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I am trying to wrap my head around sterile neutrinos. As far as I understand it, sterile neutrino is a name for every particle that feels only gravitational force. The most notable example is a right-chiral neutrino, hence the name.

What i am strugguling with is this. Some sources say that sterile neutrino that we are looking for (which would explain the mass of neutrinos) is in fact only a right-chiral partner of existing flavors like elektron or moun neutrinos, but other say that we are in fact looking for a 4th flavor that is a singlet and doesnt have a left-chiral partner (opposite of 3 known types). Reading some papers on experiments like MiniBooNE, they say a model of 3+1 neutrinos would explain their results better than the existing model so that would suggest a latter explanation of a singlet.

My question is, what are we looking for and why?

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    $\begingroup$ Sterile, generically, means not active in the weak interactions, and only connected to active ones through mixing or mass propagation. Your particular sources should define the context of the terms used. $\endgroup$ – Cosmas Zachos Jun 6 at 14:14
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So I'll try to give a short explenation on the terminology "sterile neutrino" and I hope I did understand your question right:

The term sterile neutrino generally refers to either of two things:

1st: (Looking at particles in the "interaction basis") In the interaction basis a sterile neutrino is a standard model singlet i.e. a particle that does not interact via the three standard model gauge interactions.
In other words it is a particle that has no color charge, no weak isospin (hence it must have right-handed chirality) and no hypercharge.
Every such particle will mix with left handed neutrinos hence it is called a sterile neutrino.

2nd: (Looking at particles in the "mass basis") When diagonalizing the particles from the interaction basis to mass eigenstates one generally finds that through mixing with the left-handed neutrinos there will be a mostly active light neutrino state and a mostly sterile heavy neutrino state where active and sterile can be considered again in the sence of the 1st description. Since the physical propagating particles are the mass eigenstates and not the interaction eigenstates the term sterile neutrino in e.g. cosmology refers to the "mostly sterile" mass eigenstate.

If you want to see how this mixing works you should take a look at this paper which explains the standard seesaw mechanism and other neutrino mass generating models. (You can find a longer lecture about neutrino physics including neutrino mass discussions in this paper)

It depends on the context (as stated by Cosmas Zachos in the comments) which of the above is referred to and sometimes I feel like the two get mixed up although generally they are not the same (although they are connected of course and only depend on the basis you choose).

I think it is important to note that only for neutrinos we tend to consider mostly the interaction basis while for all other particles we use the mass basis to describe them (even in the quark sector where we have mixing too). This can lead to some confusion sometimes and is probably a relic from the time when neutrinos where considered massless...

You should note that due to the two different pictures above there is no difference between saying a sterile neutrino is the "right chiral partner" or a "4th flavor neutrino". Introducing a right chiral partner in the 1st sense for the left-handed neutrinos will automatically come with a new massive "4th flavor" particle into which the already known neutrinos can oscillate. Note that when oscillation experiments talk about sterile neutrinos they typically refer to the 2nd picture since oscillation is sensitive to the differences in the squared masses of the mass eigenstates.

However the term "4th flavor neutrino" and "sterile neutrino" are not the same if one would consider a complete 4th generation of particles including a 4th gen. neutrino (i.e. a 4th "active" neutrino). These 4th gen. neutrinos would need to be very heavy though (at least heavier than half of the Z-Boson mass) so they would not show up in oscillation experiments (they would violate the condition of coherency) and are not considered by e.g. MiniBooNE

I hope that cleared things up and did not make it more confusing. The important part to take with you is propaply that every standard model singlet will mix with neutrinos and will both act as a sterile neutrino in the inerteraction basis and as right chiral partner of the standard left handed neutrinos while also generating a new massive "4th flavor" particle in the mass basis.

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