Does the mass of a particle change when emitting a photon? question is in the title. It came up since we're currently dealing with 4-vectors and collision/decay processes.
At a first glance I thought that it should be possible because of the energy/mass equivalence but then again I don't think that a electron can lose mass when emitting a photon since it's such a fundamental property of it.
Would be great if someone could help me out with this question! Lot of thanks in advance!
 A: A free electron cannot emit a photon.  When we say an electron emits a photon it is usually bound in an atom and it is really the entire atom that emits the photon.  In order to do so the atom must be in an excited state, and yes its mass does change when the photon is emitted.
A: Here's an answer using some relativistic reasoning,
as well as making contact with the previously given answers.
Yes, the [invariant rest] mass (assumed nonzero) must decrease when the particle emits a photon.
This follows from the conservation of total 4-momentum, which I diagram below.
(This diagram is basically one used for the Doppler effect (think Bondi’s k-calculus).)
$$\widetilde m_1 = \widetilde m_2 +\widetilde k$$

In the example diagrammed in the frame of the initial particle,

*

*the initial particle mass (the magnitude of the 4-momentum, assumed nonzero) is 8

*the final particle mass is 4 and has velocity $(3/5)$, relative to the initial particle frame (and thus relativistic energy $5$ units,
relativistic kinetic energy $1$ unit, and relativistic momentum $3$ units in this frame)

*the emitted photon 4-momentum is directed to the left, with relativistic energy $3$ units and relativistic momentum $-3$ units in this initial frame.

Note that:

*

*the word "invariant" in "invariant mass" means "independent of frame", 
not "constant, independent of time". So, if the particle participated in a process, this quantity may change.

*the particle must change mass (and, thinking in terms of elementary particles, must change its type) to conserve total 4-momentum. Thus, the particle before and after cannot be an electron (assuming that this is the entire system---if there are other interacting particles [like nearby atoms], then their 4-momenta must be included in this diagram).

A: The invariant mass (which is meant when talking about the mass of a particle in physics) does not change, it is an inherent property of the particle. The "relativistic mass" does change (since the electron's kinetic energy changes).
As noted in a comment and the other answers, a free electron on its own can't simply emit a photon (since 4-momentum conservation can't be fulfilled for such a process).
Nevertheless, my statement remains true if we specify the right scenario. We assume a third body to make the conservation laws work out. This is, e.g. the case in the bremsstrahlung process, where an electron emits a photon while being redirected in the electric field of a nucleus. The energy and momentum transferred to the nucleus allows the conservation laws to be fulfilled. We don't care about the nucleus, and have a process where an incoming electron emits a photon, and continues to travel as a free electron.
A: Yes. For example the mass of a hydrogen atom in an excited state changes when it emits a photon and transits to a lower state.
A: The invariant mass of the electron does not change. However, free (isolated) electrons cannot emit photons. Emphasis on isolated. When we talk about electrons emitting photons, we usually mean a quantum mechanical system, that is, an atom, molecule or just a nucleus emitting a photon.

A single free electron cannot emit a photon
A single free hydrogen atom with electron in an excited state can emit a photon by spontaneous emission
A single free nucleus can emit a photon again by relaxation of its internal structure.

Can a free particle ever emit a particle?
Yes, contrary to popular belief, even nuclei (being QM systems comprised of quarks and gluons) can emit photons.
Now back to your question, whether the mass of the emitting QM system changes or not.
The answer is yes. The rest mass of the atom or molecule or nucleus (in its own rest frame) does get reduced by the amount of energy of the emitted photon (please note there are exceptions where a molecule's vibrational or rotational or translational energy is transformed into a photon).
Why? This comes from the law of conservation of energy. Now you are asking why the rest mass (of the QM system emitting the photon) decreases, and the mass energy equivalence shows you, that if the energy of the emitting QM system (atom, molecule or nucleus) decreases in its own rest frame (and it does when emitting a photon), then its rest mass has to decrease too. The electron's rest mass does not change, it is invariant.
