Does the the quantum field theoretic process of particle–antiparticle annihilation break the axioms of Special Relativity? 
$\textbf{Note that this diagram hasn't anything to do with the question directly.}$
After a particle and its antiparticle annihilate, their energy is converted into a force carrier particle, such as a gluon, $W$ or $Z$ force carrier particle or a photon.  
When I contemplate this phenomenon, it seems curious to me that a system can have a mass, and then accelerate to the speed of of a massless particle. This curiosity stems from the special relativistic notion that it takes an infinite amount of energy to make any particle with mass accelerate to the speed of light.
The equation that I am speaking of is graphed here, and demonstrates the well recognized $\textbf {Energy-Mass against Velocity}$ asymptote of Special Relativity:

Besides the fact that this process adheres to the many other conservation laws which we have discovered, is there a reasonable explanation for this phenomenon? Is it just like putting a value into a function, and the universe knows to produce a photon (or another moderator)?  Or, does something else take place, where the particles are exposed to energy in its purest form, so they accelerate tremendously together until they separate to form another particle pair?
 A: In Feynman diagrams all four vectors conform with special relativity algebra, BUT the internal lines, even though they have the name "photon" are virtual, which means the mass can be different than zero, only the quantum numbers identify them as a photon, quark, electron, etc..
Feynman diagrams are an iconic representation one to one with the integrals and functions necessary to compute the reaction depicted. In this case electron positron annihilation into two quarks which then decay into two resonances.  There are functions called propagators included under the integrals which correspond with the named  off-mass-shell virtual  "photon"  etc which contain the mass of the virtual particle in the denominator, for example. 

note that the square of the four momentum would cancel with the mass in the denominator generating an infinity if it were not for the i*epsilon.
*Edit** after edit of question 
The energy mass relationship you are quoting uses the relativistic mass, as in E=m*c^2. 

The relativistic mass is not the mass entering Feynman diagrams or a mass that characterizes the particles in the Feynman diagrams  . That is the length of the four vector describing the particle in any reference frame and is an invariant of the Lorenz transformations, it is called the rest mass or invariant mass.
The mass of the electron is always the same , it is the relativistic mass that changes and depends on the frame of reference. When an electron and a positron annihilate there is conservation of momentum and energy and the virtual force carrier will have as mass the same as the invariant mass of the incoming pair. That is how we have detected a number of mesons, including the Z .

(Go to the link for a clearer image)
As the invariant mass of the incoming pair scans the region of the mass of the Z for example, the probability of interaction goes up, as we are crossing a pole in the propagator,  and the Z appears at its invariant mass with its width.
So there is no problem with the mass of the intermediate boson, which is virtual except exactly on mass shell as it passes the resonance, and can be bigger or smaller than the mass in the data table as the energies of interaction, the invariant mass of the e+e- pair increases. The relativistic mass is irrelevant in the interactions of elementary particles and the use of the term has fallen out of fashion. 
A: 
it seems curious to me that a system can have a mass, and then accelerate to the speed of of a massless particle

There is no paradox here. Special relativity doesn't preserve velocity - in fact the opposite - it allows one to move between frames of reference in which the velocity of objects is relative to the observer. 

[do the particles] accelerate tremendously together until they separate to form another particle pair?

No- they cancel out exactly to produce the mediating boson, however it's not sensible to ask what this `looks like'. You have to bare in mind that in quantum mechanics one can only observe the initial state and the final state. The observables here are the two annihilating particles/anti-particles and the created particles/anti-particles, not the mediator. 
Edit after edit of question
The energy mass relationship you are quoting uses the relativistic mass, as in E=m*c^2.

The relativistic mass is not the mass entering Feynman diagrams or a mass that characterizes the particles in the elementary particles table  . That is the length of the four vector describing the particle in any reference frame and is an invariant of the Lorenz transformations, it is called the rest mass. The mass of the electron is always the same , it is the relativistic mass that changes and depends on the frame of reference.
When an electron and a positron annihilate there is conservation of momentum and energy and the virtual force carrier will have the mass the same as the invariant mass of the incoming pair. That is how we have detected a number of mesons, including the Z, by scanning the energy of the e+e- at rest . As the invariant mass of the incoming pair scans the region of the mass of the Z for example, the probability of interaction goes up and the Z appears at its invariant mass with its width.
So there is no problem with the mass of the intermediate boson, which is virtual except exactly on mass shell as it passes the resonance, and can be bigger or smaller than the mass in the data table. The relativistic mass is irrelevant in the interactions of elementary particles and the use of the term has fallen out of fashion.
A: From the OP's comments I have a feeling that the real issue is not the virtual particle, but an actual annihilation of a positron and electron into 2 real photons. Like here:

And the question is, how can the system of 2 massive electrons accelerate into the speed of light, when accelerating to c requires infinite amount of energy. The answer has two parts:
1) When the you have the 2 photons, they are no longer massive particles. Photons do not accelerate. They are born moving at the speed of light. The positron and electron just cease to exist. They also don't accelerate.
2) The 2 photons are moving away from each other (in the center of mass frame of the original colliding leptons). So the center of mass of the 2-photon system is at rest in the selected frame. The center of mass did not accelerate.
Please also note that particle and antiparticle cannot annihilate into just 1 real photon. There must be at least 2 real photons due to conservation of momentum.
