KAHIBARO
Invariant Mass
Table of Contents
Calculating invariant mass
Invariant mass is a quantity that combines energy and momentum into a single number that does not change between reference frames. In particle and nuclear physics this is central to identifying particles and studying their production and decay.
In special relativity each particle is described by a four vector of energy and momentum, often represented in ROOT by the class TLorentzVector. In natural units where $c = 1$, the four vector is written as
$$
p^\mu = (E, p_x, p_y, p_z),
$$
with $E$ the energy and $p_x, p_y, p_z$ the components of the three-momentum. The invariant mass $m$ of a single particle is defined through the relation
$$
m^2 = E^2 - \vec{p}^{\,2}
= E^2 - (p_x^2 + p_y^2 + p_z^2).
$$
Key relation for a single particle:
$$
m^2 = E^2 - \vec{p}^{\,2} \quad \text{and} \quad m = \sqrt{E^2 - \vec{p}^{\,2}}.
$$
This quantity is invariant under Lorentz transformations.
In ROOT, if you have a TLorentzVector called p, you typically set it with either the components $(p_x, p_y, p_z, E)$ or with transverse momentum, pseudorapidity, azimuthal angle, and mass. Once the vector is defined, ROOT provides helper methods to get the invariant mass directly without manually applying the formula. The most common ones are
| Method | Meaning | ||
|---|---|---|---|
p.M() | Returns the invariant mass $m$ | ||
p.M2() | Returns the invariant mass squared $m^2$ | ||
p.E() | Returns the energy $E$ | ||
p.P() | Returns the magnitude of momentum $ | \vec{p} | $ |
Using M() or M2() avoids numerical errors that can appear if you subtract nearly equal numbers $E^2$ and $\vec{p}^{\,2}$ by hand.
For a system of several particles, such as two decay products in a final state, you form the invariant mass of the whole system by adding their four vectors and then taking the mass of the sum. For two particles with four vectors $p_1^\mu$ and $p_2^\mu$ the combined four vector is
$$
P^\mu = p_1^\mu + p_2^\mu = (E_1 + E_2,\ \vec{p}_1 + \vec{p}_2),
$$
and the invariant mass of the pair is
$$
M^2 = P^\mu P_\mu = (E_1 + E_2)^2 - (\vec{p}_1 + \vec{p}_2)^2.
$$
Key relation for a system of particles:
To compute the invariant mass of $N$ particles, sum their four vectors and take the mass of the result:
$$
P^\mu = \sum_{i=1}^{N} p_i^\mu, \quad
M = \sqrt{P^\mu P_\mu}.
$$
In ROOT, this is usually implemented as
TLorentzVector P = p1 + p2 + ...; followed by double M = P.M();.
In a typical ROOT analysis you construct the four vectors of the final state particles inside an event loop, sum them to build the candidate system, compute the invariant mass, and then fill a histogram with this mass value for each event that passes your selection. The resulting invariant mass distribution is the starting point for most reconstruction and resonance searches.
Reconstructing particle decays
Reconstructing particle decays means using the measured four momenta of the decay products to infer the properties of the parent particle, especially its invariant mass. Experimentally, the parent is often too short lived to be directly observed, but you can identify it through peaks in the invariant mass spectrum of its decay products.
The essential idea is straightforward. Suppose a neutral particle $X$ decays into two charged particles $a$ and $b$. In the detector you measure the momenta and, depending on the experiment, identify their particle types. For each event, you build TLorentzVector objects for $a$ and $b$ with their measured momenta and assumed masses. Then you add them to form the parent candidate:
$$
P^\mu_X = p^\mu_a + p^\mu_b.
$$
The invariant mass of the candidate is
$$
M_X = \sqrt{P_X^\mu P_{X\mu}}.
$$
In ROOT this becomes a compact calculation. For example, inside an event loop you might define two TLorentzVector objects, set them with the measured kinematics, then compute their sum and mass. For three body decays you extend the same idea and sum three four vectors. Repeating this for many events and filling a histogram of the resulting invariant masses produces a distribution where true decays of the parent create a peak at the nominal mass, while random combinations of particles form a smooth background.
The characteristic peak in the invariant mass histogram is the experimental signature of the decaying particle. For a well known resonance, such as a $J/\psi$ decaying to two muons, you expect a narrow peak near its Particle Data Group mass. The width of the peak is influenced by the intrinsic width of the resonance and by the experimental resolution modeled by detector smearing. Using ROOT histogram and fitting tools, you typically fit this peak with a model, such as a Gaussian over a polynomial background, to extract the number of signal events and the measured mass.
In more complex decays you often have several possible combinations of final state particles. For example, in multihadron events there can be many charged tracks, and different pairs or triplets can be combined to form candidate parents. ROOT TTrees store the event level information, including all tracks and their momenta, and your analysis code loops over possible combinations, builds the corresponding TLorentzVector sums, and computes their invariant masses. Selection cuts are then applied to reduce wrong combinations and enhance the true decay signal.
Central idea of decay reconstruction:
- Measure the final state particle kinematics.
- Build
TLorentzVectorobjects with appropriate mass hypotheses. - Sum the four vectors to form parent candidates.
- Compute the invariant mass of each candidate and fill a histogram.
- Identify real particles as peaks in the invariant mass distribution above background.
You can also reconstruct intermediate states in cascade decays by first combining a subset of particles to form an intermediate resonance, then combining that resonance with additional particles to reconstruct a heavier parent. Each step uses the same invariant mass principle but applied to different subsets of final state objects. ROOT simplifies this by allowing you to treat intermediate combinations as four vectors that can be added again and again.
In summary, invariant mass reconstruction is one of the main experimental tools to discover and study particles. With ROOT, the practical workflow consists of building and summing TLorentzVector objects, computing M(), and using histograms and fits to extract physics from the resulting invariant mass spectra.
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