D0 decays to K+/K- (CKM suppression)

[ChrisVer]

I was looking at the $D^0 \rightarrow K^+ \pi^-$ and $D^0 \rightarrow K^- \pi^+$.

The first is doubly Cabbibo suppressed whereas the other is Cabibbo favored.
I got the ratio:
$A= \frac{Br( D^0 \rightarrow K^+ \pi^-)}{Br(D^0 \rightarrow K^- \pi^+)} = \frac{|V_{cd}|^2 |V_{us}|^2}{|V_{cs}|^2 |V_{ud}|^2} \approx 0.002863(12)$
I used the values given for $V_{ij}$ from wikipedia .
I then checked the pdg for the appropriate decay rates :
$Br( D^0 \rightarrow K^+ \pi^-)= 1.380(28) \times 10^{-4}$
$Br( D^0 \rightarrow K^- \pi^+)= 3.88(5) \times 10^{-2}$
From which I got their ratio:
$A \approx 0.00356(9)$

I was wondering why these ratios are not equal?

 

[mfb]

For an exact calculation you have to consider higher orders. Those are very messy for charm decays.
For experimental observations, you also have to take mixing into account. The $D^0$ can go to $\overline {D^0}$ and then decay via the Cabibbo favored decay, which looks exactly like the suppressed decay. Then you also add interference between mixing and decay and you get a parabolic shape of this measured branching ratio as function of time.

LHCb has the most sensitive measurement so far.
Publication 1
Publication 2
Overview note

 

[ChrisVer]

, you also have to take mixing into account. The D0D^0 can go to D0¯¯¯¯¯¯\overline {D^0} and then decay via the Cabibbo favored decay, which looks exactly like the suppressed decay.

So you think that the pdg's values take into account the $D^0 (\rightarrow \bar{D}^0 )\rightarrow K^+ \pi^-$?
I'll have a look at your citations.

 

[mfb]

They have separate groups for the rare decay: total, via DCS, via $\overline {D^0}$. Not sure how they handle interference.

Where is the point in the non-interactive version:
http://pdg8.lbl.gov/rpp2014v1/pdgLive/Particle.action?node=S032
http://pdg8.lbl.gov/rpp2014v1/pdgLive/BranchingRatio.action?parCode=S032&desig=50

Looks like the LHCb estimate is not included yet. You can also have a look at the Heavy Flavor Averaging Group.

 

[ChrisVer]

Judging from the LHCb results, the ratio $R(t)$ has a minimum value $R_D$ which is still larger ($3.568 \times 10^{-3}$) and that gets larger with time because of the D-Dbar oscillations. So in fact the oscillations would lead in a higher $A$ than the one I obtained from pdg...and so even larger from the one I obtained from the CKM... So I guess the main difference is because of higher order contributions to those diagrams...

 

[Hepth]

If you also include approximations to the theoretical predictions, like the zero recoil form factors for the transitions (assuming factorization), then you have in addition to the CKM ratios, $\left(\frac{f_{D\pi}}{f_{DK}} \frac{f_K}{f_{\pi}}\right)^2$

from http://arxiv.org/pdf/0907.2842v1.pdf and decay constant ratios from PDG vals http://pdg.lbl.gov/2014/reviews/rpp2014-rev-pseudoscalar-meson-decay-cons.pdf

I get $0.00309508 \pm 0.000428392$

Then there's factorization violating stuff too.