Why is the neutral component of Higgs doublets in 2HDM theory different according to the document of reference? According to a reference on Higgs (anatomy of Higgs, tome 2, by Djouadi)
arXiv:hep-ph/0503173
On formula 1.55 page 30, the neutral component is on the top component of the first Higgs doublet, and in the bottom component of the second Higgs doublet.
But in https://indico.in2p3.fr/event/4003/contributions/28872/attachments/23112/28359/DEGEEJJC.pdf
page 6, the neutral component is in the bottom component for the two Higgs doublets.
Why?
 A: The question cites two references, and they have different goals:

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*Djouadi's paper (ref 1) is considering a specific two Higgs-doublet model (2HDM) that is compatible with supersymmetry (SUSY). In this model, the two Higgs doublets have opposite hypercharges, $Y=+1$ and $Y=-1$. I'll say more about the reason for this below.


*Degee's slides (ref 2) have a different purpose, namely to study the Higgs potential itself in the whole general family of 2HDMs, without paying much attention to the gauge fields or the Yukawa terms.
Section 2.3.1 in Degee's master's thesis (ref 3) gives this very brief answer to the question:

Here, we use $Y=+1$ for both doublets. In the MSSM [minimal supersymmetric standard model], $Y_1=1$, $Y_2=-1$, but our results hold up to redefinitions.

In more detail: If a Higgs doublet $H$ has hypercharge $Y=1$, then the Higgs doublet $H'$ with components $H'_a=\sum_b\epsilon_{ab}H_b^*$ has hypercharge $Y=-1$ because of using the adjoints $H_b^*$, and $H$ and $H'$ both transform the same way under $SU(2)$ because of the $\epsilon_{ab}$.
We can write things in different ways, and the most convenient way depends on what we're trying to accomplish:

*

*Degee was studying the problem of finding the miminum of the most general Higgs potential, and for that purpose it was most convenient to put the neutral component in the same position in every doublet.


*Djouadi was considering a specific SUSY model, including the Yukawa couplings that give masses to the fermions. Giving mass to both the up-type fermions and to the down-type fermions requires using Higgs doublets with different hypercharges to preserve $U(1)_Y$ gauge invariance, and with the neutral component in different positions to pair it with the different components of the left-handed fermion doublet. In the non-SUSY standard model, we do this by using the transformation described above, so that a single Higgs doublet can be used for both. But in the SUSY case with two Higgs doublets $H_1$ and $H_2$, we use $H_1$ to give mass to the up-type fermions and use $H_2$ to give mass to the down-type fermions. If we take $H_1$ and $H_2$ to have opposite hypercharges, and if we put the neutral component in different positions in each one, then we can write the superpotential without using the adjoints of $H_1$ or $H_2$. This is explained in the middle paragraph on page 30 in ref 1. (The paragraph before that also explains why we can't just use one Higgs doublet like we do in the non-SUSY standard model.)

References:

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*Djouadi, "The Anatomy of Electro–Weak Symmetry Breaking Tome II: The Higgs bosons in the Minimal Supersymmetric Model" (https://arxiv.org/abs/hep-ph/0503173)


*Degee (slides), "The most general Two-Higgs-Doublet Model" (https://indico.in2p3.fr/event/4003/contributions/28872/attachments/23112/28359/DEGEEJJC.pdf)


*Degee (master's dissertation), "Higgs mechanism in the general Two-Higgs-Doublet Model" (https://orbi.uliege.be/handle/2268/68445)
