Is Planck length constant?

Planck length $l_p$ is dependent on Newton constant $G_N$ which is related to coupling constant of interaction of gravitons, but from field theory point of view, we know that coupling constants have running with energy, so Planck length is not really a constant. In string theory, we also have the following relation : \begin{equation} G_N\sim l_p^8\sim g_s^2l_s^8 \end{equation} where $g_s$ and $l_s$ are string coupling constant and string length which are independent of each other, so Planck length goes to zero in the limit $g_s\to 0$ and $l_s\to 0$ (in some unit, for example by comparing with AdS radios $L$).

What is the meaning of Planck length, if it is not a constant? Also, can Planck length goes to zero in string theory?



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