I am reading up on the detection of the Higgs-Boson at the LHC and more specifically at CMS. I found the following graphic:
I understand that the peaks correspond to the mass of the Higgs-Boson but I struggle to interpret all the aspects of the graph in general. The CMS site itself did not yield a satisfactory answer. I was hoping someone could explain in some more detail. Thanks in advance.
The Plot:
If you have a two particle final state in $pp' \rightarrow kk'$, where and $k$ are 4 momenta, not particle types, one can look at the invariant mass of the final state:
$$ m_{\gamma\gamma}^2 =(k+k')^2 = (k+k', \vec k + \vec k)^2 $$
That is Lorentz invariant.
So on the $x$axis we have $m_{\gamma\gamma}$ in GeV.
The $y$-axis is not a production cross-section, that information can be derived from the total luminosity on the figure; rather, it is in counts per GeV bin--that's good. It allows us to look at the statistic w/o scaling. So it focuses on validity of detection, not comparison with theory.
In addition to the histogrammed 2 photon events, we have 2 curves:
The background events, $B$. That is, two photon events that are expected from the non-Higgs sector of the SM. This is calculated with a very in-depth model of physics (from proton parton distributions and hadronization simulations developed from decades of data and theory), and a detailed model of the detector apparatus, including decades of data for the material properties and passage of charge particles through matter.
The signal, $S$, which represents Higgs production. I do not know if it is normalized by an expected theoretical coupling--that should be experimentally measured, so I suspect not. Since the Higgs is unstable, its mass is uncertain, and the shape should be a Lorentzian, AKA Cauchy distribution, AKA The Breit-Wigner distribution...but maybe they just used a gaussian to fit it. Idk.
Back to the data. For simplicity, I am just going to describe the binned data in the bin in which the peak occurs:
Let's say its the bin spanning 125.5 - 126.5 GeV, which contains the Higgs peak at 125.8 GeV. Eyeballing:
$$ S + B = 5600 $$ $$ B = 500 $$
Since they are just counts, the statistical uncertainties go as the square root, so:
$$ N_{S + B} = 5600 \pm 75 $$ $$ N_B = 5000 \pm 71 $$
from which we can compute:
$$ N_S = 600 \pm 102 $$
(iirc, aren't the $B$'s correlated--better check that).
Which is already a six sigma signal--in theory, maybe I got the error wrong? I'm on a zoom call rn.
Finally, the BW fit has a centroid, a width and a magnitude, representing the Higgs mass, lifetime and coupling.
There is a fundamental question of statistics, as it used in analyzing such event count Poisson statistics along a curve: Any point of events at a fixed kinetic energy lying outside the primitive $3\sigma=3\delta N=3\left< \sqrt n\right>$ by a large amount with an error estimate of the same magnitude.
We discussed this problem in seminars in Göttingen in the 1960ties by comparing it to the experiment, to determine the distance from the station to the Instute of Theoretical Physics by counting the number footsteps a million times in order to get the distance, insecure by about 1 m by definition of the measure, with an precision goal of 1 mm.
IF you take the experimental points, the four points making up the Higgs, are well inside the experimental boundaries of Poisson noise.
