Where does the string theory’s exponentially damped propagator come from?

Question

I was reading Quantum Gravity and Quantum Cosmology (edited by Gianluca Calcagni, Lefteris Papantonopoulos, George Siopsis, and Nikos Tsamis, p. 6), and I came across this interesting statement about how the usual propagator in field theory, $\frac{1}{k^2}$, gets modified in string theory. Specifically, for a closed string propagating on a cylinder, the propagator takes the form:

$$\frac{e^{-\alpha' k^2}}{k^2}$$

where $\alpha'$ is related to the string length scale $\ell_s$. The key thing that stood out to me was how this introduces an effective cutoff, which apparently helps with divergences in quantum gravity.

Now, I’ve been trying to understand how exactly string theory deals with the problems that show up in standard covariant quantization of gravity. In the usual approach, we linearize GR and apply second quantization, leading to a non-renormalizable theory that blows up at two-loop order and beyond. But here, the string length scale seems to regulate things naturally. I want to ask two question about this specifically

1. How these cutoff cures divergences in two loop and higher diagram

   • I get that the exponential suppression tames high-momentum contributions, but does it fully remove the divergences at two loops and beyond?
   • Is there an explicit calculation somewhere that shows how string amplitudes remain finite due to this effect?

2. Origin of this exponential factor

   • A lot of books mention it, but I haven’t seen a full derivation. Is there a simple way to see why the string propagator picks up this form?
   • Is it coming from the Polyakov path integral formulation, or something more fundamental about the extended nature of strings?
   • Ashoke Sen's lectures on string theory suggested that such a suppression is seen when we express everything from spacetime point of view rather than world sheet point of view. He said that the source of these exponential suppression is the vertex factor in String Field Theory. Any source which mentions this argument with some calculation?

I’ve checked Polchinski’s String Theory, Green-Schwarz-Witten’s Superstring Theory, and Becker-Becker-Schwarz’s String Theory and M-Theory, but they all gloss over the derivation. One solution that comes to mind is suggested in the textbook String Field Theory by Erbin as: $$\frac{1}{k^2}=\int_0^\infty ds e^{-sk^2}$$ Since, string theory does not allow $s\to0$ due to modular invariance, so we replace the lower limit of integral. If I replace lower limit to $\alpha'$ then I get what I wanna see. But this seems artificial.

If anyone has a reference that actually works this out in detail, I’d really appreciate it!

Answer 1

The statement about "the propagator" is contained in all the standard texts on string theory, just in a slightly different form: They all compute the Veneziano/Virasoro-Shapiro amplitude for the tree-level four-point function and you can show that the high-energy behaviour of that amplitude is an exponential fall-off in the Mandelstam variables (see e.g. Tong's notes, section 6.2.2). Compare this to the QFT tree-level four-point functions momentum behaviour being dictated by that of the propagator, you conclude that the "string propagator" has such an exponential fall-off, as the tree-level diagram is just the propagator + some external legs.

The exponential fall-off comes from the asymptotic behaviour of the $\Gamma$-functions in the full amplitude. Ultimately the form of the amplitude (which was found long before its derivation from "modern" string theory) is strongly constrained by the conformal symmetry of the worldsheet symmetry, which is why people hand-wavingly say string-theory's UV-finiteness is due to conformal symmetry, which also more generally relates IR-divergences to UV-divergences and IR-divergences are easier to control.

Of course, just looking at this tree-level amplitude is not a proof that the full string amplitude is UV-finite - just that it is at tree-level for this particular string diagram. However, many believe that similar arguments about exponential fall-offs and conformal symmetry should work at all loop orders. For low-loop orders, many terms have been explicitly computes and shown to be finite (see again Tong's computation of the 1-loop partition function on the torus, for instance).

To what extent you believe UV-finiteness of string theory has been established at all orders depends a lot on the level of rigor you demand and what kind of divergences exactly you're worried about. See this question and its answers for discussion about the status of proving string theory is UV-finite.

Answer 2

1. TL;DR: The exponential damping is due to the Euclidean time $2\pi\tau_2$. [In Minkowskian time, it is oscillatory.] E.g. the propagator is closely tied to the 1-loop partition function on a torus $$ Z(\tau)~=~{\rm Tr}\left[\exp(2\pi i\tau_1P-2\pi\tau_2 H)\right]~=~(q\bar{q})^{-d/24}{\rm Tr}\left(q^{L_0}\bar{q}^{\tilde{L}_0}\right).\tag{P7.2.5} $$

2. Briefly speaking, modular invariance means that we can identify $\tau$ with a fundamental region $$\left\{\tau \in\mathbb{C}\mid {\rm Im}(\tau)\geq 0 \wedge |{\rm Re}(\tau)|\leq \frac{1}{2} \wedge |\tau|\geq 1, \right\},$$ cf. e.g. this related Phys.SE post. The upshot is that a short fat cylinder can be replaced by a long thin cylinder. The field theory limit corresponds to $\tau\to i\infty$.

3. For further intuition, consult the Witten reference What Every Physicist Should Know About String Theory in my Phys.SE answer here.

4. For the open & closed string propagator, see also e.g. eqs. (GSW7.1.14) & (GSW7.2.8), respectively.

References:

[P] J. Polchinski, String Theory Vol. 1, 1998; Section 7.2.

[K] E. Kiritsis, String Theory in a Nutshell, 2007; Sections 4.17 + 4.18.

[GSW] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Vol. 1, 1986; Sections 7.1 + 7.2.