**Optical rotation in magnetic field and axions**

Sean Carroll mentions three experiments:

- B-meson oscillations that we discussed here
- MINOS (neutrinos) that we described here
- Rotating light in a magnetic field

Take a one-meter-long magnet with 5 Tesla of magnetic field. Send a linearly polarized laser beam whose oscillating electric field is pointing in the direction of the external magnetic field and carefully measure the polarization plane. The authors, Zavattini et al., argue that the polarization axis was rotated by

- 3.9 +- 0.5 picoradians per pass.

Let me write a few moderately refreshing formulae. The axion explanation of the rotating plane involves a new scalar field "a", the axion. It has the ordinary kinetic and mass term:

- Lagrangian = 0.5 [(partial
_{m}a)^2 - m_{a}^{2}a^{2}]

- Lagrangian = - a E.B / M
_{a}

Now you have a new, corrected quadratic Lagrangian in which "a" and the electromagnetic field are no longer separated. If you consider an electromagnetic wave whose "E" is orthogonal to the external "B", the coupling above does not contribute, and the electromagnetic wave propagates by the usual speed of light.

However, if you consider a linearly polarized electromagnetic wave whose "E" is parallel to the external "B", for example a beam moving up with "E" in the North-South direction, things change. The electromagnetic field mixes with the axion into two new eigenstates of "omega

^{2}- p

^{2}". One of them is "mostly" the axion, and its effective mass is slightly smaller than the original mass of the axion. The other one is "mostly" the photon that however becomes slightly massive, and therefore slower. This polarization of the laser beam slows down compared to the vacuum and one can also compute the rotation of the polarization plane.

If you compute these things and think about plausible values of "m

_{a}" and "M

_{a}", you will see that the mass "m

_{a}" should be a millielectronvolt, up to a factor of two or so, while the inverse coefficient "M

_{a}" should be between 100 and 600 TeV.

You could be more conservative and say that you don't want any new field. Can you obtain the rotating plane directly by modifying the Lagrangian for the electromagnetic field? Well, the effect of the axion above can be mimicked if you integrate out the axion (for a while, imagine that the mass of the axion is higher than the frequency of your light) which will give you a term proportional to

- Lagrangian = # . ( F /\ F )
^{2}

- Lagrangian = # . ( F
_{mn}F^{mn})^{2}

A completely dumb question for the insiders: do you generate the same rotating effect by the term written below?

- Lagrangian = # . ( F
_{mn}F^{mn}) ( F /\ F )

**Axions in string theory**

Because this is a superstringy blog, I can't avoid the comment that axions are predicted by most vacua in string theory. In heterotic string theory, for example, one always has the so-called model-independent axion. It's because the heterotic string theory predicts the two-form B-field potential under which the strings are electrically charged. Its exterior derivative is a three-form; its Hodge-dual in four dimensions is a one-form that can be written as the gradient of a zero-form if you're on-shell. The zero-form from the previous sentence is a scalar field called the model-independent axion. There are usually many other, model-dependent axions, too. See, for example, a paper by Tom Banks and Michael Dine about the cosmology of string-theoretical axions.

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