Do you ever find yourself wondering how the maths you learn helps us to understand and model small things we can’t even see? Yes, I thought so. I know you might hate how physicists mess with maths but bear with me on this one. Hold on tight as we take a joyride through the fusion of mathematics and subatomic wizardry—it’s time to unlock the secrets of perturbative unitarity and those elusive gravitons you may have heard about!

## The gravity of the situation

Alright then, imagine this: particle physicists are the ultimate puzzle-solvers, working to understand the tiniest building blocks of the universe. But guess what? It’s like they’ve been handed a humongous jigsaw puzzle with no picture to refer to. The jigsaw pieces? Those are our fundamental particles, such as electrons, quarks and photons. The picture? Well, we aren’t fully sure yet, but the so-called standard model was a decent start and I will introduce it to you soon.

The standard model has some flaws in the sense that it does not include gravity, neutrino masses or dark matter. Among other issues! The standard model features the fundamental particles that exist in the universe. These are the smallest building blocks of matter, light and gravity. The standard model describes how atomic nuclei are held together via the strong force and it also describes what matter consists of. These particles are our building blocks for almost everything else in the universe.

Quarks come in six different flavours and these quarks combine in different ways to form other larger particles. The so called bosons in the standard model mediate the fundamental forces, photons which carry electromagnetism, which we see in light. The W and Z bosons are responsible for the weak nuclear force that is responsible for radioactive decay. The standard model is basically the particle physics version of the periodic table.

So far so particle-like, but I heard particles are waves too, I hear you cry. Imagine you’re looking at a nice pond, and you throw a stone into it; ripples will spread out from where the stone landed and they interact with each other. These ripples create waves on the water’s surface. Now, think of this pond as the universe, and those ripples as our particles.

Quantum field theory, or QFT for short, is a way physicists try to understand how these tiny particles work and interact with each other. Instead of treating particles like billiard balls, QFT treats all these different particles as vibrations or disturbances in quantum fields. Think of this field as an invisible substance filling all of space, and particles are like the ripples or waves in this field. Quantum field theory can successfully describe three of the four fundamental interactions: the weak interaction, electromagnetism and the strong interaction. However, it needs some extra work to describe quantum gravity.

## Gravity at large distances

When we place a bowling ball in the centre of a trampoline, it creates a dent in the trampoline’s surface. This is analogous to how massive objects, like planets, warp spacetime according to Einstein’s theory of general relativity. Now imagine our trampoline as spacetime and some kids bouncing around chaotically as our particles. They move around and interact with the dent created by the bowling ball, which is like how gravity acts. This interaction between the kids and the trampoline is what we usually describe using general relativity. Typically, when we think of gravity what comes to mind is the force that pulls things with mass together. It is what causes apples to fall from trees and keeps the planets in our solar system orbiting around the sun, but how do we describe gravity at very small length scales?

In everyday life, we experience gravity in the continuous sense, but on small scales gravity acts like a particle. Individual particles are responsible for these forces: we quantise continuous waves into particles. Quantising is like turning something continuous and smooth into tiny, discrete chunks. It’s like you are filling a glass with water, but instead of a smooth stream, the water comes out in little drops—each drop is a quantum of water. You can’t have half a drop; it’s the smallest possible amount. Quantum gravity works in the same way. We try to break up this continuous force of gravity which acts everywhere in the universe by introducing the quanta of gravity: the graviton particles.

Many people think that quantum mechanics and general relativity are incompatible, and that the solution is to find some form of a grand unified theory that can account for all the fundamental forces simultaneously. A grand unified theory, often abbreviated as GUT, is like a big puzzle-solving idea in the world of physics such that *maybe*, there’s one big theory that explains everything: how electrons and quarks behave, how gravity works, and why the universe is the way it is. But this is a bit of an outdated idea. We all accept special relativity and classical mechanics to be different theories but valid in their own regimes (high speeds for SR). For speeds much less than that of light, we arrive at the same results as Newtonian mechanics. The same situation is applied to general relativity and quantum mechanics, we take quantum gravity as the high-energy short-distance limit of general relativity. The overall point of this is because quantum mechanics and general relativity are not fully compatible when dealing with more extreme limits, we introduce a bridge between them, which we call effective field theories (EFTs).

The actual description of the universe is like a big orchestra performing some very complex music at a busy stadium with lots of musicians (the particles). But when this orchestra plays at a smaller venue, they might use a smaller group of musicians, and the music is simplified. This is how EFTs work, they simplify our theories of the universe to focus on just the important pieces at a time. With these EFTs we do not need to choose, per se, a specific theory for quantum gravity—string theory or loop quantum gravity, for example—but merely just assume one exists. In the low-energy large-distance limit, the behaviour of quantum gravity is then described by general relativity, which is a classical theory, where spacetime curvature and gravity can be calculated using Einstein’s equations (as seen in *Chalkdust*‘s iconic hall of fail). EFTs are only valid within their energy regime, but this is OK because you wouldn’t expect to model a ball rolling down a hill with special relativity (unless you are crazy or the ball is extremely fast).

## Some HEFTy techniques: perturbative unitarity to predict the Higgs boson

Maybe you have seen the standard model Lagrangian, where there are so many terms just to describe how particles move, their masses and how they interact. EFTs take terms in the Lagrangian densities which are relevant at the required energy scale to form approximations on particle interactions and scattering amplitudes. Think about a magnetic material: in principle we could describe it in terms of individual atoms and their spins, but this would be incredibly complicated. Instead we zoom out and describe the collective behaviour of the atoms through the magnetisation, a measure of how the material responds to external magnetic fields. This is an effective field theory, the effective part means averaging out unnecessary details. The aim is to average over short length scales to remove some of the intricacies of the theory, which the EFT does in terms of an action and a Lagrangian density.

In much of physics, the Lagrangian, like the energy, is a bookkeeping tool that helps us describe the behaviour of a system mathematically. Think of a mechanics problem like a ball rolling down a hill. The Lagrangian describing the ball’s motion is the difference between its kinetic and potential energies. Extremising this quantity leads to the familiar, Newtonian, equations describing the ball’s motion.The action is an integral over space and time of the Lagrangian density, a version of the Lagrangian for field theories, describing the motion of particles along different paths. Extremising the action leads to the equations of motion; it is an optimisation problem to find a particle’s most efficient trajectory.

In the quantum world all the paths that a particle can travel between two points, $A$ and $B$, are valid, those which extremise the action are just the most probable. The sum of all these probabilities must add up to $1$, which is encoded in the principle of unitarity.

## Perturbations and probabilities

Unfortunately for many QFTs, the description in terms of a Lagrangian is still very complicated, and to extract any meaningful information we need an approximation scheme. However, this approximation scheme, known as perturbation theory, can break unitarity as it involves throwing information away. We use a technique called perturbative unitarity to ensure the total probability of all possible outcomes in a particle interaction or collision—like scattering in different directions—sums to 1.

Imagine trying to solve a quadratic equation like $\varepsilon x^{2}-2x+1=0$ for $\varepsilon \ll 1$, but you didn’t know the quadratic formula. How would you go about solving this equation? We could perhaps take a guess at an expansion in powers of $\varepsilon$, and then neglect terms of higher order, $x=x_{0}+\varepsilon x_{1} +\varepsilon^{2}x_{2}+\cdots$. We can then approximate a solution of this equation by comparing coefficients of $\varepsilon$, however we will only get one root of the equation, but it’s still a good way to approximate a solution. This is the essence of perturbation theory, we pick a small parameter and expand around it.

In classical mechanics, we can think of a collision between billiard balls as being alike to the scattering of fundamental particles. Consider two balls colliding with initial momenta $\boldsymbol{p}_{1}$ and $\boldsymbol{p}_{2}$ respectively. After the collision the momenta of the individual particles will have changed; they have final momenta $\boldsymbol{p}_{3}$ and $\boldsymbol{p}_{4}$. We can use Newton’s second law to find the forces acting on these balls and predict where they will travel to after the collision. However, in quantum mechanics our particles are not modelled by solid billiard balls, and instead we describe them by wavefunctions that represent probabilities of finding the particle in different positions and states. Instead of predicting the exact path of a particle like you would with billiard balls, quantum mechanics provides probabilities. We can calculate the likelihood of a particle scattering in a certain direction or having a specific outcome, but we can’t predict precisely where it will go.

We can think of the incoming and outgoing particles as the billiard balls, and where the billiard balls collide and fuse is the part where the graviton or Higgs particle enters existence. The billiard balls then move away from each other, becoming the outgoing particles.

## S is for scattering

When particles collide, we model them using a tool called the S-matrix—the scattering matrix. Essentially we know what goes in to the collision and what comes out but we treat the collision itself as a mystery box where we do not know all the details. The S-matrix is a way of describing what we do know about the interaction. The S-matrix is such that $\boldsymbol{\mathsf{SS}}^{\dagger}=\boldsymbol{\mathsf{I}}$, where $\boldsymbol{\mathsf{S}}^{\dagger}$ is the Hermitian conjugate (complex conjugate transpose) of the S-matrix. This property gives us an equation for the scattering amplitude $\mathcal{A}$, which describes how particles interact during a collision. This matrix helped us predict the existence of the Higgs boson particle, postulated in 1964, which was eventually detected at the Large Hadron Collider in 2012. The scattering amplitude encodes the probability of a scattering interaction occurring.

Imagine that you’ve got a particle, maybe a tiny electron, and it’s moving from left to right along one direction. But suddenly, it encounters a localised potential, we’ll call it $V(x)$, at $x=0$. Our electron wants to get through this barrier and enter the right hand side where the party is kicking off. Our particle has what we call a wavefunction, $\Psi$, which is determined by the Schrödinger equation,

$$-\frac{{\hbar}^{2}}{2m}\nabla^{2}\Psi+V(x)\Psi=\mathrm{i}\hbar \frac{\partial\Psi}{\partial t}.$$ What’s the deal with this wavefunction, you ask? This wavefunction is like our particle’s ID card, holding all the information about where the particle might be hanging out and how it’s vibing with its momentum. A wavefunction encodes information about a particle’s position and momentum and allows us to calculate the probability of finding the particle in different states or locations. Solving the Schrödinger equation we obtain

$$\Psi(x)=

\begin{cases}

A\mathrm{e}^{\mathrm{i}kx} + B\mathrm{e}^{-\mathrm{i}kx} & \text{if } x<0 \\
C\mathrm{e}^{\mathrm{i}kx} + D\mathrm{e}^{-\mathrm{i}kx} & \text{if } x>0.

\end{cases}$$

If our particle is on the left side $x<0$, it's all about the coefficients $A$ and $B$, with $\Psi(x) = A\mathrm{e}^{\mathrm{i}kx} + B\mathrm{e}^{-\mathrm{i}kx}.$ $A$ and $B$ are coefficients telling us how much of the wave is coming and going. $A$ represents the incoming wave and $B$ stands for the reflected wave.
But if our particle is on the right side of the potential $x>0$, it’s more about the coefficients $C$ and $D$, and we’ve got $\Psi(x) = C\mathrm{e}^{\mathrm{i}kx} + D\mathrm{e}^{-\mathrm{i}kx}.$ With $C$ representing the outgoing wave and $D$ the wave coming in from the right. Focusing on a particle starting to the left of the potential barrier and moving to the right, we can set $D$ to zero. Then $C$ tells us how much of the wave is transmitted through the barrier, while $B$ tells us how much has been reflected. We also have a wavevector $k=\sqrt{2mE}/\hbar$, which is like the DJ at our party. Our $k$ is the tempo based on the particle’s energy and mass; if $k$ is large, the music is fast, and the particle is zipping along at high energy. On the flip side, when $k$ is small, it’s a slow jam, and the particle is moving at lower energy.

The S-matrix is like the party host that connects the vibes from the incoming guests to the outgoing ones. Then $\boldsymbol{\Psi}_{\mathrm{out}}$ is the guest list for the party, with $B$ and $C$ rocking the party. There is also $\boldsymbol{\Psi}_{\mathrm{in}}$ which is on the left side trying to enter our party; $A$ makes it in but $D$ is still chilling at home.

These are related by

$$\boldsymbol{\Psi}_{\mathrm{out}}=\boldsymbol{\mathsf{S}}\boldsymbol{\Psi}_{\mathrm{in}},$$

with

$$\boldsymbol{\Psi}_{\mathrm{out}}=\begin{pmatrix} B \\ C \end{pmatrix},\; \boldsymbol{\Psi}_{\mathrm{in}}=\begin{pmatrix} A \\ D \end{pmatrix}.$$

We use the unitarity of the S-matrix to form an upper bound on the real and imaginary parts of a scattering amplitude, which we can then convert to an upper bound on the energy, with the potential to violate unitarity for collisions above this energy. To preserve unitarity a new particle must be introduced to our theory!

## Perturbative bound for gravity

For gravity, we now take on the challenge of finding the energy scale at which unitarity breaks down, indicating the existence of some new physics in the same way as for the Higgs. We need this new particle to stop unitarity from being violated at high energies. We use a trick called partial wave decomposition. Essentially we know that the scattering amplitude solves some equation, and we also know that this equation has a basis of solutions which we can expand in. Think of this as a fancy cousin of the Fourier series known as a partial wave decomposition. This expansion gives

$$\mathcal{A}(s,\theta)=16\pi\sum\limits_{n=0}^{\infty}(2n+1)a_{n}(s)P_{n}(\cos{\theta}).$$ Where $a(s)$ are partial wave amplitudes, which is analogous to the transmission component of a wave in 1D scattering, determined by $A/C$ in our quantum mechanics example.

The same idea as what solved the Higgs case works again for gravity where we integrate the S-matrix unitarity constraint to find a bound for distinguishable (different) particles with scattering mediated by a graviton. In terms of the partial waves, the bound is

$$|a(s)|^{2}+\mathrm{Im}(a(s))=0.$$ This bound is not satisfied for every possible energy, there are values of energy $s$ where this bound is violated. But we can’t have unitarity violated, so physicists introduced correction terms and the Higgs boson to mediate the scattering process. If we apply this bound to the amplitude that can be calculated for $W^{+}W^{+}\rightarrow W^{+}W^{+}$ scattering, we find that without the Higgs particle, unitarity would be violated at an energy around 1.7TeV. This bound means at energies above 1.7TeV, our physics would stop working, and probabilities would go above 1. Now, we see the same problem with quantum gravity, except this time it is more difficult as gravity requires an infinite number of correction terms, whereas the Higgs case only needed finitely many.

We can visualise the bound by displaying it on an Argand diagram in order to see where unitarity breaks down and hence where we need to introduce a new particle. Let $y=\mathrm{Im}(a(s))$ and $x=\mathrm{Re}(a(s))$ and using the properties of complex numbers that $|a(s)|^{2}=x^{2}+y^{2}$, we obtain a circle in the complex plane which bounds the scattering amplitude. In other words for energies such that we are inside the circle unitarity is preserved, while for other energies it is violated. Our circle in this case is $x^{2}+(y+1/2)^{2}=1/4$, and the bound is $$|\mathrm{Re}(a(s))|\leq \frac{1}{2}.$$

Adding new particles to our theory changes the scattering amplitude as there are now more ways that the particles can interact which changes the bound on energies where unitarity is preserved. If this new particle has the right mass then the energy, $s$, can grow arbitrarily large without violating unitarity. The new physics that we need to preserve unitarity is the introduction of the Higgs and the graviton respectively for each case. The new particle gives a new way, or channel in particle physics language, for particles to interact.

## Strings

For quantum gravity, one new channel is not enough to preserve unitarity at every energy scale. Instead we will get a new bound and at energies greater than the new bound unitarity would still be violated. To fix this requires us to add more resonance terms to our amplitudes so that we do not violate unitarity. The difference between the Higgs case and the graviton case is that gravitons require an infinite number of correction terms, while the Higgs does not. Physicists are taking what was learnt about the Higgs case and applying it to scattering mediated by gravity.

How do we go about adding an infinite number of correction terms? A similar area of physics called string theory offers a perspective on the amplitudes of gravitons, as there we can use a specific amplitude—among others—known as the Veneziano amplitude to model graviton scattering. While at first our EFT did not assume any specific theory of quantum gravity, string theory becomes helpful in modelling graviton scattering amplitudes. Investigating these amplitudes and what they can tell us about new physics at higher energies is a current area of research.

To sum it all up, the standard model of particle physics has guided us to analyse the interactions of elementary particles, while effective field theories have opened doorways to explore the gaps in the standard model. We marvelled at the magic of perturbative unitarity, the safety net that constrains our probabilities, and the S-matrix which describes the interactions of particles, revealing the hidden secrets of the Higgs boson and paving the way for experimental discoveries at the LHC. This led onto gravitons whose scattering amplitudes help us to bridge the gap between the quantum and the cosmic scales. The next step is to actually detect these gravitons (easier said than done)… Maybe check back in 100 years?