What physical entity does not have mass-energy?
I assume the intent of the question was to ask what physical entity does not have energy at all, including mass-energy.
A photon's energy can be made arbitrarily close to zero if you redshift it enough, i.e., by boosting yourself into a frame travelling in the same direction as the photon's motion. The more you do this, the redder and redder the photon becomes. If you're ahead of the photon, and finally the photon hits your detector, you might not be able to see it at all, because its energy is too low to be detected!
On the other hand... if you boosted toward a photon, it would get blueshifted, and you can make its energy appear arbitrarily large this way.
What would it take for a particle to really appear to have no energy in every frame? Answer: its entire energy-momentum four-vector would have to vanish: no energy, no momentum.
Now a plane wave solution in quantum mechanics—a state of definite energy and momentum, looks like \[\psi \sim \exp(i(p \cdot x - Et)/\hbar)\] and if you plug in \(E = 0, p = 0\) this kills all the spatial and temporal dependence: \[\psi \sim 1\] So an entity with zero energy and zero momentum would be a completely homogeneous field—no spatial variation, no temporal variation.
If there were really such an entity, we would not speak of it as though it were a physical entity at all—it would be absorbed into the form of the laws of physics or into the values of fundamental physical constants. For example, suppose its effect were to slow down light that passes through it. [1] Then we'd just think the speed of light was a smaller number, as we'd have no observations to contradict it, since this field is assumed to be uniform across space and time—we wouldn't postulate the existence of an unobservable homogeneous background field—as Occam said, "entities should not be multiplied beyond necessity". Or suppose its effect were to exert a uniform gravitational pull on everything—then we'd just see a different value of the cosmological constant. Or suppose it behaved like a uniform electric field—then we'd just accept the curious fact that some particles drift in one direction and some particles drift in the other direction [2] and we'd work that into their Lagrangians.
What we would not do is say that we have observed a field with no energy and no momentum.
[1] This is actually excluded by experimental evidence for Lorentz invariance.
[2] This is also excluded by Lorentz invariance.
A photon's energy can be made arbitrarily close to zero if you redshift it enough, i.e., by boosting yourself into a frame travelling in the same direction as the photon's motion. The more you do this, the redder and redder the photon becomes. If you're ahead of the photon, and finally the photon hits your detector, you might not be able to see it at all, because its energy is too low to be detected!
On the other hand... if you boosted toward a photon, it would get blueshifted, and you can make its energy appear arbitrarily large this way.
What would it take for a particle to really appear to have no energy in every frame? Answer: its entire energy-momentum four-vector would have to vanish: no energy, no momentum.
Now a plane wave solution in quantum mechanics—a state of definite energy and momentum, looks like \[\psi \sim \exp(i(p \cdot x - Et)/\hbar)\] and if you plug in \(E = 0, p = 0\) this kills all the spatial and temporal dependence: \[\psi \sim 1\] So an entity with zero energy and zero momentum would be a completely homogeneous field—no spatial variation, no temporal variation.
If there were really such an entity, we would not speak of it as though it were a physical entity at all—it would be absorbed into the form of the laws of physics or into the values of fundamental physical constants. For example, suppose its effect were to slow down light that passes through it. [1] Then we'd just think the speed of light was a smaller number, as we'd have no observations to contradict it, since this field is assumed to be uniform across space and time—we wouldn't postulate the existence of an unobservable homogeneous background field—as Occam said, "entities should not be multiplied beyond necessity". Or suppose its effect were to exert a uniform gravitational pull on everything—then we'd just see a different value of the cosmological constant. Or suppose it behaved like a uniform electric field—then we'd just accept the curious fact that some particles drift in one direction and some particles drift in the other direction [2] and we'd work that into their Lagrangians.
What we would not do is say that we have observed a field with no energy and no momentum.
[1] This is actually excluded by experimental evidence for Lorentz invariance.
[2] This is also excluded by Lorentz invariance.