The low-energy effective action of the bosonic string is given by: $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\mathcal{R}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_\mu\Phi\partial^\mu\Phi\Big)$$ where $H_{\mu\nu\lambda}=\partial_\mu B_{\nu\lambda}+\partial_\nu B_{\lambda\mu}+\partial_\lambda B_{\mu\nu}$.

There are three fields: the space-time metric $G_{\mu\nu}$, the anti-symmetric tensor field $B_{\mu\nu}$ and the scalar dilaton field $\Phi$.

Does the scalar dilaton field $\Phi$ change the effective string coupling and therefore the mass of massive particles (but leaves massless particles unaffected)?

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    $\begingroup$ What particles? If you mean the masses associated to the higher-level excitations of the string, aren't the characteristic length scale $\ell_s$ and the coupling constant $g_s$ independent? $\endgroup$ Mar 6, 2021 at 7:16
  • $\begingroup$ One can define the Planck mass as a field given by $M_{Pl}=e^{-\Phi}$. All particle masses are proportional to $M_{Pl}$. $\endgroup$ Mar 6, 2021 at 17:29
  • $\begingroup$ Your answer below is better than my understanding of the subject! $\endgroup$ Mar 10, 2021 at 16:49

1 Answer 1


There appears to be some confusion here. The mass of the string spectra depends on the characteristic string length scale $\ell_s$, e.g. for the first excited states of the closed string, $M^2\sim\ell_s^{-1/2}$. On the other hand, the vacuum expectation value of the constant part of the dilaton field sets the string coupling constant $g_s=e^{\Phi_0}$. These two are independent parameters (not free parameters, obviously, but they do not depend on each other).

In the Einstein frame of the bosonic effective action, the gravitational coupling can be expressed in terms of the 26D Planck mass: $M_p^{-24}\sim\ell_s^{24}e^{2\Phi_0}$, so the VEV of the dilaton does set the Planck mass (and equivalently the Planck length scale $\ell_p=M_p^{-1}$) but this is unrelated to the mass of the string spectra, and roughly sets the scale where gravitational effects become strong. And since this VEV has no time-depenence, the question of "changing the mass" is reduced to being somewhat meaningless: change from what? The fact remains that the mass, dynamics and interactions of all the string modes are consistently derived from a single quantisation condition - so there is no notion of asking how properties change pre and post the introduction of forces or particles to augment the theory (unlike in our very modular Standard Model).

So the dilaton VEV does not change the mass of massive excitations, although it does dictate the relevance of gravitational effects at the characteristic string length scale $\ell_s$, and by extension, uniquely fixes the 26D Newton's constant.

As an aside, you need to be extremely careful while invoking the "equivalence principle" in effective actions - in the bosonic case, without a mechanism to give the dilaton mass, the equivalence principle is violated. Finally, bear in mind that while constructing the low-energy effective action, the massive modes of the string are essentially "integrated out" and appear only as effective matter content - clearly the dilaton can have no bearing on the properties of these modes in this case.

  • $\begingroup$ If the dilaton VEV sets the gravitation coupling - i.e. the gravitational "charge" of massive particles - then I would have thought by the equivalence principle that's like changing their mass. $\endgroup$ Mar 13, 2021 at 10:18
  • $\begingroup$ @JohnEastmond updated with essentially the same content in my previous comment $\endgroup$ Mar 15, 2021 at 13:27

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