Confusion about the mass operator in the RNS superstring

Question

In the RNS formulation of superstring theory we have the action:

$$S=-\frac{T}{2}\int d^2\sigma(\partial_\alpha X^\mu \partial^\alpha X_\mu + \bar{\psi}^\mu \rho^\alpha \partial_\alpha \psi_\mu)$$

where $X^\mu$ is a worldsheet scalar and $\psi^\mu$ is a worldsheet spinor. We can Fourier decompose these two. For the open string in the Ramond sector we obtain modes $\alpha_n^\mu$ for $X^\mu$ and $d_n^\mu$ for $\psi^\mu$. We can relate the zero mode of $X$ to its momentum, i.e. $\alpha_0^\mu \sim p^\mu$.

My first question is, why can't we do something similar for $\psi^\mu$, so relating $d_0^\mu$ to the momentum of the spinor?

Secondly, I know that from the anticommutation relation $\{d_0^\mu,d_0^\nu\}=\eta^{\mu\nu}$ we deduce that the $d_0^\mu$ act as gamma matrices on the states. Why then does it disappear in the mass formula for the Ramond sector open string (in the light-cone gauge, equation 4.109 in Becker, Becker and Schwarz):

$$\alpha ' M^2 = \sum^\infty_{n=1}\alpha_{-n} \cdot \alpha_{n}+\sum^\infty_{n=1} n d_{-n} \cdot d_{n}$$

I would expect a term $\sim d_0\cdot d_0=\Gamma^\mu\Gamma_\mu=10I_{32}$ on the right. Is it absorbed in the mass?

Finally, if $M^2$ only depends on $\alpha_0^\mu$, why do we consider this to represent the total mass? Doesn't it miss a mass contribution from $\psi$? Would this mean the spinor is always massless? If so, why isn't $X$ massless?

Answer 1

Very good questions. The fermionic Ramond sector should be treated with some care.

The point is that the R zero modes $d_{0}^{I}$ (absent in the NS sector) define the fermionic ground states of the theory. They define ground states because $\{\psi_{r}^{\mu},\psi_{0}^{\nu} \}=0$ for $r>0$ and because they form a representation of the ten dimensional Clifford algebra. So the answer to your questions is simply that the $d_{0}^{I}$ states are massless but $M^{2}$ does not just depend on the $\alpha_{n}^{I}$ oscillators because states $d_{r}^{I}$ with $r>0$ contribute to the mass formula.

Concerning the question of why X cannot be massless. The mass formula is the proof!

My feeling is that this issue is not discussed with enough care in the Bécker-Bécker-Schwarz book.

My recommendation is the following: Read the chapter 10.2 in the second book of Polchinski were the subtleties I pointed out are discussed,then go to the Appendix B.1 where the ten dimensional gamma matrix algebra is reviewed; workout the details of why the Dirac 32 representation of the D=10 algebra is reducible to a direct sum of two Weyl subrepresentations as 32 = 16 + 16$^{'}$. With all the latter in mind, go to the section 14.5 of the second edition of the book "A First Course in String Theory" by Zwiebach where all the of the representation 16 of degenerate Ramond ground states are explicity constructed (using the RNS formalism), the mass formula for the superstring is derived and the normal ordering on the fermionic current modes is done in extraordinary detail.