Why String Theory?

by Joseph Conlon

  Calculation shows that this process should lead to observable levels of dark radiation. The origin of the moduli as gravitational, extra-dimensional modes makes their decays democratic. Gravity talks alike to big and small, and is the most democratic of all forces. This gives the exciting possibility that if experiments over the next decade achieve their claimed sensitivity, the existence of dark radiation may be established.

  Dark radiation represents one way that apparently inaccessible physics – moduli particles with gravitational-strength interactions that ceased to be present when the universe was a microsecond old – can leave observational traces that are measurable today using the technology of today.

  It is a cosmological analogue of a dinosaur footprint. The argument involves a chain of inferences, each of which is more than reasonable by itself. The conclusion involves a quantity whose existence can be determined, and where the experimental precision on the measurement will only improve. Over the next few years, observations from a variety of either balloon-based or ground-based telescopes may determine the existence of dark radiation. These telescopes observe the sky from the rarefied air of the Atacama desert in Chile – as for the Atatcama Cosmology Telescope – or from the pristine environment of the south Pole, as for the South Pole Telescope or the Spider balloon.

  If the existence of dark radiation is established, it will be a great triumph for physics. It would be a discovery equal to the realisation that dark matter exists. As with dark matter, it would reveal the existence of a new, unknown, contribution to the energy budget of the universe.

  Of course, it may not happen. The arguments here involve reasonable assumptions, but they are not guaranteed to be correct. The scientific landscape is littered with the desiccated corpses of ideas that once seemed appealing but could not command observational support. That is the way of the subject, and the normal operation of normal science.

  However, suppose dark radiation is found to exist. Can one do better? With dark matter, we know it is there as we can weigh its total effect, but we do not know what it is. So far, every attempt to detect dark matter directly has failed. While there are many candidate ideas, we do not know whether any are correct, and we have no positive experimental evidence to guide us.

  If it exists, a similar problem would occur for dark radiation. In everything I have just said, it was crucial that dark radiation particles could survive in the dense and crowded environment of the early universe. Particles that are produced at sub-microsecond times can only survive for fifteen billion years by not interacting. The same quality that ensures their survival makes them hard to detect – one cannot be simultaneously both a social butterfly and a ninja. The feebleness of their interactions allows them to survive to the present day, but also renders it difficult for us to know what they are.

  While we may be able to measure the total amount of dark radiation through weighing its overall gravitational effect on the expansion of the universe, it is hard for us to find out more. How can we effectively ask the question: who are you, and from where do you come?

  For one special type of particle, called either an axion or an axion-like particle, this problem may be evaded. While this type of particle has never yet been observed, it arises frequently in string theory.

  10.3 AXION-LIKE PARTICLES

  The axion is the first, and so far only, particle to be named after a brand of washing-up liquid.7 As mentioned in chapter 4, it originates as a proposed solution to a specific problem: why does the charge distribution of the neutron not have a preferred direction? The neutron, a bound combination of two down quarks and one up quark, is a neutral particle, and so can be viewed as smeared-out positive charge combined with smeared-out negative charge. To the limits of experimental accuracy, however, the centroid of the positive charge distribution and the centroid of the negative charge distribution are in exactly the same location. There is no weight of positive charge on one side balanced by negative charge on the other.

  This can only happen if one particular parameter – the so-called theta angle – in the equations of the strong nuclear force is equal to zero to within around one part in ten billion. This may not be impossible, but it does not conform to good taste.

  There is a simple and elegant solution to this problem. This simple and elegant solution was found by Robert Peccei and Helen Quinn in 1977 and is now called the Peccei-Quinn mechanism. In this solution, the theta angle is not regarded as a fixed constant that the Prime Mover initialised when booting up the universe. Instead, theta is treated as a dynamical entity that is free to change its value according to the prevailing conditions. Subtle effects of the strong nuclear force lead to a preferred value for theta – which is precisely zero.8 Whatever the initial value of theta, it always ends up at zero, precisely where observation requires. This turns a problem of fine-tuning into a problem analogous to why a compass needle always points north: it aligns itself with a field.

  Even after theta has relaxed to its equilibrium value of zero, it still has residual quantum mechanical fluctuations. Soon after the publication of Peccei and Quinn’s original paper, Steven Weinberg and Frank Wilczek showed independently that these residual fluctuations were mathematically equivalent to the existence of a new particle – the axion. The particle was called the axion because, like its ablutionary namesake, it washed away a problem, in this case the problem of the charge distribution of the neutron.

  Weinberg and Wilczek demonstrated certain properties of the axion. They showed that the axion is always light and that the axion always interacts weakly. Experiments were quickly devised to look for axions that were light and interacted weakly, but with no success. If the axion exists, it is not just light and weak, but extremely light and extremely weak. As it became apparent in the 1980s that the axion, if it existed, was invisible at any conventional experiment, the expression ‘invisible axion’ was coined for such particles.

  Since then searches for the ‘invisible axion’ have continued, using dedicated experiments and novel techniques that are radically different from banging particles together at high energies. While no successful detection has yet occurred,9 new experiments such as the International Axion Observatory continue to be built, and discovery may yet occur.

  From the perspective of a particle theorist, there are three defining features to the axion. The first feature is that the axion is light. The second feature is that the axion is very weakly coupled. The third and final feature is that the axion has a special interaction with the strong force, which enables it to solve the problem of the charge distribution of the neutron.

  There is a generalisation of this to consider particles with only the first two of these features. These particles are also extremely light – for the same reasons that the axion is light. They are also weakly coupled – for the same reasons that the axion is weakly coupled. However, they lack any relation to the strong nuclear force. Such particles are, for hopefully clear reasons, known as axion-like particles, while the definitive article associated to the axion is reserved for the particle that couples to the strong nuclear force.

  I now want to explain briefly why axions and axion-like particles are so light, although this more technical discussion can be skipped if desired. The mass for the axion is not larger than one billionth of the mass of the electron, and masses for axion-like particles can be far lighter still. What could keep these particles so light, indeed so much lighter than almost any other known particle?

  The danger here is that quantum fluctuations in general give large masses to particles. There needs to be a special and powerful reason to keep a particle a flyweight in a world of cruiserweights. This special reason comes from one of the most powerful principles in physics, introduced in chapter 3: the principle of symmetry. There is a symmetry in the equations for axionic particles such that, if it were exact, the axion mass would be exactly zero.10 Roughly, the mass of a particle tells you the energy that is associated to quantum fluctuations of that particle – the larger the
mass, the greater the energy for any given fluctuation. The symmetry implies that the energy of a fluctuation is independent of its size: this is only true if the particle is massless.

  If this symmetry were flawless, the axion would be perfectly massless. However this symmetry is not perfect and it has minor blemishes. Whilst it holds for almost all calculations, it is violated by a certain class of small effects (‘instanton’ effects). These Lilliputian, exponentially small effects are the only way for axions or axion-like particles to obtain a mass, forcing this mass to be very small in magnitude.

  The tiny masses of axion-like particles are one reason to be interested in them. As they are so light, their production is not obstructed by energetic requirements. They offer an attractive target to those who balk at the scale and cost of arranging thousands of superconducting magnets in underground tunnels.

  However, another reason to care about axion-like particles is that they arise ubiquitously in string theory. If string theory is a true theory of nature, axion-like particles should exist. Why?

  Our discussion of moduli emphasised the fact that within string theory, the observed particles in four dimensions are geometric guano from ten dimensions. Like pigeon poo from above the clouds, they are visible droppings from an extra-dimensional geometry that we cannot directly access.

  Axion-like particles share this origin: they are the four-dimensional legacies of a particular feature within the geometry of the additional six spatial dimensions. What is this feature? It is the presence of subspaces within this geometry that one cannot contract away to a point.

  How can you visualise these? One way is by imagining stretched spandex under tension. Its tension makes it want to collapse to a point, and if unconstrained it will shrink like a deflated balloon. However, if there is a surface it is wrapped around – imagine a bald man’s head in a morphsuit – it cannot collapse. Instead it ends up wrapping the surface and maintaining a finite area.

  The mathematical statement is that whenever such a non-contractible surface or hypersurface appears within the extra-dimensional geometry, an axion-like particle exists in the four-dimensional theory. In fact such surfaces are common, and particular geometries may even include several hundred of them, which would lead to several hundred different axion-like particles in our four-dimensional world – all very weakly interacting.

  I want to summarise the key points of the last few pages. Axion-like particles are naturally extremely light and with extremely weak interactions. While difficult to see because of these weak interactions, they are far lighter than more familiar particles, and so there is no energetic obstruction to producing them. Axion-like particles also occur generally in string theory. If string theory is indeed a true account of nature, such particles should exist.

  The relevance of these particles is that their properties make them ideal candidates to be dark radiation. As they are so light, they behave as relativistic particles and count as radiation. As they interact so weakly, they are dark and able to propagate freely through the universe without returning their energy to the visible sector. If axion-like particles are present, moduli are expected to decay to them at a similar rate to their decays to the particles of the Standard Model. Axion-like particles are realisations of our previous generalities: they represent actual candidate particles to be ‘gravitational neutrinos’.

  10.4 COSMIC MAGNETISM

  Axion-like particles are attractive candidates to be dark radiation. So what? None of this helps if such particles are permanently dark. Experimental verification is at the heart of science. Invisible fire-breathing dragons have the same status as invisible white unicorns, and invisible axions may have the same status as invisible justmadeupons.

  As the universe expands and becomes less dense, most dark particles become harder and harder to see. Sight requires photons, and in most cases photons require interaction. Particles need to find other particles in order to interact and become visible. The fewer particles around, the smaller the chance of any interaction occurring.

  What is also interesting about axion-like particles is that they are an important exception to this rule. While the visibility of most types of dark particle falls away with the growing age of the universe, for axion-like particles the visibility can, under particular circumstances, increase. This occurs because axion-like particles have the ability, in the right environment, to turn into photons. It is clear that if this happens they become instantly visible: photons are the quantum building blocks of light, and the means through which we see the universe.

  What is the right environment? It means here a sufficiently large magnetic field stretching over a sufficiently large distance. Provided the magnetic field points at right angles to the direction of travel of the axion-like particle, there is a small but finite probability for the axion-like particle to convert into a photon. While I assert this as a true statement, I will not justify it. There is no simple way to explain it. After several graduate-level courses in quantum field theory, it becomes ‘easy’ to compute the conversion probability. This probability grows with both the square of the magnetic field and the square of the length over which the magnetic field extends. To make conversion more likely, you need to embiggen either the size of the magnetic field or the length over which it stretches.

  This principle is already used by several experiments that search for axions. For example, at CERN there is an experiment called the CERN Axion Solar Telescope. It uses a large superconducting test magnet left over from the Large Hadron Collider, pointing the magnet at the sun. It is like a conventional telescope with the roof shut. The experiment is inside a closed building, and slowly rotates the magnet so that it is always tracking the position on the roof that matches that of the sun in the sky. The experiment searches for axions that may be produced in the deep interior of the sun. Such axions would stream freely through space to us, through the roof of the building, and then reconvert to photons within the interior of the magnet. LHC magnets are used to maximise the signal. These have large magnetic fields and, at ten metres, are also relatively long.

  Large magnetic fields help because conversion grows with the square of the magnetic field. However conversion also grows with the square of the field’s extent. If axion-like particles form part of dark radiation, they have streamed through deep space during the entire history of the universe. As the universe has aged and expanded, it has made structures that are elongated over enormous scales. The best known are galaxies, beautiful patterns etched on the sky and stretching over distances greater than one hundred thousand light years. There are still larger scales though. Galaxies have come together into galaxy clusters, assemblies of hundreds or thousands galaxies orbiting one other on distances of several million light years.

  Both these gargantuan objects contain ordered magnetic fields. Within our own Milky Way galaxy, the magnetic field approximately traces the pattern and orientation of the spiral arms. This magnetic field is a large-scale structure within the Milky Way, continuing to point in the same direction across thousands of light years. The same is true of galaxy clusters. Although galaxy clusters lack the beautiful ordered spirals that can occur in galaxies, they still contain large-scale magnetic fields, which retain their orientation even across distances as large as a hundred thousand light years.

  It is true that the magnetic fields in galaxies or clusters of galaxies are not large. They are over a million times smaller than the magnetic field found on the bumpers of a child’s toy train. However, a hundred thousand light years is a jolly large distance, and conversion depends not just on this distance but on its square. While the magnetic field of a cluster of galaxies is smaller than that of the CERN Axion Solar Telescope by a factor of around ten billion, it extends over a length larger by a factor of ten billion billion. Doing the mathematics, one finds that a cluster of galaxies is more efficient than the CERN Axion Solar Telescope at converting axions to photons by a factor of ten followed by eighteen zeroes.

  Some of the be
st axion convertors in the universe, then, have been provided by nature for free. The making of galaxies or galaxy clusters cost us nothing. It took large amounts of time and energy, but it happened a long time ago, and we did not need to wait for the making. The finished products of nature’s bounty are present in the sky and available for us to look at – we just need a telescope.

  Of all places in the universe, clusters of galaxies are most efficient at converting invisible axion-like particles into visible photons. They maximise the product of size and extent of magnetic field. If dark radiation exists in the form of axion-like particles streaming through the universe, as these particles pass through a cluster of galaxies they will convert and produce an excess of photons, correlated with regions of maximal magnetic field.

  It is this behaviour within a magnetic field that makes axion-like particles special – it is a distinctive property that applies to these particles and these particles alone. It is this behaviour within a magnetic field that makes axion-like particles predictive candidates for dark radiation. Given enough knowledge about the form of the magnetic field within galaxy clusters, it is possible to predict the structure and number of the photons that are produced. The determination of the magnetic field within a cluster of galaxies is also not exotic physics. It can be determined, albeit with errors and uncertainties, using other, known techniques.

  What are the energies of the photons that are produced? These come from the initial energies of the axion-like particles. The exact calculations are not for here, but the most important fact about these is that the energy of each individual axion-like particle must be much larger than the energy of each photon that makes up the cosmic microwave background. The reason for this comes from what happened at the time the axion-like particles were formed.