24.8. Calculate Physical Quantities

Table of Contents

Clarifying the Goal

In this part of the final project you turn fit results and measured spectra into meaningful physical quantities. You start from what you already have earlier in the project: histograms, fits, and calibrated variables. You now combine them to compute things like yields, rates, cross sections, lifetimes, or other analysis-specific observables. The exact list depends on the dataset you use, so this chapter focuses on typical patterns and techniques rather than one fixed example.

The key idea is to move from “this is the shape of a distribution” to “this is a numerical statement about nature, with units and uncertainties”.

From Histograms and Fits to Numbers

Often the first step is to turn a histogram, or a fitted function on top of it, into a single number or a small set of numbers. Examples are:

Area under a peak, which represents the number of signal events after background subtraction.

Mean energy or momentum of a sample.

Width of a peak, which may be interpreted as a resolution or a lifetime.

If you have fitted a function $f(x)$ to a histogram, you typically use the fit parameters directly. For instance, for a Gaussian peak with parameters $A$, $\mu$, and $\sigma$,

$$
f(x) = A \exp\!\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right)
$$

you interpret $\mu$ as the central value of the observable and $\sigma$ as its spread or resolution. You may also integrate the fitted function to get the total yield in the peak.

When you compute a physical quantity from a fit, always:

Use the parameter values from the fit, not from visual inspection.
Use the parameter uncertainties from the fit covariance matrix.
Clearly state which parameter represents which physical quantity.

For a normalized Gaussian with area $N$ and width $\sigma$ the integral under the peak is $N$, and the fit often returns $N$ directly as a parameter. Check the function definition used in the fit to avoid misinterpreting which parameter is amplitude, which is area, and which is width.

Background Subtraction and Signal Yield

In many analyses the quantity of interest is the number of signal events in a region of a distribution. However, your histogram contains both signal and background. You usually proceed in one of two ways.

If you have fitted a signal plus background model, you obtain the signal yield from the fitted signal component. For example, if

$$
f(x) = f_{\text{sig}}(x) + f_{\text{bkg}}(x)
$$

and $f_{\text{sig}}$ has a parameter $N_{\text{sig}}$ that represents the total number of signal events, then $N_{\text{sig}}$ is your signal yield. Its uncertainty is given by the fit and is used directly in later calculations.

If you do not use a full fit, you can estimate the background with a sideband method. You choose an interval around the peak as the signal region, count all events in that interval, and subtract an estimate of how many background events fall under the peak. The background estimate is usually derived from neighboring regions that contain no signal.

Typical steps are:

Define a signal window, for example around the peak in an invariant mass distribution.

Count the number of entries in that window. This gives $N_{\text{total}}$.

Estimate the background under the peak, for example by interpolating from sidebands.

Compute the signal yield as
$$
N_{\text{sig}} = N_{\text{total}} - N_{\text{bkg}}.
$$

A signal yield used in further physics calculations must be either:

Directly taken from a signal component of a validated fit, or
Corrected for background with a clearly defined and documented method, such as sidebands.
Never use raw counts in a peak region as signal unless you can argue that the background is negligible.

The signal yield is often the core input to later steps. Many physical quantities are proportional to this yield, possibly corrected for efficiencies, acceptance, and luminosity or target thickness.

Correcting for Efficiency and Acceptance

Detectors do not see all events. Some are not triggered, some are lost due to reconstruction inefficiencies, and some never enter your kinematic acceptance. If you want a quantity that reflects the truth at the physics level, you must correct for this.

Schematically, if $N_{\text{true}}$ is the number of true signal events and $N_{\text{obs}}$ is the observed signal yield, and $\epsilon$ is the overall efficiency, then

$$
N_{\text{true}} = \frac{N_{\text{obs}}}{\epsilon}.
$$

Here $\epsilon$ is often a product of contributions such as trigger efficiency, reconstruction efficiency, selection efficiency, and geometric acceptance, each derived from control data or simulations.

You will typically have an efficiency value and its uncertainty. When you correct the yield, you must propagate the uncertainty of $\epsilon$ to the final result. This connects directly to the general rules of uncertainty propagation that you should have seen earlier in the course.

Efficiency corrections must always be applied when you report physical yields or cross sections. The corrected quantity is usually
$$
N_{\text{corr}} = \frac{N_{\text{sig}}}{\epsilon}.
$$
You must also propagate the uncertainty on $\epsilon$ into the uncertainty on $N_{\text{corr}}$.

In ROOT analyses, efficiency often depends on a variable, for example efficiency as a function of energy or angle. In that case, you may need to apply a bin-by-bin correction using a histogram of efficiencies that you have prepared earlier in the project.

Rates, Cross Sections, and Normalization

Once you have a corrected yield, you often want to express it as a rate or a cross section. The exact formula depends on the type of experiment, but the structure is similar.

For a counting experiment where you measure a process over a live time $T$, the event rate $R$ is

$$
R = \frac{N_{\text{corr}}}{T}.
$$

If your data come from a collider experiment with integrated luminosity $\mathcal{L}$, the cross section $\sigma$ for a process is often computed as

$$
\sigma = \frac{N_{\text{corr}}}{\mathcal{L}}.
$$

For a fixed target experiment with a target of areal density $n$ you use a similar expression with $n$ and the delivered beam.

No matter which case you have, the pattern is always:

physical quantity = corrected yield divided by a normalization factor.

This normalization factor encapsulates how much “opportunity” the experiment had to produce the events you see, such as total exposure time, number of incident particles, target thickness, or integrated luminosity.

When computing these quantities in ROOT you will generally:

Read the normalization inputs (for example live time, luminosity) from configuration settings, from a text file, or from a dedicated branch in your TTree.

Use simple arithmetic in your macro or RDataFrame code to divide corrected yields by the normalization.

Attach units explicitly in comments or text labels so that it is clear whether your result is, for example, events per second, barns, or some other unit.

Every reported rate or cross section must:

Use a yield that has been corrected for efficiency and acceptance.
Be divided by the appropriate normalization (time, luminosity, or beam and target parameters).
Include a clear statement of units.

This step is where your earlier work on normalization of histograms also becomes conceptually useful. However, remember that a unit normalized histogram is different from a physical quantity like a cross section. The former is a shape with total area 1, the latter is an observable with dimensions and a full uncertainty budget.

Derived Observables from Kinematics

Many physical quantities are not directly measured, but derived from measured energy, momentum, or angles. With ROOT you usually compute these observables event by event, either in a TTree loop or using RDataFrame, and then you summarize their distribution.

For example, in particle and nuclear physics you might compute:

Invariant mass of two particles, using their four-vectors.

Transverse momentum from momentum components.

Angular variables such as scattering angle or opening angle.

You usually already prepared these variables earlier in the project, possibly stored them in TTrees, and plotted their distributions. In this step, you now extract physics from them.

Examples of typical derived quantities include:

The mean value and standard deviation of an observable, computed from a histogram.

Characteristic scales such as a resonance mass from a fitted peak.

Resolution parameters from widths of response functions.

You may compute simple statistics directly from histogram methods, like the mean and RMS, or you may use fit parameters for more model based quantities. In both cases, connect the numerical value with its physical interpretation.

When you interpret derived quantities:

Use the appropriate formula for the observable, applied consistently to all events.
Make sure you know whether the quantity is computed in data space (for example histogram mean) or parameter space (for example fit parameter).
Always associate a physical interpretation and units to each quantity.

In the final report, you will not only quote numbers but explain what they mean in terms of the system under study.

Uncertainty Propagation to Physical Results

Any physical quantity you compute must come with an uncertainty. By this stage of the project, you have already encountered several sources:

Statistical uncertainty on the signal yield, from counting or from fits.

Systematic uncertainty on efficiencies.

Uncertainties on normalization factors such as luminosity or target thickness.

These uncertainties propagate to the final result. For simple combinations of quantities you can use standard error propagation. For example, if your cross section is

$$
\sigma = \frac{N_{\text{sig}}}{\epsilon \, \mathcal{L}}
$$

and you treat $N_{\text{sig}}$, $\epsilon$, and $\mathcal{L}$ as independent variables, you can write the relative variance as

$$
\left(\frac{\Delta \sigma}{\sigma}\right)^2
=
\left(\frac{\Delta N_{\text{sig}}}{N_{\text{sig}}}\right)^2
+
\left(\frac{\Delta \epsilon}{\epsilon}\right)^2
+
\left(\frac{\Delta \mathcal{L}}{\mathcal{L}}\right)^2.
$$

You can apply this in your ROOT macro by computing the relative uncertainties and combining them in quadrature.

For any final physical quantity $Q$ that depends on independent inputs $x_i$, use
$$
\left(\frac{\Delta Q}{Q}\right)^2 = \sum_i \left(\frac{\partial Q}{\partial x_i}\frac{x_i}{Q}\right)^2 \left(\frac{\Delta x_i}{x_i}\right)^2
$$
which for simple products and ratios reduces to “add relative uncertainties in quadrature.”

If some sources of uncertainty are correlated, or if your quantity depends on fit parameters with a covariance matrix, you would need a more detailed treatment. For the purposes of this beginner-level final project it is usually acceptable to treat the main inputs as independent and combine them as above. Make sure you state clearly what approximations you make.

Presenting and Documenting Physical Results

Once you have computed your physical quantities, you will present them as part of your final project output. This includes:

Stating the numerical results with appropriate significant figures and uncertainties.

Referencing the plots, fits, and histograms that led to those numbers.

Explaining the corrections and normalizations that you applied.

In practice it is useful to collect your key results in a small ROOT macro or a Python script that prints them out, and to summarize them in a table in your report.

A simple structure for presenting results is:

Quantity description, including definition and units.

Numerical value with statistical and, if possible, systematic uncertainty.

Short explanation of how it was obtained, for example from a fit to a specific histogram, or from a corrected yield and normalization.

This bridge from technical ROOT operations to clear physical statements is the main goal of this chapter. By the time you finish this step of the project, you should have one or more final numbers that answer a physics question posed at the beginning of the analysis.