# How are signal strength modifiers calculated?

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

• This is too many questions for most people to answer. Sincere advice: (1) for these kinds of situations, it is much better to ask your research supervisor for the big picture before getting this lost, (2) while asking Phys.SE might be a decent alternative, you should never try to solve this kind of problem by googling and opening 100 tabs containing random PDFs with totally different notation and levels of sophistication. You'll just get more and more confused. Instead, you should find one good book/resource (e.g. by asking your advisor for one) and read it all the way through. Mar 30, 2022 at 16:59
• Also, it's good that you've gotten rid of some of the extra questions, but in the future please refrain from making so many edits (26 is really high, the highest of any question on this site in the past month). Mar 30, 2022 at 17:09
• Yeah I don understand that, but I have been trying to solve this for the last 2 days, and I have found out a lot of new information and saw a lot of mistakes in the things I had previously written. I understand that they were a lot of questions, but often times people here give me very broad answers, and I wanted to be specific in the things I didn't know/understand. Thank you for your help though. Mar 30, 2022 at 17:12

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.

• I think I get most of this, but regarding the software part, couldn't this simply be plotted (MATLAB or Python) and the max value be found on the plot? Mar 30, 2022 at 21:09
• @user7077252 Yes, as I said you can plot it for a likelihood function in one parameter. Maybe two. You'll probably want to plot the log likelihood if you do this, because the likelihood changes over many orders of magnitude. (Note that maximizing the log likelihood is the same as maximizing the likelihood). You will not be able to do this for more complicated cases with many parameters, though.
– Chris
Mar 30, 2022 at 21:19
• That makes sense, regarding the last point in your explanation, if I understand this correctly, does this mean that: $$L_{ggF} =\frac{ L_{ggF \hspace{0.5 mm} SR1} + L_{VBF \hspace{0.5 mm} SR2}}{2}$$ but when doing it for both parameters jointly: $$L = L_{ggF} \cdot L_{VBF}$$ Does this make sense? Mar 30, 2022 at 21:44
• @user7077252 The second expression makes some sense. The first does not. $L_{ggF}$ should be something like $L_{ggF,SR1}\cdot L_{ggF,SR2}$. (NB: you expect some pollution of ggF into SR2 as well as VBF into SR1!) Likelihoods are proportional to probabilities, and the probability of X and Y happening is $P(X)P(Y)$, not $\frac{P(X)+P(Y)}{2}$.
– Chris
Mar 30, 2022 at 21:55
• @user7077252 I mean that your signal regions are not pure: both contain some amount of ggF and some amount of VBF. You shouldn't just assume that SR1 contains only ggF events and that SR2 contains only VBF events. You can get the expected ratio of these from the ratio of expected events in the respective regions.
– Chris
Mar 30, 2022 at 22:32