_{ij},Y^{d}_{ij},Y^{e}_{ij}$,
and that these are automatically diagonal and real in the quark/lepton mass
eigenbases (neglecting neutrino masses). We have also seen that SUSY with
generic $c_{ij}$’s (and generic BSM more generally) at collider-accessible
scales is typically ruled out by the powerful flavor and CP experimental
constraints we already have. This strongly suggests that:

A viable SUSY (or BSM) theory should retain the feature of having the Yukawa
couplings as the sole source of flavor- and CP-violation, and that there
should be some theoretical structure that plausibly enforces this.161616 This
should at least be the case to good approximation.

We next describe an attractive structure of this type.171717The ansatz of
using copies of the Yukawa matrices in multiple places in a Lagrangian to
violate flavor/CP in a manner that extends the GIM mechanism is called Minimal
Flavor Violation (MFV), reviewed in [28], but here we are seeking a dynamical
explanation of such an extension of GIM rather than an ansatz.

##  16 Combining “bosonic” ($x_{5}$) and “fermionic” ($\theta,\bar{\theta}$)
extra dimensions

Figure 17: The higher-dimensional framework for a toy model of “gaugino-
mediated SUSY breaking”, with boundary-localized MSSM chiral superfields
sequestered from a boundary-localized hidden sector. The gauge superfields
propagate in the 5D bulk and mediate SUSY breaking in flavor-independent
manner that solves the SUSY Flavor Problem.

I want to first focus on a toy model, but it an almost realistic one. It is a
synthesis of supersymmetric and extra-dimensional dynamics, again of the type
that might plausibly emerge from something like superstring quantum gravity
below the Planck scale. It is depicted in figure 17. It is critical that the
extra dimension discussed from here on is not the same as the extra dimension
discussed in section 6 as being responsible for the generation of SM Yukawa
(fermion mass and CKM) hierarchies. For modularity, we can think of these as
two different extra dimensions. I will assume here that the Yukawa coupling
hierarchies have been achieved by some mechanism, perhaps that of subsection
6.2, but make no further reference to this. I will continue to refer to the
relevant extra dimension as the “$5$th” (and not the “$6$th”). In particular,
in the current context and higher-dimensional spacetime, the MSSM chiral
superfields, containing quarks, leptons, Higgses and their superpartners, are
all 4D fields localized on the left-hand boundary of 5D. We call this the
“visible boundary”. The hidden sector responsible for spontaneous SUSY
breaking, which minimally is just the $\Sigma$ superfield model introduced
earlier, is localised to 4D on the right-hand boundary. We call this the
“hidden boundary”. The MSSM gauge supermultiplets are however elevated to 5D
superfields propagating in the “bulk” of the higher-dimensional spacetime.
This model is called “gaugino-mediated SUSY breaking” ($\tilde{\rm g}$mSB) [2]
for reasons we will see below.

The visible boundary dynamics in isolation is described by the 4D Kahler
potential and superpotential for the MSSM chiral superfields detailed in
section 11, while the hidden boundary dynamics in isolation is described by
the 4D Kahler potential and superpotential described in section 10. But the
boundaries are not isolated because they can both interact with the gauge
superfields. The leading such couplings to the charged MSSM matter at
$x_{5}=0$ is just given by their supersymmetric gauge couplings as described
in Eq. (75), and unpacked in subsection 11.4. Since the hidden field $\Sigma$
is a gauge singlet, it can only have couplings at $x_{5}=L$ to supersymmetric
gauge field strengths, such as the $c/M_{Pl}$ of Eq. (12). We can focus on the
4D EFT below the KK scale $1/L$. 5D gauge invariance forces the gauge fields
to have $m_{5}=0$, so that their $0$-modes are $x_{5}-$independent. Therefore
there are no extra exponentials $e^{\pm m_{5}L}$ incurred in deriving the 4D
EFT. It must just have the form given by eqs. (11.3) and (37).

But by the locality of all couplings in relativistic theories (no