String Theory Demystified

CHAPTER 13 D-Branes 235
Tachyons can actually describe D-brane decay, so let’s say a little bit about that
since it shows how they can fi t into the overall theory. Consider the action for a
scalar fi eld. Suppose that:
D
µ
S = ∫ d x(
∂ ϕ∂ ϕ + λϕ )
µ
2
Quadratic terms in the potential identify mass terms. In the above, we have:
λ = m 2
Now, notice that the quadratic terms indicates a harmonic potential. We can use this
to see why the presence of a tachyon indicates an instability of the vacuum. If
m 2
> 0,
then the potential V( ϕ ) opens upward, with the minimum located at ϕ = 0.
On the other hand, if m 2
< 0 , the parabola opens downward. This means that the
point ϕ = 0 is unstable. It’s like placing a ball at the top of a hill—a small perturbation
will cause it to roll down the hill. These potentials are illustrated in Fig. 13.3.
We can expand the potential energy V( ϕ ) about its critical points, which tell us
where the maxima and minima are, to determine its behavior. To second order it’s
going to assume the form
V(f )
Tachyons and D-Brane Decay
* *
V( ϕ) = V(
ϕ ) + λ( ϕ− ϕ ) +
2
m 2 > 0 m 2 > 0
Figure 13.3 A comparison of the potential for m 2
V(f )
f f
> 0 and m 2
< 0 .