Consider the CFT that corresponds to a gauge-fixed closed bosonic string.

Ground level string states are described by vertex operators such as $$V(p) = :\exp(i p_{\mu} X^{\mu}(z, \bar{z})):$$ which are conformal primaries with weight $$ h = \bar{h} = \frac{\alpha'}{4} p^2. $$

The physical states of the strings must have $h = \bar{h} = 1$, therefore, the physical ground state is the tachyon.

Consider the 3-point function of three ground state operators:

$$ G_{p_1, p_2, p_3}(z_1, z_2, z_3; \bar{z}_1, \bar{z}_2, \bar{z}_3) = \left< V(p_1)(z_1, \bar{z}_1) V(p_2)(z_2, \bar{z}_2) V(p_3)(z_3, \bar{z}_3) \right>. $$

Because we're dealing with a free quantum field theory, it isn't hard to calculate this function exactly. Afaik this is called the "Coulomb gas" representation, and the expression is

$$ G_{p_1, p_2, p_3}(z_1, z_2, z_3; \bar{z}_1, \bar{z}_2, \bar{z}_3) = |z_{12}|^{\alpha' p_1 \cdot p_2} \cdot |z_{13}|^{\alpha' p_1 \cdot p_3} \cdot |z_{23}|^{\alpha' p_2 \cdot p_3}, $$

where $z_{ij} = z_i - z_j = -z_{ji}$, and $|z_{ij}| = (z_{ij} \cdot \bar{z}_{ij})^{1/2} = |z_{ji}|$.

However, I expect a general 3-point function to be completely fixed by global conformal symmetries – the Mobius group $SL(2, \mathbb{C})$. The general form for three primary fields with weights $h_i, \bar{h}_i$ is:

$$ \left< \phi_1(z_1, \bar{z}_1) \phi_2(z_2, \bar{z}_2) \phi_3(z_3, \bar{z}_3) \right> = C_{123} z_{12}^{h_3-h_1-h_2} z_{13}^{h_2-h_1-h_3} z_{23}^{h_1-h_2-h_3} \bar{z}_{12}^{\bar{h}_3-\bar{h}_1-\bar{h}_2} \bar{z}_{13}^{\bar{h}_2-\bar{h}_1-\bar{h}_3} \bar{z}_{23}^{\bar{h}_1-\bar{h}_2-\bar{h}_3}, $$

where $C_{123}$ is the 3-point structure constant of the CFT.

For any physical string state, $h = \bar{h} = 1$, therefore Mobius invariance requires that

$$ \left< \phi_1(z_1, \bar{z}_1) \phi_2(z_2, \bar{z}_2) \phi_3(z_3, \bar{z}_3) \right> = C_{123} z_{12}^{-1} z_{13}^{-1} z_{23}^{-1} \bar{z}_{12}^{-1} \bar{z}_{13}^{-1} \bar{z}_{23}^{-1} = C_{123} |z_{12}|^{-2} |z_{13}|^{-2} |z_{23}|^{-2}. $$

This seems to suggest that for the physical ground states participating in scattering (tachyons),

$$\forall i, j, i \neq j: \quad \alpha' p_i \cdot p_j = -2. $$

This restriction seems very odd to me, I've never seen anything like it before.

My questions are:

  1. Have I missed something crucial?
  2. If not, what is the origin of the constraint on the values of tachyon momenta?
  • 1
    $\begingroup$ hint: momentum conservation .. $\endgroup$ May 24, 2019 at 14:18
  • 1
    $\begingroup$ @Wakabaloola of course, that’s what I was missing! Three spacelike vectors of the same length that sum to zero can only be arranged such that the angle between any two is $2 \pi /3$. That totally makes sense, thank you. Now how about writing this up as an actual answer so that I can upvote and accept it? :) $\endgroup$ May 24, 2019 at 14:33

1 Answer 1


Using momentum conservation and onshell conditions, $$ p_1+p_2+p_3=0, \qquad p_1^2=p_2^2=p_3^2=\frac{4}{\alpha'}, $$ it follows that for any $i\neq j\neq k$, \begin{equation} \begin{aligned} 2p_i\cdot p_j &= (p_i+p_j)^2-p_i^2-p_j^2 \\ &=p_k^2-p_i^2-p_j^2\\ &=-\frac{4}{\alpha'}, \end{aligned} \end{equation} so this relation is really just kinematics and onshell conditions.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.