Consider a massive scalar diagram such as

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The loop momentum enters and exits the tadpole vertex, so that in the first diagram the momentum in the propagator connecting the two vertices is zero due to overall momentum conservation. This is ok if the fields are massive.

However, in the second diagram the propagator connecting the two vertices has exactly the same momentum as the rightmost external leg, and is therefore on-shell and blows up!

I know that tadpole loop momentum integration develops a divergence and is e.g. dimensionally regularized. But the non-loop propagator being on-shell simply makes the result infinite regardless of dimensional regularization!

How to make sense of this?

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    $\begingroup$ These specific diagrams are sometimes known as slugs. More generally, a self-loop is known as a tadpole. These are eliminated by normal-ordering your operators. This is explained somewhere in Itzykson & Zuber, see also physics.stackexchange.com/search?q=normal+ordering+tadpole $\endgroup$ – AccidentalFourierTransform Jun 25 at 1:03
  • $\begingroup$ @AccidentalFourierTransform Thanks for the hint! I'll look up Itzykson & Zuber. $\endgroup$ – Kagaratsch Jun 25 at 1:40

That is why, in the LSZ prescription, you "amputate" your diagrams by multiplying by the full (or "dressed") inverse propagator

$$ \mathcal{A}(\{k_i\}\to\{p_j\}) = \prod_i Z_\varphi^{-1}(k^2_i + m_R^2) \prod_j Z_\varphi^{-1}(p^2_{j} + m_R^2) \,\langle 0 |\mathrm{T} \,\varphi(k_1)\cdots \varphi(p_1) \cdots|0\rangle\,. $$ The presence of $Z_\varphi$$-$the wave function renormalization$-$and the renormalized mass fix all these issues.

Then again, normally one deals with $1\mathrm{PI}$ diagrams, which are sufficient for all computations as you can write the effective action $\Gamma[\Phi]$ with them (e.g. in order to compute anomalous dimension and $\beta$ functions one only needs $1\mathrm{PI}$). The diagrams with the pathology you illustrated are not of that kind.

  • $\begingroup$ I see, so is it correct to say that any sort of tadpole diagram is always a pathology and is removed in renormalization one way or another? Can one always remove them with a counter-term renormalization scheme? In which case just dropping them from any amplitude calculation is fine? $\endgroup$ – Kagaratsch Jun 25 at 2:31
  • $\begingroup$ I'd say that it's slighly different conceptually. Counterterms remove $1\mathrm{PI}$ divergences. Once you have dealt with those, you still have the external legs as you observed. The LSZ presciption takes care of them. In practice however you never actually compute the diagrams with tadpoles because those are canceled identically by LSZ (by design). You simply have to write diagrams where the external legs (together with all possible radiative corrections) are replaced by $1$. $\endgroup$ – MannyC Jun 25 at 3:00
  • $\begingroup$ Ahha, but what about a tadpole contribution like for instance from a 6-point vertex in which two legs are joined into 1 loop? This tadpole does not look like a radiative correction to an external leg. Does LSZ still take care of it? $\endgroup$ – Kagaratsch Jun 25 at 4:51
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    $\begingroup$ Nono, that's a $1\mathrm{PI}$ piece. It's zero in dim reg for massless ($\propto \int dp \,p^{-2}$), and is otherwise renormalized by $Z_\varphi$ or $Z_m$. Also, in all diagrams with more than one leg it should be a subdivergence, so if you remove it at one loop you're all set (I'm not 100% sure about this). $\endgroup$ – MannyC Jun 25 at 4:59
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    $\begingroup$ I see, thank you for all the details! $\endgroup$ – Kagaratsch Jun 25 at 5:00

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