D* -> D + gamma BR calculation

[Hepth]

Can anyone walk me through this calculation? I'm not completely sure how to do it (merely a student).

So is this considered a 1 body radiative decay? So the factorized amplitude would be something like
$$M \propto <D^0|\bar{u}\gamma_\mu\left(1-\gamma_5\right)c|0>[A_1 <\gamma|\bar{u}\gamma_\mu\left(1-\gamma_5\right)c|D^{*0}>+A_2<\gamma|\bar{u}i \sigma_{\mu \nu} p_\nu \left(1+\gamma_5\right)c|D^{*0}>$$

And each matrix can be decomposed into some constants/vectors/polarizations etc.

$$<D^0|\bar{u}\gamma_\mu\left(1-\gamma_5\right)c|0> = i f_{D^0} P_{D^{0}\mu}$$

and the second and third terms can be similarly decomposed into linear combinations of terms depending only on:
Photon Polarization $$\epsilon^{*}_{\gamma \alpha}$$
$$D^{*0}$$ Polarization $$\epsilon^{*}_{D^{*0} \beta}$$
Momentums of the photon and D* (q and p respectively)
And the constants can depend on p^2.

So any sort of combination of these such that the end result yields one remaining index.

Now my first question, for those that do these sort of things, am I handling the D* matrices correctly? Am I missing any effective terms/form factors?

 

[Vanadium 50]

This is over my head, because I am an experimenter, but what are all the gamma_5's doing there? This is an electromagnetic decay, so why are you projecting out the left handed state?

 

[sir-pinski]

Hello,

Yes, this is indeed a 1-body radiative decay process. The calculation for the branching ratio (BR) of this process involves several steps, which I can walk you through.

First, we need to consider the decay amplitude, which is given by the matrix element of the weak interaction between the initial D* meson and the final state photon. This can be written as the product of two matrix elements, as you have correctly shown:

M \propto <D^0|\bar{u}\gamma_\mu\left(1-\gamma_5\right)c|0>[A_1 <\gamma|\bar{u}\gamma_\mu\left(1-\gamma_5\right)c|D^{*0}>+A_2<\gamma|\bar{u}i \sigma_{\mu \nu} p_\nu \left(1+\gamma_5\right)c|D^{*0}>

The first matrix element on the right-hand side is the same as the one you have written, and it can be decomposed as you have shown. The second matrix element, however, is slightly different. It involves the spin-1 polarization vector of the D* meson, which can be written as:

\epsilon^{*}_{D^{*0} \beta} = \frac{\epsilon_{\mu\nu\alpha\beta}p^\mu q^\nu}{m_{D^*}q^2} \epsilon^{*\alpha}

where \epsilon^{*\alpha} is the polarization vector of the photon and q is the momentum of the photon.

Next, we need to consider the decay rate, which is related to the branching ratio via the total width of the D* meson. The decay rate is given by:

\Gamma = \frac{1}{2m_{D^*}} \int |M|^2 d\Phi

where d\Phi is the phase space of the decay. This can be written in terms of the invariant mass of the photon, q^2, and the form factors of the D* meson, F_1(q^2) and F_2(q^2):

\Gamma = \frac{1}{8\pi m_{D^*}^2} \int_{q^2_{min}}^{q^2_{max}} |M|^2 dq^2

= \frac{1}{8\