# Regularization and renomalization in the lightcone quantization of bosonic string

This question relates to this link. But I still don't understand it >_<

In Polchinski's string theory vol I, p. 22, there is a divergence term (when $\epsilon \rightarrow 0$) in the zero point energy of open-string (1.3.34), $$\frac{D-2}{2} \frac{ 2l p^+ \alpha'}{ \epsilon^2 \pi}$$

It is said

The cutoff-dependent first term is propotional to the length $l$ of the string and can be canceled by a counterterm in the action proportional to $\int d^2 \sigma (-\gamma)^{1/2}$. In fact, Weyl invariance requires that it be cancelled.

I have a couple of questions related to this statement and I don't know whether it is good to split into several threads.

1. About the underlying logic of regularization and renomalization. In quantum field theory, we meet a couple of divergences. One is in quantizing the scalar field. There is an infinity, we argue that is the zero-point energy. We throw it away since energy is a relative quantity. And we left the cosmological constant problem.

If the divergence in (1.3.34) is in this sense, string-theory is a quantum gravity theory. We cannot simply throw it away like QFT.

Later in QFT, we meet other divergences in the loop calculations. The reason is, the presented QFT is a low-energy theory. It has certain scale that the theory is not applicaple. Therefore we met the divergence. We regularalize it and renormalize it, where we met the terminology "counterterm".

But, string theory is regarded as a final theory(?!), if I follow the logic of QFT, why there is still divergence in (1.3.34)? What is the scale string theory is not applicable?

2. How the counterterm works in $\int d^2 \sigma (-\gamma)^{1/2}$?

3. Why "Weyl invariance requires that it be cancelled"?

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For 2), all terms in the action have this form (see $1.2.13$ or $1.2.31$ for instance) For 3) the theory is conformal invariant, so scale invariant, you cannot accept that the theory depends on a scale $\frac{1}{\epsilon}$ –  Trimok Aug 17 at 17:15
Thank you very much! –  user26143 Aug 17 at 17:20
Hi user26143, +1, and I'm posting this comment here just in case you didn't notice this comment by Dilaton yet, as you are not very active on TeX.SE (nor am I). –  DIMension10 Nov 11 at 15:47
@DIMension10, thank you very much! I have noticed and replied the comments from Dilaton! –  user26143 Nov 11 at 15:52