What is the mathematically simplest particle field? I read the total equation for the standard model is something like this https://www.symmetrymagazine.org/sites/default/files/images/standard/sml.png
It's quite complicated. My understanding is that the fields representing each particle can be modeled separately in a fairly straightforward* way (to give an incomplete / non physical theory).
Is this correct, and can a model of a simpler universe containing just one type of particle be described in this way? If so, which particle in the Standard Model would be the simplest? I realise this could be a subjective question but I am hoping it may not be. Please assume no existing knowledge of physics (only maths) and no real-world experience, just assess the simplicity from a mathematical perspective.
You don't have to give a justification (unless it is contentious) or the mathematical form of the field, unless you would like to. I hope to address those kind of topics in future.
*Mathematically but not conceptually! It can be argued whether interaction terms are intrinsic or extrinsic to an individual field. For this question I am excluding all interaction terms that involve another field. To be clear, that includes even phenomena like self-interactions mediated by virtual particles of other fields.
 A: The simplest quantum particle, at least on the formal, mathematical level, is described by a free real scalar field $\phi(x^\mu)$. This corresponds to an abstract particle that has no charge and no intrinsic angular momentum (spin), and that does not interact with anything, just freely drifting through the universe. The simplicity of this field is the reason why most courses on quantum field theory (particle physics) start with discussing scalar fields.
However, the behavior of this particle is nothing like any real particle of the Standard model. In particular, the Higgs boson, also a particle without charge or spin, is in fact interacting in many complex ways with many particles. As a result, the Higgs boson has a lifetime of about $1.6×10^{−22} s$. In other words, write a zero, comma, then 21 more zeros, and then a  one and a six, and that's the kind of time you have a chance to register a Higgs boson for! After that it decays into a spray of other particles and in practice we never detect the Higgs directly. That is nothing like the free scalar particle described above!
So, the Higgs boson is on some arbitrary formal level very simple, but in flesh it is not. If you are looking for the simplest mathematical model which describes to high accuracy the actual behavior of real particles, then you can consider quantum electrodynamics. Quantum electrodynamics is the theory of how electrons and positrons propagate through space and interact with each other through the quantized electromagnetic field (photons). Another very simple theory which effectively describes the behavior of real particles is the Yukawa interaction, which approximates the interaction between protons and neutrons by letting them exchange pions, a quantized scalar field.
The understanding of the behavior of quantum electrodynamics and the Yukawa interaction is essentially a prerequisite to understanding the more fundamental and complicated theories of electroweak interactions and quantum chromodynamics (which are already the two building blocks of the Standard model). This is the reason why after discussing scalar fields, a typical quantum-field-theory textbook passes to free spinor fields (corresponding to spinning particles without interactions), and finally to the aforementioned quantum electrodynamics and Yukawa interactions.  
A: Based on the comments on my question, I believe the answer is the Higgs field.
