---
abstract: 'The minimal supersymmetric standard model involves a rather restrictive Higgs potential with two Higgs fields. Recently, the full set of classes of symmetries allowed in the most general two Higgs doublet model was identified; these classes do not include the supersymmetric limit as a particular class. Thus, a physically meaningful definition of the supersymmetric limit must involve the interaction of the Higgs sector with other sectors of the theory. Here we show how one can construct basis invariant probes of supersymmetry involving both the Higgs sector and the gaugino-higgsino-Higgs interactions.'
author:
- 'P. M. Ferreira'
- 'Howard E. Haber'
- 'João P. Silva'
title: 'Basis invariant conditions for supersymmetry in the two-Higgs-doublet model'
---

SCIPP-10/01\

\[sec:intro\]Introduction
=========================

The Standard Model (SM) of electroweak interactions has provided an extraordinarily successful description of currently observed particle physics phenomena. Nevertheless, there are strong reasons to expect that new physics beyond the Standard Model must emerge, ranging from the hierarchy problem and the unification of all coupling constants, to baryogenesis and dark matter. One of the leading candidates for physics beyond the SM incorporates supersymmetry near the scale of electroweak symmetry breaking in order to provide a natural explanation for the existence of the Higgs boson. Much attention has been devoted to the minimal supersymmetric extension of the standard model (MSSM), which requires two complex Higgs doublets and superpartners for all Standard Model particles [@MSSM].

In general, there is no fundamental reason why the SM should possess only one complex Higgs doublet. The most well-studied extended Higgs sector is that of the two-Higgs-doublet model (THDM) [@THDM]. The scalar potential of the most general THDM involves 14 parameters. Of these parameters, only eleven combinations are physical, as three degrees of freedom can be absorbed into a redefinition of the Higgs fields [@Lav; @DavHab]. This number may be further reduced by imposing some symmetry requirements on the Higgs Lagrangian. But, identifying such symmetries is complicated by the fact that one may perform a basis transformation of the Higgs fields. A symmetry that looks simple in one basis may be completely obscured in another basis. Hence, it is important to develop basis-invariant signals of such symmetries, which can identify the physically meaningful and experimentally accessible parameters in the theory. The need to seek basis invariant observables in models with many Higgs bosons was pointed out by Lavoura and Silva [@LS], and by Botella and Silva [@BS], stressing applications to CP violation. Refs. [@BS; @BLS] indicate how to construct basis invariant quantities in a systematic fashion for any model, including multi-Higgs-doublet models. A number of recent articles concerning symmetries and/or basis invariance in the THDM include Refs. [@Gun; @BRSM; @GK; @DavHab; @Haber; @oneil; @Nishi; @NachMani; @Ivanov1; @Ivanov2; @Gerard; @FS2; @FHS].

It is remarkable that there are exactly six classes of symmetries that one may impose on the scalar sector of the most general THDM. This was shown by Ivanov [@Ivanov1] and expanded upon by us in Ref. [@FHS]. Since the Higgs sector of the MSSM is a particular case of the THDM, one would expect that the constraints satisfied by the Higgs sector of the MSSM would correspond to one of the six classes of symmetries identified in the scalar sector of the THDM. This is *not* the case. The correct conclusion is that a physically meaningful definition of the supersymmetric limit must involve the interaction of the Higgs sector with other sectors of the supersymmetric theory. In this article we construct basis-invariant probes of supersymmetry involving both the Higgs sector and the gaugino-higgsino-Higgs interactions.

This article is organized as follows. In section \[sec:notation\] we introduce our notation. In section \[sec:MSSM\], we construct the basis invariant quantities that identify the supersymmetric limit of the scalar sector of the THDM. We draw our conclusions in section \[sec:conclusions\].

\[sec:notation\]The scalar sector of the THDM
=============================================

The scalar potential
--------------------

Let us consider a $SU(2) \otimes U(1)$ gauge theory with two hypercharge-one Higgs-doublets, denoted by $\Phi_a$, where $a=1,2$. The scalar potential may be written as $$\begin{aligned}
V_H
&=&
m_{11}^2 \Phi_1^\dagger \Phi_1 + m_{22}^2 \Phi_2^\dagger \Phi_2
- \left[ m_{12}^2 \Phi_1^\dagger \Phi_2 + \textrm{h.c.} \right]
\nonumber\\[6pt]
&&
+ \tfrac{1}{2} \lambda_1 (\Phi_1^\dagger\Phi_1)^2
+ \tfrac{1}{2} \lambda_2 (\Phi_2^\dagger\Phi_2)^2
+ \lambda_3 (\Phi_1^\dagger\Phi_1) (\Phi_2^\dagger\Phi_2)
+ \lambda_4 (\Phi_1^\dagger\Phi_2) (\Phi_2^\dagger\Phi_1)
\nonumber\\[6pt]
&&
+ \left[
\tfrac{1}{2} \lambda_5 (\Phi_1^\dagger\Phi_2)^2
+ \lambda_6 (\Phi_1^\dagger\Phi_1) (\Phi_1^\dagger\Phi_2)
+ \lambda_7 (\Phi_2^\dagger\Phi_2) (\Phi_1^\dagger\Phi_2)
+ \textrm{h.c.}
\right],
\label{VH1}\end{aligned}$$ where $m_{11}^2$, $m_{22}^2$, and $\lambda_1,\lambda_2,\lambda_3,\lambda_4$ are real parameters, and $m_{12}^2$, $\lambda_5$, $\lambda_6$ and $\lambda_7$ are potentially complex. An alternative notation, useful for the construction of invariants and championed by Botella and Silva [@BS] is $$\begin{aligned}
V_H
&=&
Y_{ab} (\Phi_a^\dagger \Phi_b) +
\tfrac{1}{2}
Z_{ab,cd} (\Phi_a^\dagger \Phi_b) (\Phi_c^\dagger \Phi_d),
\label{VH2}\end{aligned}$$ where Hermiticity implies $$\begin{aligned}
Y_{ab} &=& Y_{ba}^\ast,
\nonumber\\
Z_{ab,cd} \equiv Z_{cd,ab} &=& Z_{ba,dc}^\ast.
\label{hermiticity_coefficients}\end{aligned}$$ One should be very careful when comparing Eqs. (\[VH1\]) and (\[VH2\]) among different authors, since the same symbol may be used for quantities that differ by signs, factors of two, or complex conjugation. Here we follow the definitions of Davidson and Haber [@DavHab]. With these definitions: $$\begin{aligned}
Y_{11}=m_{11}^2, &&
Y_{12}=-m_{12}^2,
\nonumber \\
Y_{21}=-(m_{12}^2)^\ast && Y_{22}=m_{22}^2,
\label{ynum}\end{aligned}$$ and $$\begin{aligned}
Z_{11,11}=\lambda_1, && Z_{22,22}=\lambda_2,
\nonumber\\
Z_{11,22}=Z_{22,11}=\lambda_3, && Z_{12,21}=Z_{21,12}=\lambda_4,
\nonumber \\
Z_{12,12}=\lambda_5, && Z_{21,21}=\lambda_5^\ast,
\nonumber\\
Z_{11,12}=Z_{12,11}=\lambda_6, && Z_{11,21}=Z_{21,11}=\lambda_6^\ast,
\nonumber \\
Z_{22,12}=Z_{12,22}=\lambda_7, && Z_{22,21}=Z_{21,22}=\lambda_7^\ast.
\label{znum}\end{aligned}$$

Basis transformations
---------------------

The scalar potential can be rewritten in terms of new fields $\Phi^\prime_a$, obtained from the original ones by a simple (global) basis transformation $$\Phi_a \rightarrow \Phi_a^\prime = U_{ab} \Phi_b,
\label{basis-transf}$$ where $U\in U(2)$ is a $2 \times 2$ unitary matrix. Under this unitary basis transformation, the gauge-kinetic terms are unchanged, but the coefficients $Y_{ab}$ and $Z_{ab,cd}$ are transformed as $$\begin{aligned}
Y_{ab} & \rightarrow &
Y^\prime_{ab} =
U_{a \alpha}\, Y_{\alpha \beta}\, U_{b \beta}^\ast ,
\label{Y-transf}
\\
Z_{ab,cd} & \rightarrow &
Z^\prime_{ab,cd} =
U_{a\alpha}\, U_{c \gamma}\,
Z_{\alpha \beta,\gamma \delta}\, U_{b \beta}^\ast \, U_{d \delta}^\ast .
\label{Z-transf}\end{aligned}$$ Thus, the basis transformations $U$ may be utilized in order to absorb some of the degrees of freedom of $Y$ and/or $Z$, which implies that not all parameters of Eq. (\[VH2\]) have physical significance.

\[subsec:symmetries\]The six classes of symmetries in the THDM
--------------------------------------------------------------

Symmetries leaving the scalar Lagrangian unchanged may be of two types. On the one hand, one may relate $\Phi_a$ with some unitary transformation of $\Phi_b$: $$\Phi_a \rightarrow \Phi_a^S = S_{ab} \Phi_b,
\label{S-transf-symmetry}$$ where $S$ is a unitary matrix. These are known as Higgs Family symmetries, or HF symmetries. As a result of this symmetry, $$\begin{aligned}
Y_{a b} & = &
S_{a \alpha}\, Y_{\alpha \beta}\, S_{b \beta}^\ast ,
\label{Y-S}
\\
Z_{ab,cd} & = &
S_{a \alpha}\, S_{c \gamma}\,
Z_{\alpha \beta, \gamma \delta}\, S_{b \beta}^\ast \, S_{d \delta}^\ast .
\label{Z-S}\end{aligned}$$ On the other hand, one may relate $\Phi_a$ with some unitary transformation of $\Phi_b^\ast$: $$\Phi_a \rightarrow \Phi^{\textrm{GCP}}_a
= X_{a \alpha} \Phi_\alpha^\ast,
\label{GCP}$$ where $X$ is an arbitrary unitary matrix.[^1] These are known as generalized CP symmetries, or GCP symmetries [@GCP1; @GCP2]. The potential is invariant under this symmetry if and only if $$\begin{aligned}
Y_{ab}^\ast
&=&
X_{\alpha a}^\ast Y_{\alpha \beta} X_{\beta b}
\nonumber\\
Z_{ab,cd}^\ast
&=&
X_{\alpha a}^\ast X_{\gamma c}^\ast
Z_{\alpha \beta, \gamma \delta} X_{\beta b} X_{\delta d}.
\label{YZ-CPtransf}\end{aligned}$$ Under the basis transformation of Eq. (\[basis-transf\]), the specific forms of the HF and GCP symmetries are altered, respectively, as follows: $$\begin{aligned}
S^\prime &=& U S U^\dagger ,
\label{S-prime}
\\
X^\prime &=& U X U^\top.
\label{X-prime}\end{aligned}$$ Hence, a basis-invariant treatment is critical for distinguishing between two potentially different symmetries.

Of course, one may combine several HF symmetries and/or GCP symmetries. Ivanov [@Ivanov1] has proved that, whatever combination one chooses, one will end up in one of six distinct classes of symmetries. In a recent article we have clarified this issue showing how to construct such classes with simple examples [@FHS]. The result is shown in Table \[master1\].

   symmetry   $m_{11}^2$   $m_{22}^2$    $m_{12}^2$   $\lambda_1$   $\lambda_2$   $\lambda_3$         $\lambda_4$                        $\lambda_5$                   $\lambda_6$    $\lambda_7$
  ---------- ------------ ------------- ------------ ------------- ------------- ------------- ------------------------- -------------------------------------------- ------------- ---------------
    $Z_2$                                    0                                                                                                                              0              0
    $U(1)$                                   0                                                                                                0                             0              0
   $SO(3)$                 $ m_{11}^2$       0                      $\lambda_1$                 $\lambda_1 - \lambda_3$                       0                             0              0
     CP1                                    real                                                                                             real                         real           real
     CP2                   $m_{11}^2$        0                      $\lambda_1$                                                                                                      $- \lambda_6$
     CP3                   $m_{11}^2$        0                      $\lambda_1$                                           $\lambda_1 - \lambda_3 - \lambda_4$ (real)        0              0

  : Impact of the symmetries on the coefficients of the Higgs potential in a specified basis. See Ref. [@FHS] for more details.

\[master1\]

Five of the symmetry classes may be imposed by the following single requirements: $$\begin{aligned}
Z_2: &\hspace{1cm} &
S =
\left(
\begin{array}{cc}
1 & 0\\
0 & -1
\end{array}
\right),
\\
U(1): &\hspace{1cm} &
S =
\left(
\begin{array}{cc}
e^{i \alpha} & 0\\
0 & e^{-i \alpha}
\end{array}
\right)_{\alpha\neq \pi/2},
\\
CP1: &\hspace{1cm} &
X =
\left(
\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}
\right),
\\
CP2: &\hspace{1cm} &
X =
\left(
\begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}
\right),
\\
CP3: &\hspace{1cm} &
X =
\left(
\begin{array}{cc}
\cos{\theta} & \sin{\theta}\\
-\sin{\theta} & \cos{\theta}
\end{array}
\right).\end{aligned}$$ Here $0<\alpha<\pi$ (but $\alpha\neq \pi/2$, since the case of $\alpha=\pi/2$ corresponds to the $Z_2$ symmetry), and $0<\theta<\pi/2$. Invariance under the full $U(2)$ global symmetry[^2] is obtained by requiring the invariance of the scalar potential under Eq. (\[S-transf-symmetry\]), for *all* unitary matrices $S$.

\[sec:MSSM\]Basis invariant probes of the MSSM
==============================================

\[sec:Higgs\_MSSM\]The Higgs sector of the MSSM
-----------------------------------------------

The Higgs potential of the MSSM (prior to including soft-supersymmetry-breaking dimension-two squared-mass terms) is a particular case of Eq. (\[VH1\]), with $$\begin{aligned}
m_{11}^2 &=& m_{22}^2,
\label{MSSM_m11_m22}
\\
m_{12}^2 &=& 0,
\label{MSSM_m12}
\\
\lambda_1 = \lambda_2 &=&
\tfrac{1}{4} (g^2 + {g^\prime}^2),
\label{MSSM_L1_L2}
\\
\lambda_3 &=&
\tfrac{1}{4} (g^2 - {g^\prime}^2),
\label{MSSM_L3}
\\
\lambda_4 &=&
- \tfrac{1}{2} g^2,
\label{MSSM_L4}
\\
\lambda_5=\lambda_6=\lambda_7 &=& 0,
\label{MSSM_L5_L6_L7}\end{aligned}$$ where $g$ and $g^\prime$ are the $SU(2)_L$ and $U(1)_Y$ gauge coupling constants, respectively. In this case, Eqs. (\[ynum\]) and (\[znum\]) become $$\begin{aligned}
Y_{11}=Y_{22}, &&
Y_{12}=Y_{21}=0,
\label{yESL}\end{aligned}$$ and $$\begin{aligned}
Z_{11,11}=\lambda_1, && Z_{22,22}=\lambda_1,
\nonumber\\
Z_{11,22}=Z_{22,11}=\lambda_3, && Z_{12,21}=Z_{21,12}=- \lambda_1 - \lambda_3,
\label{zESL}\end{aligned}$$ with $\lambda_1$ given by Eq. (\[MSSM\_L1\_L2\]), $\lambda_3$ given by Eq. (\[MSSM\_L3\]), and all other components of the $Z$ tensor equal to zero.

Comparing Eqs. (\[MSSM\_m11\_m22\])–(\[MSSM\_L5\_L6\_L7\]) with Table \[master1\], we see that these requirements are almost the same as in the THDM with the full $U(2)$ flavor symmetry. The difference is that the $U(2)$-symmetric case implies $\lambda_4 = \lambda_1 - \lambda_3$, while the supersymmetry limit implies $\lambda_4 = - \lambda_1 - \lambda_3$. As shown by Ivanov [@Ivanov1] and by us [@FHS], the former relation can come from a symmetry requirement that exclusively involves the Higgs potential, while the latter relation cannot. In particular, there are no basis changes one can perform on the THDM to obtain, from one of the six symmetries listed in table \[master1\], the SUSY condition $\lambda_4 = - \lambda_1 - \lambda_3$.

This can also be seen by examining the renormalization group equations that control the evolution of the $\lambda_i$. Here, we focus only on those terms arising from the Higgs potential and the gauge couplings.[^3] The relevant expressions can be found, for example, in Refs. [@CEL; @HH; @GL; @FJ]. Using $\lambda_1=\lambda_2$ and $\lambda_5=\lambda_6=\lambda_7=0$, we find $$\begin{aligned}
{\cal D} \lambda_1 &=&
6 \lambda_1^2 + 2 \lambda_3^2 + \lambda_4^2 + 2 \lambda_3 \lambda_4
- \tfrac{1}{2} \left( 9 g^2 + 3 {g^\prime}^2 \right) \lambda_1
+ \tfrac{1}{8} \left( 9 g^4 + 6 g^2 {g^\prime}^2 + 3{g^\prime}^4 \right),
\nonumber\\
{\cal D} \lambda_3 &=&
2 \lambda_3^2 + \lambda_4^2 + 2 \lambda_1 (3 \lambda_3 + \lambda_4)
- \tfrac{1}{2} \left( 9 g^2 + 3 {g^\prime}^2 \right) \lambda_3
+ \tfrac{1}{8} \left( 9 g^4 - 6 g^2 {g^\prime}^2 + 3{g^\prime}^4 \right),
\nonumber\\
{\cal D} \lambda_4 &=&
2 \lambda_4^2 + 2 \lambda_1 \lambda_4 + 4 \lambda_3 \lambda_4
- \tfrac{1}{2} \left( 9 g^2 + 3 {g^\prime}^2 \right) \lambda_4
+ \tfrac{3}{2} g^2 {g^\prime}^2, \label{D134}\end{aligned}$$ where ${\cal D} = 16 \pi^2 \mu (d/d\mu)$, and ${\cal D} \lambda_5 = {\cal D} \lambda_6 = {\cal D} \lambda_7 = 0$. Hence, given the constraints $\lambda_1=\lambda_2$ and $\lambda_5=\lambda_6=\lambda_7=0$, $$\begin{aligned}
{\cal D} (\lambda_4 + \lambda_3 - \lambda_1 ) &=&
\tfrac{1}{2}
(\lambda_4 + \lambda_3 - \lambda_1 )(12 \lambda_1 + 4 \lambda_4 - 9 g^2 - 3 {g^\prime}^2)
\label{D_O3}
\\*[2mm]
{\cal D} (\lambda_4 + \lambda_3 + \lambda_1 ) &=&
2 \left(3 \lambda _1^2+\left(3 \lambda _3+2 \lambda _4\right) \lambda _1+2 \lambda _3^2+2
   \lambda _4^2+3 \lambda _3 \lambda _4\right)
\nonumber\\
&&
- \tfrac{1}{2} \left( 9 g^2 + 3 {g^\prime}^2 \right)
\left(\lambda_4 + \lambda_3 + \lambda_1 \right)
+ \tfrac{1}{4} \left( 9 g^4 + 6 g^2 {g^\prime}^2 + 3{g^\prime}^4 \right).
\label{D_MSSM}\end{aligned}$$ The first equation vanishes if $\lambda_4 = \lambda_1 - \lambda_3$; the second does not vanish if $\lambda_4 = - \lambda_1 - \lambda_3$. That is, the condition $\lambda_4 = \lambda_1 - \lambda_3$ is renormalization group invariant, whereas the condition $\lambda_4 = - \lambda_1 - \lambda_3$ is not. Note that we have not yet imposed the specific relations between the $\lambda_i$ and the gauge couplings required by the MSSM. If we impose the MSSM constraints specified by Eqs. (\[MSSM\_L1\_L2\])–(\[MSSM\_L4\]) on the right hand side of , we obtain \[RGINV\] [D]{} (\_4 + \_3 + \_1 ) = 3 g\^4 + 2 g\^2 [g\^]{}\^2 + [g\^]{}\^4, i.e., $\lambda_1 = - \lambda_3 - \lambda_4$ is still not RGE invariant.

The latter result is not unexpected. After all, the gauge boson–Higgs boson sector considered by itself can never be supersymmetric, as the corresponding superpartners are not included. Consequently, the SUSY limit of the gauge boson–Higgs boson sector can only be defined in a manner invariant under Higgs basis changes if the corresponding gaugino and higgsino superpartners are taken into account. The gaugino and higgsino interactions generate additional terms on the right hand side of . In the supersymmetric limit, these effects yield [@HH] \_[SUSY]{}(\_1)&=&-g\^4-g\^2 g\^[2]{} -g\^[4]{},\
\_[SUSY]{}(\_3)&=&-g\^4+g\^2 g\^[2]{} -g\^[4]{},\
\_[SUSY]{}(\_4)&=& 2g\^4-2g\^2 g\^[2]{}. \[deltaD\] Hence, $$\delta_{\rm SUSY}\{\mathcal{D}(\lambda_4+\lambda_3+\lambda_1)\}=
-(3 g^4 + 2 g^2 {g^\prime}^2 + {g^\prime}^4)\,.$$ Indeed, when the latter is added to , we see that ${\cal D} (\lambda_4 + \lambda_3 + \lambda_1 )=0$ as expected. Thus, the SUSY relation $\lambda_1=-\lambda_3-\lambda_4
=\tfrac{1}{4}(g^2+g^{\prime\,2})$ is renormalization group invariant when all the Higgs/higgsino/gauge/gaugino interactions are included.

\[sec:necessary\]Basis-independent conditions for the MSSM Higgs potential that are necessary but not sufficient
----------------------------------------------------------------------------------------------------------------

Based on the arguments of section \[sec:Higgs\_MSSM\], no basis-invariant conditions constructed solely from the $Y_{ab}$ and $Z_{ab,cd}$ exist that can guarantee that are satisfied for some choice of basis. Nevertheless, we can establish a weaker invariant condition that is necessary (although not sufficient) for the existence of a supersymmetric THDM scalar potential.[^4]

Consider a basis choice in which: \[weaker\] m\_[11]{}\^2=m\_[22]{}\^2,\_1=\_2, m\_[12]{}\^2=\_5=\_6=\_7=0. Clearly is satisfied by the MSSM Higgs potential in the standard MSSM basis choice for the Higgs fields. However, is not sufficient since the condition $\lambda_4=-\lambda-\lambda_3$ is imposed, nor are the conditions relating the quartic Higgs couplings to gauge couplings imposed. Nevertheless, suppose that one could establish a basis-independent condition that was equivalent to the statement that a basis exists in which is satisfied. Such a basis-independent condition would then be a necessary condition for a supersymmetric Higgs sector.[^5]

The conditions of do not match any single condition listed in Table \[master1\]. Nevertheless, given a scalar potential whose parameters satisfy , one can *always* transform to a new basis that explicitly satisfies the CP3-symmetry conditions of Table \[master1\], and vice versa. That is, a basis-invariant characterization of a CP3-symmetric THDM would guarantee that some basis exists in which holds. It follows that *the Higgs sector of the MSSM is a CP3-symmetric THDM.*

The proof of the assertions above have been given in Ref. [@FHS]. For completeness, we review the arguments here. We first define the discrete flavor symmetry $\Pi_2$ which corresponds to a THDM scalar potential that is symmetric under the interchange of the two Higgs fields. In this case, the scalar potential parameters satisfy: \[pi2\] m\_[11]{}\^2=m\_[22]{}\^2,m\_[12]{}\^2 ,\_1=\_2,\_5 , \_6=\_7. In Ref. [@FHS], we showed that given a $\Pi_2$-symmetric scalar potential, there exists another basis which is $Z_2$-symmetric. Consequently, we did not list $\Pi_2$ as a separate symmetry in Table \[master1\]. However, if we simultaneously impose $\Pi_2$ and $U(1)$ *in the same basis*, then one easily sees that the conditions of are satisfied. The same conclusion also follows if we simultaneously impose $U(1)$ and CP2 in the same basis. Moreover, starting from a scalar potential in which the conditions of are satisfied, a basis transformation can be found (see Ref. [@FHS] for the details) such that $\lambda_5=\lambda_1-\lambda_3-\lambda_4$ in the new basis, as required in the CP3-symmetric THDM \[cf. Table \[master1\]\].

Finally, we indicate the basis-independent conditions that guarantee that a basis exists in which the CP3-symmetric conditions are satisfied. Defining $Z_{ab}^{(1)}\equiv Z_{ac,cb}$, we first require that [@DavHab] \[erps\] Y\_[ab]{}=m\_[11]{}\^2\_[ab]{} \[Z\^[(1)]{}\]\^2=([Tr]{} Z\^[(1)]{})\^2. If is satisfied, then \[erps2\] m\_[11]{}\^2=m\_[22]{}\^2,m\_[12]{}\^2=0,\_1=\_2,\_7=-\_6, must hold in *all basis choices*. This is the so-called exceptional region of parameter space (ERPS) of the THDM. In Ref. [@FHS], we then constructed a second invariant quantity $D$ built out of the $Y_{ab}$ and $Z_{ab,cd}$ with the following property: if $D=0$ in the ERPS, then there exists a basis for the scalar fields such that the CP3-conditions of Table \[master1\] are satisfied.[^6] Thus, the conditions for the ERPS plus $D=0$ provide necessary (although not sufficient) invariant conditions that must be satisfied by the MSSM Higgs scalar potential.

\[sec:gaugino\_Higgs\_MSSM\]The gaugino-higgsino-Higgs interactions
-------------------------------------------------------------------

In the MSSM, the tree-level Lagrangian describing the interactions of the gauginos with the Higgs-doublets may be written as \[susygaugino\] [L]{}\_\^=\_[ij]{}\^i\_[H\_D]{} \^j\_[H\_U]{}+ \^\^\_[ij]{}(\^j\_[H\_U]{}\_2\^[i]{} +\^[ik]{}\^j\_[H\_D]{}\_1\^k) +\^(\^i\_[H\_U]{}\_2\^[i]{} -\^[ik]{}\^i\_[H\_D]{}\_1\^k)+[h.c.]{}, where $\lambda^\alpha$ and $\lambda'$ are the two-component spinor gaugino fields that are superpartners to the SU(2) and U(1)-hypercharge gauge bosons, and $\psi_{H_D}$ and $\psi_{H_U}$ are, respectively, the hypercharge $-1$ and hypercharge $+1$ weak doublet two-component spinor higgsino fields. The indices $i$, $j$ and $k$ label the two components of the weak doublet, and the index $\alpha$ is the adjoint index of the SU(2) gaugino field. We have included a supersymmetric Majorana mass term for the two-component higgsino fields, which defines the parameter $\mu$. As usual, $\epsilon^{12}=
-\epsilon^{21}=+1$ and $\epsilon^{11}=\epsilon^{22}=0$.

If we relax the constraints imposed by supersymmetry, the coupling strengths of the gaugino-higgsino-Higgs interaction above are no longer constrained to be gauge couplings as in . Moreover, four additional dimension-four interaction terms are possible, consistent with SU(2)$\times$U(1) gauge invariance. These terms are the so-called “wrong-Higgs” couplings of Ref. [@HM], and are obtained from those of by interchanging $\Phi_1$ and $\Phi_2$. In our analysis below, we consider the most general dimension-four gauge invariant couplings between the gaugino, higgsino and Higgs fields. We shall write these couplings in a form that is manifestly independent of the choice of basis for the Higgs fields: \[gensusygaugino\] [L]{}\_= \^\^\_[ij]{}(\^j\_[H\_U]{}f\_U\^a\_a\^[i]{} +\^[ik]{}\^j\_[H\_D]{}f\_D\^[a]{}\_a\^k) +\^(\^i\_[H\_U]{} f\^[a]{}\_U\_a\^[i]{} -\^[ik]{}\^i\_[H\_D]{}f\^[a]{}\_D\_a\^k)+[h.c.]{}, where the couplings $f^a_U$, $f^a_D$, $f^{\prime\,a}_U$, and $f^{\prime\,a}_D$ transform covariantly under a Higgs basis U(2)-transformation,[^7] f\^a\_[U,D]{}U\_[ab]{}f\^b\_[U,D]{},f\^[a]{}\_[U,D]{}U\_[ab]{}f\^[b]{}\_[U,D]{}. In the supersymmetric limit, there is a natural choice of basis for the Higgs fields, henceforth called the *SUSY basis*, in which: \[spbasis\] f\_U\^a=g\^[a2]{},f\_D\^[a]{}=g\^[a1]{},f\_U\^[a]{}=g’\^[a2]{},f\_D\^[a]{}=g’\^[a1]{}. In particular, in the SUSY basis, the so-called “wrong-Higgs interactions” of Ref. [@HM] are absent in the supersymmetric limit. However, under a general Higgs basis transformation, the supersymmetric gaugino-higgsino-Higgs Lagrangian will transform into a linear combination of supersymmetric and wrong-Higgs interaction terms. Thus, in a generic basis choice for the Higgs fields, the supersymmetry is not manifest. One of the goals of this section is to determine a set of basis-independent conditions that guarantees the existence of a basis choice in which is satisfied. Such basis-independent conditions would constitute an invariant signal for manifestly supersymmetric Higgs interactions.

The couplings $f^a_U$, $f_D^a$, $f^{\prime\,a}_U$, and $f^{\prime\,a}_D$ are complex vectors that live in the two-dimensional Higgs flavor space. It is convenient to define the corresponding unit vectors, $\hat f^a\equiv f^a/|f|$, where $|f|\equiv (f^{a\,\ast} f^a)^{1/2}$ is the length of the complex vector $f^a$. Next, we introduce vectors that are orthogonal to $f^a_U$, $f_D^a$, $f^{\prime\,a}_U$, and $f^{\prime\,a}_D$, respectively, g\_U\^a&&f\_U\^[b]{} \^[ba]{},g\_D\^af\_D\^[b]{} \^[ba]{},\
g\_U\^[a]{}&&f\_U\^[b]{} \^[ba]{}, g\_D\^[a]{}f\_D\^[b]{} \^[ba]{}. These are pseudo-vectors with respect to U(2) Higgs basis transformations,[^8] \[gtransform\] g\^a\_[U,D]{}([det]{} U)\^[-1]{}U\_[ab]{} g\^b\_[U,D]{},g\^[a]{}\_[U,D]{}([det]{} U)\^[-1]{}U\_[ab]{} g\^[b]{}\_[U,D]{}, due to the appearance of the complex phase, ${\rm det}~U$.

We now define U(2)-invariant, hypercharge-one Higgs fields as follows: H\_U& &f\_U\^[a]{}\_a,H’\_Uf\_U\^[a]{}\_a,\[invH1\]\
H\_D& & f\_D\^[a]{}\_a,H’\_Df\_D\^[a]{}\_a.\[invH2\] One can also define a corresponding set of hypercharge $-1$ fields, e.g., \[invH3\] H\_D\^i\^[ij]{}H\_D\^[j]{},H\_D\^[i]{}\^[ij]{}H\_D\^[j]{}. It is also convenient to define U(2) pseudo-invariant Higgs fields (denoted by calligraphic fonts), \_U& &g\_U\^[a]{}\_a,\^\_Ug\_U\^[a]{}\_a,\[invH4\]\
\_D& & g\_D\^[a]{}\_a,\^\_Dg\_D\^[a]{}\_a.\[invH5\] It then follows that: \_a&=& f\_U\^a H\_U +g\_U\^a \_U= f\_D\^a \_D +g\_D\^a \_D \[phia1\]\
&=& f\_U\^[a]{} H’\_U +g\_U\^[a]{} ’\_U= f\_D\^[a]{} ’\_D +g\_D\^[a]{} ’\_D,\[phia\] after using $\hat f_{U,D}^a \hat f_{U,D}^{a\,*}=
\hat g_{U,D}^a \hat g_{U,D}^{a\,*}=1$ and $\hat g_{U,D}^a \hat f_{U,D}^{a\,*}=\hat f_{U,D}^a \hat g_{U,D}^{a\,*}=0$.

There is some motivation for this proliferation of Higgs field definitions. In particular, as we show later in , the choices of \[four\] {\_U,H\_U}, {\^\_U,H\^\_U}, {\_D,\_D},[and]{} {\^\_D,\^\_D}, correspond to four different basis choices for the hypercharge-one Higgs doublet fields.

One can express the gaugino-higgsino-Higgs interaction Lagrangian in a manifestly U(2)-invariant form. For example, using the definitions of the invariant Higgs fields $H_U$, $H_D$, $H_U^\prime$ and $H_D^\prime$ \[defined by \], can be rewritten as: \[gensusygauginoinv\] [L]{}\_= \^\^\_[ij]{}(|f\_U|\^j\_[H\_U]{}H\_U\^[i]{} +|f\_D|\^j\_[H\_D]{}H\_D\^[i]{}) +\^(|f\^\_U|\^i\_[H\_U]{} H\_U\^[i]{} -|f\_D\^|\^i\_[H\_D]{}H\_D\^[i]{})+[h.c.]{} However, this form is not particularly useful outside of the supersymmetric limit, since $\{H_U\,,\,\widetilde H_D\}$ and $\{H^\prime_U\,,\,\widetilde H^\prime_D\}$ are *not* orthogonal pairs of hypercharge-one Higgs doublet fields in the general case. Of course, one can always rewrite in terms of one of the four basis choices of hyper-charge one doublet Higgs fields listed in by inserting the appropriate form for $\Phi_a$ given in into .

\[sec:ESL\]Basis-invariant probes of the supersymmetric Higgs interactions
--------------------------------------------------------------------------

Supersymmetry imposes strong constraints on the scalar Higgs potential and the gaugino-higgsino-Higgs interactions. These constraints must involve basis-independent combinations of the scalar potential parameters $Y_{ab}$, $Z_{abcd}$, and the gaugino-higgsino-Higgs couplings $f^a_U$, $f_D^a$, $f^{\prime\,a}_U$, $f^{\prime\,a}_D$. It is straightforward to find the necessary relations. First we exhibit the basis invariant relations that enforce supersymmetric gaugino-higgsino-Higgs couplings:

$$\begin{aligned}
f_U^a f_D^{a \ast} = 0,
&\hspace{1cm}&
f_U^a f_D^{\prime a \ast} = 0,
\nonumber\\
f_U^{\prime a} f_D^{a \ast} = 0,
&\hspace{1cm}&
f_U^{\prime a} f_D^{\prime a \ast} = 0,
\nonumber\\
f_U^a f_U^{a \ast} = f_D^a f_D^{a \ast}= g^2,
&\hspace{1cm}&
f_U^{\prime a} f_U^{\prime a \ast}
= f_D^{\prime a} f_D^{\prime a \ast} = g^{\prime\,2}\,,
\nonumber \\
f_U^a f_U^{\prime\,a \ast} = gg^\prime,
&\hspace{1cm}&
f_D^{a \ast} f_D^{\prime a} = gg^{\prime}\,.
\label{f_ESL}\end{aligned}$$

To establish U(2)-invariant conditions that enforce a supersymmetric scalar Higgs potential, we first construct basis-independent quantities that involve both the scalar potential parameters and the gaugino-higgsino-Higgs couplings. For example,[^9] $$\begin{aligned}
{\cal Y}_{DD} &=& \hat f_D^a\, \hat f_D^{b \ast}\ Y_{ab},
\qquad\qquad
{\cal Y}_{UU} = \hat f_U^a\, \hat f_U^{b \ast}\ Y_{ab},
\nonumber\\
{\cal Y}_{DU} &=& \hat f_D^a\, \hat f_U^{b \ast}\ Y_{ab},
\qquad\qquad
{\cal Y}_{UD} = \hat f_U^a\, \hat f_D^{b \ast}\ Y_{ab},
\label{calY}\end{aligned}$$ provide basis-invariant quantities involving the quadratic coefficients of the Higgs potential. Likewise, $${\cal Z}_{\alpha \beta, \gamma \delta} =
\hat f_\alpha^a\, \hat f_\beta^{b \ast}\,
\hat f_\gamma^c\, \hat f_\delta^{c \ast}\, Z_{ab,cd}\,,
\label{calZ}$$ where the indices $\alpha$, $\beta$, $\gamma$, and $\delta$ can take the values $D$ or $U$, provide basis-invariant quartic coefficients for the Higgs potential.

Evaluating the invariant quantities introduced in in the supersymmetric basis defined by Eqs. (\[yESL\]), (\[zESL\]) and (\[spbasis\]), it follows that \_[DD]{} = [Y]{}\_[UU]{},\_[DU]{} = [Y]{}\_[UD]{} = 0, \[calY\_ESL\] and $$\begin{aligned}
{\cal Z}_{DD,DD}&=& {\cal Z}_{UU,UU}  = \tfrac{1}{4}\left[ f_U^{a\,\ast} f_U^a+
 f_D^{a\,\ast} f_D^a\right],
\nonumber\\
{\cal Z}_{DD,UU} &=& {\cal Z}_{UU,DD} = \tfrac{1}{4}
\left[ f_U^{a\,\ast} f_U^a - f_U^{\prime\,a\,\ast} f_U^{\prime\,a}\right],
\nonumber\\
{\cal Z}_{DU,UD} &=& {\cal Z}_{UD,DU} = - {\cal Z}_{DD,DD} - {\cal Z}_{DD,UU},
\nonumber\\*[2mm]
{\cal Z}_{DU,DU} &=& {\cal Z}_{UD,UD} = 0, \nonumber \\
{\cal Z}_{UD,DD}&=&{\cal Z}_{DU,DD}={\cal Z}_{DD,UD}={\cal Z}_{DD,DU}=0,\nonumber\\
{\cal Z}_{DU,UU}&=&{\cal Z}_{UD,UU}={\cal Z}_{UU,DU}={\cal Z}_{UU,UD}=0.
\label{calZ_ESL}\end{aligned}$$ Since these equations are basis invariant, they must hold in any theory made up of the MSSM fields with exact supersymmetry, regardless of the exact basis choice made for the Higgs fields.

That is, independently of the choice of basis for the Higgs fields, the supersymmetric limit of the Higgs/higgsino/gauge/gaugino sectors holds if and only if Eqs. (\[f\_ESL\]), (\[calY\_ESL\]) and (\[calZ\_ESL\]) hold. If Eqs. (\[f\_ESL\]) and (\[calZ\_ESL\]) hold but Eq. (\[calY\_ESL\]) does not, then supersymmetry is softly broken due to the quadratic terms of the Higgs potential. The above results fully resolves the question of the basis-invariant form for the supersymmetric limit of the THDM.

\[sec:preferred\]A preferred basis in the MSSM
----------------------------------------------

The gaugino-higgsino-Higgs interactions provide a means for defining a *quasi-physical* choice of basis. In this context, a quasi-physical basis is one in which the Higgs fields are invariant, up to a possible rephasing of one of the Higgs fields, under U(2) transformations. That is, the coefficients of the Higgs potential are either U(2)-invariants or pseudo-invariants.

There are four possible independent quasi-physical bases, corresponding to the four normalized gaugino-higgsino-Higgs couplings, $\hat f_U^a$, $\hat f_D^a$, $\hat f_U^{\prime\,a}$ and $\hat f_D^{\prime\,a}$. Each basis is defined by imposing the condition that one of the two components of the corresponding coupling vanishes, while setting the non-vanishing component to unity. This defines the quasi-physical basis up to an arbitrary rephasing of the Higgs field that lies in the direction of the vanishing component of $f$. The latter is a pseudo-invariant field, whereas the Higgs field that lies in the direction of the non-vanishing component of $f$ is a U(2)-invariant field.[^10]

The four quasi-physical bases and their corresponding Higgs fields are: \_U: &&  f\_U\^a=(0,1),:  (\_U, H\_U),\[bu\]\
\_D: &&  f\_D\^a=(1,0),:  (H\_D, \_D),\[bd\]\
\^\_U: && f\^[a]{}\_U=(0,1),:  (\^\_U,H\^\_U),\[bup\]\
\^\_D: && f\^[a]{}\_D=(1,0),:  (H\^\_D,\^\_D),\[bdp\] where the fields denoted by (calligraphic) Roman fonts are (pseudo-)invariant with respect to U(2) basis transformations. The Higgs fields with a $U$ ($D$) subscript are hypercharge $+1$ ($-1$) fields. The coefficients of the scalar potential in the quasi-physical basis are easily constructed. For example, in basis $\mathcal{B}_D$, Y\_[D1]{}&& f\_D\^[a]{} f\_D\^b Y\_[ab]{}, Y\_[D2]{}g\_D\^[a]{} g\_D\^b Y\_[ab]{},\[cov1\]\
Y\_[D3]{}&&f\_D\^[a]{} g\_D\^b Y\_[ab]{}, Z\_[D1]{}f\_D\^[a]{} f\_D\^b f\_D\^[c]{} f\_D\^d Z\_[abcd]{},\[cov2\]\
Z\_[D2]{}&& g\_D\^[a]{} g\_D\^b g\_D\^[c]{} g\_D\^d Z\_[abcd]{},Z\_[D3]{}f\_D\^[a]{} f\_D\^b g\_D\^[c]{} g\_D\^d Z\_[abcd]{},\[cov3\]\
Z\_[D4]{}&& g\_D\^[a]{} f\_D\^b f\_D\^[c]{} g\_D\^d Z\_[abcd]{},Z\_[D5]{}f\_D\^[a]{} g\_D\^b f\_D\^[c]{} g\_D\^d Z\_[abcd]{},\[cov4\]\
Z\_[D6]{}&& f\_D\^[a]{} f\_D\^b f\_D\^[c]{} g\_D\^d Z\_[abcd]{},Z\_[D7]{}f\_D\^[a]{} g\_D\^b g\_D\^[c]{} g\_D\^d Z\_[abcd]{}.\[cov5\] That is, starting in a generic basis with a Higgs potential given by , one can always transform to the basis $\mathcal{B}_D$ with a Higgs potential given by: V\_H &=& Y\_[D\_1]{}H\_D\^H\_D + Y\_[D\_2]{} \_D\^\_D +\
&& + Z\_[D\_1]{} (H\_D\^H\_D)\^2 + Z\_[D\_2]{} (\_D\^\_D)\^2 + Z\_[D\_3]{} (H\_D\^H\_D) (\_D\^\_D) + Z\_[D\_4]{}(H\_D\^\_D) (\_D\^H\_D)\
&& + . \[VHD1\] The parameters $Y_{D1}, Y_{D2}, Z_{D1}, Z_{D2}, Z_{D3}$ and $Z_{D4}$ are manifestly real U(2)-invariants, whereas the parameters $Y_{D3}, Z_{D5}, Z_{D6}$ and $Z_{D7}$ are (potentially) complex pseudo-invariants with respect to U(2) basis transformations. The physical Higgs couplings and masses of the theory can be expressed in terms of the invariant parameters and invariant combinations of the pseudo-invariant parameters.

The discussion above is completely general. But, suppose we impose supersymmetry on the Higgs-higgsino-gaugino system. Supersymmetry imposes a very strong constraint on the quasi-physical bases, \[susyb\] \_U=\_D=\^\_U=\^\_[D]{}. That is, supersymmetry picks out a physically meaningful basis, namely the SUSY basis, in which the gaugino-higgsino-Higgs couplings take the form given by . impose covariant relations among the normalized gaugino-higgsino-Higgs couplings: \[susyfg\] g\_D\^a=g\_D\^[a]{}=e\^[i]{}f\_U\^a=e\^[i]{}f\_U\^[a]{},g\_U\^a=g\_U\^[a]{}=-e\^[i]{}f\_D\^a=-e\^[i]{}f\_D\^[a]{}, where $e^{i\eta}$ is a pseudo-invariant quantity \[i.e., $e^{i\eta}\to (\det~U)^{-1}e^{i\eta}$ under a U(2) transformation\] that is equal to $1$ in the SUSY basis. implies that the invariant and pseudo-invariant Higgs fields defined in are related in the supersymmetric limit, H\_U=H\_U\^&=&e\^[i]{}\_D=e\^[i]{}\_D\^,\[mssmhu\]\
H\_D=H\_D\^&=&e\^[-i]{}\_U=e\^[-i]{}\_U\^,\[mssmhd\] for the hypercharge $+1$ and hypercharge $-1$ MSSM Higgs fields, respectively. If we now insert into , we find that the coefficients of the Higgs potential in the SUSY basis are $\mathcal{Y}_{DD},\mathcal{Y}_{UU},e^{i\eta}\mathcal{Y}_{UD},
\mathcal{Z}_{UU,UU},\mathcal{Z}_{DD,DD},\mathcal{Z}_{DD,UU},\mathcal{Z}_{DU,UD},e^{2i\eta}\mathcal{Z}_{DU,DU},
e^{i\eta}\mathcal{Z}_{DD,DU},e^{i\eta}\mathcal{Z}_{UU,UD}$, subject to the conditions of . Note that when the conditions of are imposed, the phase $\eta$ drops out completely. Indeed, the nonzero Higgs potential parameters in the SUSY basis, $\mathcal{Y}_{DD},\mathcal{Y}_{UU},\mathcal{Z}_{UU,UU},\mathcal{Z}_{DD,DD},\mathcal{Z}_{DD,UU},\mathcal{Z}_{DU,UD}$ are physical observables that are real and invariant with respect to U(2)-basis transformations The existence of a physical SUSY basis is not surprising. Consider the higgsino Majorana mass term, which arises from a supersymmetric term in the MSSM superpotential, $$\mu \epsilon_{ij} \psi^i_{H_D} \psi^j_{H_U}.
\label{fix}$$ A THDM basis transformation mixes $\Phi_1$ and $\Phi_2$, which corresponds to a transformation between the MSSM Higgs fields, $H_U$ with $H_D^\dagger$. But the Higgs fields belong to Higgs superfields whose scalar and fermionic components are $(H_U, \psi_{H_U})$ and $(H_D, \psi_{H_D})$. As a result, one cannot apply a general U(2)-basis transformation to the full Higgs supermultiplets consistent with supersymmetry. Thus, Eq. (\[fix\]) effectively defines a preferred basis for $\Phi_1$ and $\Phi_2$ within the MSSM. This is why the SUSY basis has a physical significance.

\[sec:tanb\]Invariant $\boldsymbol{\tan\beta}$-like parameters
--------------------------------------------------------------

Until this subsection, we have made no assumptions about the nature of the Higgs vacuum. In the supersymmetric limit of the MSSM, the minimum of the tree-level potential resides at zero field and there is no electroweak symmetry breaking. After dimension-two soft-supersymmetry-breaking terms are included, it becomes possible to break the SU(2)$\times$U(1) symmetry of the scalar potential down to U(1)$_{\rm EM}$. In the generic basis, we define vacuum expectation values, \^0\_a= v\_avv\_a, where $\hat v_a$ is a complex unit vector in the two-dimensional Higgs flavor space and $v\simeq 246~{\rm GeV}$. In the MSSM, the parameter $\tan\beta$ is defined in the SUSY basis,[^11] . However, in the most general THDM, $\tan\beta$ is a basis-dependent concept that does not correspond to any physical observable. In Ref. [@oneil], a number of invariant parameters connected to the Higgs-fermion Yukawa couplings are introduced that play the role of $\tan\beta$ in constrained THDMs. Here, we shall indicate how to define $\tan\beta$-like parameters associated with the gaugino-higgsino-Higgs interactions.

The basic idea is to define $\tan\beta$ as the ratio of vacuum expectation values in the quasi-physical bases introduced in . One can then define four invariant $\tan\beta$-like parameters,[^12] \_U&& -e\^[-i]{} =-e\^[-i]{}, \_De\^[-i]{} =e\^[-i]{},\[tb1\]\
\^\_U&& -e\^[-i]{}= -e\^[-i]{}, \^\_De\^[-i]{}=e\^[-i]{}.\[tb2\] Without the factors of $e^{i\eta}$, the corresponding $\tan\beta$-like parameters are complex pseudo-invariants, whose magnitudes are basis-independent. The interpretation of these $\tan\beta$-like parameters is simplest in the Higgs basis, which is defined by $\hat v=(1\,,\,0)$. In the Higgs basis, the parameters in are (up to an overall phase) simply the ratios of gaugino-higgsino-Higgs couplings. One can now investigate the limit in which all dimension-four couplings respect supersymmetry. In this case, the invariant and pseudo-invariant Higgs fields are constrained by , in which case, \[susytb\] =\_U=(\_D)\^=\_U\^=(\_D\^)\^\*,. Corrections to the above relations at the few percent level are expected at the one-loop level due to supersymmetry-breaking effects that enter into the loops. In principle, one could extract the $\tan\beta$ parameters of from collider data (assuming gaugino-higgsino-Higgs couplings could be measured with sufficient accuracy), and check to see whether holds.

\[sec:conclusions\]Conclusions
==============================

The THDM includes the Higgs sector of the MSSM as a particular case. In the THDM only quantities that are invariant under a unitary basis change, $\Phi_a\to U_{ab}\Phi_b$ (where $a,b=1,2$ and $U$ is an arbitrary unitary matrix) can have physical meaning. Thus, it is natural to look for basis invariant definitions of the supersymmetric limit. We have shown that there is no basis-invariant definition of a supersymmetric THDM based on invariants defined exclusively in terms of parameters of the scalar Higgs potential. We have constructed basis invariant probes of the supersymmetric limit and soft-supersymmetry-breaking by employing both the Higgs potential and the gaugino-higgsino-Higgs interactions. We also observe that the usual basis choice of $H_U$ and $H_D$ in the MSSM does have a physical significance, due to the $\mu$-term of the MSSM superpotential \[cf. Eq. (\[fix\])\].

Finally, we addressed the physical significance of the parameter $\tan\beta$. In the MSSM, the tree-level parameter $\tan\beta$ is well-defined because it is defined in terms of vacuum expectation values of the neutral Higgs fields, which are given in a physical basis. However, in the most general THDM Higgs sector coupled to new particles with electroweak quantum numbers that coincide with the gauginos and higgsinos of the MSSM, the parameter $\tan\beta$, which is defined in terms of a ratio of Higgs vacuum expectation values, is a basis dependent quantity and hence unphysical. We have shown that four invariant $\tan\beta$-like parameters can be defined that are basis-independent and hence physical. One can verify that in the supersymmetric limit, these four invariant parameters coincide (at tree-level) and are equal to the $\tan\beta$ parameter of the MSSM.

The work of P.M.F. is supported in part by the Portuguese *Fundação para a Ciência e a Tecnologia* (FCT) under contract PTDC/FIS/70156/2006. The work of H.E.H. is supported in part by the U.S. Department of Energy, under grant number DE-FG02-04ER41268. The work of J.P.S. is supported in part by FCT under contract CFTP-Plurianual (U777) and under project CERN/FP/109305/2009, and by the EU RTN project Marie Curie: MRTN-CT-2006-035505. H.E.H. is most grateful for the kind hospitality and support of the Centro de Física Teórica e Computacional at Universidade de Lisboa during his visit to Lisbon. He also acknowledges conversations with Sacha Davidson and John Mason, which inspired a number of ideas presented in this paper.

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[^1]: The space coordinates of the fields, which we have suppressed, are inverted by a generalized CP transformation.

[^2]: The $SO(3)$ Higgs flavor symmetry listed in Table \[master1\] is orthogonal to the $U(1)_{\rm Y}$ hypercharge invariance (under which the THDM potential is always invariant). In the $SO(3)$-symmetric case, the *full* Higgs flavor symmetry group is $U(2)\iso SO(3)\otimes 
    U(1)_{\rm Y}$ [@FHS]. \[fn\]

[^3]: In order to include fermions in the analysis, one would have to investigate the constraints of the THDM symmetries on the Higgs-fermion Yukawa couplings.

[^4]: We thank the anonymous referee for encouraging us to consider the necessary symmetry constraints that govern the MSSM Higgs scalar potential.

[^5]: The additional conditions necessary to establish necessary *and* sufficient basis-independent probes of supersymmetry require consideration of the gaugino-higgsino-Higgs interactions. This is the subject of section \[sec:gaugino\_Higgs\_MSSM\] that follows.

[^6]: The explicit expression for $D$ is rather complicated and can be found in eqs. (39)–(41) and (44) of Ref. [@FHS].

[^7]: Note that the global U(1) Higgs flavor transformation corresponding to $\Phi_a\to e^{i\chi}\Phi_a$ ($a=1,2$) is distinguished from the global U(1) hypercharge transformation, since the higgsino fields *do not* transform under the rephasing of the Higgs fields.

[^8]: Starting from the transformation law $\hat f^a\to U_{ab}\hat f^b$, where $U^\dagger=U^{-1}$, it follows that $\hat g^c\to U^{-1}_{ab}g^d\epsilon^{ad}\epsilon^{bc}$. If we now recognize that $U^{-1}_{ab}\epsilon^{ad}={\rm det}(U^{-1})
    U_{cd}\epsilon^{bc}$, the results of easily follow.

[^9]: Assuming is satisfied, it is not necessary to construct additional invariants that involve $f^\prime_U$ and $f^\prime_D$.

[^10]: In the most general 2HDM after spontaneous symmetry breaking, the so-called Higgs basis in which $\hat{v}^a=(1\,,\,0)$ is an example of a quasi-physical basis. The Higgs fields in this basis, $H_1\equiv \hat{v}_a^*\Phi_a$ and $H_2\equiv \epsilon_{ba} \hat{v}_b \Phi_a$, are respectively invariant and pseudo-invariant with respect to U(2) basis transformations. See Refs. [@LS; @BS; @BLS; @DavHab; @Haber; @oneil] for details.

[^11]: In the SUSY basis, the overall phase of the Higgs fields can be chosen such that $v_U$ and $v_D$ are real and positive.

[^12]: The signs in have been conveniently chosen so that no extraneous signs appear in the SUSY limit \[cf. \].
