by Brian Cox
We are now going to move into the realm of speculation – although by the time you read this book the theory we are about to outline may have been verified. The LHC is currently busy colliding protons together with a combined energy of 7 TeV. ‘TeV’ stands for Tera electron volts, which corresponds to the amount of energy an electron would have if it were accelerated through a potential difference of 7 million million volts. To get a sense of how much energy this is, it’s roughly the energy that subatomic particles would have had about a trillionth of a second after the Big Bang and it is enough energy to conjure out of thin air a mass equal to 7,000 protons (via Einstein’s E = mc2). And this is only half the design energy; if needed, the LHC has more gas in the tank.
Figure 11.5. Particles of increasing mass propagating from A to B. The more massive a particle is the more it zig-zags.
One of the primary reasons that eighty-five countries around the world have come together to build and operate this vast, audacious experiment is to hunt for the mechanism that is responsible for generating the masses of the fundamental particles. The most widely accepted theory for the origin of mass works by providing an explanation for the zig-zagging: it posits a new fundamental particle that the other particles ‘bump into’ on their way through the Universe.
That particle is the Higgs boson. According to the Standard Model, without a Higgs the fundamental particles would hop from place to place without any zig-zagging and the Universe would be a very different place. But if we fill empty space with Higgs particles then they can act to deflect particles, making them zig-zag and, as we have just learnt, that leads to the emergence of ‘mass’. It is rather like trying to walk through a crowded pub – one gets buffeted from side-to-side and ends up taking a zig-zag path towards the bar.
The Higgs mechanism is named after Edinburgh theorist Peter Higgs and it was introduced into particle physics in 1964. The idea was obviously very ripe because several people came up with the idea at the same time – Higgs of course, and also Robert Brout and François Englert working in Brussels and Gerald Guralnik, Carl Hagan and Tom Kibble in London. Their work was itself built on the earlier efforts of many others, including Heisenberg, Yoichiro Nambu, Jeffrey Goldstone, Philip Anderson and Weinberg. The full realization of the idea, for which Sheldon Glashow, Abdus Salam and Weinberg received the Nobel Prize in 1979, is no less than the Standard Model of particle physics. The idea is simple enough – empty space is not empty, and this leads to zig-zagging and therefore mass. But clearly we have some more explaining to do. How can it be that empty space is jammed full of Higgs particles – wouldn’t we notice this in our everyday lives, and how did this strange state of affairs come about in the first place? It certainly sounds like a rather extravagant proposition. We have also not explained how it can be that some particles (like photons) have no mass while others (like W bosons and top quarks) weigh in with masses comparable to that of an atom of silver or gold.
The second question is easier to answer than the first, at least superficially. Particles only ever interact with each other through a branching rule and Higgs particles are no different in that regard. The branching rule for a top quark includes the possibility that it can couple to a Higgs particle, and the corresponding shrinking of the clock (remember all branching rules come with a shrinking factor) is much less than it is in the case of the lighter quarks. That is ‘why’ a top quark is so much heavier than an up quark. This doesn’t explain why the branching rule is what it is, of course. The current answer to that is the disappointing ‘because it is’. It’s on the same footing as the question ‘Why are there three generations of particles?’ or ‘Why is gravity so weak?’ Similarly, photons do not have any branching rule that couples them to Higgs particles and as a result they do not interact with them. This, in turn, means that they do not zig-zag and have no mass. Although we have passed the buck to some extent, this does feel like some kind of an explanation, and it is certainly true that if we can detect Higgs particles at the LHC and check that they couple to the other particles in this manner then we can legitimately claim to have gained a rather thrilling insight into the way Nature works.
The first of our outstanding questions is a little trickier to explain – namely, how can it be that empty space is full of Higgs particles? To get warmed up, we need to be very clear about one thing: quantum physics implies that there is no such thing as empty space. In fact, what we call ‘empty space’ is really a seething maelstrom of subatomic particles and there is no way to sweep them away and clean it up. Once we realize that, it becomes much less of an intellectual challenge to accept that empty space might be full of Higgs particles. But let’s take one step at a time.
You might imagine a tiny region of deep outer space, a lonely corner of the Universe millions of light years from a galaxy. As time passes it is impossible to prevent particles from appearing and then disappearing out of nothing. Why? It is because the process of the creation and annihilation of particle–anti-particle pairs is allowed by the rules. An example can be found in the lower diagram in Figure 10.5: imagine stripping away everything except for the electron loop – the diagram then corresponds to an electron–positron pair spontaneously appearing from nothing and then disappearing back into nothing. Because drawing a loop does not violate any of the rules of QED we must acknowledge that it is a real possibility; remember, everything that can happen does happen. This particular possibility is just one of an infinite number of ways that empty space can fizz and pop, and because we live in a quantum universe the correct thing to do is to add all the possibilities together. The vacuum, in other words, has an incredibly rich structure, made up out of all the possible ways that particles can pop in and out of existence.
That last paragraph introduced the idea that the vacuum is not empty, but we painted a rather democratic picture in which all of the elementary particles play a role. What is it about the Higgs particle that makes it special? If the vacuum were nothing other than a seething broth of matter–antimatter creation and annihilation, then all of the elementary particles would continue to have zero mass – the quantum loops themselves are not capable of delivering it.3 Instead, we need to populate the vacuum with something different, and this is where the bath of Higgs particles enters. Peter Higgs simply stipulated that empty space is packed with Higgs particles4 and didn’t feel obliged to offer any deep explanation as to why. The Higgs particles in the vacuum provide the zig-zag mechanism and they are working overtime by interacting with each and every massive particle in the Universe, selectively retarding their motion to create mass. The net result of the interactions between ordinary matter and a vacuum full of Higgs particles is that the world goes from being a structureless place to a diverse and wonderful living world of stars, galaxies and people.
The big question of course is where those Higgs particles came from in the first place? The answer isn’t really known, but it is thought that they are the remnants of what is known as a phase transition that occurred sometime shortly after the Big Bang. If you are patient and watch the glass in your window as the temperature falls on a winter’s evening, you’ll see the structured beauty of ice crystals emerge as if by magic from the water vapour in the night air. The transition from water vapour to ice on cold glass is a phase transition – water molecules rearranging themselves into ice crystals; the spontaneous breaking of the symmetry of a formless vapour cloud triggered by a drop in temperature. Ice crystals form because it is energetically more favourable to do so. Just as a ball rolls down the side of a mountain to take up a lower energy in a valley, or electrons rearrange themselves around atomic nuclei to form the bonds that hold molecules together, so the sculpted beauty of a snowflake is a lower energy configuration of water molecules than a formless cloud of vapour.
We think that a similar thing happened early on in the Universe’s history. As the hot gas of particles that was the nascent Universe expanded and cooled, so it transpired that a Higgs-free vacuum was energetically disfavoured and a vacuum filled with H
iggs particles was the natural state. The process really is similar to the way that water condenses into droplets or ice forms on a cold pane of glass. The spontaneous appearance of water droplets when they condense on a pane of glass creates the impression that those droplets simply emerged out of ‘nothing’. Similarly for the Higgs, in the hot stages just after the Big Bang the vacuum is seething with the fleeting quantum fluctuations (those loops in our Feynman diagrams), as particles and anti-particles pop out of nothing before disappearing again. However, something radical happens as the Universe cools and suddenly, out of nothing, just as the water drops appear on the glass, a ‘condensate’ of Higgs particles emerges, all held together by their mutual interactions in an ephemeral suspension through which the other particles propagate.
The idea that the vacuum is filled with material suggests that we, and everything else in the Universe, live out our lives inside a giant condensate that emerged as the Universe cooled down, just as the morning dew emerges with the dawn. Lest we think that the vacuum is populated merely as a result of Higgs particle condensation, we should also remark that there is even more to the vacuum than this. As the Universe cooled still further, quarks and gluons also condensed to produce what are, naturally enough, known as quark and gluon condensates. The existence of these is well established by experiments, and they play a very important role in our understanding of the strong nuclear force. In fact, it is this condensation that gives rise to the vast majority of the mass of protons and neutrons. The Higgs vacuum is, however, responsible for generating the observed masses for the elementary particles – the quarks, electrons, muons, taus and W and Z particles. The quark condensate kicks in to explain what happens when a cluster of quarks binds together to make a proton or a neutron. Interestingly, whilst the Higgs mechanism is relatively unimportant when it comes to explaining the mass of protons, neutrons and the heavier atomic nuclei, the converse is true when it comes to explaining the mass of the W and Z particles. For them, quark and gluon condensation would generate a mass of around 1 GeV in the absence of a Higgs particle, but their experimentally measured masses are closer to 100 times this. The LHC was designed to operate in the energy domain of the W and Z, where it can explore the mechanism responsible for their comparatively large masses. Whether that is the eagerly anticipated Higgs particle, or something hitherto undreamt of, only time and particle collisions will tell.
To put some rather surprising numbers on all of this, the energy stored up within 1 cubic metre of empty space as a result of quark and gluon condensation is a staggering 1035 joules, and the energy due to Higgs condensation is 100 times larger than this. Together, that’s the total amount of energy our Sun produces in 1,000 years. To be precise, this is ‘negative’ energy, because the vacuum is lower in energy than a Universe containing no particles at all. The negative energy arises because of the binding energy associated with the formation of the condensates, and is not by itself mysterious. It is no more glamorous than the fact that, in order to boil water (and reverse the phase transition from vapour to liquid), you have to put energy in.
What is mysterious, however, is that such a large and negative energy density in every square metre of empty space should, if taken at face value, generate a devastating expansion of the Universe such that no stars or people would ever form. The Universe would literally have blown itself apart moments after the Big Bang. This is what happens if we take the predictions for vacuum condensation from particle physics and plug them directly into Einstein’s equations for gravity, applied to the Universe at large. This heinous conundrum goes by the name of the cosmological constant problem and it remains one of the central problems in fundamental physics. Certainly it suggests that we should be very careful before claiming to really understand the nature of the vacuum and/or gravity. There is something absolutely fundamental that we do not yet understand.
With that sentence, we come to the end of our story because we’ve reached the edge of our knowledge. The domain of the known is not the arena of the research scientist. Quantum theory, as we observed at the beginning of this book, has a reputation for difficulty and downright contrary weirdness, exerting as it does a rather liberal grip on the behaviour of the particles of matter. But everything we’ve described, with the exception of this final chapter, is known and well understood. Following evidence rather than common sense, we are led to a theory that is manifestly able to describe a vast range of phenomena, from the sharp rainbows emitted by hot atoms to fusion within stars. Putting the theory to use led to the most important technological breakthrough of the twentieth century – the transistor – a device whose operation would be inexplicable without a quantum view of the world.
But quantum theory is far more than a mere explanatory triumph. In the forced marriage between quantum theory and relativity, anti-matter emerged as a theoretical necessity and was duly discovered. Spin, the fundamental property of subatomic particles that underpins the stability of atoms, was likewise a theoretical prediction required for the consistency of the theory. And now, in the second quantum century, the Large Hadron Collider voyages into the unknown to explore the vacuum itself. This is scientific progress; the gradual and careful construction of a legacy of explanation and prediction that changes the way we live. And this is what sets science apart from everything else. It isn’t simply another point of view – it reveals a reality that would be impossible to imagine, even for the possessor of the most tortured and surreal imagination. Science is the investigation of the real, and if the real seems surreal then so be it. There is no better demonstration of the power of the scientific method than quantum theory. Nobody could have come up with it without the most meticulous and detailed experiments, and the theoretical physicists who built it were able to suspend and jettison their deeply held and comforting beliefs in order to explain the evidence before them. Perhaps the conundrum of the vacuum energy signals a new quantum journey, perhaps the LHC will provide new and inexplicable data, and perhaps everything in this book will turn out to be an approximation to a much deeper picture – the exciting journey to understand our Quantum Universe continues.
When we began thinking about writing this book, we spent some time debating how to end it. We wanted to find a demonstration of the intellectual and practical power of quantum theory that would convince even the most sceptical reader that science really does describe, in exquisite detail, the workings of the world. We both agreed that there is such a demonstration, although it does involve some algebra – we have done our best to make it possible to follow the reasoning without scrutinizing the equations, but it does come with that warning. So, our book ends here, unless you want a little bit more: the most spectacular demonstration, we think, of the power of quantum theory. Good luck, and enjoy the ride.
Epilogue: the Death of Stars
When stars die, many end up as super-dense balls of nuclear matter intermingled with a sea of electrons, known as ‘white dwarves’. This will be the fate of our Sun when it runs out of nuclear fuel in around 5 billion years time. It will also be the fate of over 95% of the stars in our galaxy. Using nothing more than a pen, paper and a little thought, we can calculate the largest possible mass of these stars. The calculation, first performed by Subrahmanyan Chandrasekhar in 1930, uses quantum theory and relativity to make two very clear predictions. Firstly, that there should even be such a thing as a white dwarf star – a ball of matter held up against the crushing force of its own gravity by the Pauli Exclusion Principle. Secondly, that if we turn our attention from the piece of paper with our theoretical scribbles on it and gaze into the night sky then we should never see a white dwarf with a mass greater than 1.4 times the mass of our Sun. These are spectacularly audacious predictions.
Today, astronomers have catalogued around 10,000 white dwarf stars. The majority have masses around 0.6 solar masses, but the largest recorded mass is just under 1.4 solar masses. This single number, ‘1.4’, is a triumph of the scientific method. It relies on an understanding of nucle
ar physics, of quantum physics and of Einstein’s Theory of Special Relativity – an interlocking swathe of twentieth-century physics. Calculating it also requires the fundamental constants of Nature we’ve met in this book. By the end of this chapter, we will learn that the maximum mass is determined by the ratio
Look carefully at what we just wrote down: it depends on Planck’s constant, the speed of light, Newton’s gravitational constant and the mass of a proton. How wonderful it is that we should be able to predict the uppermost mass of a dying star using this combination of fundamental constants. The three-way combination of gravity, relativity and the quantum of action appearing in the ratio (hc/G)1/2 is called the Planck mass, and when we put the numbers in it works out at approximately 55 micrograms; roughly the mass of a grain of sand. So the Chandrasekhar mass is, rather astonishingly, obtained by contemplating two masses, one the size of a grain of sand and the other the mass of a single proton. From such tiny numbers emerges a new fundamental mass scale in Nature: the mass of a dying star.
We could present a very broad overview of how the Chandrasekhar mass comes about, but instead we’d like to do a little bit more: we’d like to describe the actual calculation because that is what really makes the spine tingle. We’ll fall short of actually computing the precise number (1.4 solar masses), but we will get close to it and see how professional physicists go about drawing profound conclusions using a sequence of carefully developed logical steps, invoking well-known physical principles along the way. There will be no leap of faith. Instead, we will keep a cool head and slowly and inexorably be drawn to the most exciting of conclusions.
Our starting point has to be: ‘what is a star?’ The visible Universe is, to a very good approximation, made up of hydrogen and helium, the two simplest elements formed in the first few minutes after the Big Bang. After around half a billion years of expansion, the Universe was cool enough for slightly denser regions in the gas clouds to start clumping together under their own gravity. These were the seeds of the galaxies, and within them, around smaller clumps, the first stars began to form.