The angular momentum of a particle is represented by a vector along the axis ofspin. The vector can either point in the same direction, or the opposite direction as the direction of motion. Let's say, it's in the same direction as the motion. Hold up your right hand, and curl your fingers. If the fingers of your hand are wrapped around the particle in the same sense as the spin, the thumb indicates the direction of motion. Therefore such a particle is called right-handed. If you hold up your left hand in the same way, it will represent a particle in which the vector of the angular momentum is in the opposite direction as the motion, and is thus called left-handed. Among the neutrinos, there only exist left-handed neutrinos and right-handed antineutrinos. At least this was the long held view. Recently, it has been suggested that there exist sterile neutrinos where the handedness is reversed from normal.
States of different handedness must be distinguished because the weak force actsdifferently on left-handed and on right-handed particles. Like the other forces, the weak force is associated with a charge, and the intrinsic strength of the weak interaction can be defined by means of a dimensionless coupling constant. The weak charge is unusual in that it is assigned on the basis of handedness. Only left-handed particles and right-handed antiparticles have weak charge. Right-handed particles and left-handed antiparticles are neutral with respect to the weak force and do not participate in these interactions.
The weak force acts on doublets of particles. The theory that describes it is an SU(2) theory in which the two members of each doublet can be transformed into each other. For example, the left-handed neutrino and left-handed electron make up one doublet. They are assigned weak charges of +1/2 and -1/2 respectively. The left-handed up quark and left-handed down quark compose another doublet, or three other doublets if you count each color separately.
Three particles associated with the weak SU(2) symmetry mediate transitionsbetween the members of each doublet. The intermediary particles are the W+, with both a weak and an electric charge of +1, the W-, with weak and electric charge of -1, and the W0, which is neutral with respect to both the weak and electromagnetic forces. The W+ and W- transform the flavors of particles. A left-handed electron can emit a W- and thereby be converted into a left-handed neutrino. In the process, the electric charge changes from -1 to 0, and the weak charge goes from -1/2 to +1/2. Do you see how if the W's -1 electric charge is taken away from the electron, the electron's electric charge goes up one, and when the W's -1 weak charge is taken away, the electron's weak charge goes up one from -1/2 to +1/2?
Now let's look at the unification of the electromagnetic force and the weak force. Let's say you represent a weak interaction on a two by two matrix, with the electron a neutrino, as both the rows and columns. The lower left-hand square, in the neutrino row, electron column, is the W+. The upper right-hand square, in the electron row, neutrino column, is the W-. The diagonals are W0. Now let's say that two electrons interact by exchanging a V0 particle, as do two neutrinos. This will symbolize electromagnetism. Now let's superimpose the electromagnetism squares onto the weak matrix. In the upper left-hand square, W0 + V0 = the Z0 particle and the photon. In the lower left-hand square, the W0 + V0 + Z0. This new two by two matrix is called SU(2) x U(1) symmetry, and describes all possible electromagnetic and weak interactions between an electron and neutrino.
There is a parallel between looking back to the origin of the Universe, and looking at very small distance scales. In the beginning of the Universe, there were much higher energies. Today, at extremely small distance scales, there is also much more energy available. The shorter the distance, the shorter the length of time, it takes a particle to travel that distance. The shorter the length of time, the more energy can be borrowed via the Heisenberg Uncertainty Principle. Therefore looking at distance approaching the Planck length today is like looking at time approaching the Planck time after t = 0. You can illustrate this electroweak symmetry breaking. At distances much smaller than 10-16 centimeters, the full symmetry is expressed. At such close range, the massive W and Z particles are exchanged as readily as massless photons. Therefore the weak and electromagnetic forces are effectively unified. Another way of saying this, is that an experiment that according to the Heisenberg Uncertainty Principle, the energy needed to probe a certain distance is inversely proportional to the distance. An experiment that examined the structure of a particle at a range less that 10-16 cm would have to be done at energies more than 100 GeV. At this energy, W and Z can be freely created, as freely created as the photon, and the mass difference between them and the photon is negligible. At distances of about 10-16 cm, the complex phenomena responsible for breaking the SU(2) x U(1) symmetry begin to intrude. W and Z particles are still observed but look quite different from the photon. At still larger distances, there is insufficient energy to create real W and Z particles, so we only see the effects of the exchange of virtual ones.
Now let's combine the strong force with electromagnetism and the weak force.We need a larger group that contains both SU(3) and SU(2) x U(1) as component structures. Many groups have this property, but the one with the most advantages, including simplicity, is SU(5), which is a group of all possible transformations of five distinct objects. This is shown on a five by five matrix. Imagine a grid with five rows and five columns. It will have 5 x 5 = 25 squares. Imagine the five rows are a red, blue, and green quark, say a down quark, and then an electron and an antineutrino. In the upper left-hand corner, where the rows and columns are quarks, you have the three by three matrix of the strong force. Therefore the SU(3) symmetry of QCD is contained within SU(5). In the lower right-hand corner, where the rows and columns are the electron and neutrino, you have the two by two matrix that we created earlier that combined electromagnetism and the weak force. Therefore the SU(2) x U(1) symmetry of the electroweak is contained within SU(5). You can imagine the particles representing therows emitting the particles in the squares and becoming the particles in the columns. Youcan imagine the particles representing the columns absorbing the particles in the squares and becomes the particles in the rows. This five by five matrix describes all possible electromagnetic. weak, and strong interactions between red, blue, and green down quark, an electron, and an antineutrino. All the particles in the diagonals have no charges at all, and cause no transformations.
Notice that the five by five matrix has a bunch of squares that did not appear in either the previous two by two or three by three groups. There are six new squares in the lower left, and six more new squares in the upper right. These two squares which have leptons for rows and quarks for columns, or vice versa, would transform leptons to quarks, or vice versa. The SU(5) theory postulates 12 new intermediary particles, labeled X. Each X particle carries weak charge, color charge, and electric charge. The electric charges have values of plus or minus 1/3 and plus or minus 4/3.
As with the distribution of color charges in SU(3), the table of charge assignments in SU(5) has some intriguing regularities. For each kind of charge, the sum of the charges assigned to the five particles is zero. For example, each of the three quarks has an electric charge of -1/3 but these are balanced out by the positron's electric charge of +1. A related observation is that all four varieties of charge are carried by at least some of the SU(5) intermediary particles. The gluons have color, the W+ and W- have both weak charge and electric charge, and the X particles carry all four forms of charge.
From these two facts, it can be deduced that all the charges are necessarily quantized. All electric charges must be multiples of 1/3. If a particle with some different charge were accepted into the family, the SU(5) carrier particles could not be emitted or absorbed by it without violating the conservation of charge. Moreover, it is not just the minimum interval between charges that is fixed. The actual values of the charges are determined by the requirement that the total charge be zero. Here at last is an explanation of the quantization of electric charge. The same requirement explains the exact commensurability of the lepton and quark charges, which in turn implies the exact neutrality of the atom. In addition, the intriguing coincidence that all color neutral systems of particles have integral electric charge follows from the organization of the family.
If quarks can be converted into leptons, as in SU(5), then you can have failure of baryon number conservation. Let's say you have a proton forming the nucleus of a hydrogen atom. The proton consists of two up quarks and a down quarks, and the colors of the three quarks is red, blue, and green. If two quarks happen to approach within 10-29 centimeter, an X particle can pass between them. For example, a right-handed red down quark can emit an X with an electric charge of -4/3 and color charges corresponding to red. The down quark, having lost its color charge, and having changed its electric charge from -1/3 to 1, would thereby become a positron. Meanwhile, the X particle could be absorbed by a left-handed green up quark, which would be converted into a left-handed up antiquark with the color antiblue. The new up antiquark would combine with the remaining up quark to form a neutral pi meson. The baryon numbers of both the positron and pi meson are zero, so that the total baryon number went from +1 to 0. The positron would then meet an electron, perhaps that which was part of the original hydrogen atom, and annihilate each other. The up quark and up antiquark would also annihilate each other. Therefore an entire hydrogen atom, all by itself, would be converted into photons.
In the early Universe, and at very small distances today, X particles would exist freely, so leptons and quarks could be freely converted into each other. In that world, it's meaningless to make a distinction between quarks and leptons, since they are so freely interchanged, and so there would only exist one particle. At a range of 10-29 centimeter, the world may be a very simple place, with just one kind of elementary particle and only one force, two counting gravity. In this world, all matter would be unstable, with quarks and leptons being eventually converted to photons.
One problem is the hierarchy problem, which is why is the electroweak scale at such lower energies than the Planck scale, and why is gravity so much weaker than the other forces? An attempt to explain this was supersymmetry. Supersymmetry was invented in 1973 by Wess and Zumino, and earlier in a nonlinear realization by Volkov and Akulov. Supersymmetry is based on the idea that for every ordinary particle there exists a superpartner having similar properties, except for a quantity known as spin.
There are two kinds of ordinary particles: basic constituents of matter and those that mediate forces. Constituents of matter are leptons and quarks. They are fermions, which are particles that carry a spin equal to half-integer units. Particles that mediate forces, such as photons, are bosons, which means that their spins are integer units such as 0, 1, 2, etc. Bosons can occupy the same energy state while fermions can not. Therefore, fermions occupy different energy states while bosons clump together in the lowest energy state.
Supersymmetry relates particles with different spins, namely those with the adjacent spins. Any fermion and boson with adjacent spins can be manifestations of a single superparticle, like an arrow in auxiliary space. Supersymmetric transformations result in a change in the orientation of a particle.
Supersymmetry is the name given to a hypothetical symmetry of nature. Basically, it is a symmetry which relates fermions and bosons. Just as there are operators that change neutron -> proton, e -> v, we can postulate the existence of operators that change bosons to fermions, Qb> = f which a conjugate operator going the opposite way. Q leaves all quantum numbers unchanged except spin. It has been shown that mathematically consistent, supersymmetric quantum field theories can be constructed. The motivations for studying supersymmetric theories is quite strong. However, today there is not yet any experimental evidence that the universe is supersymmetric.
According to supersymmetry, every fermion is associated with a boson that is identical except for spin, and every boson is associated with a fermion that is identical except for spin. The supersymmetric partner of a fermion has a spin of 1/2 less than the fermion. The supersymmetric partner of a boson has a spin on 1/2 less than the boson. Supersymmetric partners are denoted by putting a ~ above the symbol of the particle. The ~ is supposed to be put above the letter but here I'll put it before since I'm limited by the keyboard. They are named by attaching an -ino for a gauge boson or an s- for a fermion. Thus the supersymmetric partner of the photon is the photino, which has the symbol [~gamma], and a spin of 1/2. The supersymmetric partner of the electron is the selectron, which has a spin of 0. The supersymmetric partner of the up quark is the up squark, which has a spin of 0. The supersymmetric partner of the gluon is the gluino, which has a spin of 1/2. The supersymmetric partner of the muon neutrino is the muon sneutrino, which has a spin of 0. The supersymmetric partners of W and Z intermediate vector bosons are winos and zinos. The supersymmetric partner of the graviton is the gravitino. Supersymmetric particles are called sparticles. The supersymmetric partners of fermions and bosons, are sfermions and bosinos. I've noticed that the word "sfermion" is one of the only words in the English language that has an "s" followed by an "f". Usually that sound is written "sph".
If there were an unbroken supersymmetry, then many phenomena would occur. There would be a super-hydrogen atom with ~e bound to a proton. The chemistry of multi-selectron atoms, with bosons rather than fermions bound to the nucleus, would be very different. There would be additional weak interactions with ~W and ~Z exchanged. Obviously, we don't live in a universe with an unbroken supersymmetry.
Since we know about the broken symmetry of the electroweak theory, perhaps there is a similarly broken supersymmetry. Just as with the fermion masses in the Standard Model, a supersymmetric theory can be written that allows the superpartners to have arbitrary masses. But no one has found a way to calculate the masses. Currently, we can only search for supersymmetric particles at whatever mass range is accessible to experiment. Just as in the Standard Model, once you assume mass values for the superpartners, the theory is fully predictive. All rates can be calculated.
To calculate in supersymmetry, you need the Feynman rules. You just take the rules for the Standard Model, and replace the particles by their partners in pairs, keeping the coupling strengths the same. The replacement has to be in pairs since otherwise the number of half-integral spin particles would be odd, and it would be impossible to conserve angular momentum in a transition.
In addition to the interaction of a photon with quarks, there is a quark-squark-photino interaction, and a photon-squark-squark interaction. The strengths of all of the gauge couplings are just the measured ones we already know, because the measured couplings would know about the existence of the supersymmetric theory even if we don't. Because the couplings change with momentum transfer, if the superpartners were very much heavier than Mw, there would be differences in the couplings. There is a space-time dependence in the vertices of the Feynman diagrams which changes as the spin changes. If it were necessary to know the space-time dependence, you would have to go back and construct the full Lagrangian, which would then generate the appropriate space-time dependence. It is usually the simplest possibility that occurs.
You can draw three important conclusions for a normal supersymmetric theory.
1. Supersymmetric partners will be produced in pairs starting from normal particles.
2. The decay of supersymmetric particles will contain a supersymmetric partner.
3. The lightest supersymmetric partner will be stable.
Because of this last conclusion, the lightest supersymmetric particle (LSP) is one of the candidates for the missing mass in the Universe.
Starting from beams of quarks and leptons, you can draw a variety of diagrams to superpartners. The production cross sections involve the same couplings we are used to, so the cross sections are typical of production rates for W's, quarks, etc., except that there is a phase space suppression if the superpartners are heavy. Next, you have to ask how the partners would act once they are produced. For simplicity, let's assume that gluinos are heavier than squarks, so the decay ~q -> q(~g) is not allowed by energy conservation, and that photinos are lighter than squarks, winos, and zinos. Then the dominant decays for any sfermion with electric charge will be ~f -> f + photino, such as smuon -> muon + photino, or down squark -> down quark + photino. Typical decay widths for a superpartner of mass M will be a multiple of M, [gamma] ~ [alpha]M, where [gamma] is the decay width. If M is in the tens of GeV, then [gamma] is of the order 0.1 - 1 GeV. The associated lifetimes are short compared to 10-20 seconds, so only the decay products would enter the detector.
To complete the analysis, it is necessary to decide which will be the lightest supersymmetric particle (LSP), since all the others will decay into it. There are several possibilities, but it's usually assumed to be the photino for simplicity. If some other superpartner were lighter than the photino, you could go through a similar analysis. Details would change but the qualitative conclusions would not.
Since all the superpartners that are produced will decay in a very short time, only normal particles plus the photino will enter the detector. To detect the presence of supersymmetry, we must be able to detect the photino. To see how to do that, you have to study how it interacts. The photino will interact by hitting a quark in the detector, which it reacts with to form a squark. The squark could be real or virtual, depending on the available energy. For simplicity, let's assume the squark is real. The cross section is
[sigma] = [summation over q][integral] dx q(x) [^sigma](^s)
where [sigma] is the cross section, x is the fraction of the proton's momentum carried by the quark, q(x) is the quark structure function, and [^sigma] is the constituent cross section for [~gamma] + q -> ~q. There is a sum over all the quarks in the proton. The square of the center of mass energy of the [~gamma} is (^s), so (^s) = M2, where M is the squark mass. Also, (^s) = xs, where s is the square of the center of mass energy of the photino and proton.
The matrix element is approximately M ~ (eq)e(u bar)u, where (eq) is the quark charge (2/3 or -1/3). As usual, you replace the spinors by the appropriate mass. If you go through the rest of the calculations, you end up with
[sigma]([~gamma]p) = ((4([pi]2)[alpha])/(M2))(F((M2)/s))
Notice that, although we are working in a hypothetical theory, we have calculated the photino interaction cross section in terms of familiar quantities, plus an assumed squark mass. To estimate [sigma]([~gamma]p) numerically, you have to pick a value for M. Analyses such as this have been done and currently imply that a signal for a squark would have been seen if M< 70 GeV, so let's assume the mass of the squark is about that of a W particle. Looking up F in the Particle Data Tables, we find that over a range of x in the region x~ 0.1, F is about 0.15. Then [sigma]([~gamma]p) ~ 2.5 x 10-33 cm. This istypical of a neutrino cross section, about 10-7 of a pion cross section.
A typical photino will not interact in a detector. It will escape, carrying away momentum. Thus, the experimental signature of supersymmetry is an event where apparently momentum is not conserved. Such events can also occur if neutrinos are produced, for example in decays of W's or heavy quarks, but then a charged lepton is also produced. If events are ever discovered with apparent failure of conservation of momentum and no charged leptons, they could be a signal of supersymmetry. Then, detailed analysis can establish whether they could, in fact, come from production of superpartners. The relative rates for various processes, the distribution of missing momentum from large to small, and a number of other quantitative predictions can all test whether a supersymmetric interpretation is possible.
To see why symmetry between bosons and fermions is of interest to the study of elementary particle physics, I point out that renormalizable quantum field theories with scalar particles, such as the Higgs sector of unified gauge theories, have the unfortunate feature that the scalar masses have quadratic divergences in one- and higher-loop orders. Unlike the logarithmic divergences associated with fermion masses, which can be eliminated by taking advantage of chiral symmetries, there is no apparent symmetry that can control the divergences associated with scalar field masses.
If you assume that the loop integrals are cut off at a scale [lambda](/\) >> Mw, where new physics appears, a natural value for the scalar mass would be [lambda], and it's hard to see why the Higgs mechanism leads to a mass scale of Mw/g. In fact, this problem gets worse if there is no new physics until all the way down to the Planck scale, since in that case [lambda] = Mpl, and extremely fine tuning is needed to understand the electroweak scale. In the technicolor model, the scale of technicolor interaction provides a natural cut off for [lambda], but without that, you need some other way of eliminating the quadratic divergences. If you have a theory that couples fermions and bosons, the scalar masses have two sources for their quadratic divergences: one from the scalar loop which comes with a positive sign, and one from the fermion loop with a negative sign. This suggests that if there was a symmetry that related the couplings and masses of fermions and bosons, all divergences from scalar field masses could be eliminated.
One of the first requirements of supersymmetry is an equal number of bosonic and fermionic degrees of freedom in one multiplet. I'll demonstrate this with a single example. Let's say you have two pairs of creation/annihilation operators: (a, [a dagger]) and (b, [b dagger]), with a being bosonic and b being fermionic. They satisfy the following commutation and anticommutation relations, respectively:
[a, [a dagger]] = { b, [b dagger]} = 1
The Hamiltonian for this system can be written as:
H = (wa)[a dagger]a + (wb)[b dagger]b
If you define the fermionic operator:
Q = [b dagger]a + [a dagger]b
then
Q, [a dagger]] = +[b dagger] {Q, [b dagger]} = [a dagger]
Thus if [a dagger] 0> and [b dagger] 0> represent bosonic and fermionic states respectively, Q will turn bosons into fermions and vice versa.
[Q, H] = ((wa) - (wb))Q
So, for (wa) = (wb), meaning there is equal energy for the bosonic and fermionic states, H is supersymmetric. In this case,
{Q, [Q dagger]} = (2/w)H
Therefore, the algebra of Q, [Q dagger], and H closes under anticommutation. If there is more than one a and b, then there must be an equal number of them, otherwise the two above equations can't be satisfied together.
One point that distinguishes supersymmetry from other known symmetries is that the anticommutator of Q, [Q dagger] involves the Hamiltonian. For any other bosonic symmetry, the charge commutation never involves the Hamiltonian.
I will spare you the long derivation of the supersymmetric Lagrangian,
L = -1/2[([partial derivative of A with respect to u]2) + ([partial derivative of B with respect to u]2) + [psi bar][gamma u][partial derivative of psi with respect to u] - (F2) - (G2)]
which leads to new field equations F = G = 0> in addition to the usual ones for A, B, and [psi]. F and G are auxiliary fields. They are added to make the Lagrangian invariant for arbitrary values of the fields.
Soon after the discovery of supersymmetry by Wess and Zumino, Salem and Strathdee proposed the concept of the superfield as the generator of supersymmetric multiplets. You want to maintain symmetry between ordinary space and fermionic space, so you introduce four extra dimensions. You can describe the fermionic coordinates as elements of a Majorana spinor or as a pair of two-component Weyl spinors. Points in superspace are then identified by the coordinates
zM = ((xu), ([theta]a), ([theta bar]a))
where [theta]'s are anticommuting spinors. Salam and Strathdee proposed that a function [phi](x, [theta], [theta bar]) of the superspace coordinates, called superfield, which has a finite number of terms in its expansion in terms of [theta] and [theta bar] due to their anticommuting property, be considered as the generator of the various components of the supermultiplets.
Often, in physics, we notice a pattern in what we observe, and then try to think up something that could account for it. In medieval Europe, a few alchemists noticed that some irreducible substances had similar characteristics and could be grouped together. This evolved over time until the modern periodic chart was developed independently by Dmitry Mendeleyev in 1869, and Julius Myer in 1870.
So, then, we had this pattern in the elements, and we were motivated to think up something which could explain the pattern. The final conclusion of this process was the atomic shell theory, in which atoms with the same number of valence electrons in their outer shell have similar properties. A similar process started in the 1930's, when a large number of new particles were discovered. These particles were grouped into Eightfold Way patterns, developed independently by Murray Gell-Mann and Yuval Ne'eman in 1961. This illustrated a pattern in the characteristics of baryons and mesons. These patterns were used to think up the idea of quarks, developed independently by Murray Gell-Mann and George Zweig.
Today, we notice patterns in the characteristics of quarks and leptons, which we call Standard Model, and we're in the process of trying to think up something that could account for it. With supersymmetry, we're trying to do something similar, except in that case, we do not observe a pattern in the characteristics of fermions and bosons. We are simply imagining that one exists.
Actually, Plato did something similar. He theorized that each of the elements was associated with a platonic solid. The problem with this was that there were four elements and five platonic solids. Therefore, he just invented another element, which he imagined was the element that celestial bodies were made out of.