# How can long range forces be described in string theory?

Perturbative string theory interactions are modeled as strings coming into direct contact and then joining, or one string splitting apart. But what is the right way to think about long range or inverse square interactions using only the basic conceptual building blocks of string theory?

For example, if we wanted to be extremely pedantic and describe the orbit of Earth around the Sun in the language of perturbative string theory, can we say that the (strings comprising) each body are exchanging virtual gravitons/off shell gravitational excitations? Does such a concept of virtual strings (or any way of talking about anything off shell) even make sense when we are not working with the underpinning of spacetime filling fields, and there is no object to "host" the off shell states? If this is indeed not possible, how do we explain this without just going into a QFT limit and thereby using different concepts?

In the case of gauge interactions between open strings attached to branes, I believe this question has a natural answer. We can say that the long range interactions are mediated by the Born-Infeld theory on the brane itself. But when closed strings are involved, either gauge interactions in a heterotic theory or for gravity generally, I don't know what "runs between" massive or charged strings, so as to permit a local account of long range interactions in this manner.

• One string emits a closed string and the other one absorbs it... Is there a problem with that? – Mitchell Porter Dec 21 '17 at 23:00
• @mitchell it doesn't seem right to me that what are virtual particles in a QFT (always stressed as mere mathematical constructs representing un-particle-like field configurations, which are in principle undetectable) could be real particles in string theory. We think a QFT like the Standard Model (plus a low energy graviton field) is just a low energy limit of a string theory model. So I don't see how something real in the deeper theory could be demoted to something virtual in the effective theory. – Felix Wotter Dec 22 '17 at 0:02
• @mitchell, also before posting, I had actually read your comment here where you say that virtual photon strings make sense in the context of Born-Infeld theory in an intersecting D6 type model, and I agree that makes sense. But for gravity, with no brane or space filling field, I just don't get where the virtual process can occur, as I think they need something field-like to "live" on. physicsforums.com/threads/… – Felix Wotter Dec 22 '17 at 0:10
• The right treatment is emission and absorption of closed strings, even though in the low energy limit one should use the language of supergravity. I guess that your question is basically how to go from supergravity (or more simply, general relativity) to a description in terms of QFT forces mediated by a graviton. – Rexcirus Dec 22 '17 at 7:42
• @rexcirus I am more trying to ask: if we don't go to a SUGRA or GR limit, how would we explain long range interactions – Felix Wotter Dec 22 '17 at 17:20

Long range interactions arise naturally in string theory by the exchange of open and closed strings that are at the massless state. For long distance, the correlation produced by massive strings will drop out exponentially with typical length $L\gg\hbar c/m_s\sim l_s$, where $l_s$ is the string length and $m_s$ the lowest massive state. So, for large distances, just the massless string state will be important, and for bosonic closed string these states will be effectively described by the following action:

$$S=\frac{1}{2\kappa_0}\int d^Dx(-G)^{1/2}e^{-2\Phi}\{-\frac{2(D-26)}{2\alpha'}+R-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_{\mu}\Phi\partial^{\mu}\Phi+\mathcal{O}(\alpha')\}$$

Where $\Phi$ is the dilaton, $R$ is the Ricci scalar and $H_{\mu\nu\lambda}$ is the field strength of the $B$-form field. This theory describes interactions that does fall as $r^{1-d}$, where $d$ is the number of non-compact dimensions.

For bosonic open strings, with Chan-Paton degrees of freedom, we have Yang-Mills theory for large distance:

$$S=\frac{1}{g_o'^2}\int d^{26}x\{-\frac{1}{2}Tr(D_{\mu}\varphi D^{\mu}\varphi)+\frac{1}{2\alpha'}Tr(\varphi^2)+\frac{2^{1/2}}{3\alpha'^{1/2}}Tr(\varphi^3)-\frac{1}{4}Tr(F_{\mu\nu} F^{\mu\nu})\}$$

where $\varphi$ is the tachyon, $F_{\mu\nu}$ is the strength tensor of the gauge $A_{\mu}$.

There much more sources of such long range interactions, and they are all described by the massless states of a given string theory. This is so because the exchange of massless objects produces, at tree-level, S-matrices of the type: $$\langle p',k \rvert S \lvert p,k \rangle \rvert_{conn} = -\mathrm{i}\frac{e^2}{\lvert \vec p -\vec p'\rvert^2 - \mathrm{i}\epsilon}(2m)^2\delta(E_{p,k} - E_{p',k})(2\pi)^4\delta(\vec p - \vec p')$$ with is equivalent to a Coulomb potential. See this for more information.

Actually, in the case of superstrings, the long range features of the theory are used to distinguish each of the five consistent superstring theories:

and all this long range theories does have the interactions you are wondering.

In the case of D-Branes we can have massless states trapped on a stack of D-branes, leading to gauge theories inside. See this and this for more information.

Now, if you are wondering if we can describe this interactions by doing a full stringy calculation, i.e. calculating the scattering amplitude via perturbative string theory, the answer is obviously yes. The only thing that changes is that there is annoying massive terms that will shows up in each scattering amplitude, terms that does not contribute for long distance (low energy), terms proportional to $\alpha'$.

Example: the scattering amplitude at tree level of three gauge bosons from Chan-Paton is given by $$\mathcal{A}(k_1,a_1,e_1;k_2,a_2,e_2;k_3a_3,e_3)=ig_o'(2\pi)(2\pi)^{26}\delta^{26}(\sum_{i} k_i)((e_1\cdot k_{23}) (e_2\cdot e_{3})+\\+(e_2\cdot k_{31}) (e_1\cdot e_{3})+(e_3\cdot k_{12}) (e_2\cdot e_{1})+\frac{\alpha'}{2}(e_1\cdot k_{23})(e_2\cdot k_{31})(e_3\cdot k_{12}))$$ the last term does not contribute for long distance.

If you want to know about off-shell amplitudes, there is String Field Theory.