I am trying to understand some calculations of $B$ meson decays and just stumbled upon the low energy effective weak Hamiltonian describing $\Delta S = 1$ $B$ decays:

$$\mathcal{H}_\text{eff} = \frac{G_F}{\sqrt 2} \left[ V_{ub}^* V_{us} \left( \sum_1^2 c_i Q_i^{us} + \sum_3^{10} c_i Q_i^s \right) \; + \; V_{cb}^* V_{cs} \left( \sum_1^2 c_i Q_i^{cs} + \sum_3^{10} c_i Q_i^s \right) \right],$$ where $c_i$ are scale-dependent Wilson coefficients and the flavor structure of the various four-quark operators is $Q_{1,2}^{qs} \sim \overline{b}q\overline{q}s$, $Q_{3,\ldots,6}^s \sim \overline{b}s \sum \overline{q}{}'q'$, $Q_{7,\ldots,10}^s \sim \overline{b}s \sum e_{q'} \overline{q}{}'q'$ ($q' = u,d,s,c$).

(cf. arXiv:hep-ph/0008292). The former part seems to be the tree and the latter the penguin contribution. However, I don't understand why there is no contribution coming from the $t$ quark? In my naive little world, such loops are dominated by the heaviest fermion and I am / was (?) quite sure that this is the case for box diagrams. Continuing reading I find that Gronau keeps arguing with these charm contribution and actual this is somehow a crucial point in this paper since this leads to some cancellation...

My question: Is there a rule of thumb for identifying which quark dominates a loop calculation? Is it really the case, that for instance in the $b \to s$ transition the charm contribution is larger than the top contribution? Why isn't this the case for box diagrams (e.g. in $B^0$-$\overline{B}{}^0$ mixing)?

  • $\begingroup$ Minor comment to the post (v1): In the future please link to abstract pages rather than pdf files. $\endgroup$ – Qmechanic Sep 7 '18 at 16:51

In the meantime I found an answer: The top contribution wasn't neglected, but shifted by utilizing CKM's unitarity $V_{tb} V_{ts}^* = -V_{ub} V_{us}^* - V_{cb} V_{cs}^*$, i.e.: both parts of $\mathcal{H}_\text{eff}$ now have top contributions, specifically the former part now has tree and penguin contributions.

Rule of thumb for identifying dominant quark: it depends

  • QCD penguins: contributions from $u$ and $c$ are larger than the contribution coming from the top quark
  • $b \to s$: $V_{ub} V_{us}^*$ is negligibly small; charm contribution is roughly two times larger than the top contribution (top-loop dominance is a myth in this case!)
  • $Z$ penguins and boxes: top loop is dominant (at least)

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