Invariant mass

The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total energy and momentum that is the same in all frames of reference related by Lorentz transformations.[1] If a center-of-momentum frame exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass (a.k.a. relativistic mass) of the system is greater than the invariant mass, but the invariant mass remains unchanged.

Due to mass–energy equivalence, the rest energy of the system is simply the invariant mass times the speed of light squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.

Systems whose four-momentum is a null vector (for example a single photon or many photons moving in exactly the same direction) have zero invariant mass, and are referred to as massless. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized tachyon), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass.

If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by c2. See mass–energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy.

For an isolated massive system, the center of mass of the system moves in a straight line with a steady sub-luminal velocity (with a velocity depending on the reference frame used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c2. This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames.

Note that for reasons above, such a rest frame does not exist for single photons, or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.

Rest mass 0 and 1
Possible 4-momenta of particles. One has zero invariant mass, the other is massive

Sum of rest masses

Because the invariant mass includes the mass of any kinetic and potential energies which remain in the center of momentum frame, the invariant mass of a system can be greater than sum of rest masses of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).

Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it.

The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound).

They will often also interact through one or more of the fundamental forces, giving them a potential energy of interaction, possibly negative.

For an isolated massive system, the center of mass moves in a straight line with a steady sub-luminal velocity. Thus, an observer can always be placed to move along with it. In this frame, which is the center of momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which always exists, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by c2.

As defined in particle physics

In particle physics, the invariant mass m0 is equal to the mass in the rest frame of the particle, and can be calculated by the particle's energy E and its momentum p as measured in any frame, by the energy–momentum relation:

or in natural units where c = 1,

This invariant mass is the same in all frames of reference (see also special relativity). This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector (E, p), calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle. The Dirac quantum operator corresponds to the particle four-momentum vector.

Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from the general formula:


is the invariant mass of the system of particles, equal to the mass of the decay particle.
is the sum of the energies of the particles
is the vector sum of the momentum of the particles (includes both magnitude and direction of the momenta)

The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows (in natural units):

If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.

In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy) the Transverse mass is used.

Example: two-particle collision

In a two-particle collision (or a two-particle decay) the square of the invariant mass (in natural units) is

Massless particles

The invariant mass of a system made of two massless particles whose momenta form an angle has a convenient expression:

Collider experiments

In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle  and pseudorapidity . Additionally the transverse momentum, , is usually measured. In this case if the particles are massless, or highly relativistic ( ,) then the invariant mass becomes:

Rest energy

The rest energy of a particle is defined as:


where is the speed of light in vacuum.[2] In general, only differences in energy have physical significance.[3]

The concept of rest energy follows from the special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See background for mass–energy equivalence.

On the other hand, the concept of the equivalent Dirac invariant rest mass may be defined in terms of the self energy corresponding to the product of a geometric matter current and a generalized potential [4] as part of a single definition of mass in a geometric unified theory.

See also


  • Landau, L.D., Lifshitz, E.M. (1975). The Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2. Butterworth Heinemann. ISBN 0-7506-2768-9.CS1 maint: Multiple names: authors list (link)
  • Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.


  1. ^ Physics for Scientists and Engineers, Volume 2, page 1073 - Lawrence S. Lerner - Science - 1997
  2. ^
  3. ^ Modell, Michael; Robert C. Reid (1974). Thermodynamics and Its Applications. Englewood Cliffs, NJ: Prentice-Hall. ISBN 0-13-914861-2.
  4. ^ González-Martín, Gustavo R. (1994). "A geometric definition of mass". Gen. Rel. Grav. 26: 1177. Bibcode:1994GReGr..26.1177G. doi:10.1007/BF02106710.
ARGUS distribution

In physics, the ARGUS distribution, named after the particle physics experiment ARGUS, is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.

Bare mass

In quantum field theory, specifically the theory of renormalization, the bare mass of an elementary particle is the limit of its mass as the scale of distance approaches zero or, equivalently, as the energy of a particle collision approaches infinity. It differs from the invariant mass as usually understood because the latter includes the 'clothing' of the particle by pairs of virtual particles that are temporarily created by the force-fields around the particle. In some versions of QFT, the bare mass of some particles may be plus or minus infinity. In the theory of the electroweak interaction using the Higgs boson, all particles have a bare mass of zero.

This allows us to write , where m denotes the experimentally observable mass of the particle, its bare mass, and the increase in mass owing to the interaction of the particle with the medium or field.

Conservation of mass

The law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as system's mass cannot change, so quantity cannot be added nor removed. Hence, the quantity of mass is conserved over time.

The law implies that mass can neither be created nor destroyed, although it may be rearranged in space, or the entities associated with it may be changed in form. For example, in chemical reactions, the mass of the chemical components before the reaction is equal to the mass of the components after the reaction. Thus, during any chemical reaction and low-energy thermodynamic processes in an isolated system, the total mass of the reactants, or starting materials, must be equal to the mass of the products.

The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. Historically, mass conservation was demonstrated in chemical reactions independently by Mikhail Lomonosov and later rediscovered by Antoine Lavoisier in the late 18th century. The formulation of this law was of crucial importance in the progress from alchemy to the modern natural science of chemistry.

The conservation of mass only holds approximately and is considered part of a series of assumptions coming from classical mechanics. The law has to be modified to comply with the laws of quantum mechanics and special relativity under the principle of mass-energy equivalence, which states that energy and mass form one conserved quantity. For very energetic systems the conservation of mass-only is shown not to hold, as is the case in nuclear reactions and particle-antiparticle annihilation in particle physics.

Mass is also not generally conserved in open systems. Such is the case when various forms of energy and matter are allowed into, or out of, the system. However, unless radioactivity or nuclear reactions are involved, the amount of energy escaping (or entering) such systems as heat, mechanical work, or electromagnetic radiation is usually too small to be measured as a decrease (or increase) in the mass of the system.

For systems where large gravitational fields are involved, general relativity has to be taken into account, where mass-energy conservation becomes a more complex concept, subject to different definitions, and neither mass nor energy is as strictly and simply conserved as is the case in special relativity.

Electron rest mass

The electron rest mass (symbol: me) is the mass of a stationary electron, also known as the invariant mass of the electron. It is one of the fundamental constants of physics. It has a value of about 9.109×10−31 kilograms or about 5.486×10−4 atomic mass units, equivalent to an energy of about 8.187×10−14 joules or about 0.5110 MeV.

Energy–momentum relation

In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum:

holds for a system, such as a particle or macroscopic body, having intrinsic rest mass m0, total energy E, and a momentum of magnitude p, where the constant c is the speed of light, assuming the special relativity case of flat spacetime.The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy-momentum equation.


In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime. The contravariant four-momentum of a particle with relativistic energy E and three-momentum p = (px, py, pz) = γmv, where v is the particle's three-velocity and γ the Lorentz factor, is

The quantity mv of above is ordinary non-relativistic momentum of the particle and m its rest mass. The four-momentum is useful in relativistic calculations because it is a Lorentz covariant vector. This means that it is easy to keep track of how it transforms under Lorentz transformations.

The above definition applies under the coordinate convention that x0 = ct. Some authors use the convention x0 = t, which yields a modified definition with p0 = E/c2. It is also possible to define covariant four-momentum pμ where the sign of the energy is reversed.

Gyrovector space

A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry. Ungar introduced the concept of gyrovectors that have addition based on gyrogroups instead of vectors which have addition based on groups. Ungar developed his concept as a tool for the formulation of special relativity as an alternative to the use of Lorentz transformations to represent compositions of velocities (also called boosts - "boosts" are aspects of relative velocities, and should not be conflated with "translations"). This is achieved by introducing "gyro operators"; two 3d velocity vectors are used to construct an operator, which acts on another 3d velocity.

Mandelstam variables

In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion. They are used for scattering processes of two particles to two particles. The Mandelstam variables were first introduced by physicist Stanley Mandelstam in 1958.

If the Minkowski Metric is chosen to be , the Mandelstam variables are then defined by

Where p1 and p2 are the four-momenta of the incoming particles and p3 and p4 are the four-momenta of the outgoing particles, and we are using relativistic units (c=1).

s is also known as the square of the center-of-mass energy (invariant mass) and t is also known as the square of the four-momentum transfer.

Mass in special relativity

Mass in special relativity incorporates the general understandings from the laws of motion of special relativity along with its concept of mass–energy equivalence. The word mass is given two meanings in special relativity: one (rest or invariant mass, and its equivalent rest energy) is an invariant quantity which is the same for all observers in all reference frames; the other (relativistic mass or the equivalent total energy of the body) is dependent on the velocity of the observer. The term relativistic mass tends not to be used in particle and nuclear physics and is often avoided by writers on special relativity. They do, however, talk about the (total) energy of a body, which is the equivalent to its relativistic mass, rather than the rest energy equivalent to its rest mass. The measurable inertia and gravitational attraction of a body in a given frame of reference is determined by its relativistic mass, not merely its rest mass. For example, light has zero rest mass but contributes to the inertia (and weight in a gravitational field) of any system containing it.

For a discussion of mass in general relativity, see mass in general relativity. For a general discussion including mass in Newtonian mechanics, see the article on mass.

Massless particle

In particle physics, a massless particle is an elementary particle whose invariant mass is zero. The two known massless particles are both gauge bosons: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, gluons are never observed as free particles, since they are confined within hadrons. Neutrinos were originally thought to be massless. However, because neutrinos change flavor as they travel, at least two of the types of neutrinos must have mass. The discovery of this phenomenon, known as neutrino oscillation, led to Canadian scientist Arthur B. McDonald and Japanese scientist Takaaki Kajita sharing the 2015 Nobel prize in physics.

Mass–energy equivalence

In physics, mass–energy equivalence states that anything having mass has an equivalent amount of energy and vice versa, with these fundamental quantities directly relating to one another by Albert Einstein's famous formula:

This formula states that the equivalent energy (E) can be calculated as the mass (m) multiplied by the speed of light (c = ~3×108 m/s) squared. Similarly, anything having energy exhibits a corresponding mass m given by its energy E divided by the speed of light squared c2. Because the speed of light is a very large number in everyday units, the formula implies that even an everyday object at rest with a modest amount of mass has a very large amount of energy intrinsically. Chemical, nuclear, and other energy transformations may cause a system to lose some of its energy content (and thus some corresponding mass), releasing it as the radiant energy of light or as thermal energy for example.

Mass–energy equivalence arose originally from special relativity as a paradox described by Henri Poincaré. Einstein proposed it on 21 November 1905, in the paper Does the inertia of a body depend upon its energy-content?, one of his Annus Mirabilis (Miraculous Year) papers. Einstein was the first to propose that the equivalence of mass and energy is a general principle and a consequence of the symmetries of space and time.

A consequence of the mass–energy equivalence is that if a body is stationary, it still has some internal or intrinsic energy, called its rest energy, corresponding to its rest mass. When the body is in motion, its total energy is greater than its rest energy, and equivalently its total mass (also called relativistic mass in this context) is greater than its rest mass. This rest mass is also called the intrinsic or invariant mass because it remains the same regardless of this motion, even for the extreme speeds or gravity considered in special and general relativity.

The mass–energy formula also serves to convert units of mass to units of energy (and vice versa), no matter what system of measurement units is used.


Oops-Leon is the name given by particle physicists to what was thought to be a new subatomic particle "discovered" at Fermilab in 1976. The E288 experiment team, a group of physicists led by Leon Lederman who worked on the E288 particle detector, announced that a particle with a mass of about 6.0 GeV, which decayed into an electron and a positron, was being produced by the Fermilab particle accelerator. The particle's initial name was the greek letter Upsilon (). After taking further data, the group discovered that this particle did not actually exist, and the "discovery" was named "Oops-Leon" as a pun on the original name and the first name of the E288 collaboration leader.

The original publication was based on an apparent peak (resonance) in a histogram of the invariant mass of electron-positron pairs produced by protons colliding with a stationary beryllium target, implying the existence of a particle with a mass of 6 GeV which was being produced and decaying into two leptons. An analysis showed that there was "less than one chance in fifty" that the apparent resonance was simply the result of a coincidence. Subsequent data collected by the same experiment in 1977 revealed that the resonance had been such a coincidence after all. However, a new resonance at 9.5 GeV was discovered using the same basic logic and greater statistical certainty, and the name was reused (see Upsilon particle).

Today's commonly accepted standard for announcing the discovery of a particle is that the number of observed events is 5 standard deviations (σ) above the expected level of the background. Since for a normal distribution of data, the measured number of events will fall within 5σ over 99.9999% of the time, this means a less than one in a million chance that a statistical fluctuation would cause the apparent resonance. Under this standard, the Oops-Leon "discovery" might have gone unpublished.

Particle decay

Particle decay is the spontaneous process of one unstable subatomic particle transforming into multiple other particles. The particles created in this process (the final state) must each be less massive than the original, although the total invariant mass of the system must be conserved. A particle is unstable if there is at least one allowed final state that it can decay into. Unstable particles will often have multiple ways of decaying, each with its own associated probability. Decays are mediated by one or several fundamental forces. The particles in the final state may themselves be unstable and subject to further decay.

The term is typically distinct from radioactive decay, in which an unstable atomic nucleus is transformed into a lighter nucleus accompanied by the emission of particles or radiation, although the two are conceptually similar and are often described using the same terminology.

Quality (philosophy)

In philosophy, a quality is an attribute or a property characteristic of an object. In contemporary philosophy the idea of qualities, and especially how to distinguish certain kinds of qualities from one another, remains controversial.

Relativistic mechanics

In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of moving objects are comparable to the speed of light c. As a result, classical mechanics is extended correctly to particles traveling at high velocities and energies, and provides a consistent inclusion of electromagnetism with the mechanics of particles. This was not possible in Galilean relativity, where it would be permitted for particles and light to travel at any speed, including faster than light. The foundations of relativistic mechanics are the postulates of special relativity and general relativity. The unification of SR with quantum mechanics is relativistic quantum mechanics, while attempts for that of GR is quantum gravity, an unsolved problem in physics.

As with classical mechanics, the subject can be divided into "kinematics"; the description of motion by specifying positions, velocities and accelerations, and "dynamics"; a full description by considering energies, momenta, and angular momenta and their conservation laws, and forces acting on particles or exerted by particles. There is however a subtlety; what appears to be "moving" and what is "at rest"—which is termed by "statics" in classical mechanics—depends on the relative motion of observers who measure in frames of reference.

Although some definitions and concepts from classical mechanics do carry over to SR, such as force as the time derivative of momentum (Newton's second law), the work done by a particle as the line integral of force exerted on the particle along a path, and power as the time derivative of work done, there are a number of significant modifications to the remaining definitions and formulae. SR states that motion is relative and the laws of physics are the same for all experimenters irrespective of their inertial reference frames. In addition to modifying notions of space and time, SR forces one to reconsider the concepts of mass, momentum, and energy all of which are important constructs in Newtonian mechanics. SR shows that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated. Consequently, another modification is the concept of the center of mass of a system, which is straightforward to define in classical mechanics but much less obvious in relativity – see relativistic center of mass for details.

The equations become more complicated in the more familiar three-dimensional vector calculus formalism, due to the nonlinearity in the Lorentz factor, which accurately accounts for relativistic velocity dependence and the speed limit of all particles and fields. However, they have a simpler and elegant form in four-dimensional spacetime, which includes flat Minkowski space (SR) and curved spacetime (GR), because three-dimensional vectors derived from space and scalars derived from time can be collected into four vectors, or four-dimensional tensors. However, the six component angular momentum tensor is sometimes called a bivector because in the 3D viewpoint it is two vectors (one of these, the conventional angular momentum, being an axial vector).

Rest frame

In special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest.

The rest frame of compound objects (such as a fluid, or a solid made of many vibrating atoms) is taken to be the frame of reference in which the average momentum of the particles which make up the substance is zero (the particles may individually have momentum, but collectively have no net momentum). The rest frame of a container of gas, for example, would be the rest frame of the container itself, in which the gas molecules are not at rest, but are no more likely to be traveling in one direction as another. The rest frame of a river would be the frame of an unpowered boat, in which the mean velocity of the water is zero. This frame is also called the center-of-mass frame, or center-of-momentum frame.

The center-of-momentum frame is notable for being the reference frame in which the total energy (total relativistic energy) of a particle or compound object, is also the invariant mass (times the scale-factor speed of light squared). It is also the reference frame in which the object or system has minimum total energy.

In both special relativity and general relativity it is essential to specify the rest frame of any time measurements, as the time that an event occurred is dependent on the rest frame of the observer. For this reason the timings of astronomical events such as supernovae are usually recorded in terms of when the light from the event reached Earth, as the "real time" that the event occurred depends on the rest frame chosen. For example, in the rest frame of a neutrino particle travelling from the Crab Nebula supernova to Earth the supernova occurred in the 11th Century AD only a short while before the light reached Earth, but in Earth's rest frame the event occurred about 6300 years earlier.

Transverse mass

The transverse mass is a useful quantity to define for use in particle physics as it is invariant under Lorentz boost along the z direction. In natural units it is:

where the z-direction is along the beam pipe and so
and are the momentum perpendicular to the beam pipe and
is the (invariant) mass.

This definition of the transverse mass is used in conjunction with the definition of the (directed) transverse energy

with the transverse momentum vector . It is easy to see that for vanishing mass () the three quantities are the same: . The transverse mass is used together with the rapidity, transverse momentum and polar angle in the parameterization of the four-momentum of a single particle:

Using these definitions (in particular for ) gives for the mass of a two particle system:

Looking at the transverse projection of this system (by setting ) gives:

These are also the definitions that are used by the software package ROOT, which is commonly used in high energy physics.

Upsilon meson

The Upsilon meson (ϒ) is a quarkonium state (i.e. flavourless meson) formed from a bottom quark and its antiparticle. It was discovered by the E288 experiment team, headed by Leon Lederman, at Fermilab in 1977, and was the first particle containing a bottom quark to be discovered because it is the lightest that can be produced without additional massive particles. It has a lifetime of 1.21×10−20 s and a mass about 9.46 GeV/c2 in the ground state.


The Y(4140) particle is an electrically neutral exotic hadron candidate that is about 4.4 times heavier than the proton. It was observed at Fermilab and announced on 17 March 2009. This particle is extremely rare and was detected in only 20 of billions of collisions.Since it decays into J/ψ and φ mesons, it has been suggested that this particle is composed of charm quarks and charm antiquarks, possibly even a four quark combination.

The existence of the particle has been confirmed by members of the CMS collaboration at the Large Hadron Collider on November 14, 2012 and by the DØ experiment at the Tevatron on September 25, 2013. The Belle experiment has searched for this particle but found no evidence for its existence.

The LHCb experiment observes a peak at the same position in the J/ψϕ invariant mass, but it is best described as a Ds±Ds∗∓ cusp, and is much broader than the previous measurements of the Y(4140).The Particle Data Group has renamed Y(4140) to follow naming conventions to X(4140).

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