renormalization - Divergent sum in lightcone quantization of bosonic string theory

I had the following question regarding lightcone quantization of bosonic strings - The normal ordering requirement of quantization gives us this infinite sum $\sum_{n=1}^\infty n$. This is regularized in several ways, for example by writing $$ \sum_{n=1}^\infty e^{- n \epsilon } n = \frac{1}{\epsilon^2} - \frac{1}{12} + {\cal O}(\epsilon^2) $$ Most texts now simply state that the divergent part can be removed by counterterms. David Tongs notes (chap. 2 page 29) specifically state that this divergence is removed by the counterterm that restore Weyl invariance in the quantized theory (in dimensional regularization).

I would like to see this explicitly. Is there any note regarding this? Or if you have any other idea how one would systematically remove the divergence above, it would be great!

Answer

Note that $n$ is really the momentum in the $\sigma$ direction so it has the units of the world sheet mass. The exponent $-n\epsilon$ in the regulator has to be dimensionless so $\epsilon$ has the units of the world sheet distance.

Consequently, the removed term $1/\epsilon^2$ has the units of the squared world sheet mass. This are the same units as the energy density in 1+1 dimensions. If you just redefine the stress energy tensor on the world sheet as $$T_{ab} \to T_{ab} + \frac{C}{\epsilon^2} g_{ab}$$ where $C$ is a particular number of order one you may calculate (that depends on conventions only), it will redefine your Hamiltonian so that the ground state energy is shifted in such a way that the $1/\epsilon^2$ term is removed.

This "cosmological constant" contribution to the stress-energy tensor may be derived from the cosmological constant term in the world sheet action, essentially $C\int d^2\sigma\sqrt{-h}$. Classically, this term violates the Weyl symmetry. However, quantum mechanically, there are also other loop effects that violate this symmetry – your regulated calculation of the ground state energy is a proof – and this added classical counterterm is needed to restore the Weyl (scaling) symmetry.

It's important that this counterterm and all the considerations above are unable to change the value of the finite leftover, $-1/12$, which is the true physical finite part of the sum of positive integers. This is the conclusion we may obtain in numerous other regularization techniques. The result is unique because it really follows from the symmetries we demand – the world sheet Weyl symmetry or modular invariance.