## Friday, March 25, 2011

### Phase Fun

Recall that a convenient way to express a phase factor is as

$\theta (x) = - i \textrm{log} \frac{-1 + x i}{1 + x i}$

where $x$ is a typical mixing matrix parameter. For example, when $x = 24$, we have $\theta^{-1} = 12.007$. In fact, for any positive integer $x > 2$, $\theta^{-1}$ is close to $x/2$. If $x$ is in the form $\textrm{tan} \phi$, we can get rid of the tangent using the Weierstrass rule

$-i e^{2 i \phi} = \frac{- \textrm{tan} \phi + i}{- \textrm{tan} \phi - i}$

Now for the first two CKM parameters $b = 24$ and $a = -0.231$, the phase $\theta (a) + \theta (b)$ is close to $\theta_{W} - \pi$, where $\theta_{W}$ is the Weinberg angle, as measured by the coincidence $\textrm{sin}^{2} \theta_{W} = \textrm{tan} \theta_{C}$ $= 0.231 = a$.

1. Some time ago I guessed that the small CKM parameter $c = 0.0035$ might be $2/b^2$. This seems a little more plausible now, given the rule
$-2 i \textrm{log} ((-1 + xi)/(1 + xi)) = x + 2/x^2 + \cdots$
2. oops, there should be a $-1$ power on the logarithm