15.5. Signal and Background

Identifying signal

In real experimental data you rarely measure the quantity of interest in isolation. Instead, the events that truly correspond to your physics process of interest, the signal, are mixed with other processes, the background. The goal of analysis is to isolate the signal as well as possible without distorting it.

In ROOT-based analyses, signal identification usually begins with understanding the characteristic features of the signal in one or more observables. These observables can be simple quantities such as energy, time or position, or more complex derived quantities such as invariant mass or missing transverse energy. You typically start from plots created earlier in the workflow, for example histograms or TTree draws, and visually inspect them for structures that can correspond to signal.

A common situation is a narrow peak sitting on top of a broader distribution. For instance, if you plot a histogram of reconstructed invariant mass for particle decays, genuine decays of a specific particle often produce a peak near the particle mass. The underlying combinatorial or random combinations of tracks produce a smoother background. In time spectra from nuclear or detector calibration data, a signal can appear as a sharp timing peak on top of a flat or slowly varying accidental background.

To identify signal you first need a signal region. This is a region of the observable space where signal is expected to be enhanced compared to background. It can be defined by simple cuts, such as an energy range, a mass window, or a time interval. ROOT makes it easy to implement such definitions at the histogram level with axis ranges and cuts in TTree::Draw, or in RDataFrame through Filter calls. The exact choice of the signal region is usually guided by physics expectations, detector resolution and sometimes by fits to simulated signal distributions.

You often also make use of additional discriminating variables to improve signal purity. For example, you might require that tracks satisfy quality criteria, or that energies in different detector components fall within expected ranges. In ROOT, these conditions are applied as logical expressions when filling histograms or selecting entries from TTrees. Combining several cuts can significantly increase the signal to background ratio, at the cost of reducing the total signal yield.

An important practical consideration is to avoid designing cuts by looking only at statistical fluctuations. Where possible, you should rely on simulations, control samples or well understood calibration data to understand how signal should look. ROOT allows you to overlay histograms from simulation and data, or from known “pure” samples, so you can verify that your assumed signal shape and position match observations. This comparison is a key step before you define your final signal region.

Finally, keep in mind that signal identification is often iterative. You might start with a broad region, check how the distribution behaves, refine the region, and examine how the apparent peak or feature changes. All along, ROOT tools such as histogram fits, projections and profile plots help you characterize both the signal and the surrounding background.

Estimating background

Once you have a signal region, the next task is to estimate how many of the events in that region are actually background. The difference between the total number of events in the signal region and the estimated number of background events is your estimated signal yield. A reliable background estimate is essential for any quantitative result such as cross sections, branching ratios or discovery claims.

There are two broad strategies for background estimation in ROOT analyses. The first uses models fitted to the data itself. The second uses separate control regions or external information such as simulations or independent measurements.

A very common, and conceptually simple, method is the sideband technique. Suppose your signal appears as a peak in a mass histogram. You define a central window around the peak as the signal region, and two sideband regions on either side where you expect mainly background. The key assumption is that the background varies smoothly across the region of interest. In ROOT, you can integrate the histogram in the sidebands and extrapolate under the peak. For a flat background you might take the average bin content in the sidebands and multiply by the number of bins in the signal window. For slowly varying backgrounds, you can fit the sidebands with a simple function using TF1 and TH1::Fit, for example a polynomial of low order, and then integrate this function over the signal region to get the expected background.

Fitting background shapes is often more robust than simple counting. You choose a function that is flexible enough to describe the observed distribution but not so flexible that it starts to mimic the signal peak. In ROOT, you can fit the whole histogram with a model that includes both signal and background components, or you can exclude the signal region from the fit and fit only the sidebands. For example, you may fit the background with an exponential or polynomial, then examine the fit quality by looking at the chi square value and residuals. If the fit is acceptable, you integrate only the background component of the fitted function over the signal window to obtain the background estimate.

Another powerful approach relies on control regions. A control region is chosen such that the signal contribution is negligible, while the background composition is similar to that in the signal region. In ROOT, you often define control regions by different cuts in TTree::Draw or RDataFrame::Filter. For instance, you might invert one of the signal selection criteria, or select a different mass range dominated by background. You then count events or fit distributions in the control region and transfer this information to the signal region, sometimes with scaling factors derived from simulation or theory. This is particularly common in more complex analyses where multiple background processes contribute.

Simulated samples can also help to estimate background. After validating the simulation against data in background-dominated regions, you can use the simulated yields and shapes to predict the background in the signal region. In ROOT this typically means reading in simulated TTrees, applying the same selection cuts, and filling histograms that represent each background component separately. These histograms can then be normalized to known cross sections, luminosities or control data, and summed to give a total background prediction.

Uncertainties on the background estimate are as important as the central value. Statistical uncertainties arise from the finite size of the sidebands, control samples or simulated samples. ROOT histogram tools such as bin errors and integrals with error propagation help you quantify these. Systematic uncertainties, such as the choice of background fit function or the assumption that sidebands represent the background under the peak, are often evaluated by varying analysis choices and repeating the estimate. Comparing results from alternative methods, for example sidebands and fits, is also a valuable cross check.

Background subtraction

After you have determined how many background events are expected in your signal region, you can subtract this background to obtain an estimate of the pure signal distribution. This step is called background subtraction. In ROOT, background subtraction can be as simple as subtracting histogram contents, or as sophisticated as decomposing several components with a simultaneous fit.

The simplest case involves a single observable, such as invariant mass, and a histogram of all events in the signal selection. Suppose you have another histogram that represents the background in the same observable and selection, normalized to the expected background yield. The signal histogram is then obtained by subtracting the background histogram from the data histogram. In ROOT, this is implemented with TH1::Add using a negative scale factor for the background histogram. If $H_{\text{data}}$ is the data histogram and $H_{\text{bkg}}$ is the normalized background histogram, the signal-only histogram is
$$
H_{\text{sig}} = H_{\text{data}} - H_{\text{bkg}}.
$$

ROOT automatically propagates statistical uncertainties when you subtract histograms, provided the histograms carry bin errors. For each bin, if $N_{\text{data}}$ and $N_{\text{bkg}}$ are the contents and $\sigma_{\text{data}}$ and $\sigma_{\text{bkg}}$ are their errors, the signal bin content is $N_{\text{sig}} = N_{\text{data}} - N_{\text{bkg}}$ and the statistical uncertainty is
$$
\sigma_{\text{sig}} = \sqrt{\sigma_{\text{data}}^2 + \sigma_{\text{bkg}}^2},
$$
assuming the data and background estimates are statistically independent. If your background histogram comes from another dataset or a fit, you should ensure that bin errors are correctly set. For histograms filled with weights or scaled by factors, calling Sumw2 before filling guarantees that ROOT tracks the sum of squared weights and computes meaningful uncertainties.

Direct histogram subtraction is particularly useful when you want the background subtracted shape of a distribution, for example the energy spectrum of signal events. It allows you to feed the signal-only histogram into later analysis steps, such as fits or integration. However, you need to check for unphysical negative bin contents after subtraction. Small negative values consistent with zero within uncertainties are usually harmless, but large negative bins indicate a problem with normalization or background shape.

In many analyses, background subtraction is embedded in a fit rather than performed explicitly on histograms. For example, you might fit a mass histogram with a function that is the sum of a signal model and a background model. The fit returns the signal and background yields and their uncertainties directly, without you subtracting histograms by hand. ROOT fits can also be extended to multidimensional distributions or unbinned likelihoods. In such cases, background subtraction is conceptually performed at the level of model parameters instead of bin contents.

When background subtraction uses sidebands, the basic idea is that you estimate the background under the signal region from the sidebands and subtract this estimate from the total number of events in the signal window. If $N_{\text{SR}}$ is the number of entries in the signal region and $N_{\text{bkg,SR}}$ is the estimated background there, the signal yield is
$$
N_{\text{sig}} = N_{\text{SR}} - N_{\text{bkg,SR}}.
$$
You can implement this in ROOT by integrating the histogram over the relevant ranges and using the integrals and their errors. This approach is straightforward when you are mostly interested in the total signal count rather than the detailed shape.

In advanced scenarios, you may have several background sources with different shapes. A common technique is template fitting. You create a set of template histograms for each component, including signal and each background, usually from simulation or control regions. You then fit the data histogram as a linear combination of these templates, leaving the component normalizations as free parameters. ROOT supports this approach through fits with custom TF1 functions that read template bin contents, or with dedicated statistical tools. The effective result is that the fit separates and subtracts the background contributions to reveal the signal.

Whatever method you choose, it is essential to propagate both statistical and systematic uncertainties on the background estimate through to the subtracted signal. Statistical uncertainties are handled naturally in ROOT through bin errors and error propagation. Systematic uncertainties, such as variations in sideband choice or background model, are explored by repeating the subtraction with altered assumptions and comparing the resulting signal distributions or yields. The spread in these results gives an estimate of the systematic contribution to your uncertainty.

Finally, background subtraction is not always the end of the story. Removing background can alter the statistical properties of your sample, and in some contexts it is preferable to fit models directly to the un-subtracted data. In this course, however, background subtraction serves as a useful tool to visualize and quantify the underlying signal, and ROOT provides a flexible set of operations for performing and checking it.

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