# What is the value of the fine structure constant at Planck energy?

1. Jun 23, 2009

### heinz

At low energy, the value is around 1/137.036. What is the value at the Planck energy? Is it listed somewhere?

After all, the renormalization of the charge (as it is also called) is mentioned everywhere. But numbers are rare ...

Hz

2. Jun 23, 2009

### Naty1

No specific answer but this is interesting:

via wikipedia at http://en.wikipedia.org/wiki/Fine-s...the_fine_structure_constant_truly_constant.3F

3. Jun 23, 2009

### Avodyne

It depends on whether there are any presently-undiscovered charged particles with masses below the Planck scale. Also, above the W mass (~80 GeV), it makes more sense to talk about the SU(2) X U(1) coupling constants g2 and g1, rather than the electromagnetic coupling e, though we can still formally define it via 1/e^2 = 1/g2^2 + 1/g1^2. Because of all this, no one bothers to compute the formal value of alpha at the Planck scale.

4. Jun 23, 2009

### Civilized

The answer is definitely that no one knows because it would involve summing all the diagrams to all orders of perturbation theory. I can, however, give you the second order correction:

$$\alpha(q^2) = \alpha(0)(1 + \frac{\alpha(0)}{3\pi}f(\frac{-q^2}{m^2c^2}))$$

where:

$$q^2 = -4|p|^2 sin(\theta / 2)$$
$$f(x) = 6 \int_0^1 z(1-z) ln(1 + x z(1 - z)) dz$$

It's irritating that we have to leave f(x) in the form of an integral that cannot be reduced to elementary functions, but the asymptotic behaviors are f(x) = x/5 when x << 1 and f(x) = ln(x) when x >> 1. As you can see, 1/137 is still pretty much as valid as ever, although the corrections would be measurable if we could get up to the Planck energy.

Last edited: Jun 23, 2009
5. Jun 23, 2009

### heinz

Thank you! And what are p and theta?

Hz