1. The Standard Model

You have to learn the rules of the game, and then you have to play better than anyone else - Albert Einstein

What has come to be known as the "Standard Model" (SM) of particle physics has been enormously successful [1]. This quantum theory of the fundamental particles and their interactions has been crafted through decades of developments, both major and minor, from a crucial interplay of theory and experiment. Particle physics has given us a description of nature so accurate as to agree with all experimental tests put to it. Without exaggeration, it is a triumph of human endeavour. But despite the accuracy and all the understanding that the Standard Model gives, from a certain perspective it is simply the foundation of a much grander and all encompassing theory that must describe nature to an even better degree. This is not in terms of calculation accuracy, for the accuracy of certain predictions is one thing, but a puzzle isn’t complete until every last hole is filled, and although the pieces of the Standard Model fit perfectly with one another, the bigger picture has some significant gaps remaining. It is accepted that the paradigm of accepted physical law is going to have to shift again.

So just what are the issues? Perhaps first and foremost is gravity. For such a familiar concept, indeed what is really the only fundamental force that is obvious on the human scale, in some respects, it continues to elude description of its behaviour on the smallest scales, the quantum realm. It has been 100 years now since Einstein’s general relativity revolutionised our description of gravity [2], indeed of the very relationship between space and energy, but it applies only to the cosmological scale, and says nothing of how gravity operates on a more fundamental level. The Standard Model, on the other hand, is a quantum theory that deals very well indeed with very small scales, but the shapes of the Standard Model puzzle pieces don’t match those of general relativity. Something has to give.

There are other questions rooted closer to the domain of the Standard Model itself though, without looking so far afield as the cosmological domain, which is perhaps the least surprising area that a quantum framework could be found to be at odds with. There are a number of things that seem somewhat arbitrary in the Standard Model, and arbitrary does not seem natural from a physics perspective. A description of nature should, one would hope, seem like the most natural thing there is, with no open questions as to why things are as they are. But currently this is just not so.

The last several decades have revealed what seem to be the fundamental particles of nature, in both their experimental appearance and in the rules that govern their interactions - the Standard Model. We find that all particles can be categorised in terms of the charges associated with the forces they obey, namely their electric charge (for the electromagnetic (EM) force), their weak charge (for the weak force), their colour charge (for the strong force), and their mass (the "charge" of the gravitational force, in some sense). One further important property is the so-called spin, which plays a key role in determining how particles interact with each other regardless of the forces they obey. The particles that make up matter each carry a half-integer multiple of the basic unit of spin, and the particles that transmit the force messages between matter particles each carry a whole-integer multiple. The former a referred to as "fermions" and the latter as "bosons".

An immediate mystery presents itself when the fermions are organised according to their charge properties. It is apparent that there are three generations of fermions, each generation being identical to the others in terms of their charges, but different only in relative mass. There are three particles that make up the familiar world, namely the electron the up-quark and the down-quark, all of which are from the lightest generation of fermions, along with the electron-neutrino which is also present in abundance, but which is not directly observed due to interacting only weakly. The other two generations are replicas of this lightest generation, but with different relative masses, and these particles decay to the lightest generation after a very short time, hence they are not seen in the stable state of the natural world. So why, then, are there three families of particles? Why not four? Or a hundred? The Standard Model cannot say. And why are the particle masses exactly as they are? Other than the approximate mass scaling, the Standard Model cannot answer this either.

The Standard Model describes the EM force, the weak force and the strong force in terms of an underlying symmetry that corresponds to the mathematical group SU(3)×SU(2)×U(1). So why are there only these forces, obeying this symmetry, along with gravity? Why not a more simple symmetry with fewer forces, or a more complex symmetry with more forces? And, aside from gravity as the other fundamental aspect of nature we are directly aware of, it is clear that there are three spatial dimensions and one of time. But dimensionality is a trick of mathematics, and there’s no known reason for the universe to have the dimensionality that is seen. So why is it so? Or is it even so? Perhaps this is only apparently so from our scale and perspective. The Standard Model leaves us hanging without comment here too.

Furthermore, the SM has at least 19 seemingly arbitrary values [3], a number of which seem to require very specific tuning for nature to work as it does. Slight deviations from these values would result in a universe that could not form galaxies, stars, planets and, ultimately, life. The SM does not explain why the various force charges and quantum numbers are as they are, not the electric charge, weak isospin, hyper charge, or colour charge that appear in the theory. There are three gauge couplings that dictate the strengths of the three SM forces, which seem completely independent and arbitrary. There is possibly a strong interaction parameter that may violate the combined operations of charge conjugation and parity. And there are a number of mixing parameters and angles that appear without any obvious reason.

For a theory that prides itself on being based on principles of symmetry, the Standard Model is certainly a bit out of shape, like a half-formed crystal. It could well be that this arbitrariness is just how nature is, nothing more to it than that. But this is an unsatisfactory way to accept things, and from what we believe we know of the character of physical law, it is also unlikely. Thankfully there are a number of options available that offer improvements, many based on the guiding principle that seems to be at the very heart of nature - symmetry.

2. Making Symmetry Super

I’m certainly not someone who believes Susy has to be right, I just can’t think of anything better - Stephen Martin

A physical conservation law is one that is invariant under a certain mathematical transformation. In other words, the laws of nature are rooted in symmetry, and it is this principle, firmly established by Emmy Noether in 1918 [4], that has become a sure guide in theoretical physics ever since. No more so has this been the case than in the development of the particle physics Standard Model, which has found layer upon layer of mathematical symmetry in the rules of the universe. There are essentially two types of symmetry in physics - "spacetime" and "internal" [5]. Spacetime symmetry is that which ensures the laws of nature are invariant for all positions and times in space, and so for all speeds too, which is necessarily limited by the speed of light. Internal symmetry is that associated with particle physics interactions, enabling the particular transitions to occur in the way that they do by conserving various currents of quantum numbers.

In the mathematics of the Standard Model there are fermion components and boson components, and one feature of the model is that, when manipulating these components, like can only be exchanged for like. That is, fermions can only be exchanged with fermions, and bosons can only be exchanged with bosons [6]. As fermions are identified with matter and bosons with force, this is in some sense unsurprising at first sight. However, the two types of particles are essentially very similar in many ways except for their spin. So there is no real reason to suppose that you couldn’t, by changing the spin type, change a fermion into a boson and vice versa, so that matter particles would become force particles and vice versa. This reversal of spin type is yet another type of symmetry, and it has, in fact, been proven to be the one and only fundamental symmetry that is not present in the existing Standard Model [7]. This is what is referred to as "Supersymmetry", or "Susy" for short.

Mathematically, the basic definition of the Susy operator Q is such that [8]:

Q|Boson⟩ = |Fermion⟩

Q|Fermion⟩ = |Boson⟩

In more technical terms, Q is in fact an anti-commuting fermionic spinor operator [8]. The Standard Model tells us that fermions have so-called chiral behaviour, that is that they can be distinguished by properties termed as left-handed and right-handed, each type behaving differently under gauge transformations. As such, the following commutation relations must be obeyed, where α and β are spin indices (un-dotted meaning the first two components of a Dirac bi-spinor and dotted meaning the last two components), σ^μ are the Pauli spin matrices, and p_μ is relativistic 4-momentum [5, 8]:

Applying two successive Susy transformations on a certain type of particle returns the original type of particle, but shifted in spacetime. We have that the commutator of two Susy transformations is therefore a spacetime translation [6]. This all goes to show a deep connection between supersymmetry and spacetime, that supersymmetry in fact "extends the concept of spacetime itself" by joining internal and spacetime symmetries, which is a theoretically appealing prospect [3, 9]. This would seem to go against what is known as the Coleman-Mandula theorem, but that work in its original form was designed only for bosons, and a modified version that caters also for fermions provides Susy with the justification it needs [3, 9]. The mathematics of relativistic spacetime symmetry is known as "Poincaré algebra", which can be extended to cater for supersymmetric effects and then dubbed "super-Poincaré algebra" [5].

Further from the basic notion of changing the spin type of a particle, the real symmetry involved is present in the particles’ quantum field description. The Susy algebra and the Susy field theory were originally created independently of each other, and did not resemble the modern theory in their first casting [8]. As with other quantum field theories, which Susy must merge with and extend, the symmetry is in the action formula, or, equivalently, in the Lagrangian density which is integrated over spacetime to give the action, and from there the equations of motion are given and we are back to a more explicit description of physics. With the Standard Model Lagrangian as it is, exchanging the role of fermionic and bosonic terms does not yield an equivalent expression - the Lagrangian is not invariant under this transformation. To make it so we must introduce new fermionic and bosonic terms, which transform in conjunction with the original terms in just such a way as to give the desired invariance. In other words, such an expression is then symmetric under the exchange of roles taken by the different spin types, which is the essence of supersymmetry [10].

It is important to realise that supersymmetry is a principle rather than a theory, in that there are many theories in which it may be implemented, with the disproval of one or more such theories not being the same as the disproval of Susy itself [10]. Indeed, there are theories which use Susy as a building block with the intention of describing something else, as opposed to simply being geared towards Susy itself. First and foremost of these are superstring theory and the closely related M-theory, which offer the best and actually the only current option that could be a Theory Of Everything (TOE), describing all of the forces and particles in a single framework [3, 5]. Supersymmetry is an essential aspect of these ideas, meaning that if they are to be the much sought ultimate description of nature, then nature must be supersymmetrical.

Regardless of its role as a possible feature of the universe, the mathematics of supersymmetry is interesting in its own right. The concept of "superfields" existing in "superspace" was developed to condense the Susy maths, and although being a further level of abstraction that is not absolutely necessary, as such, and despite the apparent complexity, the superspace approach is actually simpler to deal with when working on Susy problems for real [5, 8]. The superspace approach also makes the supersymmetric transformations manifestly invariant, rather than this being imposed [8]. This would indicate, if Susy is found to be true of nature, that superspace may not just be a mathematical convenience but rather a higher level of reality.

From all considerations it can be seen that supersymmetry is a wide reaching concept, and would be a true revolution in physics if proven correct. To quote theorist John Schwarz, if Susy is found then "this would be one of the most profound achievements in the history of humankind. It would be more profound, in my opinion, than the discovery of life on Mars" [11].

3. Reflections

It has long been an axiom of mine that the little things are infinitely the most important - Arthur Conan Doyle

As far as the Standard Model is concerned, there is no real benefit from being supersymmetric per sé, but the extra symmetry brings with it extra particle states, and it is these states that do make a significant difference to the calculations. The effect of the extra particles is so significant, and altogether so beneficial, that many hold on to the hope that supersymmetry will yet be proven true despite not one of the extra particles having been found experimentally, as of early 2015.

For every input fermion to the Susy operator, there is predicted to be a unique output boson partner, and vice versa for input bosons. The list of known particles currently accepted as fundamental then doubles under the Susy principle. Not that this is justification for the truth of Susy, but this is reminiscent of the history of antimatter. The development of the Dirac fermion equation in 1928 suggested the existence of new particles that had opposite charges to those that were known at the time, but that were otherwise equivalent [13, 14]. This had no basis in the experimental data available when the theory was produced, but these particles were indeed subsequently discovered and labelled as antimatter, thereby doubling the number of known fundamental entities. In fact the Susy situation is almost exactly parallel to the antimatter situation from another consideration too. The Dirac equation arises from a coupling of quantum mechanics and special relativity, and supersymmetry, in its most desirable incarnation, allows for a coupling of quantum mechanics and general relativity [15]. This point will be returned to later, the message here being one of a real world historical symmetry that may be unfolding before our eyes.

Figure 1: Standard Model and basic Susy particles [12]

The Susy particles come with the practical issue of requiring names. Already having been carried away with the naming of the Standard Model particles, those in the naming department kept it simpler for Susy. For each of the SM fermions, the corresponding Susy boson takes the same name but with a prefix "s", which stands for "scalar" rather than "supersymmetric", for these particles are spin-0 just like the currently known Higgs boson [8]. And for each of the SM bosons, the corresponding Susy fermion takes the same name but with a suffix "ino", with any awkward sounds given a bit of rounding off. We then have that SM fermion "quarks" translate to Susy boson "squarks", with the "top" going to the "stop", for example; "leptons" go to "sleptons", with the "electron" going to the "selectron", for example; "photon" goes to "photino"; "gluon" goes to "gluino", and so on. Furthermore, the Higgs particle is taken to have five variants on the SM side in supersymmetric theory, and there can be quantum mechanical mixing of the Susy versions of these particles with the other neutral Susy bosons, giving "neutralinos", and with the charged Susy bosons, giving "charginos". Collectively, all of the Susy particles are referred to as "sparticles", which are subdivided in to "sfermions" (Susy fermions) and "sbosons" (Susy bosons) i.e. here the "s" does stand for "supersymmetric". Just to be clear, the terms "particlino" and "bosino" for the Susy bosons are not used, even though that might be thought to follow from the naming rules previously stated.

Figure 2: Supersymmetric particle labels 1 [8]
Figure 3: Supersymmetric particle labels 2 [8]

This one-to-one correspondence of SM particles to Susy particles gives the very minimum particle set needed for supersymmetry, and this is referred to as the "Minimum Supersymmetric Standard Model", or "MSSM" [3]. The gauge interactions of the MSSM are the same as those of the Standard Model, and the Yukawa interactions are similar. There are, however, other Susy variants that propose more than one superpartner for every current SM particle, for example with each SM particle associated with a supersymmetric pair, so giving a one-to-two correspondence between SM and Susy [16]. The next most complicated variant of Susy is termed the "Next to Minimal Supersymmetric Standard Model", or "NMSSM", or even "(M+1)SSM", and so it goes [8]. Whatever the number of supersymmetric particles, the important thing is that each Susy particle differs from its SM counterpart in spin type, but is otherwise identical in all other quantum numbers.

What is not a quantum number, though, is mass, which is a good thing. Theoretically it would not be a problem if SM masses and Susy masses were identical, just like the non-spin quantum numbers. But this would mean that Susy particles would already have been copiously produced and likely seen in particle accelerators for many years now, but this is very much not the case. There have been precisely zero confirmed observations of supersymmetric particles, meaning that, if they exist at all, they must each be heavier than their SM partner and have yet to be produced. Either that or their production rate is far less common than predicted by theory, in which case their signals are simply being swamped and hidden by other signals, but this is thought to be less likely than the mass explanation. Supersymmetry, if it exists, must therefore not be so symmetrical regarding mass, and is said to be a "broken" symmetry [16–18].

Symmetries require some mechanism to break them, there has to be a reason for a broken symmetry to be in such a state, which usually comes down to reasons of energy. If a complete symmetry can go to an overall lower energy state by becoming a broken symmetry, then it will do so as long as there is nothing to inhibit the breaking. It is not known what the exact breaking mechanism could be in the case of Susy, which is a prominent question and one of the most crucial outstanding tasks in theoretical development [8]. What is known is that the breaking would appear as extra terms in the Lagrangian density, aside from those that enable the Susy transformation in the first place [8]:

Deducing what the breaking terms must be is work in progress. There is enough of a pattern to suggest that the breaking is not arbitrary [8], and that it fits with being "spontaneous" breaking due to a non-supersymmetric vacuum state [3]. This is analogous to the Higgs breaking mechanism of the electroweak force. In the Higgs mechanism, a massless Goldstone boson interacts with a gauge boson to gain mass and become the Higgs boson, which then interacts with the electroweak bosons in different ways, breaking the symmetry of this interaction and giving rise to what are seen as separate EM and weak interactions. It is thought that a similar mechanism could be at play in the Susy case, whereby the Susy equivalent (which is not the Susy partner) of the Goldstone boson, namely a Goldstone fermion or "Goldstino", interacts with a gravitino to gain mass and become the Susy equivalent (again, not the Susy partner) of the Higgs boson. This particle would then act differently with SM and Susy particles, thus bestowing each SM particle with a different mass to its Susy partner or partners, and the symmetry is broken [3].

It turns out that this breaking mechanism is only valid for Susy theories that are locally symmetric, rather than only globally symmetric [3]. It also turns out that no other mechanisms work quite as well. In fact, there are no other alternatives that could even work at all for globally symmetric theories, so we simply must use locally symmetric theories, and we are strongly encouraged to use the above mechanism [3].

Going to a local symmetry, another interesting this happens - the equations of general relativity are automatically invoked [3, 8, 19]. So from particle physics motivations the path naturally leads us to consider gravity, which is theoretically appealing, to say the least. The resulting system that involves all these elements is, of course, called "Super- gravity" or "Sugra". That’s not to say that this is a quantum theory of gravity though, but such can be given by incorporating string theory, leading to "Superstring" theory or, going up one final notch, to the closely related "M-theory", which is seen as an overarching framework. Superstring/M-theory is the best, and some would say the only realistic option on the table for a "Theory Of Everything" (TOE) that couples all the known forces [3]. It is this connection, to what could be said to be the ultimate goal of theoretical physics, that some regard as the most important role of Susy, which is a key component [11]. Furthermore, just to cap things off, the introduction of string theory into the mix offers a possible explanation as to why there are three and only three generations of SM fermions, which was one of the original set of outstanding issues of the Standard Model.

It is gravity considerations that put further restrictions on Susy, in terms of its dimensionality N. The Standard Model is an N = 0 theory, but there are a number of Susy variants with N > 0, with N > 1 theories referred to as "extended" Susy [5]. For renormalisation we must have Susy with N ≤ 4, but we can still consider non-renormalisable Susy of higher N as effective field theories. There is a harder restriction that suggests we must have N ≤ 8 though, which is the limit for having only one type of graviton as is believed to be so, and otherwise for N > 8 we would have to consider massless particles with a helicity modulus greater than 2, which is not theoretically desirable and strongly believed not to exist [5].

All of these more involved considerations are not without further abstraction, going beyond what was needed with our original goal of simply trying to invoke the Susy principle. Superstring theory requires extra spatial dimensions, that are not observed on the macroscopic scale due to being "Kaluza-Klein compactified", which makes them only apparent on the quantum scale, essentially as an extra quantum degree of freedom [5]. M-theory requires 11 extra dimensions, for example, and if this is ever proven to be true, then Susy itself won’t seem anything like a far fetched idea at all from a future perspective.

Slowly but surely, we can see that through this process of model extension, abstraction and feedback, a number of issues become mutually tied and possibly solved. Regardless of the above grand ideas and extrapolation to further models and their own benefits, the direct benefits of Susy itself have actually yet to be discussed. There are three main points to be made, each of which are significant in their own right, and if they can together be struck down by one supersymmetric stone then all the better.

4. Susy’s Promise

I don’t know where I’m going from here, but I promise it won’t be boring - David Bowie

Why do the four fundamental interactions behave so differently, and in particular why do they have such different strengths? The greatest jump between force strengths is that between gravity and the weak force, the former being some 10^{32} orders of magnitude weaker than the latter, which is truly a colossal difference [20]. The question should perhaps not be why gravity is so weak but, despite the misleading nomenclature, why the weak force is so strong.

There are only a handful of fundamental constants in nature: the speed of light c, the Planck constant h, and the gravitational constant G. From various mixtures of these constants we can produce values with any combination of the basic units of mass length and time, which are therefore the values associated with those units that you would think nature would be interested in, coming as they do straight from its own basic tool set. These values are referred to as the Planck mass, the Planck length and the Planck time, respectively.

Why the masses of the SM fermions are exactly what they are is not well understood, which is one of the shortcomings of the Standard Model, but it is understood why these particles occupy the particular mass range that they do, which comes down to each particle’s mass being subject to quantum loop corrections from other SM particles [21]. An open question, though, lies with the now known mass of the Higgs particle [22, 23], as at ~125GeV it is much lighter than it should be based on calculations involving SM loop corrections only, which places it on the order of 10^{15}GeV, which is on the scale of the Planck mass [24]. And it is the Higgs mass which in turn dictates the W and Z boson masses, and that affects the range, and therefore the strength, of the weak force. The lightness of the Higgs particle, in other words, allows the weak force to be as strong as it is. A much heavier Higgs particle, giving much heavier W and Z particles and so a much smaller range and strength of the weak force, seems like the more natural state that should be. But this is not so, and it is difficult to exaggerate just how disparate the hierarchy of force strengths is.

This is referred to as the "hierarchy problem" [3]. The Higgs particle is the only particle whose mass is not even approximately determined by use of SM quantum loop corrections, and there is no other apparent mechanism for its known value to arise with known physics, which means that its value seems to be set arbitrarily. This is one of a number of so-called fine-tuning problems of the Standard Model, namely that certain values appear in the theory and give us the physics that we experience, and that if they were not much different then our very existence could not even be so. For example, if the Higgs mass were about five times its known value of ~125GeV, then the formation of atoms other than hydrogen would be completely suppressed [25]. Without explicit mechanisms that demand all values to be as they are, some definite clockwork in which all components mutually justify one another, then things just don’t seem natural.

Enter supersymmetry. Bosonic quantum loop corrections have the opposite effect on a particle’s mass to fermionic quantum loop corrections, so if Susy exists, then for every SM fermion that acts to drive the Higgs mass up, there would be at least one Susy boson that acts to bring it back down [24, 26, 27]. If Susy were an exact and unbroken symmetry, with each SM fermion having a one-to-one correspondence with a Susy boson of equal mass, then the loop contributions of all the SM-Susy pairs would exactly cancel, leaving the Higgs mass dictated by self interactions only [19, 24]. As already discussed, it is highly unlikely that Susy is an unbroken symmetry if it does exist, but rather that each SM particle has a different mass from its Susy counterpart. This would mean that it is the residual difference in the SM-fermion and Susy-boson loop corrections that contributes to the Higgs mass that we see.

Figure 4: Higgs mass loop corrections. Left: SM-fermion loop; Right: Counteracting Susy-boson loop. [8]

This actually gives us a predictive handle on what the masses of the Susy particles might be, at least in terms of the approximate range. We know the approximate range that the Higgs mass should be driven to by SM-fermion loop corrections, and so we can calculate the opposing effect from Susy-boson loop corrections as we now have a known target Higgs mass to aim for. While not resulting in precise values for the masses of each Susy particle one by one, the total of the full set can be obtained in this way, and then some educated guesswork and averaging lets us divide that total amongst the individual particles.

Even before the Higgs particle was discovered it was known what its approximate mass should be, from considering the known masses of the W and Z particles, so the Susy masses could also still be guessed at even then. In fact the Higgs and Susy masses were coupled predictions, with a particular Susy model leading to a particular Higgs prediction as well as the other way around. The MSSM, for example, gave the Higgs mass an upper bound of ~135GeV and a lower bound of around the Z mass at ~90GeV [8]. If the Higgs mass was found to be towards the middle of this range at around ~115GeV, it would have been very strong evidence for supersymmetry being at play, or at least some other beyond-the-standard-model (BSM) physics with very similar effects [3]. Finding the Higgs mass of ~125GeV does not work so well in Susy’s favour, being as it is towards the upper bound of the MSSM prediction [28].

All things considered, what is known is that the Susy masses cannot be too dissimilar from their SM partners, that is they cannot differ by multiple orders of magnitude if they are responsible for bringing the Higgs mass into line. The greater the Susy particle masses then the greater the degree to which Susy is said to be broken, and the less the effect of its beneficial reduction of the Higgs mass [20]. We say that Susy should not be broken "too badly", or that it is only broken "softly", for the Susy particles to sit in the range required for good Higgs correction [29, 30].

In terms of the maths, as the breaking mechanism itself is presently unknown, the breaking terms in the Lagrangian density must be inserted "by hand" in order to give the desired effects. These manually implemented terms are not themselves supersymmetric though, and the greater the mass of the particles they represent then the less supersymmetric they are. This returns us to the same problem of not having Lagrangian invariance under the exchange of fermionic and bosonic terms, and the whole point of the exercise is defeated [30]. In other words, the greater the difference in the SM and Susy particle masses, then the more that Susy is broken, and the more the equations literally break in terms of what we want them to do. So we need "soft" breaking not only for giving an acceptable Higgs mass correction, but also to retain the very essence of supersymmetry.

For things to work out for the best, Susy particles should have masses between 100 and 1000 times the proton mass, with a "sweet spot" between ~0.8TeV and ~1TeV [11, 17, 26, 31]. Whatever the exact predictions, most Susy models agree that all Susy masses should at least be within an order of magnitude of each other [8]. But none of these predictions are guaranteed, as it may well be that other factors are at play with the Higgs mass. There are no certain lower or upper Susy mass limits, the latter point meaning that Susy is unlikely to be experimentally falsified in any absolute sense, only ruled out as unlikely if more and more powerful machines are built and Susy particles persist in their absence. Only if the Planck energy scale was ever probed could there be a definitive word on Susy, or on physics itself for that matter. As previously mentioned, lack of a supersymmetric signal could always be due to the event rates being so low as to not be noticed, rather than not being present at all for want of greater energy, and this too is a persistent issue that cannot be absolutely ruled out. Experimental particle physics is, at the end of the day, a statistical game, and for many things you can always say there is a chance, however low that may be.

Not to lose hope at this stage though, there are other things to consider that support the existence of Susy, with particles in just the same range as that required for the Higgs correction as described. We know that the EM and weak forces can be unified at a certain energy scale, above which they essentially act as one, and below which they differ due to electroweak symmetry breaking via the Higgs mechanism. So going to an even higher energy scale, could such unification be true of all the forces? Could the electroweak force merge with the strong force, at least, and perhaps even with gravity? These are questions that are answered in terms of force coupling strength as a function of interaction energy, as these quantities are not actually independent, although this is approximately so at low energies. If we look at the running of the force strengths with energy, we see that the EM force does indeed match both the weak force and the strong force at certain points, but these points are not coincident under Standard Model conditions. Again, like the Higgs mass, these trends are affected by quantum loop corrections, and when the additional corrections for some Susy variants are taken into account something remarkable happens - the three force strengths do then match at ~10^{16}GeV [11]. It doesn't have to be so that all forces match in this way, but that it just so happens to occur for the same level of Susy corrections as needed for the Higgs mass, to many this seems more than a little coincidental. One more point to Susy, perhaps.

Figure 5: Inverse force coupling strengths (relative scale). Dashed lines: Without Susy; Solid lines: With Susy. [8]

One catch with supersymmetry is that, in many models, it forces us to impose an additional conservation rule that isn't automatically invoked. This is because certain processes could otherwise occur that we know aren't true. It is known that the proton must be stable with a lifetime of at least 5.9×10^{33} years [32], but Susy in its unrestricted form would allow channels by which the proton could decay much more rapidly, with a lifetime on the order of a second [33]. To inhibit such processes, it has been proposed that there is a further conserved quantum number that differs between regular SM particles and Susy particles, called "R-parity" or "matter parity" [8, 26]. By definition, the regular SM particles have an R-number of +1, and the Susy particles have an R-number of -1, and, rather than being an additive conservation rule, R-parity works multiplicatively, with the product of R-numbers at each Feynman vertex of a process demanded to be +1 [19, 26]. With B being baryon number, L being lepton number, and S being spin number, the R-number for any particle is given by [3, 19, 30]:

It can be seen immediately that processes involving non-Susy particles automatically satisfy R-parity conservation, as all R-numbers are positive. The consequence for Susy particles themselves is that there must always be an even number of them joined to any one vertex, which essentially means that: 1) they are only produced in pairs; 2) they decay to an odd number of Susy particles (usually one) and an arbitrary number of SM particles; and 3) the lightest supersymmetric particle (LSP) is stable, as there are no other Susy particles that it can decay to [8, 26, 30, 34]. The ultimate consequence of R-parity conservation, then, is that there should be at least one or two immortal LSPs in the universe. Actually, if it exists at all, then the LSP should be present in copious quantities, as relics from the big bang when both SM and Susy particles would have been prevalent [3]. This means that the LSP must not be an electrically or strongly interacting particle, otherwise its effects would be readily apparent today [34]. So if it exists then it can only be weakly and/or gravitationally interacting, and that comes down to it being either: a) the sneutrino; b) the lightest neutralino; or c) the gravitino. The sneutrino has actually been ruled out by both terrestrial experiments and cosmological observations, and although the gravitino is not ruled out as a possibility, it is the lightest neutralino (Higgsino + neutral-electroweak-Susy-boson admixture) which is the favourite candidate [3, 8, 26, 30].

Figure 6: Proton decay via the supersymmetric particle \tilde{S} if R-parity is violated [8]

A particle such as the LSP nicely fits the bill for solving yet another physics mystery - dark matter. Observations of galactic rotation curves, gravitational lensing, and the Cosmic Microwave Background Radiation (CMBR), are all suggestive of the universe containing a large quantity of gravitationally interacting, long-lived, non-baryonic particles [19]. The proposed LSP big bang relic density, again for the sort of Susy particle mass range suggested by the unification and Higgs particle corrections, fits well with the idea that the LSP is indeed dark matter [3, 11]. Susy scores again.

On many theoretical fronts Susy appears to be winning, and everything looks like it could fit very well together, with resolved issues spanning from the quantum to the cosmological. But it is always experiment that has the final say, as it doesn't matter how good something looks on paper at the end of the day - if it's not true of nature, then that's that. And experiment, so far, has not had much to say about Susy at all.

5. Hide and Seek

There is no certainty, there is only adventure - Roberto Assagioli

There have been many experiments over the years looking for signs of supersymmetry, but the best that has been given are particle energy ranges that have been ruled out as highly unlikely, rather than any kind of positive sighting. Experiments that have been involved in such work include the Brookhaven anomalous muon magnetic moment experiment [35], the WMAP satellite [36], the XENON-100 dark matter experiment [37], the Tevatron collider at Fermilab [38], and the LEP, SPS and LHC colliders at CERN [39]. At the time of writing CERN is preparing for run-II of LHC experiments, and what is clear from run-I data is that most of the best studied Susy theories are now practically off the cards [40].

However, the number of theories that are or are not options is not the most important thing, as only one correct theory is needed. It must be realised that we have yet to explore most of the energy range that is applicable to Susy, which is the crucial point, and this offers hope yet for the principle to be discovered in some guise or another [3]. Indeed, there are 15 orders of magnitude from current experimental energies to the Planck scale at ~2.4×10^{18}GeV, leaving room to explore far beyond the LHC capabilities, and it is almost certain that there is more to be found in this energy range [8]. It would be unsurprising to perhaps find a few surprises too, and it has been said that even if Susy is not found, it is likely for something beyond current understanding to be found that satisfies the same issues [17].

It could be that Susy events are so infrequent that even at the LHC they won’t stand out and won’t be seen, even if they are present. But calculations are possible that predict what the Susy event rates should be for a given Susy mass spectrum, and it has been claimed that if Susy exists within the scope of LHC energies that finding signs of it should be "easy" [19]. Some would even say that the LHC will "almost certainly" find direct evidence for Susy if it is the reason behind the hierarchy problem, that is to say if it is the reason for the low Higgs mass [8]. Not all Susy particles may be found at the LHC, but if any at all are, studies suggest that the squarks and gluinos will at least be seen, even if not some other entities such as neutralinos, charginos and sleptons [3]. It was studies like this that suggested the lightest Higgs boson would be seen, as was the case. But there are also some predictive models that say no Susy particles will be seen at all, nor will any other types of heavier Higgs boson. No-one can say for sure what will happen when the run-II data starts to flow later in 2015, and we will just have to wait and see.

What could we potentially see though? What sort of signals would distinguish Susy events from the multitude of other events? There are two main search categories: 1) free Susy particles; and 2) Susy quantum loop corrections of SM particles. Let’s look at the former type first. As discussed previously, all Susy particles except the LSP itself will decay to the LSP, providing that R-parity is conserved. The LSP, being only weakly interacting, will behave very much like a neutrino and so will pass right through the detector without direct trace, only being noticeable from the missing momentum when the accounting for the event is done [31, 34]. So a first thing to look for is events with missing momentum - but there are plenty of those already with the known neutrino events. However, what would distinguish an LSP event from a neutrino event is what’s on the other side of the missing momentum, as such Susy events are expected to be rich in high energy SM quark and gluon jets, which come from the end of complex Susy decay chains that are not directly seen, as they would occur within the beam-pipe and not the active regions of the detector [19, 31, 34].

Another class of free Susy signal would be given by events that are not individually distinguishable from SM events, but that can be distinguished statistically. The wealth of SM calculations and data collected thus far tell us very well how often certain known final states occur, so any deviations from this existing knowledge suggests new processes being at play. For example, if any b-quarks are found in the final state of an SM-only event, it is likely that this will be as a single bb pair. It is possible for some SM-only events to result in two bb pairs, although this is much less common than the single pair outcome. But there are Susy processes in which the double bb final state is relatively frequent, due to the frequency with which gluinos decay to quarks, and so an increase of this type of outcome would be strongly suggestive of Susy being involved [41]. Furthermore, for all types of free Susy searches, a key telltale sign is of course the spin, which can be inferred from the angular distributions of the final state particles with respect to the beam axis [8]. In conjunction with the other readings mentioned, a corroborating spin would act very strongly in Susy’s favour.

The Higgs mass and the force strengths are not the only things that may be affected by Susy quantum loop corrections, which brings us to the second type of Susy search. In the previous type of search the root process being sought was Susy based, only finishing with an SM state that could be used to infer the root process. But we can also scrutinise processes that we are sure both begin and end only with SM particles, and now look for statistical differences that could be attributed to Susy quantum loop corrections. Calculations show that even such SM-only processes would be affected by Susy quantum corrections if Susy exists at all, and that with enough data the difference would be seen when compared to the Standard Model prediction. This would be picked up on most significantly with processes that are otherwise very rare under SM conditions alone, as well as in precision measurements like those needed for the magnetic moment of the muon [19].

We can see that there are plenty of ways in which Susy may be seen, and with the coming jump in LHC centre of mass energy from the run-I 8TeV to the run-II 13TeV, and, importantly, with the increased luminosity too, the near future is sure to be an interesting time for the Susy search. If Susy is found, then there are a number of predictions about the relationships between the new particles that can be tested. For example, one prediction is that, unlike the trend in the SM particles where the tau is the heaviest of the leptons, the tau Susy equivalent of the stau is actually thought to be the lightest of the sleptons, and whereas the top and bottom are the heaviest of the quarks, the stop and sbottom are actually thought to be the lightest of the squarks [8]. What is retained is the mass relationship between leptons and quarks, with the sleptons thought to be lighter than the squarks [8]. All of this and more will be put to the test if Susy shows up.

6. The Past

Those who do not remember the past are condemned to repeat it - George Santayana

The ideas of supersymmetry arose as a natural consequence of particle physics development. There were a number of complimentary and, in some cases, virtually identical ideas that were produced independently of one another, suggesting that the concepts presented thus far were somewhat inevitable. The body of work related to supersymmetry constitutes over 10,000 scientific publications to date [10, 24], so it is a concept that has clearly already made its mark on history. This section gives a brief and by no means complete overview of some of the main steps that led to where we are today, in terms of their historical order. As to whether or not supersymmetry has such a prominent position in the future remains to be seen.


  • John Archibald Wheeler introduced the S-matrix, a tool for calculating scattering and decay cross-sections [42]


  • Werner Heisenberg further developed S-matrix theory [43]


  • S-matrix theory was influential

  • Symmetry concepts came to be more appreciated as powerful theoretical tools, particularly with the SU(2)×U(1) group that was realised for electroweak interactions and the SU(3) group that was realised for strong interactions. It was seen that particles could be organised in SU(3) multiplets, in which all particles had the same spin type, but questions started to be asked about what it would mean to have larger multiplets of mixed spin type [5].


  • Philip Anderson proposed non-relativistic electroweak symmetry breaking (EWSB) [44]


  • Peter Higgs proposed relativistic EWSB [45]


  • Hironari Miyazawa first proposed the basic ideas of supersymmetry, but the work had little impact at the time [46]

  • Supersymmetry was proven to be the only realistic extra symmetry that could be added to those of relativity [16]

  • Theorists tried to merge internal and spacetime symmetries [3]


  • Coleman and Mandula showed that more than just bosonic charges were needed for non-trivial combinations of internal and external symmetries [3, 47]. This became known as the "no-go theorem" [5].

1968 - 1971

  • Gol’fand and Likhtman formally developed supersymmetric ideas in Russia, giving the first non-trivial use of fermionic charges to extend Poincaré algebra via spinor generators [3, 5, 48]. The work was unknown to the wider world at the time due to Cold War communication restrictions [6].


  • S-matrix theory lost popularity in favour of QCD, which better described strong interactions. S-matrix theory did not go quietly though, and led to the ideas of string theory, which was specifically only for bosons at first but associated with quark confinement.


  • Neveu and Schwarz [49], Gervais and Sakita [50], also Ramond [51], independently made 2D Susy models trying to make baryonic fermion string theories [3, 5]

1972 - 1974

  • Wess and Zumino created supersymmetry independently of Gol’fand and Likhtman, again from a string theory basis, with the sole intention of unifying the descriptions of fermions and bosons, rather than trying to solve any particular problem [3, 6, 15, 52]. This theory, a renormalisable Susy QFT in 4D spacetime for mesons and baryons, is often regarded as the true beginning of the supersymmetry we know today [3, 5]. Soon after its development it was realised, in work with Iliopoulos and Ferrara [53, 54], that field theories did not suffer from certain divergences when coupled with Susy in this way [3]. Although it wasn’t picked up on at the time, this is the root cause of the beneficial Susy Higgs mass correction [3].


  • Volkov and Akulov used a 4D non-linear Susy approach, in conjunction with the concept of massless goldstone particles, to describe a massless fermion they hoped would represent the neutrino [3, 5, 55]. Their intention was not satisfied, but this was nonetheless an important step in Susy’s history [3].


  • Haag, Lopuszanski and Sohnius generalised the Coleman-Mandula no-go theorem to include spinor generators [3, 5, 56]

  • Pran Nath and Dick Arnowitt developed "gauge supersymmetry", the first local Susy theory [57]


  • Freedman, Van Nieuwenhuizen and Ferrara created the first theory of supergravity, and it was hoped that the resulting graviton would be able to join the framework of lower spin particles [3, 58]


  • Fayet and Ferrara proposed the first realistic supersymmetric Standard Model, now called the MSSM, in order to explain the hierarchy problem. This theory led to predicting superpartner masses between 100GeV and 1TeV [59].


  • Grisaru, Siegal and Rocek used supergraphs (Susy Feynman diagrams) to further determine the properties of Susy quantum corrections [5, 60]


  • Most physicists believed that superpartners would only be slightly heavier than SM particles [20]


  • Cremmer, Ferrara, Girardello and Van Proeyen coupled supergravity and matter via brute force tensor calculus methods [5, 61]

7. The Future

Only those who will risk going too far can possibly find out how far one can go - T.S. Eliot

If Susy is positively identified as a factor of the statistics from the LHC, then the work does not end there. The new particles will all require proper dedicated study on a more refined level than simple identification, looking carefully at the kinematics to produce better mass values, cross sections, decay ratios and so on, and comparing everything with theory to shore up any gaps in understanding [8]. Looking further ahead, precision measurements will benefit from having a lepton collider, possibly linear, in order to provide the cleanest collisions and data [3]. Predicting anything about the future of particle physics after the LHC is not really possible at this stage though, as theory has taken us as far as possible and the coming data are expected to dictate what direction things will take.

That’s not to say that there are no more theoretical ideas, there are plenty, but some contrast so much that they cannot be mutually compatible, and grounding in the reality of experiment is needed to allow the necessary evolution of ideas to occur. As previously mentioned, the need for Susy in superstring/M-theory is sometimes hailed as one of Susy’s best features, and perhaps the one that really sells it to some physicists, as the benefits of superstring/M-theory are so strong. That’s not to say these theories must be correct but, like Susy, nothing else as appealing on so many fronts has so far been developed.

As far as mechanisms for providing certain values such as the Higgs mass, as well as all other apparently fine-tuned variables, there is actually one theory that does offer an explanation without need of Susy at all. It could simply be that the universe we see is one of a vast spectrum in a multiverse, in which there are many universes that don’t have the same variables as ours, and which then don’t have the same types of composite particles, if any, or even the potential for life. Our universe may just be present by virtue of there necessarily being a point, on an infinite spectrum, that provides just the right conditions to allow for the existence of all that we see. With no other strong contending ideas, then at least as far as the Higgs mass goes, if Susy is not found or is ruled out then the concept of the multiverse gets some real support [25]. But the multiverse would be something that we may never have a real means of testing for, and is an unsatisfactory idea in many ways.

There are also other theories that offer a means by which quantum mechanics and gravity may be coupled, such as Loop Quantum Gravity (LQG), in which neither Susy nor extra spatial dimensions are needed [62]. LQG does not offer a means by which the forces may be unified though, unlike Susy-related theories, but then again there’s nothing to say that such unification must be so, it’s just that it would be a neat feature of nature it it were. It might be the case that there are more fragmented facets of nature than our mathematics says are possible - nature might just not be as neat as it possibly could be if we could design it from scratch. As far as quantum gravity goes, the superstring/M-theory explanation is still considered by many to be the more likely when compared to LQG, and that means everything that comes with superstring/M-theory too, Susy included [6].

In some of the most recent Susy related developments, some intriguing patterns have been found in the underlying mathematics that describe what are seen elsewhere as error correcting codes. Such codes are otherwise present in nature solely in genetics, and in the man-made world as part of computing technology, in order to reduce the corruption of signals that occurs as they propagate [63]. Why such patterns should exist in the fundamental laws of physics is currently a very deep mystery, but it can be speculated that they force the laws of nature to evolve, in some sense, into what we see [63]. In other words, this may be the key, or at least part of it, to the fine tuning problem.

So despite the continuing lack of Susy evidence, it’s still an idea with a lot going for it, and also a lot riding on it. It might turn out that the energy range and conditions of the second LHC run may not be adequate to observe Susy effects, for indeed it could be either statistically or energetically out of reach, depending on what parts of theory turn out to be not quite right, even if Susy is. If nothing at all is discovered, not Susy nor anything else, then the chances of funding an even more powerful machine will become very slim indeed, and this may then be the end of the road for terrestrial particle physics, at least with today’s technology. It is not widely believed that such overall null results will be the case though, and that even if Susy is not found something interesting almost certainly will be. Based on past experience, there’s simply so much that suggests that as we seek, we will find.



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