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The Nambu-Goto action is given by $$ S = -\frac{T_0}{c} \int_{-\infty}^{+\infty} d\tau \int_{0}^{\sigma} d\sigma \sqrt{ \Bigg(\frac{\partial X^\mu}{\partial \tau} \frac{\partial X_\mu}{\partial \sigma}\Bigg)^2 - \Bigg(\frac{\partial X^\mu}{\partial \tau} \frac{\partial X_\mu}{\partial \tau}\Bigg)\Bigg(\frac{\partial X^\nu}{\partial \sigma} \frac{\partial X_\nu}{\partial \sigma}\Bigg) }.\tag{6.39} $$ This action can be written in a particular parametrization as (chapter 6 of the book A First Course in String Theory by Barton Zwiebach) $$S = -T_0 \int dt \int ds \sqrt{1-\frac{v_\perp^2}{c^2}}\tag{6.88}$$ where $v_\perp$ is the velocity of the string in a direction perpendicular to its spacial length and $ds$ represents a small string length element. So, the above Nambu-Goto action does not have any longitudinal degrees of freedom. This is expected as this is an action for a fundamental string (so all the points on the string are the same).

My question is how can one modify this action (say by adding higher curvature corrections) to include the longitudinal degrees of freedom to model composite strings?

PS - David Tong mentions a small comment in his notes on string theory about this (pg. no. 17), but does not give any details.

"There will typically be additional terms in the action that depend on the width of the string. The form of these terms is not universal, but often includes a rigidity piece of form $L \int K^2$ , where $K$ is the extrinsic curvature of the worldsheet. Other terms could be added to describe fluctuations in the width $L$ of the string".

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The second-order term involving the extrinsic curvature was first introduced by Polyakov$^1$ (and also studied by Kleinert), who augmented the Nambu-Goto action as: $$ S_\text{NG}=\mu_0\int\mathrm d^2\sigma\sqrt{-\det g_{\alpha\beta}} \\S_{(2)}=\frac{1}{\alpha_0}\int\mathrm d^2\sigma\sqrt{-\det g_{\alpha\beta}}\ (K^i_{\rho\sigma})^2 \\S'=S_\text{NG}+S_{(2)} $$ where $\alpha_0$ is the (free) rigidity parameter and $K^i_{\rho\sigma}$ is the second fundamental form of the worldsheet, and so this additional term represents fluctuations in the width of the worldsheet. It is defined by: $$ \nabla_\beta\partial_\alpha X^\mu\equiv \sum_i K^i_{\alpha\beta} n^\mu_i $$

One might also think to add a Ricci scalar term of the same order, but this is seen to be equivalent to $S_{(2)}$ up to a total divergence. In fact, all geometric terms, such as the normals and tangents to the worldsheet, can be constructed out of the induced metric $g_{\alpha\beta}$ (the first fundamental form) and the second fundamental form. Indeed, $S_{(2)}$ is the unique extension of the NG action (up to total divergences) that is Poincaré-invariant, worldsheet reparameterisation-invariant, and also scale-invariant. Polyakov's motivation here was to study the phase structure of string theory in the IR. Interestingly, this second-order correction does not change the equations of motion for an isolated string.

For our purposes, if we write the tension $\mu_0$ as $\mu_0\sim\ell^{-2}$, then we see that $S_{(2)}$ is suppressed by the mass scale induced by the string length. This suggests that we can form an effective field theory using local geometric invariants of the worldsheet of all orders. In particular, we can construct higher-derivative terms using derivatives and torsion-part of the $K_{\alpha\beta}$, suppressed by factors of $\ell^{2}$. $$ S_\text{eff}=\int\mathrm d^2\sigma\sqrt{-\det g_{\alpha\beta}}\left(\ell^{-2}+\frac{1}{\alpha_0}(K_{\sigma\rho})^2+\ell^2(...)\right) $$

The relevance of this model is most prominent not in string theory itself, but in flux tubes, vortices and cosmic strings, where it is known as the "long string effective action". Essentially, in the absence of the constraints of string theory, the Nambu-Goldstone modes due to the breaking of $D-2$ translational symmetries$^2$ by the string are the only massless states in the spectrum, so we can integrate out the massive modes due to transverse fluctuations and obtain an effective description of the longitudinal fluctuations of the NG bosons along the string. Note that qualitatively this is very similar to the Coleman-Callan-Wess-Zumino Construction for pions.

There are more modern constructions$^3$ for this effective action, but the simplest method, as above, is to work in a coordinate-independent framework and construct higher-order terms that are $\mathrm{ISO}(1,1)\times\mathrm{SO}(D-2)$ invariant (even if this symmetry is realised non-linearly), though this theory will not be manifestly unitary. Such terms are, for instance $\int\sqrt{-g}R^2$, $\int\sqrt{-g}g^{\alpha\beta}g^{\gamma\delta}g^{\epsilon\zeta}g^{\eta\kappa}K^i_{\alpha\beta}K^i_{\gamma\delta}K^j_{\epsilon\zeta}K^j_{\eta\kappa}$, etc. with the appropriate factor of $\ell$. However, remember that determining the precise numerical prefactor for each term is a matter of matching with the UV theory (as with other EFTs) or, depending on the specific model under consideration, using explicit lattice computations.


$^1$A. M. Polyakov, Fine Structure of Strings, Nucl. Phys. B268 (1986) 406–412.

$^2$Specifically, the symmetry breaking is $\mathrm{ISO}(D-1, 1)\to\mathrm{ISO}(1,1)\times\mathrm{SO}(D-2)$

$^3$Polchinski, Strominger, Effective String Actions, Phys. Rev. Lett. 67, 1681

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