# Is electron/positron annihilation to two gravitons forbidden?

This follows from the question Can stress energy tensor vanish in general relativity?. What I'm really asking is whether electron/positron annihilation to two gravitons is allowed, but the obvious answer is that we don't know because no theory of quantum gravity exists.

So I'm asking whether, based on our current knowledge, we have reasons to believe that the annihilation to gravitons is forbidden. For example are there any obvious conservation laws that would be violated? If not we can presumably appeal to the totalitarian principle and conclude that it is (probably) allowed.

Although a QFT treatment of gravity is not renormalisable effective field theories exist and since annihilation to gravitons is a relatively low energy process it can presumably be described using an effective field theory. Do we have any evidence from this as to whether the process is forbidden?

And of course there is the obvious question of whether string theory has anything to say on the subject.

• Hmm interesting question indeed... – Horus Sep 23 '15 at 12:39
• Excellent question. If it is allowed, wouldn't an electron/positron gas be producing more gravitons than a free electron gas? If that is so, wouldn't that violate the equivalence principle? – CuriousOne Sep 23 '15 at 12:44
• The amplitude for this process is p. 21 here in covariant gravity : blogs.umass.edu/grqft/files/2014/11/… – Slereah Sep 23 '15 at 12:47
• @Slereah: It's as simple as this: I will never be able to understand this paper. :-) What's the actual branching ratio? What does it mean for the equivalence principle that an electron-positron gas decays (at least partially) into high energy gravitons while a free electron gas can never decay like that because of lepton number conservation. Or does the paper suggest that gravity breaks lepton number conservation? But then, what am I supposed to do with charge? – CuriousOne Sep 23 '15 at 12:58
• @CuriousOne: This is probably very simple but can you explain to me the connection with the equivalence principle? An electron and a positron annihilate to gravitons (if possible), but an electron and an electron can never do that. Ok, but what has that to do with the equivalence principle? On a related note why should lepton number be conserved? – MBN Sep 23 '15 at 15:59

To my knowledge it is not forbidden by the standard model laws. The point is that at low energy scale this process is strongly subleading with respect to other decay's channels, for instance to an electromagnetic decay in two photons. At high energy the strength of gravitational force grow and that process could be one of the main channels of decay.

To be more precise we need a theory of quantum gravity in which we can do calculations. Actually we already have such a theory, is string theory. In string theory this is a very simple scattering amplitude (at least at tree level) and in particular this is a mixed open-closed string amplitudes. The open strings lives on a Dp-brane and join to emit a closed string. This is the stringy realization of the Hawking radiation (the time reversal of the picture on the left).

On the right of the image you can see the process from the viewpoint of the worldsheet. Closed strings vertex operators must be inserted in the interior of the disk (the disk is the worldsheet's topology that in this case gives the leading contribution in a perturbative $g_s$ string coupling expansion), while open vertex operators are inserted on the boundary.

More explicitly the amplitude is something like:

$$A=\int dx <V_F(x) V_{NS-NS}(z=i,\bar{z}=-i) V_F(y=-x)>$$

In which $V_F$ and $V_{NS-NS}$ are respectively the open vertex operators for fermions and closed vertex for Neveu-Schwarz bosons (so for instance gravitons). The explicit form of the vertex operators I think is too technical and it doesn't add much to the discussion. (A clarification: particles like electrons of the standard model emerge in string theory from complicated intersecting brane models. This is very simplified situation, and these fermions are the more exotic gravitini, dilatini or gaugini, that are superpartners respectively of gravitons, dilaton and gauge bosons.)

The picture is taken from: Scattering of strings from D-branes, (Akikazu Hashimoto, Igor R. Klebanov), Mar 1996, hep-th/9611214