I've been asked to measure a likelihood function , and to measure "simultaneously parameters" $\mu^{ggF}$ and $\mu^{V \hspace{0.5 mm} BF} $ , and to measure the parameters $\mu^{ggF}$ and $\mu^{V \hspace{0.5 mm} BF}$, for the decay channel below:
"A scalar boson $\phi$ is observed in the decay channel $\phi \to \gamma \gamma$. $\phi$ is created via the gluon fusion and via the vector boson fusion processes, which have distinct experimental signatures. We want to measure the signal strength modifiers of $\mu^{ggF}$ and $\mu^{V\hspace{0.5mm} BF}$ $$\tag{1}\mu = \frac{\sigma\cdot BR}{\sigma_{SM} BR_{SM} }.$$ An analysis with two signal regions was prepared. Each is optimised to select one of the two methods shown below:"
$\sigma$ is the cross-section, and $BR$ is the Branching ratio.
SR1 | SR2 | |
---|---|---|
$N^{obs}$ (observed events for each signal region) | 24 | 8 |
$n^{ggF}$ ( expected signal events from the SM for this mechanism ) | 16.2 | 2.1 |
$n^{V \hspace{0.5 mm} BF}$ (expected signal events from the SM for this mechanism ) | 0.9 | 4.2 |
$n^b$( expected background) | 0.2 | 0.9 |
I don't have a background on this, so the following questions might sound too simple:
Questions:
- How does one calculate $BR$ and $\sigma$?
- What difference is there between measuring the parameters $\mu^{ggF}$ and $\mu^{V \hspace{0.5mm} BF}$, and between measuring the "simultaneously parameters"?
Edit:
I've found that for a simple counting experiment :
$$L = \prod_{i=1}^N \frac{(\mu S_i + n_i^B) ^{N_i^{obs}} e^{-(\mu S_i + B_i)}}{N_i^{obs}!}, \tag{2} $$
where $S_i = n^{ggF} + n^{VBF}$, resulting in,
$$\tag{3} L = \frac{17.1 \mu +0.2}{24!} e^{-(17.1 \mu +0.2)} + \frac{25.3 \mu +0.9}{8!} e^{-(25.3 \mu +0.9)}. $$
How can this help me determine the requested parameters $\mu^{ggF}$ and $\mu^{V \hspace{0.5 mm} BF}$?