Flip-flopping contexts

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I was working on this article for Women in Science day, and then missed my deadline and again tried to finish this article for International Women’s day and again I missed my deadline, then Peter Higgs passed away and I am again late, ugh!

I guess this is just a recurring sign that something doesn’t need to be perfect before it is shared, and hopefully your audience also recognizes this. That being said, please don’t judge this article too hard, but do point out any mistakes! This is not meant for physicists by the way, you should instead check out the literature directly.

This article will outline somewhat poetically what the introductory chapters of my thesis are trying to say, by highlighting some of the work that was inspired and built off several incredible people and their ideas over the 20th century.

Note that this particular article was quite challenging for me to write, it’s hard gauging what a general audience would comprehend but I’ll do my best and imagine I am explaining this all to a friend who is scientifically inclined but is not in particle physics.

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Beauty and poetry in mathematics and physics

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Some people take the viewpoint that modern physics is really the study of symmetries. And a lot of people, myself included, have romanticized the role of symmetry in physical theories at some point in their lives. I could write an article poeticizing symmetry in physics, but there are so many other articles and videos out there that do this — I’ll avoid it. add links!

Instead, I will try to introduce you to the truly awesome humans that have had a deep impact on the ideas that lay the foundation for what ends up as the introduction of my research.

Here is an introduction to the theory and a tribute to some of my heroes.

Emmy Noether (c. 1915)

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In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.

— Albert Einstein, 1935 letter to the New York Times

Our story begins at the turn of the 20th century with Emmy Noether, a mathematician who produced two theorems that revolutionized the way we interpret the mathematics behind physical theories. Noether’s theorems (in broad-strokes) state the following:

For every differentiable symmetry of the action of a physical system there is a corresponding conservation law associated with it.

Okay, that was a lot of jargon so let’s break it down.

What is action?

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The “action” in physics is a running sum over energies.¹ So what the jargon in the above statement is trying to say is that, for given initial and final conditions, if you take this running sum over energies and vary it by some value at each point and it leaves the overall action unchanged then there is a conservation law associated with this symmetry. For those of you that have heard of it: the principle of least (or stationary) action is associated, or maybe you’ve heard of the “path of least resistance”. These are similar concepts associated with Noether’s theorem.

Now, what Noether did was mathematically prove that there is an association between symmetries of an abstract physical quantity that we can calculate (the action) and observable conservation laws, e.g., the conservation of electric charge, momentum, and the conservation of energy. These conservation laws dictate what is and is not possible in our Universe! I.e., the reason I can’t just pop a new tennis ball into my hand is because conservation of energy.

What is symmetry?

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Okay, but what exactly do we mean by symmetry here? In everyday life we think of symmetries as things where there is something we can change, but the object’s representation is unchanged. For example, if I take a circle and spin it around its central axis it’s still a perfect, identical, circle no matter how much I spin it around. Similarly, an equilateral triangle, can be turned exactly 120 degrees and will look exactly the same. These are two types of symmetries: the circle’s case is continuous (I can choose any number of degrees to rotate by) but in the triangle case, I can only choose a multiple of 120 degrees.

In physics, there are continuous and discrete symmetries too. For example, theories that are written down where time can be modified, i.e. t -> t + dt, i.e. the action is said to exhibit time-invariance and the associated conservation law with this symmetry is the conservation of energy. Spatial invariances lead to momentum conservation, and rotational invariances to conservation of angular momentum, e.g., the reason a figure skater who pulls in their arms begins to spin much more rapidly is due to conservation of angular momentum.

In particle physics, we have a few more symmetries to play with and these are much more abstract.

An interesting note to highlight here is that as experimentalists we actually see these conservation laws. We observe their effects, but we do not a priori know what symmetry (if any) is associated with them.

So how do neutrinos and radioactivity fit into this…?

Maria Goeppert-Mayer (c. 1935)

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Nuclear shell structure and Double Beta Decays

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Maria Goeppert-Mayer was a brilliant nuclear physicist who won the Nobel prize for her work in the development of the nuclear shell model. i.e., she recognized that the protons and neutrons making up the nuclei of atoms, can only be in specified energy states and exhibit a shell structure analogously to the electrons in an atom (think back to your highschool chemistry).

What this means is that the processes that go on inside a nucleus emit quanta of energy (and not a continuous energy distribution). Because of these quantized states, the radiation emitted by a nucleus must come at specific energies and not have a continuous distribution. In the same way that the sodium street lamps in Oman are orange because they only emit a few very specific lines of orange when excited and we can visibly see this; for the case of nuclei the emitted photons are X-rays or gamma rays of much higher energy but also come in specific packets of energy.

Goeppert-Mayer was also known for predicting the time scale that would be expected for double beta decays, that is the half-lives of nuclei who are energetically unable to undergo single beta decay but can “tunnel” through a potential barrier and undergo the simultaneous ejection of two electrons and two anti-electron neutrinos.

When we measure the energy of beta decays however, we see these spectra as continuous because we can only easily measure the energy of charged particles, i.e., the electron in this case. We cannot measure the energy of the neutrinos that are emitted as they fly away from our detector and do not ionize any of the material in it. Hence, if we sum up the energies of the electrons observed for many events we see something that looks like this.

Madame Wu (c. 1956)

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The Universe is left-handed

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Over the Christmas break in 1956, Madame Dr. Wu stayed holed up in her lab designing and proving Lee and Yang’s theory that the weak force breaks parity (right-left symmetry). This is weird, why would nature by asymmetric in right- and left-handed processes?

The first thing to ask yourself is this: how do you objectively define a “left” and a “right” in the world? Well, it turns out that by taking cross-products of vectors, these do/don’t change sign under a change of variables in your equations like flipping the sign of all positions in your equation: \(x \rightarrow -x \).

Well, through clever experimental design she actually showed that indeed, the weak force (one of the four fundamental forces of nature) actually breaks parity. Moreover, not only does it break parity slightly, it is fully left-handed! No other known subatomic forces have this feature as far as we can tell.

Ettore Majorana: why do we need antimatter? (c. 1937)

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Ettore Majorana was a strange Italian physicist who proposed a relativistic wave equation for massive particles which are identical to their charge-conjugated states (think of flipping electrical charges). In the case of photons (particles of light) there are no charges, and so its charge-conjugate is itself. However, a photon is massless hence its infinite range and why you can see with your eyeballs the light from stars spread out across our galaxy, the Milky Way. If the photon was massive electromagnetism would not be an infinitely long-ranged force, and we would not see the stars in the sky.

Now, particles of matter have mass and so instead of being described by a massless equation, they are described by the Dirac equation for electrons (actually, this is because they are spin-1/2 and not spin-1 like the photon, but if you don’t know what spin is then just ignore this). This is awesome, and was what Paul Dirac proposed in 1928 (?!!).

Now, the interesting contribution from Majorana was that there is a way to describe how a spin-1/2 massive particle would behave, but does such a fundamental particle exist?

It turns out that yes, maybe(?!)

A strange lad: Majorana disappeared after taking a ferry trip in 1938.

The origin of mass

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Okay it turns out that particles gaining a mass term was quite difficult to present mathematically. The way we do this in the Standard Model is to glue the right- and left-handed particles of the same type together via the Higgs field. This flip-flopping between right- and left-handed states gives rise to the physical massive particles we know, e.g. an electron.

Higgs et al. and the “God” particle.

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Clearly, mass exists. We have mass. The Earth has mass. But it turns out that just writing down a mass term in the equations of the Standard Model breaks the beautiful symmetries that make the theory work in the first place. This was a real headache for theorists in the 1960s.

The solution came from several people working somewhat independently: Robert Brout, François Englert, and Peter Higgs (among others — hence the et al.). The idea is that there is a field that permeates all of space, and particles that interact with this field acquire mass. The more strongly a particle interacts with the Higgs field, the heavier it is. Particles that don’t interact with it at all, like the photon, remain massless.

You can think of it a bit like wading through a swimming pool. If you’re slim and slippery (like a photon) you pass right through unimpeded. If you’re bulky (like a top quark) you get slapped left and right and slowed — that resistance is, in some rough sense, mass.

The “God particle” nickname came from a book title and most physicists really dislike it, but it stuck in the media. In 2012, the Higgs boson (the quantum of the Higgs field, i.e., the particle associated with it) was finally discovered at CERN’s Large Hadron Collider. Peter Higgs and François Englert won the Nobel Prize in 2013. Sadly, Robert Brout had passed away in 2011 and Nobel Prizes are not awarded posthumously.

The Big Bang

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From the famous E=mc2 equation we can tell two things. What this equation is saying is that energy in some sense can “condense” into mass. But what mass? Well, if you are creating mass from a lot of energy then you must create equal parts matter and antimatter, positive and negative. You create an electron, you must create an anti-electron (positron) which is the thing we use in PET scans to search for cancerous tissue. Yes, this stuff is real.

Matter-antimatter asymmetry

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We live in a world filled with matter! Look around you — everything you see, your coffee mug, the air in your lungs, the Sun, distant galaxies — it’s all matter. But if the Big Bang created equal parts matter and antimatter, where did all the antimatter go? Why didn’t it all just annihilate back into pure energy and leave us with nothing but light?

This is one of the biggest open questions in physics. Something happened in the early Universe that tipped the scales ever so slightly in favour of matter over antimatter. For every billion antimatter particles there were a billion and one matter particles, and after everything annihilated, that tiny remainder is what makes up, well, everything.

In 1967, the Soviet physicist Andrei Sakharov laid out three conditions that any explanation for this asymmetry must satisfy. One of them is that there must be processes that violate something called baryon number conservation, i.e., processes where the total amount of matter is not preserved. Another is that the symmetry between matter and antimatter (called CP symmetry, for charge-parity, the parity part being what Wu was doing her experiment on mentioned earlier) must be violated. And the third is that all of this has to happen out of thermal equilibrium — otherwise any asymmetry that builds up just gets washed back out.

We’ve observed CP violation in certain particle decays, but not nearly enough to explain the amount of matter we see. So something else must be going on. This is where it gets interesting and where my research comes in.

Weinberg, Seesaws and a pathway to our matter-filled Universe?

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Okay if you’ve gotten this far down I’ll hint at the punchline.

Remember Majorana’s equation from earlier? The one that describes a massive particle that is its own antiparticle? Well, there is one known particle that could be a Majorana particle: the neutrino. Neutrinos have no electric charge, which means there’s no obvious way to tell a neutrino from an anti-neutrino other than their right- or left-handedness. If the neutrino is indeed a Majorana particle, then the neutrino and anti-neutrino are the same thing, just viewed from a different vantage point.

Steven Weinberg showed how to write down an elegant operator that, if neutrinos are Majorana particles, connects naturally to what is called the seesaw mechanism — a way to explain why neutrinos are so absurdly light compared to all other massive particles (for context, a neutrino is at least 500,000 times lighter than the next lightest particle, the electron). The idea is that there are very heavy neutrino-like particles that we can’t see directly, and through a sort of balancing act (like a seesaw), their heaviness pushes the masses of the neutrinos we observe down to the tiny values we measure. The heavier the unseen partner, the lighter the neutrino we detect.

And here’s the kicker: those same heavy Majorana particles, in the hot early Universe, could have decayed in just the right way to produce a slight excess of matter over antimatter. This process is called leptogenesis, and it beautifully ties together the tiny masses of neutrinos with the existence of all the matter around us. It satisfies all three of Sakharov’s conditions.

So how do we test this? Well, if the neutrino is truly a Majorana particle, then there should exist a process called neutrinoless double beta decay — a nuclear decay where two neutrons convert into two protons and emit two electrons, but no neutrinos. This would violate the conservation of something called lepton number, and would be the smoking gun for Majorana neutrinos. This is what experiments like nEXO (the one my thesis is about!) are searching for. We haven’t seen it yet, but the hunt is very much on.

Conclusion

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That was a super long article. It was really difficult to write, it’s hard trying to convey the intricacies of mathematics without actually showing the notational machinery. But I hope you got something out of it.

What I really wanted to share is how these ideas connect. Emmy Noether showed us that symmetries are the backbone of physics. Goeppert-Mayer revealed the structure inside the nucleus. Madame Wu proved that the Universe plays favourites with left and right. Majorana proposed that a particle could be its own antiparticle. Higgs and friends explained how particles get their mass. And Weinberg tied it all together with a mechanism that might explain why we exist at all.

These ideas span almost a century, built by people from different countries, backgrounds, and circumstances — many of whom faced enormous institutional barriers. The fact that their work connects so deeply is, to me, one of the most beautiful things about the study of physics and the human pursuit of knowledge.

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