# What is the mass gap?

In a relativistic theory, whenever there is a state with definite 4-momentum p, there is also one with definite momentum p' = Lambda p obtained by applying a Lorentz transform Lambda. The orbit of 4-momenta obtained in this way forms a hyperboloid in the future cone (because of causality), characterized by a mass m &ge 0. Its equations are

p^2=m^2, p_0 &ge 0.
This includes as a limiting case massless states with m=0, where the orbit consists of the future light cone with 0 excluded.
Therefore the possible values of p are characterized by the possible values of m, which defines the mass spectrum of the theory. The mass spectrum is the relativistic analogue of the energy spectrum of the Hamiltonian in a nonrelativistic theory, shifted such that the ground state has E=0.

The only state with zero momentum is the ground state, usually called the vacuum. If the values of p^2 for the realizable nonzero p is bounded below by a positive number, the theory is said to have a mass gap. The largest value of m>0 for which m^2 is such a lower bound defines the precise value of the mass gap. Usually there is a state for which p^2=m^2; this is then interpreted as the state of a single 'dressed' particle.
In general, the mass spectrum consist of a discrete and a continuous part. The discrete part of the spectrum corresponds to bound states, the continuous part to scattering states.
The continuous spectrum starts when there is the possiblity of scattering. which means that the energy is large enough that two asymptotically independent systems can exist. Given a state of mass m, one expects to have states with two almost independent systems of mass m and an arbitrary relative momentum, giving a continuous spectrum of scattering states with all possible squared momenta exceeding (2m)^2, as a simple calculation reveals: If p is the sum of two timelike vectors p1,p2 of mass m then

p^2 = (sqrt(p1^2+m^2)sqrt(p2^2+m^2))^2 - (p1+p2)^2 = 2m^2 + 2 sqrt((p1^2+m^2)(p2^2+m^2)) -2p1 dot p2
By making p2=-p1 one gets arbitrarily large values of p^2, hence part of the continuous spectrum. The minimum of p^2 must occur by Cauchy/Schwarz for p2=p1, and is then (2m)^2, independent of the spatial momentum.
Thus the continuous spectrum extends from mass 2m to infinity, where m is the mass gap.
There may be bound states with mass m_b < 2m, forming the discrete spectrum. These are not scattering states, hence not obtained by simply adding momenta. For bound states of k particles with masses m_1,...,m_k, one needs to subtract from (m_1+...+m_k)c^2 the binding energy of the bound particles. There might be bound states with mass m_b>2m embedded in the continuous spectrum, but these are possible only if there are selection rules that forbid the decay into particles with smaller mass.
In particular, the state of minimal mass m, if it exists, is always a bound state (including the case of a single particle).

If there is no mass gap, one expects massless dressed particles to be present. This corresponds to the limiting case m --> 0 of the above discussion.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ