1 Introduction

The Two-Higgs Doublet Model (2HDM) [1,2,3] is one of the most popular extensions of the Higgs sector. It naturally stems in prominent beyond-the-Standard-Model (BSM) scenarios, such as grand unification [4, 5], supersymmetry [6] and axion models [7]. Besides, it has the potential to improve the performance of the conventional Higgs sector for important issues, e.g. the appealing possibility of Higgs-portal dark matter (see [8,9,10,11,12,13]). On top of all this, the 2HDM represents the simplest non-trivial extension of the SM Higgs sector, and thus a test bed of BSM physics, which could be probed in present and future experiments.

The scalar sector of the 2HDM consists of two scalar doublets, which leads to five physical scalar fields (\(h, H^0, H^\pm , A\)), unlike the sole SM Higgs boson. On the other hand, the current experimental data require that the properties of one of these fields (typically the lightest CP-even Higgs mass eigenstate, h), such as quantum numbers, mass and couplings, are equal or very similar to those of the SM Higgs boson. This is called the alignment limit [14,15,16,17,18,19], which is essentially imposed by the observation.

Apart from its theoretical appeal, the 2HDM has two potential fine-tuning problems. The first one is related to the electroweak (EW) breaking, when the magnitude of the VEV, \(v^2=(246~{\textrm{GeV}})^2\), is much smaller than the squared-mass terms entering the theory. The second one is related to the alignment condition. The main goal of this paper is to explore these fine-tunings, looking for regions of the parameter space where both are mild or irrelevant.

In this analysis we will adopt an “agnostic” point of view, in the sense that we will not consider additional symmetries which constrain the model (beyond the usual \({\mathbb {Z}}_2\) parity to avoid FCNC). There have been several analyses exploring this alternative direction, in particular to get an exact or approximate alignment [20,21,22,23,24,25,26,27]. As pointed out in Ref. [24], an approximate alignment can arise from a softly broken global symmetry of the scalar potential, but this requires to extend the Yukawa sector, e.g. with vector-like top quark partners.

In Sect. 2 we present the generic 2HDM, fixing the notation and providing analytical expressions for the Higgs VEVs and the alignment parameter. In Sect. 3 we explain the method to evaluate the fine-tunings and discuss the expected regions where they become severe. In Sect. 4 we provide analytical expressions for all the potential fine-tunings, both in terms of the initial 2HDM parameters and the physical ones (masses, mixing angles, etc.) We also explain how to remove the “double counting” of tunings by projecting the variations of the alignment onto the constant-\(v^2\) hypersurface. In addition, simple approximate expressions are also provided. The analysis shows that the electroweak and the alignment fine-tunings become severe in different regions of the parameter space. In Sect. 5 we illustrate all these trends by numerically analyzing several representative scenarios. This allows to identify the regions where both fine-tunings are acceptably mild or even irrelevant. Finally, in Sect. 6 we present our conclusions. The complete analytical expressions for all the fine-tunings are given in the Appendix.

2 The two-Higgs doublet model

Let us briefly review the general formulation of the 2HDM following the notation and conventions of Ref. [28]. Denoting by

$$\begin{aligned} \Phi _1=\left( \begin{array}{c} \Phi _1^+\\ \\ \Phi _1^0 \end{array}\right) ,\quad \Phi _2=\left( \begin{array}{c} \Phi _2^+\\ \\ \Phi _2^0 \end{array}\right) \end{aligned}$$
(1)

the two complex \(Y=1/2,\, SU(2)_L\) doublet scalar fields, the most general gauge-invariant renormalizable scalar potential is given by:

$$\begin{aligned} {V}= & {} m_{11}^2\Phi _1^\dagger \Phi _1+m_{22}^2\Phi _2^\dagger \Phi _2 -[m_{12}^2\Phi _1^\dagger \Phi _2+{\mathrm{h.c.}}]\nonumber \\{} & {} +\frac{1}{2}\,\lambda _1(\Phi _1^\dagger \Phi _1)^2 +\frac{1}{2}\,\lambda _2(\Phi _2^\dagger \Phi _2)^2\nonumber \\{} & {} +\lambda _3(\Phi _1^\dagger \Phi _1)(\Phi _2^\dagger \Phi _2) +\lambda _4(\Phi _1^\dagger \Phi _2)(\Phi _2^\dagger \Phi _1)\nonumber \\{} & {} +\Bigg \{\frac{1}{2}\,\lambda _5(\Phi _1^\dagger \Phi _2)^2 +\big [\lambda _6(\Phi _1^\dagger \Phi _1)\nonumber \\{} & {} +\lambda _7(\Phi _2^\dagger \Phi _2)\big ] \Phi _1^\dagger \Phi _2+{\mathrm{h.c.}}\Bigg \}, \end{aligned}$$
(2)

where the \(m_{11}^2,\, m_{22}^2\) mass terms and the \(\lambda _{1,2,3,4}\) couplings are real, while \(m_{12}^2\) and \(\lambda _{5,6,7}\) could be complex. The absence of flavour changing neutral currents (FCNCs) essentially requires each type of quark to couple to just one scalar doublet [29, 30]. This is accomplished by means of a softly-broken \({\mathbb {Z}}_2\) symmetry, which implies \(\lambda _6 = \lambda _7=0\), while a non-zero \(m_{12}^2\) value is still possible, see Table 1 below.

Furthermore, to avoid dangerous CP-violating phenomena, we will assume throughout the paper that \(m_{12}^2\), \(\lambda _5\) are real. In addition, stability bounds on the quartic coefficients imply (see e.g. [31])

$$\begin{aligned} \lambda _1, \lambda _2> & {} 0, \nonumber \\ \lambda _3> & {} - \sqrt{ \lambda _1 \lambda _2}, \nonumber \\ \lambda _3 +\lambda _4- |\lambda _5 |> & {} - \sqrt{ \lambda _1 \lambda _2}. \end{aligned}$$
(3)

We have also imposed constraints on the size of the various couplings from perturbativity and the requirement that unitarity is not violated in 2HDM scalar scattering processes, along the lines of Refs. [32,33,34].

As long as the Higgs mass matrix possesses at least one negative eigenvalue, the scalar fields develop non-zero vacuum expectation values (VEVs):

$$\begin{aligned} \langle \Phi _1 \rangle =\frac{1}{\sqrt{2}} \left( \begin{array}{c} 0\\ v_1\end{array}\right) , \quad \langle \Phi _2\rangle = \frac{1}{\sqrt{2}} \left( \begin{array}{c}0\\ v_2 \end{array}\right) , \end{aligned}$$
(4)

which must satisfy \(v_1^2+v_2^2=v^2\simeq (246~\text {GeV})^2\). As usual, we define the \(\beta \) angle such that

$$\begin{aligned} v_1 = v \cos \beta = v \, c_\beta ,\quad v_2 = v \sin \beta = v \, s_\beta . \end{aligned}$$
(5)

The vacuum is CP-conserving provided \(|m_{12}^2|\ge \lambda _5|v_1||v_2|\) [14]. Besides, the fields can be redefined so that \(v_1, v_2\ge 0\).

The minimization conditions \(\partial _{\Phi _i} {V}|_{(v_1,v_2)}=0\) read

$$\begin{aligned} 2 m_{11}^2 v_1 - 2 m_{12}^2 v_2 + v_1^3 \lambda _1 + v_1 v_2^2 \lambda _{345}= & {} 0, \end{aligned}$$
(6)
$$\begin{aligned} - 2 m_{12}^2 v_1 + 2 m_{22}^2 v_2 + v_2^3 \lambda _2 + v_1^2 v_2 \lambda _{345}= & {} 0, \end{aligned}$$
(7)

where \( \lambda _{345} = \lambda _3 + \lambda _4 + \lambda _5\).

A special instance takes place when either \(v_1=0\) or \(v_2=0\), i.e. \(t_\beta =\tan \beta =0,\infty \). This requires \(m_{12}^2=0\) and corresponds to the so-called inert-doublet model [8]. This case is protected by the symmetry \(\Phi _1\rightarrow \Phi _1\), \(\Phi _2\rightarrow -\Phi _2\), and represents a somehow trivial situation where one of the two Higgs fields plays exactly the SM-Higgs role while the other one does not couple to the rest of the SM fields.

For the non-trivial cases (\(t_\beta \ne 0,\infty \)), we can write the previous conditions in terms of v and \(t_\beta \),

$$\begin{aligned} {\mathscr {P}}_1\equiv & {} 2 m_{11}^2 (1+ t_\beta ^2) - 2 m_{12}^2 ( t_\beta + t_\beta ^3) \nonumber \\{} & {} + v^2 \left( \lambda _1 + t_\beta ^2 \lambda _{345} \right) = 0, \end{aligned}$$
(8)
$$\begin{aligned} {\mathscr {P}}_2\equiv & {} - 2 m_{12}^2 (1+ t_\beta ^2) + 2 m_{22}^2 ( t_\beta + t_\beta ^3) \nonumber \\{} & {} + v^2 \left( t_\beta ^3 \lambda _2 +t_\beta \lambda _{345} \right) = 0. \end{aligned}$$
(9)

Eliminating v from Eqs. (8) and (9), we get a quartic equation for \(t_\beta \):

$$\begin{aligned} {\mathscr {P}}_\beta\equiv & {} m_{12}^2 (-\lambda _1+ t_\beta ^4 \lambda _2) - m_{11}^2 \left( t_\beta ^3 \lambda _2 + t_\beta \lambda _{345} \right) \nonumber \\{} & {} + m_{22}^2 \left( t_\beta \lambda _1 +t_\beta ^3 \lambda _{345} \right) = 0, \end{aligned}$$
(10)

which allows the replacement of either \({\mathscr {P}}_1=0\) or \({\mathscr {P}}_2=0\) by \({\mathscr {P}}_\beta =0\) in the couple of minimization equations (8, 9).

The \(\beta \) angle introduced before relates the initial basis (in which the \({\mathbb {Z}}_2\)-symmetry is apparent) to the so-called Higgs-basis

$$\begin{aligned} H_1= & {} \left( \begin{array}{c} H_1^+\\ \\ H_1^0 \end{array}\right) \equiv \Phi _1 c_\beta +\Phi _2 s_\beta ,\quad \nonumber \\ H_2= & {} \left( \begin{array}{c} H_2^+\\ \\ H_2^0 \end{array}\right) \equiv -\Phi _1 s_\beta +\Phi _2 c_\beta , \end{aligned}$$
(11)

which satisfies \(\langle H_1^0\rangle =v/\sqrt{2}\), \(\langle H_2^0\rangle =0\). The associated quadratic (quartic) parameters of the Higgs scalar potential in this basis are denoted by \(Y_i\) (\(Z_j\)), where \(i\in \{1,2,3\}\) (\(j\in \{1,\dots , 7\}\)) (expressions for \(Y_i, Z_j\) in terms of the initial parameters, \(m_{ij}\), \(\lambda _j\) can be found in Ref. [28]).

From the original eight scalar degrees of freedom in the two Higgs doublets, three Goldstone bosons are absorbed by the electroweak bosons, \(W^\pm \) and Z, and the remaining degrees of freedom correspond to five physical Higgs particles: two CP-even scalars (h and H, with the convention \(m_h\le m_H\)), one CP-odd scalar (A) and one pair of charged Higgses (\(H^\pm \)).

The A and \(H^\pm \) fields stem directly from the above \(H_2\) doublet, namely \(A=\sqrt{2}\ {\textrm{Im}}(H_2^0)\), \(H^+=H_2^+\), \(H^-=(H_2^+)^\dagger =H_2^-\), with masses given by

$$\begin{aligned}{} & {} m_A^2=m_{12}^2\dfrac{1+t_\beta ^2}{t_\beta }-v^2 \lambda _5, \end{aligned}$$
(12)
$$\begin{aligned}{} & {} m_{H^\pm }^2=m_A^2+\dfrac{1}{2}v^2 (\lambda _5-\lambda _4). \end{aligned}$$
(13)

On the other hand, in the initial basis the CP-even neutral Higgs fields, \((\sqrt{2} {\textrm{Re}}\Phi _1^0-v_1), (\sqrt{2} {\textrm{Re}}\Phi _2^0-v_2)\) mix through the squared-mass matrix

$$\begin{aligned} {\mathcal {M}}^2\!=\!\left( \begin{array}{cc} m_{11}^2 \!+\!\frac{1}{2}v^2(3\lambda _1c_\beta ^2 \!+\! \lambda _{345}s_\beta ^2) &{} m_{12}^2\!+\!\lambda _{345}v^2s_\beta c_\beta \\ m_{12}^2\!+\!\lambda _{345}v^2s_\beta c_\beta &{} m_{22}^2 \!+\!\frac{1}{2}v^2(3\lambda _2s_\beta ^2 \!+\! \lambda _{345}c_\beta ^2) \end{array} \right) , \nonumber \\ \end{aligned}$$
(14)

which is diagonalized by

$$\begin{aligned} \left( \begin{array}{cc} m_H^2 &{} 0 \\ 0 &{}m_h^2\end{array}\right) = \left( \begin{array}{cc} c_\alpha &{} s_\alpha \\ -s_\alpha &{} c_\alpha \end{array}\right) \left( \begin{array}{cc} {\mathcal {M}}_{11}^2 &{} {\mathcal {M}}_{12}^2 \\ {\mathcal {M}}_{12}^2 &{}{\mathcal {M}}_{22}^2\end{array}\right) \left( \begin{array}{cc} c_\alpha &{} -s_\alpha \\ s_\alpha &{} c_\alpha \end{array}\right) \end{aligned}$$
(15)

(with the usual notation, \(c_\alpha =\cos \alpha \), \(s_\alpha =\sin \alpha \)).

Obviously, for the trivial cases \(s_\beta =0\) (\(c_\beta =0\)), i.e. \(t_\beta =0\) (\(\infty \)), the mass matrix (14) is diagonal from the beginning, since \(m_{12}^2=0\). Then the SM-Higgs corresponds to \(\Phi _1\) \((\Phi _2)\) and \(\alpha =-\pi /2\) (0) . Otherwise, \(\alpha \) is determined by the equation

$$\begin{aligned} {\mathscr {P}}_\alpha\equiv & {} m_{12}^2 (1+ t_\beta ^2) ( t_\alpha - t_\beta )(1 + t_\alpha t_\beta ) \nonumber \\{} & {} + v^2 t_\beta \left( t_\alpha ( - \lambda _1 + t_\beta ^2 \lambda _2) + t_\beta (1- t_\alpha ^2 ) \lambda _{345} \right) =0,\nonumber \\ \end{aligned}$$
(16)

where we have used Eqs. (8, 9) to eliminate \(m_{11}^2, m_{22}^2\) in terms of the derived quantities \(v^2\), \(\tan \beta \). The \(\alpha \) angle is defined modulo \(\pi \) and we have chosen the convention \(-\pi /2\le \alpha \le \pi /2\), thus \(c_\alpha \ge 0\). The mass eigenvalues (15) for the light and heavy neutral Higgses are then given by

$$\begin{aligned} m_H^2\!= & {} \!\frac{m_{12}^2 (t_\alpha \!-\! t_\beta )^2 (1\! +\! t_\beta ^2) \!+\! t_\beta v^2 (\lambda _1 \!+\! t_\alpha t_\beta (t_\alpha t_\beta \lambda _2\!+\! 2 \lambda _{345}))}{(1 \!+\! t_\alpha ^2)t_\beta (1 \!+\! t_\beta ^2)},\nonumber \\ \end{aligned}$$
(17)
$$\begin{aligned} m_h^2\!= & {} \!\frac{m_{12}^2(1 \!+\!t_\alpha t_\beta )^2 (1 \!+\!t_\beta ^2) \!+\! t_\beta v^2 (t_\alpha ^2 \lambda _1\!+\! t_\beta ^2 \lambda _2 \!-\! 2 t_\alpha t_\beta \lambda _{345})}{(1 \!+\! t_\alpha ^2)t_\beta (1 \!+\! t_\beta ^2)},\nonumber \\ \end{aligned}$$
(18)

corresponding to the physical mass-eigenstates

$$\begin{aligned} H= & {} (\sqrt{2} Re\ \Phi _1^0-v_1) c_\alpha +(\sqrt{2} Re\ \Phi _2^0-v_2) s_\alpha , \end{aligned}$$
(19)
$$\begin{aligned} h= & {} -(\sqrt{2} Re\ \Phi _1^0-v_1) s_\alpha +(\sqrt{2} Re\ \Phi _2^0-v_2) c_\alpha . \end{aligned}$$
(20)

This is usually called the physical basis, which is related to the Higgs-basis (11) by a \(\beta -\alpha \) rotation:

$$\begin{aligned} H= & {} c_{\beta -\alpha }(\sqrt{2} Re\ H_1^0-v) -s_{\beta -\alpha }(\sqrt{2} Re\ H_2^0) , \end{aligned}$$
(21)
$$\begin{aligned} h= & {} s_{\beta -\alpha }(\sqrt{2} Re\ H_1^0-v) +c_{\beta -\alpha }(\sqrt{2} Re\ H_2^0) . \end{aligned}$$
(22)

The couplings of these Higgses to the gauge bosons, \(V=W^{\pm },\,Z\), are usually parametrized by the coefficients \(C_V^h=\sin (\beta -\alpha )=s_{\beta -\alpha }\), \(C_V^H=\cos (\beta -\alpha )=c_{\beta -\alpha }\), which relate the former to the SM Higgs couplings

$$\begin{aligned} g_{hVV}=C_V^h\ g_{h_{SM} VV},\quad g_{HVV}=C_V^H\ g_{h_{SM} VV}, \end{aligned}$$
(23)

where \(h_{SM}\) is the SM Higgs.

Similarly, the couplings of these Higgses to the fermions are parametrized by the \(C_F^h,\, C_F^H\) coefficients, which in turn depend on the initial couplings of the two \(\Phi _1, \Phi _2\) doublets to the fermions. The latter are strongly restricted by the requirement of the absence of dangerous FCNCs. As mentioned above, this is guaranteed by imposing a softly-broken \({\mathbb {Z}}_2\) symmetry, which affects both the Higgs doublets and the fermion fields. The possible \({\mathbb {Z}}_2\) charge assignments lead to the well-known four types of 2HDM, shown in Table 1 [35].

Table 1 \({\mathbb {Z}}_2\) charge assignments that forbid tree-level Higgs-mediated FCNC effects
Full size table

Then, the corresponding \(C_V\) and \(C_F\) coefficients are given in Table 2 for the four types of 2HDM.

Table 2 Tree-level vector boson couplings, \(C_V\), and fermionic couplings, \(C_F\), normalised to the SM-Higgs couplings for the h, H, A fields in the four 2HDM types; \(s_a, c_a\) stand for \(\sin a, \cos a\)
Full size table

2.1 Radiative corrections

Let us briefly comment the impact of radiative corrections in the previous description of the 2HDM. The effective potential (2) receives the usual radiative corrections [36], which involve contributions of all the particles coupled to the Higgs fields. A useful way to handle them is to consider a leading-log expansion. As has it has been shown in Refs. [37,38,39], the N-loop effective potential improved by (N+1)-loop renormalization-group equations resumes all Nth-to-leading logarithm contributions. In particular, the tree-level potential (2), with all parameters (\(m_{ij}, \lambda _k\)) understood as (1-loop) running parameters at a certain renormalization scale \(\mu \), resumes all 1-loop logarithmic radiative corrections.

Obviously, such expression is to be corrected by next-to-leading log contributions. These are expected to be small, provided one chooses a judicious \(\mu \) scale. A sensible choice is to take \(\mu \) of the order of the largest tree-level mass with sizeable dependence on the scalar fields [37,38,39,40]. In our case, this means \(\mu \sim {\mathcal {O}} (m_H, m_A, m_{H^\pm })\). Therefore, all the previous expressions, in particular the minimization equations (8, 9) should be understood at that scale.

The physical parameter that is most affected by the radiative corrections is, by far, the mass of the lightest Higgs boson, \(m_h\). This should be run from the high-scale, \(\sim m_H\) till the low scale \(\sim m_{\textrm{top}}\). This amounts a correction \(\Delta m_h^2\propto \log (m_H/m_{\textrm{top}})\), which for \(m_H=1\) TeV is \(\sim (69~{\textrm{GeV}})^2\) [41, 42]. In other words, for \(m_H=1\) TeV, the value of \(m_h\), given in Eq. (18) should be 104 GeV, rather than 125 GeV.

We will comment later on the (mild) impact of this remark on the size of the fine-tuning.

2.2 The alignment limit

This limit occurs when the light CP-even Higgs, h, behaves as the SM Higgs, i.e. it presents the same couplings as the latter to all the SM fields. From the couplings of h to vector bosons and fermions shown in Table 2, it is clear that this is achieved for

$$\begin{aligned} \text {Alignment limit:}\quad \quad c_{\beta -\alpha }\rightarrow 0. \end{aligned}$$
(24)

Since \(0\le \beta \le \pi /2\), \(-\pi /2\le \alpha \le \pi /2\), this is accomplished for \(\alpha =\beta -\pi /2\). In this limit the Higgs-basis and the physical basis are equivalent. Namely, from Eq. (22), h becomes aligned with \((\sqrt{2} Re\ H_1^0-v)\) and \(H_1\) plays the role of the ordinary SM doublet. Note also that in this limit the first term in the numerator of Eq. (18) vanishes, so \(m_h^2={\mathcal {O}}(\lambda v^2)\), as in the SM.

Up to now, the Higgs boson observed in the LHC looks remarkably similar to the SM one [43]. This is why the 2HDM is usually considered at the (either exact or approximate) alignment limit [16]. More precisely, ATLAS [44] and CMS [45] Run 2 Higgs data allow to determine \((C_V^h, C_F^h)\) within a \(\sim 10\%\) error (reduced to \(\sim 7\%\) when combining both analyses). These bounds are more restrictive when evaluated in a given model since \(C_F\)’s are in general non-universal and correlated with \(C_V\). In the 2HDM, these correlations are given in Table 2. In particular, for type I, deviations from \(C_V^h=1\) are \(\sim 1\%\) and [46, 47]

$$\begin{aligned} |c_{\beta -\alpha }| \lesssim 0.15. \end{aligned}$$
(25)

The range is even more reduced in type II.

It is worth commenting that there are symmetries of the potential [48,49,50,51] that lead to an exact alignment, i.e. \(c_{\beta -\alpha }=0\). A rather trivial instance is the above-mentioned inert-doublet scenario, where \(t_\beta =0,\infty \) and there is an exact \(\Phi _1\rightarrow \Phi _1\), \(\Phi _2\rightarrow -\Phi _2\) symmetry, so that one of the Higgs doublets precisely corresponds to the SM-Higgs, coupled to all fermions, while the other one is completely decoupled.

Beside this special case, the existence of symmetries that lead to alignment can be explored by eliminating \(t_\beta \) in \(\{ {\mathscr {P}}_\alpha \ , {\mathscr {P}}_\beta \}=0 \) evaluated at the alignment limit, \(t_\alpha = - 1/{t_\beta } \); namely

$$\begin{aligned} t_\beta = \sqrt{\frac{\lambda _1-\lambda _{345}}{\lambda _2-\lambda _{345}}}, \end{aligned}$$
(26)

subject to the consistency condition

$$\begin{aligned}{} & {} \left( \lambda _1 \lambda _2 -\lambda _{345}^2 \right) \left( m_{12}^2 ( \lambda _1 - \lambda _2) \right. \nonumber \\{} & {} \quad \left. + (m_{11}^2 -m_{22}^2) \sqrt{(\lambda _1 - \lambda _{345} ) (\lambda _2 - \lambda _{345} )} \right) = 0. \end{aligned}$$
(27)

Whenever Eq. (27) is fulfilled by the initial parameters of the potential, there exists a minimum with \(t_\beta \) given by Eq. (26) where the alignment is exact. Notice, in particular, that Eq. (27) is satisfied for some simple relations between the potential parameters, such as \(\{ \lambda _1=\lambda _2 \; \hbox {and} \; m_{11}^2 =m^2_{22} \} \), which can be achieved by imposing certain global symmetries in the potential [20, 22, 24, 25]. However, for phenomenological reasons these symmetries cannot be exact. Still, an approximate alignment can arise if the symmetry is softly broken in the scalar sector. Nevertheless, as pointed out in Ref. [24], this requires to extend the Yukawa sector, e.g. with vector-like top quark partners coupled to the Higgses, which goes beyond the scope of this paper.

In any case, even if the parameters are in a suitable combination to get exact or approximate alignment, if the underlying symmetry is not exact one must vary the parameters in a free way to evaluate the fine-tuning. This is exactly what we have done in the present analysis, which we expose below, so these possibilities are taken into account.

3 Fine-tuning in the 2HDM: decoupling and non-decoupling regimes

A theoretical model presents fine-tuning (or, equivalently, absence of naturalness) when some observable quantity depends critically on a fine adjustment (or “conspiracy”) of the fundamental parameters. Such adjustment in the parameter space is conceptually problematic, as it is implausible unless it can be explained from the theory itself.

Regarding the 2HDM, we distinguish two potential fine-tunings. The first one is related to the EW breaking, when the magnitude of the vacuum expectation value, \(v^2\), is much smaller than the other squared-mass terms entering the theory. Namely, from Eq. (8) or Eq. (9):

$$\begin{aligned} v^2\backsim \frac{\sum {\mathcal {O}} (m_{ij}^2)\text {-terms}}{{\mathcal {O}} (\lambda _k)}. \end{aligned}$$
(28)

Since \(\lambda _i{\mathop {{}_\sim }\limits ^{<}}1\), if some \(m_{ij}^2\gg v^2\) (as required for \(m_H^2\gg v^2\)), then \(v^2\) is likely to be fine-tuned. This is sometimes called the “little hierarchy problem” [52]. Given that in the 2HDM there are two expectations values, \(v_1\) and \(v_2\), or equivalently \(v^2\), \(\tan \beta \); one can wonder if there are two independent fine-tunings, i.e. if \(\tan \beta \) might also be a fine-tuned parameter. We will examine this issue throughout the paper. Let us note that the mass of the SM-like Higgs, \(m_h\), has electroweak size, however it is not a fine-tuned parameter since, as mentioned above, \(m_h^2={\mathcal {O}}(\lambda v^2)\), as in the SM. Therefore, once the value of \(v^2\) has been set, \(m_h^2\), similarly to \(M_W^2, M_Z^2\), is naturally of electroweak size.

The second potential fine-tuning is related to the magnitude of \(c_{\beta -\alpha }\) in the alignment regime, i.e. \(|c_{\beta -\alpha }|\ll 1\), since, in principle, there is no reason why the initial parameters should yield such small value. To see this more closely, let us use the expression given in Ref. [28] that links \(c_{\beta -\alpha }\) with \(m_h^2\), \(m_H^2\) and the \(Z_{1,6}\) parameters:

$$\begin{aligned} c_{\beta -\alpha }=\frac{-Z_6 v^2}{\sqrt{(m_H^2-m_h^2)(m_H^2-Z_1 v^2)}}, \end{aligned}$$
(29)

where

$$\begin{aligned} Z_6= & {} -\frac{1}{2}\, s_{2\beta } \left[ \lambda _1 c_{\beta }^2 -\lambda _2 s_{\beta }^2 - \lambda _{345} c_{2\beta } \right] ,\nonumber \\ Z_1= & {} \lambda _1 c_\beta ^4 + \lambda _2 s_\beta ^4 +\frac{1}{2}\,\lambda _{345}s_{2\beta }^2. \end{aligned}$$
(30)

Demanding an approximate (exact) alignment limit is equivalent to demand a small (vanishing) value of \(|c_{\beta -\alpha }|\). This can be achieved in two ways:

  1. 1.

    \(m_H^2\gg v^2\). Since \(Z_{1,6}=\sum _i {\mathcal {O}} (\lambda _i) {\mathop {{}_\sim }\limits ^{<}}1\), the denominator of Eq. (29) becomes in this case much larger than the numerator, leading to a small \(|c_{\beta -\alpha }|\). This instance is called in the literature alignment by decoupling [14, 17, 19]. It is worth-noticing that this limit requires large \(m_{12}^2\gg v^2\) (see Eq. (17)). Then, from Eqs. (12, 13, 17), all the extra Higgs states get similar masses:

    $$\begin{aligned} m_H^2\simeq m_A^2\simeq m_{H^\pm }^2\simeq \dfrac{1+t_\beta ^2}{t_\beta }m_{12}^2. \end{aligned}$$
    (31)

    Actually, in this regime one can integrate out the heavy Higgs states and the resultant effective theory is essentially the SM (more precisely, an SMEFT), with h playing the role of the SM-like Higgs boson. This may seem quite natural, but, as discussed above, it implies a potential fine-tuning related to the EW breaking.

  2. 2.

    \(Z_6\ll 1\). Then, \(|c_{\beta -\alpha }|\ll 1\) without requiring large Higgs masses. This regime is commonly called alignment without decoupling [16, 18, 53]. However, as it is clear from Eq. (30), this typically requires a precise cancellation inside \(Z_{6}=\sum _i {\mathcal {O}} (\lambda _i) \). A possible exception, however, occurs when \(s_{2\beta }\simeq 0\), i.e. for \(t_\beta \gg 1\). Note here that the \(t_\beta \ll 1\) case is excluded because it leads to a non-perturbative Yukawa coupling for the top quark (which, by definition, is coupled to the \(\Phi _2\) doublet).

Hence, the alignment generically requires fine-tuning, whether it is achieved by decoupling or not. Thus, evaluating both potential fine-tunings in detail may lead to a better understanding of the parameter space, and to find regions in which none of them is too large. This is the main goal of this paper.

In order to quantify the fine-tuning associated with a generic observable, \(\Omega \), with respect to an initial parameter, \(\theta _i\), we will use the somewhat standard criterion proposed by Ellis et al. [54] and Barbieri and Giudice [55]. Namely, we define the fine-tuning parameters as

$$\begin{aligned} \Delta _{\theta _i}\Omega =\frac{d \ln \Omega }{d \ln \theta _i }=\frac{\theta _i }{\Omega }\frac{d \Omega }{d \theta _i },\quad \Delta \Omega \equiv \max |\Delta _{\theta _i}\Omega |. \end{aligned}$$
(32)

Then \(\Delta \Omega \sim 10\) (100) denotes a fine-tuning of about \(10\%\) \((1\%)\), etc. [55].

As discussed above, the set of (potentially fine-tuned) observables that we will consider isFootnote 1

$$\begin{aligned} \Omega = \{v^2,t_\beta , c_{\beta -\alpha } \}, \end{aligned}$$
(33)

while the set of initial parameters is

$$\begin{aligned} \theta _i = \{ m_{11}^2 , m_{22}^2 , m_{12}^2 , \lambda _1, \lambda _2, \lambda _3, \lambda _4, \lambda _5 \}. \end{aligned}$$
(34)

Our aim is to find analytical expressions for \(\Delta _{\theta _i} v^2\), \(\Delta _{\theta _i} t_\beta \) and \(\Delta _{\theta _i} c_{\beta -\alpha }\) in terms of the initial parameters or (most conveniently) the physical parameters, and discuss their magnitude in different regimes.

4 Analytical expressions for the fine-tuning

4.1 Fine-tuning in terms of the initial parameters

The most straightforward way to get analytical expressions for the fine-tuning on any observable, \(\Omega \), is to start with its explicit dependence on the initial parameters, \(\Omega (\theta _i)\), and evaluate the derivatives \(d \Omega /{d \theta _i }\) involved in Eq. (32). However, in our case these kinds of expressions are cumbersome; e.g. \(t_\beta \) is given by the quartic equation (10). Fortunately, we can still extract analytical expressions for the fine-tuning using the constraints \({\mathscr {P}}_{1,2,\beta ,\alpha }=0\) (see Eqs. (8, 9, 10, 16)), together with the Implicit Function Theorem.

Next we do this for the three potentially fine-tuned observables, \(v^2\), \(t_\beta \), \(c_{\beta -\alpha }\).

4.1.1 \(\underline{\text {Fine-tuning in }t_\beta }\)

For convenience, we start with the fine-tuning in \(t_\beta \). Taking derivatives in Eq. (10),

$$\begin{aligned} \frac{d {\mathscr {P}}_\beta }{d\theta _i} = 0\implies \frac{\partial {\mathscr {P}}_\beta }{\partial \theta _i }+\frac{\partial {\mathscr {P}}_\beta }{\partial t_\beta }\frac{\partial t_\beta }{\partial \theta _i }=0, \end{aligned}$$
(35)

so

$$\begin{aligned} \Delta _{\theta _i}t_\beta =-\frac{\theta _i}{t_\beta }\frac{\partial {\mathscr {P}}_\beta }{\partial \theta _i}/\frac{\partial {\mathscr {P}}_\beta }{\partial t_\beta } . \end{aligned}$$
(36)

Hence, replacing here the explicit expression (10) for \({\mathscr {P}}_\beta \) we get analytical formulas for all \(\Delta _{\theta _i}t_\beta \).

The above expression is enough to see that, in general, \(\Delta _{\theta _i}t_\beta {\mathop {{}_\sim }\limits ^{<}}{\mathcal {O}} (1)\), and thus \(t_\beta \) is not a fine-tuned parameter. To check this, note that the size of the denominator

$$\begin{aligned} \frac{\partial {\mathscr {P}}_\beta }{\partial t_\beta }= & {} m_{12}^2(4t_\beta ^3\lambda _2) - m_{11}^2(3t_\beta ^2\lambda _2+\lambda _{345}) \nonumber \\{} & {} + m_{22}^2(\lambda _1 + 3t_\beta ^2\lambda _{345}) \end{aligned}$$
(37)

is typically \({\mathop {{}_\sim }\limits ^{>}}{\mathcal {O}}(m_{ij}^2)\), barring accidental cancellations; and this is also the expected size of the numerator for any \(\theta _i\). E.g. let \(\theta _i\) be one of the initial mass-parameters (the ones more directly involved in the electroweak fine-tuning), say \(\theta =m_{12}^2\). Then Eq. (36) reads

$$\begin{aligned}{} & {} \Delta _{m_{12}^2}t_\beta \nonumber \\{} & {} \quad =\!-\frac{m_{12}^2(-\lambda _1\!+\!t_\beta ^4\lambda _2)}{m_{12}^2(4t_\beta ^4\lambda _2) \!-\! m_{11}^2(3t_\beta ^3\lambda _2\!+\!t_\beta \lambda _{345}) \!+\! m_{22}^2(t_\beta \lambda _1 \!+\! 3t_\beta ^3\lambda _{345})},\nonumber \\ \end{aligned}$$
(38)

which is typically \({\mathop {{}_\sim }\limits ^{<}}{\mathcal {O}}(1)\). This is obviously the case for \(\lambda _i, t_\beta = {\mathcal {O}}(1)\), and also for \(t_\beta \gg 1\). (In the last instance \(|\Delta _{m_{12}^2}t_\beta | \sim 1/4\).)

The fact that \(t_\beta \) is not fine-tuned in most cases can be traced back to the origin of this observable. If \(m_{11}^2>0\), the VEV of \(\Phi _1\) is triggered by that of \(\Phi _2\) through the linear term in the potential (2),

$$\begin{aligned} v_1 \sim \frac{m_{12}^2}{m_{11}^2}v_2. \end{aligned}$$
(39)

Hence, \(v_1\) can have a similar size as \(v_2\) with no need of tunings or cancellations (whether or not \(v_2\) has electroweak size); thus \(t_\beta \) can be \({\mathcal {O}}(1)\), or actually any size, with no fine-tuning. Note that a very large (or small) \(t_\beta \) implies a strong hierarchy between \(m_{11}^2\) and \(m_{12}^2\) . This may be considered as an odd fact (as it is e.g. the hierarchy of masses of the fermionic generations), but it is not a fine-tuning.

4.1.2 \(\underline{\text {Fine-tuning in }v^2}\)

For the fine-tuning in \(v^2\), we proceed in a similar way. From Eqs. (8, 9), we get

$$\begin{aligned} \dfrac{d {\mathscr {P}}_{1, 2} }{d\theta _i}= & {} 0\implies \dfrac{\partial {\mathscr {P}}_{1, 2} }{\partial \theta _i }+\dfrac{\partial {\mathscr {P}}_{1, 2}}{\partial t_\beta }\dfrac{\partial t_\beta }{\partial \theta _i } \nonumber \\{} & {} +\dfrac{\partial {\mathscr {P}}_{1, 2}}{\partial v^2}\dfrac{\partial v^2 }{\partial \theta _i }=0, \end{aligned}$$
(40)

which leads to:

$$\begin{aligned} \Delta _{\theta _i}v^2 =-\frac{\theta _i}{v^2}\left( \frac{\partial {\mathscr {P}}_{1, 2}}{\partial \theta _i}+ \frac{t_\beta }{\theta _i}\frac{\partial {\mathscr {P}}_{1, 2}}{\partial t_\beta }\Delta _{\theta _i}t_\beta \right) /\frac{\partial {\mathscr {P}}_{1, 2}}{\partial v^2}. \end{aligned}$$
(41)

Thus, replacing the explicit expressions (8, 9) for \({\mathscr {P}}_{1, 2}\), we get two alternative but equivalent analytical forms for each \(\Delta _{\theta _i}v^2\).

Now it is easy to see that \(v^2\) is generically fine-tuned with respect to the initial mass-parameters. E.g. for \(\theta _i=m_{11}\), we replace

$$\begin{aligned}{} & {} \frac{\partial {\mathscr {P}}_1}{\partial t_\beta }=4t_\beta m_{11}^2 -2(1+3t_\beta ^2)m_{12}^2 +2t_\beta \lambda _{345}v^2, \end{aligned}$$
(42)
$$\begin{aligned}{} & {} \frac{\partial {\mathscr {P}}_1}{\partial v^2}=\lambda _1 +t_\beta ^2\lambda _{345}, \end{aligned}$$
(43)
$$\begin{aligned}{} & {} \frac{\partial {\mathscr {P}}_1}{\partial m_{11}^2}=2(1+t_\beta ^2), \end{aligned}$$
(44)

into Eq. (41), getting

$$\begin{aligned}{} & {} \Delta _{m_{11}^2}v^2 =-\frac{m_{12}^2}{v^2} \frac{1}{\lambda _1+t_\beta ^2 \lambda _{345}} \Bigg [ 2(1+t_\beta ^2)+\frac{t_\beta }{m_{11}^2}\nonumber \\{} & {} \quad \left( 4t_\beta m_{11}^2-2 (1+3t_\beta ^2)m_{12}^2 +2t_\beta \lambda _{345}v^2\right) \Delta _{m_{11}^2}t_\beta \Bigg ] , \end{aligned}$$
(45)

which is typically \({\mathop {{}_\sim }\limits ^{>}}{\mathcal {O}}(m_{ij}^2/v^2)\). Hence, \(\Delta _{\theta }v^2\) captures the whole fine-tuning associated to the scale of the EW breaking (the little hierarchy problem).

4.1.3 \(\underline{\text {Fine-tuning in }c_{\beta -\alpha }}\)

Let us finally consider the fine-tuning in the alignment, i.e. \(\Delta _{\theta _i} c_{\beta -\alpha }\). Using the trigonometric identity

$$\begin{aligned} c_{\beta -\alpha }=\frac{1+t_\alpha t_\beta }{\sqrt{(1+t_\beta ^2)(1+t_\alpha ^2)}} \end{aligned}$$
(46)

in Eq. (32) (for \(\Omega = c_{\beta -\alpha }\)) we can express the fine-tuning in \(c_{\beta -\alpha } \) as

$$\begin{aligned} \Delta _{\theta _i} c_{\beta -\alpha } = \frac{1}{c_{\beta -\alpha }}\left( t_\alpha \frac{\partial c_{\beta -\alpha }}{\partial t_\alpha }\Delta _{\theta _i}t_\alpha +t_\beta \frac{\partial c_{\beta -\alpha }}{\partial t_\beta }\Delta _{\theta _i}t_\beta \right) , \end{aligned}$$
(47)

where \(\Delta _{\theta _i}t_\alpha \equiv \dfrac{\theta _i }{t_\alpha }\dfrac{d t_\alpha }{d \theta _i }\) is the “fine-tuning in \(\alpha \)”, which can be obtained from Eq. (16) following similar steps as for the other fine-tunings:

$$\begin{aligned} \dfrac{d {\mathscr {P}}_\alpha }{d\theta _i}= & {} 0\implies \dfrac{\partial {\mathscr {P}}_\alpha }{\partial \theta _i }+\dfrac{\partial {\mathscr {P}}_\alpha }{\partial t_\beta }\dfrac{\partial t_\beta }{\partial \theta _i }\nonumber \\{} & {} +\dfrac{\partial {\mathscr {P}}_\alpha }{\partial v^2}\dfrac{\partial v^2}{\partial \theta _i }+\dfrac{\partial {\mathscr {P}}_\alpha }{\partial t_\alpha }\dfrac{\partial t_\alpha }{\partial \theta _i }=0. \end{aligned}$$
(48)

Hence,

$$\begin{aligned} \Delta _{\theta _i} t_\alpha= & {} - \frac{\theta _i}{t_\alpha }\left( \frac{\partial {\mathscr {P}}_\alpha }{\partial \theta _i}+ \frac{t_\beta }{\theta _i}\frac{\partial {\mathscr {P}}_\alpha }{\partial t_\beta }\Delta _{\theta _i}t_\beta \right. \nonumber \\{} & {} \left. + \frac{v^2}{\theta _i}\frac{\partial {\mathscr {P}}_\alpha }{\partial v^2 }\Delta _{\theta _i}v^2\right) /\frac{\partial {\mathscr {P}}_\alpha }{\partial t_\alpha }. \end{aligned}$$
(49)

Replacing here the explicit expression (16) for \({\mathscr {P}}_\alpha \), beside Eqs. (36, 41), we get analytical formulas for all \(\Delta _{\theta _i}t_\alpha \). Finally, the alignment fine-tuning reads

$$\begin{aligned} \Delta _{\theta _i} c_{\beta -\alpha }=-\frac{t_\alpha -t_\beta }{1+t_\alpha t_\beta }\left( \frac{t_\alpha }{1+t_\alpha ^2}\Delta _{\theta _i}t_\alpha -\frac{t_\beta }{1+t_\beta ^2}\Delta _{\theta _i}t_\beta \right) , \end{aligned}$$
(50)

which, upon replacement of Eqs. (36, 49), has an explicit analytical expression for all \({\theta _i}\). Note that, in the alignment limit, \(1+t_\alpha t_\beta \rightarrow 0\), thus we expect \(c_{\beta -\alpha }\) to be generically fine-tuned.

It is worth-noticing that the term proportional to \(\Delta _{\theta _i}t_\alpha \) has “inside” the electroweak fine-tuning, \(\Delta _{\theta _i}v^2\), through Eq. (49). Actually, it can be checked from the previous expressions that for \(m_{ij}^2\gg v^2\) (decoupling limit) and \(1+t_\alpha t_\beta \rightarrow 0\) (alignment limit) the coefficient of \(\Delta _{\theta _i} v^2\) is 1:

$$\begin{aligned} \Delta _{\theta _i} c_{\beta -\alpha }\ \supset \ \Delta _{\theta _i}v^2. \end{aligned}$$
(51)

This remarkably simple result can be understood as follows. Let us recall from Eq. (29) that

$$\begin{aligned} c_{\beta -\alpha }=\frac{-Z_6 v^2}{\sqrt{(m_H^2-m_h^2)(m_H^2-Z_1 v^2)}}. \end{aligned}$$
(52)

Hence, in absolute value,

$$\begin{aligned} \Delta _{\theta _i}c_{\beta -\alpha }= & {} \frac{d\ \log c_{\beta -\alpha }}{d\ \log \theta _i} \nonumber \\= & {} \frac{d}{d\ \log \theta _i}\left[ \log v^2 + \log Z_6 \right. \nonumber \\{} & {} \left. - \frac{1}{2}\log ((m_H^2-m_h^2)(m_H^2-Z_1 v^2))\right] . \end{aligned}$$
(53)

The first term is exactly \(\Delta _{\theta _i} v^2\), the second one represents the fine-tuning in \(c_{\beta -\alpha }\) when there is no decoupling and the third term is \({\mathcal {O}}(1)\) since \((m_H^2-m_h^2)(m_H^2-Z_1 v^2) \simeq m_H^4\), which is not a fine-tuned quantity.

Equations (52, 53) show in a transparent way why the electroweak fine-tuning, \(\Delta v^2\), appears involved in the alignment one. The definition of fine-tuning takes into account all the adjustments required to get the desired value of the observable under consideration. In our case, if the alignment is achieved through decoupling, there is a fine-tuning price, namely the one required to get \(v^2\) much smaller than the mass parameters of the potential.

From the above discussion it is clear that the fine-tuning in \(v^2\) and \(c_{\beta -\alpha }\) are not independent. Thus, it would be incorrect to multiply them to get the “total” fine-tuning. In particular, in the decoupling limit, once the initial parameters have been adjusted to get the right size of \(v^2\), there is no need of extra adjustments to get alignment. Hence, a practical way to remove the “double counting” in the evaluation of the \(c_{\beta -\alpha }\) fine-tuning is to discard variations in the \(\theta _i\) parameters that change the value of \(v^2\). In the \(\{\log \theta _i\}\) space, this is equivalent to project the gradient \(\Delta _{\theta _i} c_{\beta -\alpha }=\partial \log c_{\beta -\alpha }/\partial \log \theta _i\) onto the constant-\(v^2\) hypersurface [56]:

$$\begin{aligned} \Delta _{\theta _i} c_{\beta -\alpha } \rightarrow \Delta _{\theta _i} c_{\beta -\alpha }-\left( \sum _j \Delta _{\theta _j} c_{\beta -\alpha }\cdot n_j\right) n_i \end{aligned}$$
(54)

where \(n_i=\Delta _{\theta _i} v^2/\sqrt{\sum _j(\Delta _{\theta _j} v^2)^2}\) is the unitary vector normal to the constant-\(v^2\) hypersurface. Obviously, in doing so, all the \(\Delta _{\theta _i} v^2\) pieces in Eq. (50) are removed.

4.2 Fine-tuning in terms of the physical parameters

Very often it is convenient and more meaningful to express the fine-tuning in terms of the physical parameters of the model, rather than in terms of the initial ones. To this end, we have to trade the eight initial parameters, \(\theta _i\), of Eq. (34) for eight physical observables, \(\chi _i\). Specifically, we have chosen

$$\begin{aligned} \chi _i = \{v, t_\beta , t_\alpha , m_h, m_H, m_A, m_{H^\pm } , \lambda _5\}. \end{aligned}$$
(55)

Although \(\lambda _5\) is not strictly an observable, it is related to the triple scalar self-couplings, e.g.

$$\begin{aligned} g_{Hhh}=\frac{m_H^2+2m_h^2-4(m_A^2+v^2\lambda _5)}{v}\ c_{\beta -\alpha }+{\mathcal {O}} (c_{\beta -\alpha }^2). \end{aligned}$$
(56)

Note that the eight physical quantities of Eq. (55) are not input parameters in a strict sense (unlike the eight parameters of Eq. (34)), but they are equally useful to label a particular model, as they are in one-to-one correspondence with them (see Eq. (57) below). Of course, the fine-tuning has to be evaluated with respect to the initial parameters (34).

Now the fine-tunings \(\Delta _{\theta _i} t_\beta , \Delta _{\theta _i} v^2, \Delta _{\theta _i} c_{\beta -\alpha }\), given by Eqs. (36, 41, 50), can be written as functions of the physical parameters, \(\chi _i\), by replacing the initial parameters in terms of the latter, i.e. \(\theta _i(\chi _j)\). This can be done with the help of the relations obtained in Sect. 2. Namely, from Eqs. (12, 13) we derive \(m_{12}^2(\chi _i)\), \(\lambda _4(\chi _i)\). Then, from Eqs. (16, 17, 18) we derive \(\lambda _{1,2,3}(\chi _i)\); and finally, from the minimization conditions (8, 9) we get \(m_{11}^2(\chi _i)\), \(m_{22}^2(\chi _i)\). Altogether, the explicit expressions for \(\theta _i(\chi _j)\) read

$$\begin{aligned} m_{11}^2= & {} \frac{m_h^2t_\alpha (t_\beta - t_\alpha )}{2(1 \!+\! t_\alpha ^2)} \!-\! \frac{m_H^2(1\!+\! t_\alpha t_\beta )}{2(1 \!+\! t_\alpha ^2)} \!+\! \frac{t_\beta ^2}{1 + t_\beta ^2} (m_A^2 \!+\! v^2 \lambda _5), \nonumber \\ m_{22}^2= & {} \frac{m_h^2(t_\alpha -t_\beta )}{2 t_\beta (1 \!+\! t_\alpha ^2)} \!-\! \frac{m_H^2t_\alpha (1+ t_\alpha t_\beta )}{2t_\beta (1 \!+\! t_\alpha ^2)} \!+\! \frac{1}{1 \!+\! t_\beta ^2} (m_A^2 \!+\! v^2 \lambda _5), \nonumber \\ m_{12}^2= & {} \frac{t_\beta }{1 + t_\beta ^2} (m_A^2 + v^2 \lambda _5), \nonumber \\ \lambda _1= & {} \frac{m_h^2t_\alpha ^2+m_H^2}{v^2} \frac{1+t_\beta ^2}{1 + t_\alpha ^2} - t_\beta ^2 \frac{m_A^2 + v^2 \lambda _5}{v^2}, \nonumber \\ \lambda _2= & {} \frac{m_h^2+ m_H^2 t_\alpha ^2 }{v^2} \frac{1+t_\beta ^2}{(1 + t_\alpha ^2)t_\beta ^2} - \frac{m_A^2 + v^2 \lambda _5}{t_\beta ^{2} v^2}, \nonumber \\ \lambda _3= & {} \frac{2 m_{H^\pm }^2 - m_A^2}{v^2} - \frac{m_h^2- m_H^2}{v^2} \frac{ t_\alpha (1+ t_\beta ^2 )}{t_\beta (1 + t_\alpha ^2)} - \lambda _5, \nonumber \\ \lambda _4= & {} \frac{2 ( m_A^2 - m_{H^\pm }^2)}{v^2} + \lambda _5, \nonumber \\ \lambda _5= & {} \lambda _5. \end{aligned}$$
(57)

Incidentally, the change of variables from the initial parameters to the physical observables corresponds to a remarkably simple Jacobian,

$$\begin{aligned} J = {\frac{32 m_h^3 m_H^3 (m_H^2 -m_h^2 )m_A m_{H^\pm } (1 + t_\beta ^2)^2}{v^9 t_\beta ^3 (1 + t_\alpha ^2)}}. \end{aligned}$$
(58)

We present the final analytical expressions for all the fine-tunings in the Appendix. This is one of the main results of the present work.

4.3 Results in \(\epsilon \)-expansion

Due to the smallness of \(c_{\beta -\alpha }\) in the alignment regime, we can rewrite all the fine-tuning expressions of the Appendix as power series in \(c_{\beta -\alpha }=\epsilon \), using

$$\begin{aligned} t_\alpha = - \frac{1}{t_\beta } + \frac{1+t_\beta ^2}{t_\beta ^2}\epsilon +{\mathcal {O}} (\epsilon ^3 ). \end{aligned}$$
(59)

This expansion yields simpler formulas, which capture the general behaviour of the fine-tuning and allow for a closer examination.

Next we gather and discuss the expressions of the various fine-tunings at the lowest order in \(\epsilon \). We will focus the discussion on the two instances that, as argued in Sect. 3, naturally lead to alignment, namely the decoupling limit and the \(t_\beta \gg 1\) regime.

4.3.1 \(\underline{\text {Fine-tuning in }v^2}\)

$$\begin{aligned} \Delta _{m^2_{11}} v^2= & {} \frac{(1+t_\beta ^2) m_h^2 - 2 t_\beta ^2 (m_A^2 + v^2 \lambda _5)}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{m^2_{22}} v^2= & {} \frac{ t_\beta ^2 (1+t_\beta ^2) m_h^2 - 2 t_\beta ^2 (m_A^2 + v^2 \lambda _5)}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{m^2_{12}} v^2= & {} \frac{ 4 t_\beta ^2 (m_A^2 + v^2 \lambda _5)}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _1} v^2= & {} \frac{ -m_h^2 + t_\beta ^2 (m_A^2 + v^2 \lambda _5-m_H^2)}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _2} v^2= & {} \frac{t_\beta ^2 (m_A^2 + v^2 \lambda _5-m_H^2 -m_h^2t_\beta ^2 )}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _3} v^2= & {} \frac{2 t_\beta ^2 (m_A^2 + v^2 \lambda _5+m_H^2 -m_h^2 -2 m_{H_\pm }^2 )}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _4} v^2= & {} - \frac{2 t_\beta ^2 (2 m_A^2 + v^2 \lambda _5-2 m_{H_\pm }^2 )}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _5} v^2= & {} - \frac{2 t_\beta ^2 v^2 \lambda _5}{(1+t_\beta ^2)^2 m_h^2} + {\mathcal {O}} (\epsilon ). \end{aligned}$$
(60)

In the decoupling regime, \(m_H^2\simeq m_A^2\simeq m_{H^\pm }^2 \gg v^2\), so \(\Delta _{m_{ij}^2} v^2={\mathcal {O}}(\frac{m_H^2}{v^2})+{\mathcal {O}} (\epsilon )\), indicating that generically the EW fine-tuning diverges as the masses of the extra Higgses increase, as expected from the discussion in Sect. 4.1. More precisely, in that regime

$$\begin{aligned} \frac{1}{2}\Delta _{m_{12}^2}v^2\simeq & {} -\Delta _{m_{11}^2}v^2\simeq -\Delta _{m_{22}^2}v^2\nonumber \\\simeq & {} 2\left( \frac{t_\beta }{1+t_\beta ^2}\right) ^2\frac{m_A^2}{m_h^2}. \end{aligned}$$
(61)

In contrast \(\Delta _{\lambda _{i}} v^2\) remain \({\mathcal {O}}(1)\).

Let us consider now the \(t_\beta \gg 1\) regime. It can be checked from Eqs. (60) that in this case all \(\Delta _{\theta _i}v^2\) fine-tunings tend to zero, except \(\Delta _{m_{22}^2}v^2\), \(\Delta _{\lambda _2}v^2\), which become \({\mathcal {O}}(1)\), i.e. irrelevant. This remarkable absence of EW fine-tuning comes from the following. In the \(t_\beta \rightarrow \infty \) limit

$$\begin{aligned} v^2\simeq v^2_2 \simeq \dfrac{- 2m_{22}^2}{\lambda _2}, \end{aligned}$$
(62)

(this can be seen, e.g. from Eqs. (7) or (9)). Since \(\lambda _2\le {\mathcal {O}}(1)\), this situation requires \(m_{22}^2 = {\mathcal {O}} (m_h^2)\), which by itself does not entail any fine-tuning. Note that the extra Higgses can still be heavy provided \(m_{11}^2\) is large. So for \(t_\beta \gg 1\) there is a hierarchy between \(m_{ij}^2\) mass terms; namely, from Eqs. (57):

$$\begin{aligned} m_{11}^2\simeq m_A^2,\quad |m_{22}^2|\simeq m_h^2/2,\quad m_{12}^2\simeq m_A^2/ t_\beta , \end{aligned}$$
(63)

so that

$$\begin{aligned} m_{11}^2\gg |m_{22}^2|,m_{12}^2, \end{aligned}$$
(64)

while

$$\begin{aligned} |m_{22}^2|\simeq \left( \frac{m_h^2}{2m_A^2} \right) m_{11}^2. \end{aligned}$$
(65)

As already mentioned, a hierarchy of mass terms may be or may be not seen as an odd fact, but it does not imply a fine-tuning between parameters.

This situation contrasts with what happens in the supersymmetric case. In the minimal supersymmetric Standard Model (MSSM), the \(t_\beta \gg 1\) regime (which is desirable in order to reproduce the ordinary Higgs mass) implies a notorious fine-tuning. This occurs because \(m_{22}^2\) at low energy contains several contributions proportional to the soft supersymmetric masses, e.g. a large and negative contribution proportional to the mass-squared of the gluinos, and a positive contribution equal to the \(\mu -\)term (squared). Then, the smallness of \(|m_{22}^2|\) can only be achieved by a cancellation between all these large contributions, and this does imply a fine-tuning. However, this does not need to be the case in a generic 2HDM.

Of course, the previous discussion has nothing to do with the famous hierarchy problem [57, 58], which is related to the stability of the scalar mass-parameters under quadratic radiative corrections.

In summary, from Eq. (61) the electroweak fine-tuning in the 2HDM has the approximate size

$$\begin{aligned} \Delta v^2\sim \frac{4}{t_\beta ^2}\frac{m_A^2}{m_h^2} \end{aligned}$$
(66)

which shows that it increases with the masses of the extra Higgses (as expected), but it can be compensated by a large \(t_\beta \).

Let us recall here that the quantity \(m_h\) appearing in all the previous expressions, in particular the last one, corresponds to the Higgs mass at the \(m_H\) scale. This is typically \({\mathcal {O}}(10)\) GeV smaller than the physical value, \(m_h^{\textrm{phys}}=125\) GeV, see Sect. 2. Consequently, the radiative corrections slightly increase the severity of the electroweak fine-tuning. This is in fact the only noticeable impact of the radiative corrections in the degree of the fine-tuning. In particular, the other two relevant fine-tunings, \(\Delta c_{\beta -\alpha },\ \Delta t_\beta \) are quite insensitive to radiative corrections, as will be clear soon.

4.3.2 \(\underline{\text {Fine-tuning in }t_\beta }\)

$$\begin{aligned} \Delta _{m^2_{11}} t_\beta= & {} - \frac{(1+t_\beta ^2) m_h^2 - 2 t_\beta ^2 (m_A^2 + v^2 \lambda _5)}{2 m_H^2(1+t_\beta ^2) } + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{m^2_{22}} t_\beta= & {} \frac{(1+t_\beta ^2) m_h^2-2 (m_A^2 + v^2 \lambda _5)}{2 m_H^2(1+t_\beta ^2)} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{m^2_{12}} t_\beta= & {} - \frac{(t_\beta ^2-1)(m_A^2 + v^2 \lambda _5)}{m_H^2(1+t_\beta ^2)} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _1} t_\beta= & {} \frac{ m_h^2 - t_\beta ^2(m_A^2-m_H^2 + v^2 \lambda _5) }{2 m_H^2(1+t_\beta ^2)} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _2} t_\beta= & {} \frac{m_A^2 -m_H^2 -m_h^2 t_\beta ^2 + v^2 \lambda _5}{2 m_H^2(1+t_\beta ^2)} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _3} t_\beta= & {} - \frac{ (t_\beta ^2 \!-\!1) (m_A^2 \!-\! m_h^2 \!+\! m_H^2 \!-\! 2 m_{H_\pm }^2 + v^2 \lambda _5) }{2 m_H^2(1\!+\!t_\beta ^2)} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _4} t_\beta= & {} \frac{(t_\beta ^2 -1)(2 m_A^2 - 2 m_{H_\pm }^2 + v^2 \lambda _5)}{2 m_H^2(1+t_\beta ^2)} + {\mathcal {O}} (\epsilon ), \nonumber \\ \Delta _{\lambda _5} t_\beta= & {} \frac{(t_\beta ^2 -1) v^2 \lambda _5}{2 m_H^2(1+t_\beta ^2)} + {\mathcal {O}} (\epsilon ). \end{aligned}$$
(67)

It is clear from the previous expressions that \(t_\beta \) is not a fine-tuned parameter in any instance, as anticipated in Sect. 4.1. In particular, in the decoupling regime, \(m_H^2\simeq m_A^2\simeq m_{H^\pm }^2 \gg v^2\), all \(\Delta _{\lambda _i} t_\beta \rightarrow {\mathcal {O}} (\epsilon )\), whereas \(\Delta _{m_{11}^2} t_\beta , \Delta _{m_{12}^2} t_\beta = {\mathcal {O}} (1)\) and \(\Delta _{m_{22}^2} t_\beta \sim 1/(1+t_\beta ^2)\).

Something similar occurs in the large \(t_\beta \) regime, where all the fine-tuning parameters are smaller than 1, except \(\Delta _{m_{11}^2} t_\beta \sim \Delta _{m_{12}^2} t_\beta \sim 1\). This comes from the fact that if \(m_{11}^2>0\), then \(t_\beta \sim m_{11}^2/m_{12}^2\), as discussed around Eq. (39).

In summary, \(t_\beta \) is not a fine-tuned parameter and typically

$$\begin{aligned} \Delta t_\beta {\mathop {{}_\sim }\limits ^{<}}{\mathcal {O}}(1). \end{aligned}$$
(68)

4.3.3 \(\underline{\text {Fine-tuning in }c_{\beta -\alpha }}\)

In this case the lowest order in the expansion is always \(1/\epsilon \), reflecting the fact that we are evaluating the fine-tuning in \(c_{\beta -\alpha }=\epsilon \) itself, which is logically more severe as \(\epsilon \) shrinks:

$$\begin{aligned}{} & {} \Delta _{m^2_{11}} c_{\beta - \alpha } \nonumber \\{} & {} \quad =- \frac{1}{\epsilon }\; \frac{t_\beta (m_A^2 \!-\!m_H^2 \!+\! v^2 \lambda _5)(m_h^2(1\!+\!t_\beta ^2) \!-\! 2 t_\beta ^2(m_A^2 \!+\!v^2 \lambda _5)) }{m_H^2 (m_H^2\!-\!m_h^2)(1+t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{m^2_{22}} c_{\beta - \alpha } \nonumber \\{} & {} \quad = \, \frac{1}{\epsilon }\; \frac{t_\beta (m_A^2 \!-\!m_H^2 \!+\! v^2 \lambda _5) (-2 m_A^2\!+\! m_h^2(1+t_\beta ^2) \!-\! 2 v^2 \lambda _5) }{m_H^2 (m_H^2\!-\!m_h^2)(1+t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{m^2_{12}} c_{\beta - \alpha } \nonumber \\{} & {} \quad = \frac{1}{\epsilon } \; \frac{2 t_\beta (t_\beta ^2 -1) (m_A^2 -m_H^2 + v^2 \lambda _5) (m_A^2+ v^2 \lambda _5) }{m_H^2 (m_H^2-m_h^2)(1+t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{\lambda _1} c_{\beta - \alpha } \nonumber \\{} & {} \quad = - \frac{1}{\epsilon } \; \frac{ t_\beta (m_A^2+ v^2 \lambda _5) (-m_h^2 + t_\beta ^2(m_A^2-m_H^2 + v^2 \lambda _5) ) }{m_H^2 (m_H^2-m_h^2)(1+t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{\lambda _2} c_{\beta - \alpha } \nonumber \\{} & {} \quad = \, \frac{1}{\epsilon } \; \frac{ t_\beta (m_A^2+ v^2 \lambda _5)(m_A^2 -m_H^2 -m_h^2 t_\beta ^2 + v^2 \lambda _5)}{m_H^2 (m_H^2-m_h^2)(1+t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{\lambda _3} c_{\beta - \alpha } \nonumber \\{} & {} \quad = - \frac{1}{\epsilon } \; \frac{ t_\beta (t_\beta ^2 \!-\!1)(m_A^2\!+\! v^2 \lambda _5) (m_A^2 \!-\! m_h^2 \!+\! m_H^2 \!-\! 2 m_{H_\pm }^2 \!+\! v^2 \lambda _5)}{m_H^2 (m_H^2\!-\!m_h^2)(1\!+\!t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{\lambda _4} c_{\beta - \alpha } \nonumber \\{} & {} \quad = \, \frac{1}{\epsilon } \; \frac{ t_\beta (t_\beta ^2 -1) (m_A^2+ v^2 \lambda _5) (2 m_A^2 - 2 m_{H_\pm }^2 + v^2 \lambda _5)}{m_H^2 (m_H^2-m_h^2)(1+t_\beta ^2)^2 }\nonumber \\{} & {} \qquad + {\mathcal {O}} (\epsilon ^0), \nonumber \\{} & {} \Delta _{\lambda _5} c_{\beta - \alpha }= \frac{1}{\epsilon } \; \frac{ t_\beta (t_\beta ^2 -1) v^2 \lambda _5 (m_A^2+ v^2 \lambda _5) }{m_H^2 (m_H^2-m_h^2)(1+t_\beta ^2)^2 }+ {\mathcal {O}} (\epsilon ^0). \end{aligned}$$
(69)

Let us note that the \({\mathcal {O}} (\epsilon ^0)\) term, that has been dropped above, precisely contains the \(\Delta _{\theta _i}v^2\) contribution to \(\Delta _{\theta _i}c_{\beta -\alpha }\), see the discussion around Eq. (54). Actually, Eqs. (69) are a good approximation of the independent \(\Delta _{\theta _i}c_{\beta -\alpha }\) fine-tuning, i.e. once it is projected onto the constant-\(v^2\) hypersurface, according to Eq. (54). Consistently with that, we note that at this order in the \(\epsilon -\)expansion, \(\Delta _{\theta _i} c_{\beta -\alpha }={\mathcal {O}}\left( \frac{v^2}{m_H^2}\frac{1}{\epsilon }\right) \), thus the alignment becomes natural in the decoupling limit.

Likewise, if \(t_\beta \gg 1\), then \(\Delta _{\theta _i} c_{\beta -\alpha }\sim 1/t_\beta \), showing that the alignment becomes non-fine-tuned in that regime. The origin of this remarkable fact was already mentioned in Sect. 3, when discussing Eq. (29).

In summary, the alignment fine-tuning in the 2HDM is of the order

$$\begin{aligned} \Delta c_{\beta -\alpha }\sim \frac{1}{c_{\beta -\alpha }}\frac{1}{t_\beta }\frac{v^2}{m_H^2} \end{aligned}$$
(70)

which shows that it is mitigated in the decoupling limit and for large \(t_\beta \).

5 Numerical analysis

We perform in this section a numerical exploration of the fine-tuning(s) in the 2HDM parameter-space. We will focus on their dependence on the masses of the extra Higgses and on \(t_\beta \); which, as discussed above, are the main determinants of the size of the fine-tuning. We will also consider two possible circumstances: that the alignment is sharp, i.e. significant smaller than the present experimental upper bounds, or close to them.

For these purposes we have designed five scenarios which capture all the relevant features of the problem, see Table 3. We examine them in order.

Table 3 Benchmark Scenarios used in the numerical analysis to explore the electroweak and alignment fine-tunings in the 2HDM. The fact that \(m_A=m_{H^\pm }\) implies \(\lambda _4=\lambda _5\) in all cases
Full size table

5.1 Scenario 1

This scenario is chosen to show the dependence of the fine-tuning on the extra-Higgs masses in the case of sharp alignment, \(\epsilon =c_{\beta -\alpha }=0.01\) and low \(t_\beta \). As discussed above, in the alignment limit the larger the extra-Higgs masses are, the closer each other must be. This feature is trivially fulfilled here as we have set \(m_H=m_A= m_{H^\pm }\equiv m_0\). On the other hand, the value of \(\lambda _5\) has been chosen to enhance the range of \(m_0\) that preserves perturbativity and stability, but it does not play an important role in the discussion.

The electroweak fine-tuning, \(\Delta _{\theta _i}v^2\) vs. \(m_0\) is shown in Fig. 1 for the various \(\theta _i\) parameters. As expected, the only important fine-tunings are those associated with mass parameters, \(m_{ij}^2\), which, for large \(m_0\) increase according to the trend of Eq. (61), becoming as large as 250 for \(m_0 \sim 2\) TeV. In contrast, for \(m_0{\mathop {{}_\sim }\limits ^{<}}700\) GeV the electroweak fine-tuning is quite mild.

Fig. 1
figure 1

Dependence of the electroweak fine-tuning, \(\Delta _{\theta _i}v^2\), on the common mass of the extra Higgses, \(m_0\) for Scenario 1

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Concerning the alignment fine-tuning, we have projected the complete expressions for \(\Delta _{\theta _i}c_{\beta -\alpha }\) given in the Appendix onto the constant-\(v^2\) surface, according to Eq. (54), in order to show only the independent fine-tuning. As commented at the end of the previous section, this is essentially equivalent to use the lowest-order expressions for \(\Delta _{\theta _i}c_{\beta -\alpha }\), Eqs. (69). The corresponding results are shown in Fig. 2 in a pseudo-logarithmic scale. As expected, the alignment fine-tuning increases for decreasing masses of the extra Higgses, becoming as large as \(\sim 1000\) at \(m_0=200\) GeV.

Fig. 2
figure 2

The same as Fig. 1 for the independent \(\Delta _{\theta _i} c_{\beta -\alpha }\) fine-tuning, i.e. projected onto the once the \(v^2={\textrm{const}}.\) hypersurface. The vertical axis represents the magnitude of \(\Delta _{\theta _i} c_{\beta -\alpha }\) in a logarithmic scale, keeping track of its sign. Namely, we have defined \(``{\textrm{Log}}''(y)= {\textrm{sign}} (y) \log _{10} (|y|+1)\)

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By comparing Figs. 1 and 2, we see that there is an intermediate region, \(550~{\textrm{GeV}}{\mathop {{}_\sim }\limits ^{<}}m_0{\mathop {{}_\sim }\limits ^{<}}700~{\textrm{GeV}}\), where both the EW and the alignment fine-tunings remain at acceptable values, \({\mathop {{}_\sim }\limits ^{<}}{\mathcal {O}}(10)\).

Finally, the fine-tuning in \(t_\beta \) is shown in Fig. 3. As expected, \(t_\beta \) is not a fine-tuned quantity, i.e. the associated fine-tuning parameters, \(\Delta _{\theta _i}t_\beta \) are \({\mathcal {O}}(1)\) in all cases.

Fig. 3
figure 3

The same as Fig. 1 for the \(\Delta _{\theta _i} t_\beta \) fine-tuning

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5.2 Scenario 2

This is similar to Scenario 1, but with a weak alignment; namely, \(\epsilon =c_{\beta -\alpha }=0.1\), about as large as allowed by experimental constraints. An important consequence of the mild alignment is that now \(m_0\) cannot be too large. The reason is that for very large masses of the extra-Higgses we enter the decoupling regime, which automatically leads to alignment. Under such circumstances, it is very difficult to keep a certain misalignment (typically it requires non-perturbative \(\lambda \)-couplings). In consequence the available range for \(m_0\) lies below \(\sim 1\) TeV. As we will see later, for higher \(t_\beta \), this upper bound becomes stronger.

Apart from this limitation, the trends of the fine-tuning are similar to those for Scenario 1, as shown in Figs. 4, 5 and 6. The main difference has to do with the alignment fine-tuning, \(\Delta c_{\beta -\alpha }\), which is less severe than before (compare Figs. 2 and 5) because the required value of \(c_{\beta -\alpha }\) is not that small. Again, there is an intermediate region, \(500~{\textrm{GeV}}{\mathop {{}_\sim }\limits ^{<}}m_0{\mathop {{}_\sim }\limits ^{<}}700~{\textrm{GeV}}\), where both the electroweak and the alignment fine-tunings remain at acceptable values, \({\mathop {{}_\sim }\limits ^{<}}{\mathcal {O}}(10)\). Same as Fig. 1 for Scenario 2.

Fig. 4
figure 4

The same as Fig. 1 for Scenario 2

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Fig. 5
figure 5

The same as Fig. 2 (and using the same pseudo-logarithmic scale) for Scenario 2

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Fig. 6
figure 6

The same as Fig. 3 for Scenario 2

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5.3 Scenario 3

This scenario is chosen to show the dependence of the fine-tuning on \(t_\beta \) in the case of sharp alignment, \(\epsilon =c_{\beta -\alpha }=0.01\) and low extra-Higgs masses, \(\sim 250\) GeV. The value of \(\lambda _5\) has been picked to ensure perturbativity and stability in a significant range of \(t_\beta \), namely \(t_\beta \in [1,15]\).

The electroweak fine-tuning, \(\Delta _{\theta _i}v^2\) vs. \(t_\beta \) is shown in Fig. 7. As discussed in the previous section, the fine-tuning decreases with increasing \(t_\beta \). Since this scenario has a rather low value of \(m_0\), the electroweak fine-tuning is never large or even significant.

The fine-tuning in the alignment, \(\Delta _{\theta _i}c_{\beta -\alpha }\) vs. \(t_\beta \), is shown in Fig. 8. As discussed above, this fine-tuning also gets less severe as \(t_\beta \) increases, and for \(t_\beta {\mathop {{}_\sim }\limits ^{>}}10\) it becomes \(\le {\mathcal {O}} (10)\). It is worth-noticing that several fine-tuning parameters, namely \(\Delta _{m^2_{12}}c_{\beta -\alpha }\), \(\Delta _{\lambda _3}c_{\beta -\alpha }\), \(\Delta _{\lambda _4}c_{\beta -\alpha }\) and \(\Delta _{\lambda _5}c_{\beta -\alpha }\), abruptly fall to zero for \(t_\beta \rightarrow 1\). This is a general fact, as can be verified from the general expressions (69) for these quantities, which have a \((t_\beta ^2-1)\) factor. The origin of this feature is the following. It is easy to check from Eqs. (8, 9, 26) that \(t_\beta \simeq 1\) and a fine alignment require \(\lambda _1\simeq \lambda _2\), \(m_{11}^2\simeq m_{22}^2\). This corresponds to a situation where the consistency condition for perfect alignment, (27) is approximately fulfilled in an obvious way. In that case, it is clear that changes in \(m_{12}^2, \lambda _3, \lambda _4, \lambda _5\) do not spoil condition (27); in other words, the fine alignment is insensitive to these changes, thus the vanishing of the associated fine-tuning parameters at \(t_\beta =1\). This rule does not apply to the other initial parameters, \(m_{11}^2, m_{22}^2, \lambda _1, \lambda _2\). As it was commented after Eq. (27), in the \(\lambda _1= \lambda _2\), \(m_{11}^2= m_{22}^2\) limit (which corresponds to \(t_\beta =1\) and perfect alignment) there arises a symmetry in the potential, which is however broken by the Yukawa couplings.

Finally, as usual, there is no fine-tuning in \(t_\beta \), as shown in Fig. 9.

Fig. 7
figure 7

Dependence of the electroweak fine-tuning, \(\Delta _{\theta _i}v^2\) on \(t_\beta \) for Scenario 3

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Fig. 8
figure 8

The same as Fig. 7 for the independent \(\Delta _{\theta _i} c_{\beta -\alpha }\) fine-tuning (projected onto the once the \(v^2={\mathrm{const.}}\) hypersurface)

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Fig. 9
figure 9

The same as Fig. 7 for the \(\Delta _{\theta _i} t_\beta \) fine-tuning

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5.4 Scenario 4

This scenario explores the dependence of the various fine-tunings on \(t_\beta \), in a scenario with large extra-Higgs masses, \(\sim 1500\) GeV, and very sharp alignment, \(\epsilon =10^{-3}\). The results are shown in Figs. 10, 11 and 12.

Such a strong alignment is actually the natural consequence of two features that push in that direction: the decoupling due to the large extra masses, and the sizeable \(t_\beta \). This is demonstrated by the fact that, even with such alignment, there is no relevant \(\Delta c_{\beta -\alpha }\) fine-tuning (Fig. 11). For \(t_\beta \le {O}(10)\), there is still a (strong) electroweak fine-tuning, \(\Delta v^2\), see Fig. 10, due to the large extra Higgs masses. This fine-tuning is compensated by the competing effect of a large \(t_\beta \) for \(t_\beta \ge {O}(10)\).

Finally, once more, \(t_\beta \) is not fine-tuned, as shown in Fig. 12.

Fig. 10
figure 10

The same as Fig. 7 for Scenario 4

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Fig. 11
figure 11

The same as Fig. 8 for Scenario 4

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Fig. 12
figure 12

The same as Fig. 9 for Scenario 4

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5.5 Scenario 5

This scenario explores the dependence of the various fine-tunings on \(t_\beta \), for small extra-Higgs masses, \(\sim 300\) GeV, and mild alignment, \(\epsilon =0.1\) (the opposite to scenario 4). The results are shown in Figs. 13, 14 and 15. In this case, due to the low extra-Higgs masses, there is no appreciable electroweak fine-tuning, Fig. 13. Likewise, due to the (as large as possible) value of \(c_{\beta -\alpha }\), there is no fine-tuning for the alignment, Fig. 14. Of course, \(t_\beta \) is not fine-tuned either, Fig. 15.

Fig. 13
figure 13

The same as Fig. 7 for Scenario 5

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Fig. 14
figure 14

The same as Fig. 8 for Scenario 5

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Fig. 15
figure 15

The same as Fig. 9 for Scenario 5

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6 Conclusions

The Two-Higgs Doublet Model is one of the most popular and natural extensions of the Higgs sector. It arises in many BSM scenarios, such as grand unification, supersymmetry and axion models. But, besides its theoretical appeal, the 2HDM has two potential fine-tuning problems.

The first one is related to the electroweak breaking, when the magnitude of the VEV, \(v^2=(246~{\textrm{GeV}})^2\), is much smaller than the squared-mass terms entering the theory. This is the familiar electroweak fine-tuning or little-hierarchy problem. The second one is related to the alignment condition. Namely, current experimental data require that the properties of one of the Higgs fields are equal or very similar to those of the SM Higgs boson, i.e. the model must be close to the alignment limit.

In this paper we have explored how these fine-tunings arise in the 2HDM (with the usual \({\mathbb {Z}}_2\) parity implemented to avoid dangerous FCNC), looking for regions of the parameter space where both are mild or irrelevant. In order to quantify the fine-tunings we have used the fairly standard Barbieri-Giudice criterion; e.g. the fine-tuning in \(v^2\) respect to some initial parameter \(\theta \) is quantified by the fine-tuning parameter \(\Delta _\theta v^2=\partial \log v^2/\partial \log \theta \).

Concerning the electroweak fine-tuning, since there are two vacuum expectation values, \(v_1^2+v_2^2=v^2\), one may wonder if there are fine-tunings associated with both \(v_1^2, v_2^2\), or equivalently with \(v^2\) and \(t_\beta =v_2/v_1\). We have shown that, in all cases, \(t_\beta \) is not a fine-tuned parameter, so the whole electroweak fine-tuning is captured by \(\Delta v^2\). Regarding the alignment, the relevant fine-tuned observable is \(c_{\beta -\alpha } =\cos (\beta -\alpha )\), which vanishes for perfect alignment and is currently constrained by experimental bounds to be \(|c_{\beta -\alpha }|{\mathop {{}_\sim }\limits ^{<}}0.1\).

We have obtained analytical expressions for \(\Delta _{\theta _i}v^2\), \(\Delta _{\theta _i}t_\beta \), \(\Delta _{\theta _i}c_{\beta -\alpha }\) both in terms of the initial parameters of the theory, \(\{m^2_{i,j}, \lambda _i\}\) and of the physical observables, \(\{m_H, m_A, m_{H^\pm },t_\beta ,\ {\mathrm{etc.}} \}\). As already mentioned, there is no fine-tuning for \(t_\beta \), i.e. \(\Delta _{\theta _i}t_\beta = {\mathcal {O}}(1)\) in all cases. On the other hand, the electroweak and alignment fine-tunings are not independent and they should not be multiplied without further ado to get the “total” fine-tuning.

In particular, in the decoupling limit, once the initial parameters have been adjusted to get the right \(v^2\), there is no need of extra adjustments to get alignment. To remove the “double counting” in the evaluation of the \(c_{\beta -\alpha }\) fine-tuning one has to discard variations in the \(\theta _i\) parameters that change the value of \(v^2\). This is equivalent to project the gradient \(\Delta _{\theta _i} c_{\beta -\alpha }=\partial \log c_{\beta -\alpha }/\partial \log \theta _i\) onto the constant-\(v^2\) hypersurface. This is the procedure we have followed to present the \(\Delta _{\theta _i}v^2\), \(\Delta _{\theta _i}c_{\beta -\alpha }\) fine-tunings as independent parameters.

We have also obtained approximate expressions for the various fine-tunings performing an expansion in \(c_{\beta -\alpha }\). The general trend for them is:

$$\begin{aligned} \Delta v^2\sim \frac{4}{t_\beta ^2}\frac{m_A^2}{m_h^2}, \quad \Delta c_{\beta -\alpha }\sim \frac{1}{c_{\beta -\alpha }}\frac{1}{t_\beta }\frac{v^2}{m_H^2},\quad \Delta _{\theta _i}t_\beta = {\mathcal {O}}(1). \end{aligned}$$
(71)

We have investigated numerically the behaviour of the fine-tunings in the 2HDM parameter space, by using representative benchmark scenarios that explore different relevant regions.

As it is clear from the previous general trends, for moderate values of \(t_\beta \), the electroweak and the alignment fine-tunings, \(\Delta v^2\) and \(\Delta c_{\beta -\alpha }\), become severe in different regions of the parameter space, namely in the regimes of large and small extra-Higgs masses, respectively. The first one corresponds to the decoupling limit. This means that there is an intermediate region, more precisely \(500~{\textrm{GeV}} {\mathop {{}_\sim }\limits ^{<}}\{m_H, m_A, m_{H^\pm }\} {\mathop {{}_\sim }\limits ^{<}}700~{\textrm{GeV}}\), where both tunings are acceptably small.

It should be mentioned here that, if one does not take into account the electroweak fine-tuning it is not correct to say that the alignment becomes natural in the decoupling limit. In that case, one should not project \(\Delta c_{\beta -\alpha }\) onto the constant-\(v^2\) hypersurface. Then it contains a contribution of order \(\Delta _{\theta _i}v^2\) that shows that it is in fact fine-tuned. This occurs because the Barbieri-Giudice parameters incorporate all the adjustments required to get the desired size for the observable under consideration.

The analytical expressions and the numerical results show an interesting trend that is not obvious at first glance. Namely, for large \(t_\beta \) both the electroweak and the alignment fine-tunings become mitigated. We have discussed in the paper the reason for this feature. In consequence, the 2HDM becomes quite natural for \(t_\beta \ge {\mathcal {O}} (10)\), even if \(m_H, m_A, m_{H^\pm }\) are as large as 1500 GeV.

This situation contrasts with the supersymmetric case, where the \(t_\beta \gg 1\) regime implies a notorious fine-tuning. The reason for that was that the mass parameter \(m_{22}^2\) at low energy contained several contributions proportional to the soft supersymmetric masses and the \(\mu \)-term. Then, the smallness of \(|m_{22}^2|\) could only be achieved by a fine cancellation of such contributions. This does not need to be the case in a generic 2HDM.

Let us finally comment on experimental constraints that we have not considered throughout the paper. Since we have provided the complete analytical expressions for the fine-tuning in the 2HDM, it would be possible to explore the interplay among the 2HDM experimental constraints and fine-tuning. Such study, although interesting, is quite model-dependent and beyond the scope of this work. Specific, partial, studies of the allowed parameter space in the 2HDM have been done for concrete models, see e.g. [59] for Type I; or addressing particular scenarios, like possibly sizeable triple Higgs couplings (such as hHH) [47, 60, 61], or in the context of possible experimental anomalies, such as the muon \(g-2\) [62].

Still, it is possible to assess the impact of these studies on the scenarios we have described. For example, from [61] we conclude that the Scenarios 1, 2 would be in conflict with flavour constraints unless \(m_H > 1000\) GeV (\(m_H > 600\) GeV) for Type I (Type II and Type Y). Nevertheless, we stress the fact that such strict constraints derive in part from the requirement \(m_H=m_A=m_{H^\pm }\) used in the mentioned scenarios. This is a useful simplification to show the trends of the fine-tuning, but it is not intended to be realistic in a strict way. In fact, as soon as one allows slight non-degeneracies among these masses the parameter space expands extensively [61]; however, the fine-tuning figures would be essentially the same. Similarly, other benchmark scenarios considered in this work would be in conflict with experimental constraints. In particular, scenarios 3, 5 would be experimentally excluded for Type II [63] unless \(m_A\) is raised.

In summary, the 2HDM has potentially severe fine-tunings associated with the smallness of the electroweak scale and the sharpness of the alignment, but there are interesting regions in the parameter space where both remain at acceptable levels.