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however, in models where the χ fields couple to SM leptons. The simplest choice for that is to put them in triplets of G$_{LF}$ . There are four different ways to do that, which are given in Table I. We call the four resulting models as LE, LL, EE, and EL in an obvious correlation to the way that χ$_{L}$ and χ$_{R}$ transform under SU (3)$_{L}$ × SU (3)$_{E}$. 

MLFV requires that the Lagrangian terms be made of the L$_{L}$, E$_{R}$, χ$_{L}$ and χ$_{R}$ fields and of Y E spurions in a formally G$_{LF}$ invariant way. By definition, this holds for the Y E term in Eq. (2.1). On the other hand, the Y $^{χ}$, Xχ and X L involve the new fields, and consequently their structure is different in one model from the other: 

• The LE model: Y $^{χ}$, Xχ and X L must all transform as (3 , ¯ 3) and are therefore pro- portional to Y $^{E}$. Note, however, that the L$_{L}$ and χ$_{L}$ fields transform in the same way under both G$_{SM}$ and G$_{LF}$ . There is therefore freedom in choosing a basis in the ( L$_{L}$, χ$_{L}$ ) space. We choose this freedom to make X L = 0, namely we define the χ$_{L}$ fields as the three fields that have bare mass terms. 

• The LL model: Y $^{χ}$, Xχ and X L must transform as (3 , ¯ 3), (1 + 8 , 1) and (1 + 8 , 1), respectively. We thus have Y χ ∝ Y E and X χ ∝ 1 while, again, we are free to choose a basis in the ( L$_{L}$, χ$_{L}$ ) space where X L = 0. 

• The EE model: Y $^{χ}$, Xχ and X L must transform as (1 , 1 + 8), (1 , 1 + 8) and (3 , ¯ 3), respectively. We thus have Y χ ∝ 1 , X χ ∝ 1 and X L ∝ Y $^{E}$. 

