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I am currently studying QFT and came upon this question. We are dealing with a theory of a complex field $\phi$ and a real field $\chi$. The interaction Lagrangian density is given by:

$${\cal L}_{\rm int} = g \chi \phi^\dagger \phi.$$

Now the goal is to first reformulate the theory introducing counterterms and then determine the condition for the cancellation of tadpoles. I am not sure how to go about this now.

Qmechanic
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JoVel
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1 Answers1

1

Hints:

  1. The condition for cancellation of tadpoles can be formulated as vanishing of 1-point functions $$ \langle \phi(x)\rangle_{J=0}~=~0\qquad\text{and}\qquad\langle \chi(x)\rangle_{J=0}~=~0,$$ cf. e.g. my Phys.SE answer here.

  2. The first condition is automatic satisfied due to a $U(1)$-symmetry of the complex field $\phi$.

  3. The second condition can be made to hold by adding a 1-vertex interaction term $Y\chi$ to the Lagrangian density, cf. Ref. 1.

References:

  1. M. Srednicki, QFT, 2007; chapter 9. A prepublication draft PDF file is available here.
Qmechanic
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