The 5D Chirality Problem


Message from the 5th dimension: “We’re ambidexterous up here. Why are you left/right-handed?”

There are lots of neat little problems that pop up when you play with models of extra dimensions. One such problem that one might not have expected is how to write down a 4D chiral theory. For simiplicity let’s play with a flat extra dimension compactified on a circle. This way we don’t have to worry about things like vielbeins.

First off, the Clifford algebra in 5D is given by the usual 4D \gamma matrices with -i\gamma_5 appended as the fifth gamma matrix. Let us label our 5D indices by M so that our 5D gamma matrices are \Gamma^M = (\gamma^\mu, -i\gamma^5).

The fact that i\gamma_5, which 4D quantum field theorists know as the parity operator, is also the extra gamma matrix to fill out the 5D rep of the Clifford algebra a general feature in various dimensions, see for example Professor Hitoshi Murayama’s Physics 230A notes. It is this dual role of \gamma_5 that will cause trouble.

Anyway, the point is that the \gamma matrices in 5D are the usual 4\times 4 matrices that we worked with in flat, 4D quantum field theory. Thus spinors in 5D are four-component Dirac spinors.

But then there’s the problem (the 5D chirality problem): how can one ever hope to extract a 4D chiral theory, i.e. 4D Weyl spinors, from a theory that is inherently composed of non-chiral Dirac spinors? If every 4D Weyl spinor must come from a 5D Dirac spinor, then we end up doubling the number of degrees of freedom of our 4D theory because every left-handed Weyl fermion would have a corresponding right-handed Weyl fermion with exactly the same quantum numbers.

Sanity check: this is a problem because the Standard Model is a chiral 4D theory. Left-handed electrons have different quantum numbers as right-handed electrons (or left-handed positrons).

Double sanity check: why can’t we construct 5D helicity projection operators analogous to the 4D projection operators P_{L,R} = \frac{1}{2}(1\mp\gamma^5)? We cannot do this because there’s no analogue to \gamma^5 in five dimensions! The ability to write an extra gamma matrix that plays the role of a parity operator is unique to even dimensional (or perhaps even 0 mod 4 dimensional) representations of the Clifford algebra. Thus it isn’t well defined to talk about helicity projection operators for spinors in 5D. We’re stuck with the Dirac spinors, which generate left- and right-handed 4D Weyl spinors.

In my next post I’ll explore one solution to this 5D chirality problem, but to finish off let’s look at an explicit example:

An example with a surprise

Suppose one had a bulk fermion \Psi(x,\phi) that can be decomposed as

\Psi(x,\phi) = \sum^\infty_{n=-\infty} \psi^{(n)}(x)e^{in\phi} .

We’ve chosen to parameterize the extra dimension by the angular variable \phi. Suppose further that one had a bulk U(1) gauge field A_M, which also has a Kaluza Klein decomposition of the form above. The fermionic action then looks like

S = \int d^4x \int (R d\phi) \bar\Psi i D_M\Gamma^M \Psi .

Now extracting the fifth component:

S = \int d^4x \int (R d\phi) \bar\Psi i D_\mu\Gamma^\mu \Psi + \bar\Psi\gamma_5\partial_5\Psi + ig\bar\Psi A_5\gamma_5\Psi

Inserting the spinor decomposition above and performing the \phi integral:

S = 2\pi R\int d^4x \sum^{\infty}_{n=-\infty} \bar\psi^{(n)} i ( \gamma^\mu\partial_\mu - i\frac{n}{R}\gamma_5 )\psi^{(n)} + ig \bar\psi^{(n)}\gamma_5 A_5\psi^{(n)} +\cdots

So what do we see? We have a full set of both left handed and right handed fermions with the same quantum numbers. If this is not clear work in a representation where the top components of the Dirac spinor are the left-handed Weyl spinor and the bottom two components are the right-handed Weyl spinor. It’s straightforward to see that if the gauge group were more complicated, the fermions would still have the same quantum numbers.

Now here’s the surprise: our 4D theory includes an interaction with the A_5 field. The Kaluza Klein modes of this fields are 4D Lorentz scalars. (What else could they be?) And so these terms are Yukawa interactions with a scalar boson with the same coupling as the coupling to the 4D vector gauge boson. This is called “gauge-Yukawa unification.” Pretty neat! Note that this coupling is also proportional to the dreaded \gamma_5, which is now playing the role of the 4D parity operator. It’s also rather nice that we get a scalar particle popping out with no extra input. Eager model builders might immediately want to try to massage this scalar into the 4D Higgs boson.

In my next post I’ll explain one way to fix the chirality problem via a process called orbifolding. We will see, however, that this process kills the candidate Higgs boson. (I hope you didn’t get too attached to it.)

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