How are signal strength modifiers calculated?

Question

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

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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"?

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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 $\mu^{V \hspace{0.5 mm} BF}$?

Answer 1

How does one measure the branching ratio and cross section?

You cannot measure the cross section or the branching ratio directly. What you really can measure is the cross section for the overall $\rm XX\to\phi\to\gamma\gamma$. This is equal to the cross section $\sigma({\rm XX\to\phi+X})$ times the branching ratio $\operatorname{BR}({\rm\phi\to\gamma\gamma})$. In other words, your shouldn't think of $\sigma$ and $\operatorname{BR}$ as two individual parameters you can measure, but $\sigma\cdot\operatorname{BR}$ as one single parameter to be measured. Specifically, you expect that $n_{\rm events}=\mathcal{L}\left(\sigma\cdot\operatorname{BR}\right)$, where $n_{\rm events}$ is the number of signal events and $\mathcal{L}$ is the luminosity.

What difference is there between measuring the parameters $\mu^{ggF}$ and $μ^{VBF}$, and between measuring the "simultaneously parameters"?

I'm not familiar with this exact wording, but I think it's likely this means you are meant to find maximize the likelihood for the parameters individually versus maximizing the likelihood for the two parameters jointly. That is, find the likelihood function as a function of just one parameter and maximize it versus finding the joint likelihood function and maximizing that for both parameters.

I've found for a simple counting experiment

Note that $\prod_{i=1}^N$ represents a product over $i$, not a sum, and that you dropped the power of $N_i^{\rm obs}$ in the following expression. Also presumably if you want to find both $\mu$ separately, you should not just add $n^{ggF}$ and $n^{VBF}$ together, but you need to find separate likelihoods for the two.

I suggest trying to understand why that is the correct likelihood function for a simple counting experiment. Then you can adapt it to your circumstances rather than just blindly plugging things in.

Once you've found the likelihood function, you want to find the value of $\mu$ that maximizes it. For a simple likelihood function in one parameter, you can simply plot it or find points where $\frac{dL}{d\mu}=0$. In the more general case, you will want to use a software minimization package like MINUIT.