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I have a question related to the heterotic string theory. I have seen many references that are talking about heterotic tadpole but no one explains why it is an important quantity. The questions are that

  • What is a heterotic tadpole?

  • Why is it important?

  • Does a non-zero heterotic tadpole mean that we consider the heterotic theory in some background?

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I don't have a good reference to hand, but I can provide some context. Tadpoles in general are Feynman diagrams that look like this:

Tadpole Feynman Diagram

They are not specific to string theory, let alone heterotic strings and were in fact introduced by Coleman in a QFT context. The string diagram involves surfaces rather than lines but the basic idea remains the same.

They are important for the following reason: If we take the time direction to increase upwards in the diagram, then the diagram says that a pair of fermions is created out of nothing and then decays into a scalar. In other words, a stable scalar particle was just created out of nothing.

There are strong implications for theories that allow such diagrams. For example, it means that the vacuum is unstable. Particles can be created out of nothing. For that reason, "tadpole diagrams" and "vacuum instability" are often discussed together.

Whether such theories are "good" or "bad" is up to debate. For example, one viewpoint is that this should not be allowed and this can be used as a motivation to introduce supersymmetry. In supersymmetric heterotic theories there is an exact cancellation of such diagrams.

On the other hand, the contribution of these diagrams is also related to the cosmological constant $\Lambda$, so in theories were tadpoles are allowed one can calculate the cosmological constant explicitly. In most non-supersymmetric heterotic models that I am aware of, this calculation gives a cosmological constant that is wrong by many orders of magnitude. Nevertheless, one of the selling points of string theory is that at least we can perform these calculations and provide an explicit number for $\Lambda$, something which is not the case in different frameworks.

It would of course be very desirable to find ways to improve on these models and find ways to suppress the predicted cosmological constant and this is an area of active research.

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  • $\begingroup$ Thank you very much for the answer. I did not get this point "There are strong implications for theories that allow such diagrams." Is there a theory that does not allow such diagrams? (It is always possible to make all one-point 1PI diagrams to be zero by imposing some renormalization condition on current counterterm.) Also, do you mean that only scalars can be produced in such process? $\endgroup$
    – QGravity
    Oct 7, 2017 at 20:02
  • $\begingroup$ It seems to me that you can consider a general state from NS sector of heterotic string theory and compute the tadpole diagram in string perturbation theory whose value contributes to background values of the associated space-time fields. $\endgroup$
    – QGravity
    Oct 7, 2017 at 20:02
  • $\begingroup$ "Is there a theory that does not allow such diagrams?" Theories with no fundamental scalars do not have this problem (e.g. technicolor). String theory models always have at least one fundamental scalar (the dilaton), so the tadpole contribution needs to be calculated/discussed (if the model is non-supersymmetric). $\endgroup$
    – Heterotic
    Oct 8, 2017 at 15:59

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