0
$\begingroup$

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.

Improve this question
$\endgroup$
1
  • 1
    $\begingroup$ In how many spacetime dimensions? $\endgroup$
    – Qmechanic ♦
    Commented Oct 15, 2023 at 17:41

1 Answer 1

Reset to default
1
$\begingroup$

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.
Improve this answer
$\endgroup$