Post #5 Supersymmetry

Moving onto a bit of physics I'm not so familiar with now, so the potential for getting things wrong is increasing....

To get to twistor strings, I need to pass through superstring theory and supersymmetric twistor space. To accomplish this, I'll need to say a few words about supersymmetry, which is the topic of this post. As usual, this isn’t complete or rigorous – just meant to give a flavour.

Quantum theory presents us with the problem of identical particles. Assume a pair of particles has identical quantum numbers (rest mass, charge etc..) so you can't use these to distinguish them. If you keep the particles isolated, then you can continue to distinguish them, however if their wavefunctions are allowed to overlap, then you can no longer keep track of which particle is which. This is a manifestation of the fact that particles don't follow classical trajectories.

In quantum mechanics, a single particle wavefunction is traditionally represented as Ψ(x). A two particle wavefunction would be a function of two variables Ψ(x

1,x

2), where x

1 and x

2 represent the position variables of the two particles. If we define an operator P which just swaps the two particles around:

P Ψ(x

1,x

2) = Ψ(x

2,x

1)

then obviously P

2 Ψ(x

1,x

2), so there are only two possibilities.

Either P Ψ(x

1,x

2) = Ψ(x

1,x

2) or P Ψ(x

1,x

2) = - Ψ(x

1,x

2)

i.e. the two particle wavefunction is symmetric under interchange of the particles, or it's antisymmetric. Particles displaying the former behaviour are called bosons and ones displaying the latter behaviour are called fermions.

In quantum field theory, you create a particle from the vacuum by applying a creation operator. If you create more than one particle, you find that your particles are automatically indistinguishable - this is an inescapable feature of quantum field theory. If you constructed your QFT using

canonical commutation relations, you will get bosons. If you used canonical anticommutation relations, you will get fermions.

Supersymmetry is a proposal for a type of symmetry that the quantum field theory in nature should obey. To describe what it is, we need first to say what a symmetry in a quantum field theory is. The QFT is traditionally described by its Lagrangian (strictly Lagrangian density). This is a function of the all the fields present in the theory - these may include all sorts of things - scalar fields, vector/tensor fields, spinor fields etc. Suppose I just denote all the fields in the theory by the letter Ψ - just think of it as a big column vector. A symmetry of the theory is an operation (think matrix), U, which maps fields to fields

Ψ' = UΨ

such that

ℒ(Ψ', 𝜕

μΨ') = ℒ(Ψ, 𝜕

μΨ)

For example, if I take the Lagrangian for a Dirac field (here Ψ is a 4 component spinor, composed from a pair of two component spinors that we discussed in an earlier post, and γ

μ are the

Dirac gamma matrices :

ℒ(Ψ, 𝜕

μΨ) = Ψ

†(γ

0i𝜕

μγ

μ-m)Ψ

Then an example of a symmetry is provided by the charge conjugation operation, C, which swaps particles for antiparticles. C is represented by the matrix -iγ

2γ

0. C mixes up and modifes the components of Ψ. Supersymmetry does a similar thing, but this time it mixes up the components of fermionic fields with the components of bosonic fields.

Suppose we take a simple example to illustrate this - take a theory with two entities:

FERMION: a single two component spinor field Ψ

ABOSON: a complex scalar field φ

Our supersymmetry will be implemented by an operator Q

A which takes us from the boson field to the fermion field. This is known as a supercharge. This is an infinitesimal generator, in the same sense that P

μ and J

μν are the infinitesimal generators of the Lorentz group (see the appendix below). Introducing the supercharge and defining its relationship with the Poincare group infinitesimal generators allows us to define the “super Poincare algebra” relations:

[P

μ, Q

A] = 0

[J

μν, Q

A] = i(σ

μν)

ABQ

B{ Q

A, Q

barA’} = 2σ

μAA’P

μHere σ

μAA’ are the Pauli spin matrices I mentioned in a previous post. (σ

μν)

AB = (1/4)( σ

μAA’ σ

barνA’B - σ

νAA’ σ

barμA’B). (bar is complex conjugate as usual).

The last relation is interesting. It says that combining a (infinitesimal) supersymmetry transformation with its conjugate is equivalent to an infinitesimal spacetime translation. Note – if we allow the supersymmetry transformation to be dependent upon the spacetime point, rather than being global, then we end up with infinitesimal translations being spacetime point dependent, i.e. we end up generating infinitesimal general coordinate transformations. This leads us naturally to gravity and is the basis of the theories of supergravity.

It’s possible to define supersymmetric theories with more than one supercharge. The number of supercharges is traditionally denoted by 𝒩. So an “𝒩 = 4 supersymmetric theory” is a theory with 4 supercharges.

Motivations for Supersymmetry.

1 Hierarchy ProblemThe first motivation that is normally advanced for supersymmetry is the hierarchy problem. The famous mexican-hat Higgs potential is:

V = -μ

2φ

†φ +( λ/4)(φ

†φ)

2V has a minimum at |φ| = 2μ/ √ λ

λ is proportional to (the square of) the Higgs mass. Now the problems start to crop up when we include quantum corrections at the one loop level:

The term (φ

†φ)

2 defines a 4 point self interaction of the φ field. A tree level, the interaction vertex just contributes a λ factor to amplitudes. When we want to compute amplitudes to the next level in perturbation theory we must include loops like:

- HiggsSelfInt.jpg (7.07 KiB) Viewed 2542 times

This diagram represents a one loop contribution to the φ propagator. Here, we must integrate over the momentum of the internal line. In principle this internal momentum can have any value, so the integration is to infinity. Since this results in a divergent answer, the normal procedure is that we introduce a cutoff Λ at high momentum.

The one loop term resulting from the (φ

†φ)

2 interaction term results in a correction λΛ

2φ

†φ which is applied to the term -μ

2φ

†φ.

Now we know (well, phenomenologists know) the vacuum expectation value of Higgs field is of the order of 246Gev (from muon decay apparently!), and this expectation value is proportional to μ/ √λ, so consequently the physical mass μ (after correction) can't be greater than a few hundred GeV. But this is a problem: - this correction λΛ

2φ

†φ is proportional to the square of the cutoff. What is this cutoff ? If we don't expect any new physics up to the Planck scale, it's of the order of the Planck mass i.e. 10

19GeV.

So we're in the rather dubious situation where we have a bare mass term of the order of -(10

19GeV)

2 cancelled by a correction term of a similar size, leaving a result of size a piddling couple of hundred Gev

2. Imagine something like this (the numbers are fictional, but the order of magnitude is representative)

(Physical Mass)

2 = (Bare Mass)

2 - (Correction Term)

2 18769 = 49984905920593950496034950295949634709 -49984905920593950496034950295949615940

Looks very fishy doesn't it ? You have mass renormalizations in other models too, so don't they all look fishy like this ?

Ian Aitchison makes the point that it usually goes differently. For example in QED, the one loop correction would be "electron emits photon and then reabsorbs it". When worked out, this gives a correction to the electron mass of logΛ. This is a hell of a lot more benign than Λ (put some numbers in to convince yourself).

In fact in the case of interest, if we have some fermion interactions, then in addition to the one loop self interaction of φ, we also have contributions from the one loop case where a φ splits into a fermion/antifermion pair, which then recombine. The interesting thing is that this contribution is of the opposite sign to the φ self interaction contribution. You can see where this is going:

If there is the right symmetry between fermions and bosons in the Lagrangian, then the one loop corrections cancel exactly, and we don't have to worry about the "cancelling two huge numbers to get one small one" issue. The symmetry that allows this to happen is of course supersymmetry.

Note I think that there are several issues floating around in physics which sometimes have the name "hierarchy problem" attached to them. Note also, that supersymmetry isn't the only potential solution to this hierarchy problem.

2 GUTs and Gauge Coupling ConstantsThere is a technique called the

renormalization group equation that allows you to investigate the behaviour of coupling constants as a function of the energy scale in which you're working. There are 3 gauge couplings in the standard model. If you plot their behaviour as a function of the energy scale, you get something like the upper picture. If you introduce supersymmetry into the model you get something like the lower picture, i.e. they converge at the same (GUT) energy scale.

- GUTCoupling.jpg (14.59 KiB) Viewed 2544 times

The proponents of supersymmetry would argue that this is too great a coincidence to ignore. Interestingly though, if there is any new (non SM) physics between here and the GUT scale, this argument goes out of the window, because using the renormalization group to predict the scaling behaviour of the coupling constant in this regime would no longer be valid.

3 Coleman Mandula TheoremThis basically states that the symmetry of a theory (which satisfies some “reasonable” conditions) is always of the form

Spacetime symmetry X Internal symmetry

By “spacetime symmetry” I mean the usual Lorentz/Poincare symmetry, by “internal symmetry” I mean things like SU(2)xU(1) (electroweak symmetry). In other words the composite symmetry is a straightforward product, no mixing of the two symmetries is possible.

Supersymmetric theories provide a way round this – they do not have this restriction. I don’t really understand it, but this is quoted as an argument in their favour.

Appendix - Poincare Group, infinitesimal generators and all that

In an earlier post, I gave an example of a matrix representing a particular Lorentz transformation - where the relative velocity between the primed and unprimed frames is in the x direction. ( In this case, only the x and t components got screwed around, y and z remained untouched).

(ɣ ...-βɣ...0...0)

(-βɣ...ɣ....0...0)

(0 ....0.....1...0)

(0.....0 ....0...1)

where ɣ = 1/sqrt(1-v

2/c

2), and β = v/c.

For a general Lorentz transformation, we would write this relation as

x'

μ = Λ

μνx

ν, where Λ

μν is the matrix representing the transformation.

To see what infinitesimal transformations are like, take a Lorentz transformation which differs from the identity by a tiny amount

Λ

μν = 𝛿

μν + 𝛿ω

μν where 𝛿ω

μν is a very small perturbation away from the identity transformation. Since a Lorentz transformation must preserve the Minkowski metric:

η'

μν = Λ

μρ Λ

νσ η

ρσ =( 𝛿

μρ + 𝛿ω

μρ )( 𝛿

νσ + 𝛿ω

νσ ) η

ρσ = η

μν + 𝛿ω

μν + 𝛿ω

νμ + 𝒪(ω

2)

So we must have 𝛿ω

μν = -𝛿ω

νμ i.e 𝛿ω is antisymmetric.

Traditionally, we expand -𝛿ω in terms of a basis for these infinitesimal transformations, so the infinitesimal Lorentz transformation looks like

Λ

αβ(𝛿ω) = 𝛿

αβ – (i/2)𝛿ω

μν(J

μν)

αβThe J

μν)

αβ are the infinitesimal generators of the Lorentz transformations. Think of the indices μ and ν as labelling which generator we’re looking at (since μν always appears in an antisymmetric combination, there are six of these), and α and β as labelling the rows and columns of each generator, which, being a Lorentz transformation is a matrix.

The structure of the Lorentz group is characterized by the commutation relations obeyed by its infinitesimal generators (omit the matrix indices for clarity, just write the indices which label the generators):

[J

μν, J

λσ] = i(J

λνη

μσ - J

σνη

μλ - J

μλη

νσ + J

μση

νλ )

When people rabbit on about “algebras” of this that and the other, this is what they mean – the relations obeyed by the infinitesimal generators of the group in question.

Here, we've been looking at the action of Lorentz transformations on elements of Minkowski space, which are themselves Lorentz vectors. In general, we would be looking at the action on elements of any old representation of the Lorentz group – on spinors for example, but we would find the same algebra for the infinitesimal Lorentz generators. For example, for spinors, the infinitesimal generators would be of the form J

μνAB where μ and ν are the spacetime indices and A and B are the spinor indices.

So, the Lorentz group has 6 infinitesimal generators J

μν, three for the three independent rotational degrees of freedom, and three for the three independent directions you can do your boosts in. In additional to rotational and boosting symmetry, flat spacetime is also symmetric under translations. There are, fairly obviously, four translational degrees of freedom, and after expanding the Lorentz group to include these degrees of freedom, we end up with a 10 parameter group – the Poincare group.

To represent a Poincare transformation on spacetime, we two things – a matrix Λ as before, for our rotation/boost, and a vector b for our translation. So the Poincare transformation is

x'

μ = Λ

μνx

ν + b

μwhere Λ

μν is the Lorentz matrix and b is the translation. Two Poincare transformations combine like

(Λ

2, b

2) (Λ

1, b

1) = (Λ

2Λ

1, Λ

2b

1+b

2)

Of course, to do anything meaningful, we need to represent our Poincare transformations as matrices. To do this, we actually use 5x5 matrices. The matrix representing the Poincare transformation (Λ, b) is

(Λ..........b)

(0..........1)

i.e. we just stuff the vector b into the last column. In a similar way to before, we can now talk about the infinitesimal generators corresponding to the translations. There are, of course, 4 of them and they’re traditionally denoted as P

956. You can now extend the Lorentz algebra above by computing the commutation relations of the P’s with themselves and with the J’s. The full set of commutation relations becomes:

[J

μν, J

λσ] = i(J

λνη

μσ - J

σνη

μλ - J

μλη

νσ + J

μση

νλ )

[P

μ, J

λσ] = i(P

λη

μσ - P

ση

μλ)

[P

μ, P

ν] = 0

This is the Poincare algebra and lies at the heart of all of our conventional quantum field theory.