Is this diquark something realistic or is it an out-dated object, i.e. ruled out by experiments?
According to Are Diquarks Real? there is experimental evidence that diquarks do exist.
Also from the more-recent Images of the Origin of Mass: "The modern diquark correlation is nonpointlike, with the charge radius of a given diquark being typically 10% larger than its mesonic analogue. Hence, diquarks are soft components within baryons. Notably, empirical evidence in support of the presence of diquarks in the proton is accumulating [citations]"
When and why was it postulated?
According to The Diquark–Quark Approximation
"Diquarks are almost as old as quarks. The possibility that quarks might cluster pairwise in baryons, leading to a simple two-body structure, has been suggested by many authors since the early days of the quark model."
Where does the charge assignement $4/3$ [for an anti-leptodiquark] come from? On the basis of the weight I would have called it a anti-up quark, since it is the upper component of the doublet, the $1$ in the first brackets, and transforms as a $\bar{3}$ under $SU(3)$ (since $(01)$ is the corresponding fundamental weight). Hence I would assign it a charge $1/3$, which later on will be used to determined the charges of the other particles. How does he comes to the conclusion that it must be a diquark?
See table 1 of Slansky's Fun with E6 and table 20 of Slansky's appendix. An antilepto-diquark can have a 4/3 or 1/3 charge. There is also a related article by Shaw and Slansky E6 model with composite muon and τ families which discusses the antilepto-diquark.
The charge is calculated by the dot product of $\frac{1}{3}(2,1,2,0,1,0)$ and the particle E6 root. The anti-leptodiquark being discussed is (1,-1,1,-1,1,0). The dot product is 4/3.
Stansky says the antilepto-diquarks are bosons in Fun with E6 at page 10, but in Topics in beyond Standard Model Collider Phenomenology, as page 39, it is stated that:
"Another resonance called Lepto - diquark has been introduced into this picture, which
is a vector-like fermion that transforms under SU(3) the same way as the diquark and
has the same baryon number, but differs in terms of other quantum numbers. It also has to be lighter than the diquark such that the decay of diquark into lepto-diquark can be allowed by phase space."
•He comments that it mediates the proton decay. I thought that usually a force is mediated by a boson... Does he means that since the quarks will be together with the leptons in a multiplet, the baryon number does not need to be conserved. Which unable the proton to decay?
In Supermodels for early LHC the following hypothetical decay chain is presented:
two up quarks $\rightarrow$ diquark $\rightarrow$ negatively charged lepton + leptodiquark $\rightarrow$ oppositely charged lepton pair + 2 jets
In this context, it seems "mediates the proton decay" means that there is an intermediate state in a decay chain that involves the particle.
