Wrong sign and symmetric limits and non-decoupling in 2HDMs

We analyse the possibility that, in two Higgs doublet models, one or more of the Higgs couplings to fermions or to gauge bosons change sign, relative to the respective Higgs Standard Model couplings. Possible sign changes in the coupling of a neutral scalar to charged ones are also discussed. These \textit{wrong signs} can have important physical consequences, manifesting themselves in Higgs production via gluon fusion or Higgs decay into two gluons or into two photons. We consider all possible wrong sign scenarios, and also the \textit{symmetric limit}, in all possible Yukawa implementations of the two Higgs doublet model, in two different possibilities: the observed Higgs boson is the lightest CP-even scalar, or the heaviest one. We also analyse thoroughly the impact of the currently available LHC data on such scenarios. With all 8 TeV data analysed, all wrong sign scenarios are allowed in all Yukawa types, even at the 1$\sigma$ level. However, we will show that B-physics constraints are crucial in excluding the possibility of wrong sign scenarios in the case where $\tan \beta$ is below 1. We will also discuss the future prospects for probing the wrong sign scenarios at the next LHC run. Finally we will present a scenario where the alignment limit could be excluded due to non-decoupling in the case where the heavy CP-even Higgs is the one discovered at the LHC.


I. INTRODUCTION
The Large Hadron Collider (LHC) has confirmed the existence of a Higgs boson [1,2] compatible with the one predicted by the Standard Model (SM). The Higgs couplings to fermions and gauge bosons are well within the expected SM couplings. In addition, no extra scalar particles were found, leaving us with a theory that, at the present scale, is indeed very close to the SM. All extensions of the SM are therefore being pushed to some kind of SM limit. Such is the case of the simplest extensions of the scalar sector like the ones obtained by simply adding a complex singlet or a complex doublet to the SM field content, the latter designated by the two-Higgs doublet model (2HDM).
In a previous work [3] we have discussed the interesting possibility of a sign change in one of the Higgs Yukawa couplings. There, we have defined the wrong sign scenario to be such that a sign change occurs in one of the Yukawa couplings relative to the Higgs coupling to V V (V = W ± or Z). The LHC data analysed so far does not allow to differentiate between scenarios where a sign change in one of the Yukawa couplings occurs (see e.g. Refs. [4][5][6][7]). These studies were performed taking into account the measured properties of the SM-like Higgs.
In this work we will discuss all the possible sign changes in the Higgs couplings to fermions and to massive gauge bosons. The various wrong sign scenarios will all have in common the property that they are physically meaningful, that is, each of them can in principle be probed experimentally and distinguished from the limit where the model resembles the SM. In contrast to the wrong sign scenarios are the cases where all Higgs couplings to other SM particles change sign (while no significant difference occurs in the Higgs self-couplings). Since we will be interested in probing sign changes through loop induced vertices (which receive contributions from several couplings to SM particles and can only change if relative sign changes occur) the latter are not considered.
The study will be performed in the framework of the softly-broken Z 2 symmetric and CP-conserving 2HDM. The 2HDM is the simplest model that can provide wrong sign scenarios as defined above, since adding instead a (simpler) singlet field implies that the shift of the Higgs couplings to the other SM particles is the same for all such couplings. The 2HDM contains a decoupling limit and an alignment limit. In the exact decoupling limit [8] the theory is the SM while in the alignment limit the SM-like Higgs boson couplings to the SM particles are exactly the SM ones. However, the coupling structure of the 2HDM further allows for a change in the sign of the tree-level couplings to

II. THE CP-CONSERVING 2HDM
The two-Higgs double model (2HDM) is an extension of the SM where an extra complex scalar doublet is added to the field content of the SM while keeping its gauge symmetry. It was first proposed by T.D. Lee [9] as a means to explain the matter-antimatter asymmetry (see Refs. [10,11] for a detailed description of the model). With two doublet fields (denoted henceforth Φ 1 and Φ 2 ) the most general Yukawa Lagrangian gives rise to tree-level (Higgs-mediated) flavour-changing neutral currents (FCNC) which are severely constrained by experimental data.
A natural way [12] of avoiding FCNCs is to impose an extra symmetry on the scalar potential. We choose to impose a Z 2 symmetry such that the potential is invariant under Φ 1 → Φ 1 , Φ 2 → −Φ 2 (i.e the doublets are Z 2 -even and Z 2 -odd respectively). The symmetry is extended to the Yukawa sector such that a fermion of a given charge couples only to one doublet. There are four possible independent coupling choices for the Yukawa Lagrangian [13]. In the literature two of the models have been named type I and type II and the other two have been changing names over the years. We shall call them type Flipped (F) and type Lepton Specific (LS) (also called Y and X [14], respectively). The different Yukawa types are built such that: only Φ 2 couples to all fermions (type I); or Φ 2 couples to up-type quarks and Φ 1 couples to down-type quarks and leptons (type II); or Φ 2 couples to up-type quarks and to leptons and Φ 1 couples to down-type quarks (type F); or finally Φ 2 couples to all quarks and Φ 1 couples to leptons (type LS).
The scalar potential in a softly broken Z 2 symmetric 2HDM can be written as where Φ i , i = 1, 2 are complex SU(2) doublets. We choose all parameters and the vacuum expectations values to be real. This leads to an 8-parameter CP-conserving potential and we take as free parameters the four masses, the rotation angle in the CP-even sector, α, the ratio of the vacuum expectation values, tan β = v 2 /v 1 , and the soft breaking parameter m 2 12 . Without loss of generality, we choose the conventions 0 ≤ β ≤ π/2 and −π/2 ≤ α ≤ π/2. It is also instructive for our study to re-call how the two physical CP-even eigenstates, h and H, relate to the original field fluctuations (before diagonalisation) which determine the coupling to other SM particles. If we denote them by h i (for each Φ i respectively), then in our convention Now we can find a map between the couplings of the (already observed) Higgs in the scenario where it is h (the lightest CP-even state), to the case where it is instead H (the heaviest CP-even state). Due to our convention for the range of α one has to be careful. One can check that the correct map (which also preserves our convention) is Thus all expressions later obtained in the discussion of the wrong sign scenario can be transposed from the case where h is the observed Higgs boson to the case where it is H by using this map. Eq. (2) will later explain some sign flips in our results. Nevertheless, it is clear that the experimental constraints have different effects on the allowed parameter space for each scenario (light or heavy), because the various limits on new (yet to observe) scalars are not uniform in mass (thus the allowed parameter space of one scenario cannot be obtained by applying this map to the data points allowed in the other scenario).

A. Theoretical and experimental constraints
The constraints to impose on the CP-conserving 2HDM models originate from two sources (for a recent review see Ref. [15]): i) consistency with theoretical principles/conditions and ii) consistency with experimental data. Regarding the theoretical constraints it is well known, at tree level, that once a CP-conserving minimum of the potential is chosen, no additional minima that spontaneously break the electric charge and/or CP symmetry exist [16]. Furthermore we demand that the CP-conserving minimum is the global one [17], that the potential is bounded from below [18] and that tree-level unitarity [19] is obeyed.
Regarding the consistency with experimental data we impose various conditions. We require the model to satisfy electroweak precision constraints [20][21][22][23], i.e. that the S, T, U variables [20] predicted by the model are within the 95% ellipsoid centred on the best fit point to the electroweak data. There are also indirect constrains originating from loop processes that involve charged Higgs bosons, which depend on tan β through the charged Higgs coupling to fermions. They originate mainly from B physics observables [24,25] and from the R b ≡ Γ(Z → bb)/Γ(Z → hadrons) [26] measurement. They give rise to the best bound on the charged Higgs mass in a type II model which yields m H ± 340 GeV almost independently of tan β.
LEP searches based on e + e − → H + H − [27] and recent LHC results [28,29] based on pp →t t(→ H +b ) constrain the mass of the charged Higgs to be above O(100) GeV, depending on the model type. Finally, we should note that there is a 3.4 σ discrepancy between the value of B → D ( * ) τ − ν τ measured by the BaBar collaboration [30] and the corresponding SM prediction. If confirmed, this observation would exclude both the SM and the versions of the 2HDM considered in this work. So far we have described mostly pre-LHC bounds. The parameter space of the 2HDM is already very constrained by the LHC results [31]. We will now briefly re-analyse these results to find the parameter space still allowed after the 8 TeV run. We have used the ScannerS [32] program interfaced with SusHi [33] for the pp(gg + bb) → h production process at NNLO and Hdecay [34,35] for all 2HDM decays. The numbers were cross-checked with HIGLU [36] and 2HDMC [37]. The remaining Higgs production cross sections, VBF, associated production (with a Z or W ) and tth were taken from [38] at NLO. SM electroweak corrections were not considered in any production process because the 2HDM electroweak corrections can be significantly different. All 95% C.L. exclusion limits, obtained experimentally from the non-observation of new scalars in experimental searches at colliders, were applied using HiggsBounds [39]. Consistency with the observed signals of the Higgs boson at the LHC was tested with HiggsSignals [40], which computes a probability for the model point to fit all known signal data 1 . The theoretical constraints associated with vacuum stability and tree level unitarity are inbuilt in the ScannerS code for any model, whereas specific functions were developed for the 2HDM to test electroweak precision observables and B-physics observables (all constrained to be within 95% of the best fit values as discussed above).
We will use the standard definition of signal strength where σ is the Higgs production cross section and BR(h → f ) is the branching ratio of the decay into some given final state f ; σ SM and BR SM (h → f ) are the expected values for the same quantities in the SM. In the following sections we will also make predictions for the next LHC run at 13 TeV. In these predictions we will not use the present LHC data (but will use all other constraints) but instead we will ask that the rates µ h f for the final states f = W W , ZZ, γγ and τ + τ − to be within 20, 10 or 5 % of the SM predictions.
We also define which at tree-level is just the ratio of the couplings κ i = g 2HDM i /g SM i . Taking the hW + W − coupling as an example, we write and the last equality only holds for Leading Order (LO) widths. Obviously, because the decays h → γγ and h → gg are one-loop processes at LO, κ γ or κ g can only be calculated by the ratio of the 2HDM width to the respective SM width. Unless otherwise stated, the theoretical values of κ F (where F is a fermion) and κ V (where V is a massive vector boson) refer to LO widths. Note that while κ F and κ V can be either positive or negative, κ γ and κ g are strictly positive. These definitions for the couplings κ coincide with the definitions used by the experimental groups at the LHC [45], at leading order. We shall also make the simplifying assumption (which holds in the SM and in the 2HDM under consideration) that all down-type [up-type] fermion final states are governed by the same κ D [κ U ].
B. Allowed parameter space after the 8 TeV LHC and the wrong sign limit In this section we discuss some important features of the allowed parameter space of the models, after imposing all theoretical and experimental constraints mentioned above. Unless stated otherwise, we have set one of the CPeven eigenstates to a mass of 125.9 GeV and left all other masses free to run over an interval. We will refer to the case where m h = 125.9 GeV as the light Higgs scenario, and to the case where m H = 125.9 GeV as the heavy Higgs scenario. For all scans, before applying the constraints, we allow 0.1 < tan β < 50, |α| < π/2 and 2 −(900 GeV) 2 < m 2 12 < (900 GeV) 2 . All (eigenstate) masses are free in the range [50,1000] GeV, but we have also imposed that the masses of the other neutral scalars are away from the Higgs mass 125.9 GeV by more than 5 GeV. These conditions apply to all scans. Any other extra condition (such as lower or upper bounds imposed on masses) will be specified for each scan.
Let us start by discussing the light Higgs scenario where m h = 125.9 GeV. In figure 1 we present the allowed parameter space projected on the tan β vs sin(β − α) plane with all constraints applied. We have also imposed that the CP-odd scalar mass obeys m A > m h + 5 GeV. We have accepted points that explain the observed Higgs signals with a fit probability within 3σ, and also represent on top of these the points that survive at 2σ and 1σ. For the top left and middle panels (Types II and F), the allowed parameter space is centred around two lines. One is sin(β − α) = 1, the alignment limit where all Higgs couplings to fermions and massive gauge bosons are exactly the SM ones. The other one is sin(β + α) = 1 which we have called the wrong sign limit 3 in [3], to be discussed in detail in the next section. The two plots are very similar, both in the allowed range for sin(β − α) and in that large values of tan β are excluded except for sin(β − α) very close to 1. In order to understand the shape of the curves let us consider the approximation where the production occurs only via gg while the the total width is dominated by h → bb. As shown in [41] , this approximation can be written and by imposing 0.8 < µ V V < 1.2 one reproduces figure 1 for types II and F with remarkable accuracy, as shown in [42]. Hence, the bounds on µ V V alone, can explain not only the shape but also the numerical values presented in the plots in figure 1 for types II and F. Furthermore, the b-loop contribution in gg → h and bb → h grows with 2 For the scans we present it turns out that the combination of the global minimum condition with the other constraints implies m 2 12 > 0 in practice. 3 Note that in our convention the band of points associated with this line appears for α > 0. which is exactly tan 2 β when sin(β − α) = 0 but even for, say sin(β − α) = 0.8, taking tan β = 10 we get an enhancement factor of 27 relative to the respective SM contribution. As sin(β − α) approaches 1, the 2HDM lightest Higgs branching ratios (BRs) to SM particles do not differ much from the values of the respective SM Higgs decays. Therefore, the inclusion of the b-loops would just confirm the exclusion of the high tan β region except close to the alignment limit.
In the top right and bottom panels of figure 1 we show the allowed parameter space for type LS and type I. Let us focus first on the top right and bottom left panels (for which no extra cut is present). We start by observing that there is no tan β enhancement in the Higgs production cross section. In fact, the Higgs couplings to both up-type and down-type quarks are the same and the SM cross section for gg + bb → h is just multiplied by the factor that could only be large for tan β 1, which is forbidden by B-physics constraints. For type I, considering the limit where the production occurs only via gg while the total width is dominated by h → bb (similarly to type II and F), we obtain We conclude that the result for type I is a bound on sin(β − α) which is almost independent of tan β. In fact, except for the Higgs self-couplings, the type I 2HDM is similar to the model obtained by adding a singlet to the SM, because if tan β 1 (using equation 8) Hence, only constraints related to the shape of the potential (such as the ones arising from vacuum stability and perturbative unitarity) can introduce some tan β dependence.
In the case of type LS, a similar approximation needs to take into account both Γ(h → bb) and Γ(h → τ + τ − ). In fact, if we take say sin(β − α) = 0.8 the two widths are equal for tan β ≈ 6. The value of tan β for which the widths cross grows with sin(β − α) and above the crossing value h → τ + τ − dominates. Therefore, depending on the values of sin(β − α) and tan β we either have an approximate expression for µ V V that is closer to type II (when h → τ + τ − dominates -equation (6)) or to type I (when h → bb dominates -equation (9)). We can also write an approximate expression for µ V V by considering as dominant the sum of the two widths Γ(h → bb) Finally, also the measurement of pp → h → τ + τ − affects considerably more the parameter space of type LS than that of type I [43]. As this decay becomes more important with growing tan β the exclusion region increases in the large tan β region.
In the bottom panels for type I, we also present (middle and right) the effect of placing a cut on the heavy Higgs mass, m H . It is quite remarkable the effect that this cut (combined with the constraints) has on the allowed parameter space for type I (which is otherwise almost independent of tan β). This behaviour is mainly related to the theoretical constraints imposed on the 2HDM including the discriminant that forces the model to be in the global minimum [17] at tree-level. To see the effect of the latter we show in figure 2: in black, points for which the global minimum conditions was lifted; in red, the subset of the black points that survive the mass cut m H > 300 GeV at 3σ; and in yellow, the subset which survives the mass cut and the global minimum condition. One should note that the global minimum condition does not play in general a major role in constraining the parameter space. Indeed this condition does not change the allowed regions for the other models, once the LHC constraints are imposed. Moreover, as discussed in [17], the theory can still be viable in a local minimum provided that the tunnelling time to the global one is larger than the age of the universe.
The second scenario we consider is the heavy Higgs scenario of figure 3, where m H = 125.9 GeV. In this case we have also imposed the lightest CP-even scalar mass, m h to be varied in the range 70 to 120 GeV. This somewhat short range for m h was chosen mainly because we want to disallow the decay H → hh. If allowed it would be the main decay channel and the model would have to be very fine-tuned (taking g Hhh ≈ 0) [44] for H to still be the SM-like Higgs found at the LHC. This is justified because our main goal is to compare the alignment limit with the different wrong sign scenarios for the heavy Higgs case (to be discussed in detail in the next sections). One should stress that all collider bounds were taken into account, including the LEP bound on a light scalar coming from e + e − → Zh.
It is straightforward to show (using the map of Eq. (2)) that all arguments that were used above to explain the excluded (large) tan β regions in the light scenario still hold with the replacements in Eqs. (7) and (8). This also explains why the two bands that are still allowed for each model type, now appear separated, on the left (cos(β − α) → −1) and on the right (cos(β − α) → 1) respectively. Similarly the line equivalent to the sin(β + α) = 1 line was mapped to cos(β + α) = 1 (now with α < 0) (left band of each plot in figure 3). After discussing in detail all the possible wrong sign scenarios we will return to the discussion of figure 3 in Sect. IV C.

III. THE WRONG SIGN LIMITS OF THE CP-CONSERVING 2HDM
In this section we will classify the possible sign changes that can occur (for each scenario) in the Higgs couplings to fermions and massive gauge bosons, relative to the corresponding SM Higgs couplings.

A. The light Higgs scenario
We start by discussing the scenario where m h = 125.9 GeV. As discussed in the previous section, in this scenario, models II and F have two disjunct allowed regions. One corresponding to the alignment limit and the other one centred around the line sin(β + α) = 1. With our conventions, the latter corresponds to the situation where the Higgs coupling to down-type quarks changes sign relative to the SM, while couplings to up-type quarks and massive gauge bosons are the SM ones. This is the wrong sign limit [3] (see also [42,46,47] for the CP-conserving 2HDM and [48] for the complex 2HDM) and it is imposed only at tree-level. The wrong sign scenarios were first studied in the context of the 2HDM in [49,50].
We will now analyse the limit sin(β + α) = 1 for all Yukawa types. The main goal is to understand if a sign change in a given Higgs coupling can be measured at the LHC, being therefore distinguishable from the alignment limit. The Yukawa coupling signs for the different model types, when sin(β + α) = 1, are shown in Table I. In order to probe sign changes in the Higgs couplings we need processes where interference occurs. The best way to probe a sign change in the Yukawa sector is to use the effective hgg vertex. The amplitude for both the gg → h production process and the h → gg decay is the sum of two contributions (considering only the third generation), one with a top-quark loop and the other one with a bottom-quark loop. Therefore if κ U κ D < 0 the interference term changes sign relative to the SM, so, in principle, the signal rates may be substantially different from the SM value as to allow for a discrimination between the 2HDM and the SM. In fact, focusing on the types II and F, for which this wrong sign scenario may occur, the ratio between the two LO widths in the exact limit sin(β + α) = 1 is As discussed in [3], this interference effect, almost 30% relative to the SM, is not so strong in the gg → h production process, which is the main Higgs production mode at the LHC. In fact, in contrast with the LO result, at NNLO in the limit of sin(β + α) = 1, we have while the ratio of the partial widths of h → gg does not suffer any significant change in going from LO to NNLO. In order to test the stability of the ratio (15), we have performed the calculation with two PDF sets, MSTW2008nnlo68cl.LHgrid [51] and CT10nnlo.LHgrid [52] and we have varied the factorization and renormalization scales (taken equal) from m h /4 to m h (all tests were performed with HIGLU.). The maximal variation was with the scales and it ranged from 1.122 to 1.130, that is, below 1%. For a center-of-mass energy of 14 TeV, the maximal variation was from 1.107 to 1.120, about a 1% variation. Therefore, the ratio is stable and κ g can in principle be used to distinguish between the two scenarios in model types II and F if measured with enough accuracy. However, the difference in the values of (13) and of (15) is one of the reasons why this scenario is not yet excluded at the LHC. In fact, a wrong sign cross section about 30% above the SM one, would probably have already been excluded. However the enhancement of about 12 % is not enough to exclude this scenario at present energies. When sin(β + α) = 1, the tree-level coupling to massive gauge bosons can be written as Therefore, there are two distinct regimes regarding the sign of κ V : when tan β > 1, κ V > 0, while if tan β < 1, κ V < 0. Note that when κ V > 0, tan β 1 implies κ V ≈ 1; on the contrary, if κ V < 0 because tan β 1 is disallowed, κ V can never reach the alignment limit. In fact, even for very small tan β, say 0.5, we would get κ V = −0.6 and therefore a value of κ 2 V quite far from 1. We will come back to this point later. The other effective vertex with interference being measured at the LHC is the hγγ coupling. In this case, besides the fermion loops we have the W-loop and also the charged Higgs loop contribution, where a new vertex, g hH ± H ∓ , comes into play. In the notation of [3], in the wrong sign limit, the coupling g hH ± H ∓ takes the form [49] g hH ± H ∓ = − tan 2 β − 1 tan 2 β + 1 Hence, as discussed in [3], when κ V > 0 the charged Higgs contribution approaches a constant (negative) value and reduces the value of Γ(h → γγ). However, when κ V < 0, this contribution is positive and can be very close to zero (it is exactly zero when tan β = 1). Therefore, Γ(h → γγ) is no longer reduced by the charged Higgs loop contribution when κ V < 0. By examining Table I we can now enumerate the wrong sign scenarios that could in principle be probed in each model. This is shown in Table II, where κ L was left out because there is no relevant interference term contributing to either κ g or κ γ . In conclusion, the wrong sign scenario can be defined as either κ D κ V < 0 (for tan β > 1) or No κD κV < 0 κD κV < 0 No κD κU < 0 κD κU < 0 tan β < 1 κU κV < 0 κU κV < 0 κU κV < 0 κU κV < 0 κD κU < 0 κD κU < 0 κ U κ V < 0 (for tan β < 1). We can further have (for types II and F) κ D κ U < 0, in which case both κ g and κ γ are affected, otherwise, if κ D κ U > 0 (and κ V < 0) , only κ γ is affected.

B. The heavy Higgs scenario
In the scenario where we set the heaviest CP-even state, H, to be the Higgs, i.e. m H = 125.9 GeV, the alignment limit is obtained by setting cos(β − α) = 1. The Higgs couplings to fermions and massive gauge bosons are κ H F = κ H V = 1. In this scenario the wrong sign limit is obtained when cos(β + α) = 1. The Yukawa couplings for the different model types, in the limit cos(β + α) = 1, are shown in Table III. The Yukawa couplings have all changed sign relative to lightest Higgs scenario. It is clear that it is again type II and type F that can be distinguished from the corresponding alignment limit in κ g . It should be noted, however, that in each of the corresponding wrong sign scenarios (heavy Higgs scenario and last section's light Higgs scenario) the value of κ g is exactly the same. As for the Higgs coupling to massive gauge bosons, when cos(β + α) = 1, it is given by Again we have two distinct regimes regarding the sign of κ H V : when tan β > 1, κ H V < 0 while if tan β < 1, κ H V > 0. Therefore, also the sign of κ H V is reversed relative to that of κ V . As will be discussed later in detail, the case tan β < 1 is already excluded and it will not be further mentioned. Finally, the charged Higgs coupling for cos(β + α) = 1 is while in the heavy Higgs alignment limit we obtain where M 2 = m 2 12 /(sin β cos β). Therefore, there is a simultaneous change of sign in the Higgs couplings to massive gauge bosons and in the Yukawa couplings relative to the lightest Higgs case. That is, the wrong sign scenarios are exactly the same as the ones for the lightest Higgs case. This is true even for the charged Higgs coupling to the heavy Higgs. As we will see, no major difference is found in the results regarding the wrong sign limits relative to the light Higgs scenario.

IV. THE PRESENT STATUS AND THE FUTURE OF THE DIFFERENT WRONG SIGN SCENARIOS
Throughout this section we will use (as we did in our previous work [3]) the expected errors for the 14 TeV LHC, Tables 1-20 of [53], as a reference. The quoted expected errors for κ g based on fittings are 6-8% for L = 300 fb −1 and 3-5% for L = 3000 fb −1 . The predicted accuracy for κ γ is 5-7% for an integrated luminosity of L = 300 fb −1 and 2-5% for L = 3000 fb −1 . For comparison, the predicted accuracy at the International Linear Collider can be found in [54,55].

A. Light Higgs scenario for tan β > 1
This scenario was discussed in detail in a previous work [3]. There, we have analysed the case where the lightest Higgs is the SM one in a type II model. We have forced all rates to be within 20, 10 and 5% of the SM predictions. We have concluded that measurements of either κ g and κ γ with 5% accuracy would enable us to distinguish between the alignment limit and the wrong sign scenario. In this section we show the status of κ g and κ γ at the end of the 8 TeV run. In the left panel of figure 4 we show κ γ as a function of κ D and in the right panel we can see κ g as a function of κ D . All points are for type II with tan β > 1 and have passed both the pre-LHC constraints and the 7/8 TeV LHC Higgs data at 1σ (blue), 2σ (red) and 3σ (green). It is clear that, at the end of the 8 TeV run, the wrong sign scenario is still allowed at 1σ. This was expected because, as discussed in [3], we need the 13/14 TeV LHC with at least an integrated luminosity of 300 f b −1 to exclude this particular wrong sign scenario. Finally, although not exactly the same, the plots for type F look very similar, and there is no point in showing them here. As discussed in [3] there is no wrong sign scenario in types I and LS for tan β > 1.

B. Light Higgs scenario in the low tan β regime
We will now analyse the lightest Higgs wrong sign scenarios for tan β < 1 for the type II model. As discussed earlier, B-physics constraints and R b force tan β > O(1) although values of tan β slightly smaller than 1 are still  allowed depending on the charged Higgs mass. We will come back to this point later.
In order to understand if the different wrong sign scenarios are still allowed for tan β < 1 after the 8 TeV LHC, we have first generated a separate set of points with the same experimental and theoretical constraints applied as before, but where we have turned off the R b and B-physics constraints and the experimental constraints coming from colliders (LHC, Tevatron and LEP). In addition we also impose 0.5 < tan β < 1 which is the region of interest that we will discuss below in the full sample with all constraints turned on.
In figure 5, left panel, we plot µ V V as a function of µ γγ for type II without (blue) and with (green) the cut κ V κ U < 0. It is clear that the wrong sign-scenario κ V κ U < 0 will be very constrained if the values of µ V V measured at the LHC are taken into account. In fact whatever the rates for µ γγ and µ F F , µ V V is always well below 1 when κ V κ U < 0 in this region of small tan β.
Due to our conventions for the angles, κ U is always positive and therefore the region where κ V κ U < 0 corresponds to sin(β − α) < 0. This is shown in the right panel of figure 5 where sin(β + α) as a function of sin(β − α) is presented for the same scenario. Not only the green points are all in the sin(β − α) < 0 region but it is clear that when sin(β + α) ≈ 1 the allowed values of sin(β − α) are quite far from 1, forcing µ V V to be well below 1 and thus contradicting the LHC results.
The second possibility is to have κ V κ D < 0. In figure 6, left panel, we plot µ V V as a function of µ γγ using the same color key but now the green points correspond to the cut κ V κ D < 0. The latter are distributed around two regions. The first one is similar to the one in figure 5  region corresponds to larger values of sin(β − α). The W and top loops give the largest contribution to h → γγ and interfere destructively. When κ V is reduced (although larger than in the previous scenario, it is always below ≈ 0.7), because tan β < 1, κ U is enhanced and the amplitude is reduced (taking κ V = 0.7 and tan β = 0.7, κ U ≈ 1.7). Hence, it is foreseeable that both scenarios are already excluded by the LHC data analysed so far. Clearly, the scenario of low tan β in the case of κ V κ D < 0 is indeed excluded as was shown in [3]. In fact, only for large tan β does sin(β − α) approaches sin(β + α) thus leading to values of the rates closer to the SM ones. Therefore, this scenario is allowed only for large values of tan β.
Let us now turn back to the first possibility κ V κ U < 0 and investigate the points that survive in the full scan 4 . In figure 7 (left) we present tan β as a function of sin(β − α) for the type II model and tan β < 1 with all LHC data analysed so far taken into account at 2σ (red), 3σ (green) and 4σ (yellow). Only points with very low tan β survive and only from 2σ onwards. In this figure 7 we have taken into account all constraints, except the ones from the B-B mixing data. In the right panel of figure 7 we present m H ± as a function of m A . The main purpose is to show that the values that give rise to the allowed 2σ points require a large m H ± close to the unitarity limit. This is a consequence of the structure of the vertex tbH ± which gauges the new physics contributions in loop processes where the W -loop is replaced by a charged Higgs loop. Hence, the constraints coming from B-physics are typically exclusion regions in the (tan β, m H ± ) plane. As previously discussed, because κ U is positive in our convention, we have κ V < 0 and, as seen in the left panel of figure 7, its value is well below 1 meaning that it is therefore very hard to satisfy the LHC bounds on µ V V .
Let us now discuss in more detail the constraints available from B-physics. Contrary to type I and type LS, where b → sγ forces tan β > 1, in type II (and type F) values slightly below tan β = 1 are still allowed for large charged Higgs masses. The main B-physics observables that provide an exclusion of the small tan β region are R b , that was included as a filter at 95% in figure 7, and the B-B mixing data, that was not included. The constraint from B-B mixing is derived from the measurement of ∆m d and ∆m s [56]. In the SM, neutral-meson mixing occurs due to a box diagram with W -boson exchange. In the 2HDM the box contains new contributions due to the charged Higgs bosons, which are obtained by replacing one or two W -boson lines by charged Higgs lines (expressions for such contributions at leading-order can be found in [56]). The presence of the new diagrams implies that, in the 2HDM, the CKM matrix parameters should be determined from the data simultaneously with the 2HDM parameters in the diagrams (i.e. the charged Higgs mass m H ± and tan β). This modifies the SM fit to fix the CKM matrix [57,58]. When performing this simultaneous fit, the constraints on the (m H ± tan β) plane become less restrictive as shown in [25]. In figure 8 [25] we present the exclusion lines obtained from R b and from the B-B mixing data at 95 % C.L., on the (m H ± , tan β) plane.
Even if the simultaneous fit relaxes the bounds on tan β, we have concluded that the inclusion of the B-B constraints in the analysis at 95 % C.L., makes the entire region in the left plot of figure 7 vanish. Hence, the B-physics constraints exclude this particular wrong sign scenario at 95 % C.L.. However, it is clear that even without the constraints coming from loop processes, this scenario will be definitely excluded by the data obtained during the next run of the LHC, assuming that all measurements converge to the SM values with higher precision.

C. Heavy Higgs scenario
We now return to the discussion of the wrong sign cases for the heavy Higgs scenario, focusing on tan β > 1. The case where the heaviest CP-even Higgs is the scalar state that was observed at the LHC was first analysed in [44], in the context of the 2HDM and it was discussed after the LHC 8 TeV run in [59]. As previously discussed, the type of wrong sign limits are exactly the same in the light and in the heavy Higgs scenario. Therefore, the only possible difference between the two scenarios could only come from fact that the parameter spaces scanned are not exactly the same. In fact, in the heavy scenario there is a CP-even scalar with a mass below 125 GeV which alters the conditions of the scan. However, the general trend is the same and in figure 9 we present κ γ , for the heavy Higgs scenario, as a function of κ D and also as a function of κ D κ V with all points that have passed both the pre-LHC constraints and the 7 and 8 TeV LHC Higgs data at 1σ (blue), 2σ (red) and 3σ (green), for the type II model (left) and type F model (right). Qualitatively, there seems to be no major differences when we compare these results with the ones obtained for the light Higgs case. In the next section, while discussing the non-decoupling nature of this scenario we will see that even the conclusions regarding the exclusion of the wrong sign scenario with a 5% precision measurement of the rates also apply to the heavy Higgs case. Moreover, the reason for this exclusion is exactly the same in the two scenarios -a decrease in Γ(h → γγ) due to the charged Higgs loop contribution in the wrong sign limit.

V. THE NON-DECOUPLING NATURE OF THE HEAVY SCENARIO
In this section we investigate what a measurement of the rates within 5, 10 and 20% of the SM value could tell us about the heavy Higgs scenario. We start with type II (the results for type F are very similar) and our first goal is to understand if the light Higgs and heavy Higgs scenarios could be distinguished at the LHC. From now on we drop the superscript H in κ H i . In figure 10 we show the predicted allowed space for type II with all rates within 5, 10 and 20% of the SM values. In the left panel we present the light Higgs scenario while in the right panel one can see the heavy Higgs case. Because the loop integrals are exactly the same in the two wrong sign limits (heavy and light), the values of κ g are both centred at ≈ 1.12. The main difference between the two scenarios is the shape of the allowed regions which is mainly due to the reduced size of the parameter space in the case of heavy Higgs which implies smaller allowed regions. Hence, κ g can be used to distinguish between wrong and alignment scenarios but not between the heavy and light cases. The same conclusion can be drawn from figure 11 where we compare κ γ in the two wrong sign scenarios. On the left we show the light Higgs case and on the right the heavy Higgs scenario. Clearly we see that a 5% precision would allow us to distinguish wrong sign from alignment scenarios both in the heavy and in the light Higgs case but not the two different wrong sign scenarios from each other. We recall once again that it is the reduction of the width Γ(h → γγ), which is due to the charged Higgs boson loop contribution, that ultimately decreases µ γγ below 0.95.
However This result for the heavy scenario in the alignment limit is extremely interesting as it clearly shows the nondecoupling nature of the heavy scenario. In the next section we discuss the type I model that has no wrong sign limit for tan β > 1.

VI. THE SYMMETRIC LIMIT
In figure 3 we saw that in the heavy scenario there is a region analogous to the sin(β + α) = 1 region of the light Higgs scenario. Such region is now centred on the line cos(β + α) = 1. However, regardless of the scenario we are considering, the limits sin(β + α) = 1 (light case) and cos(β + α) = 1 (heavy case), are not a priori wrong sign scenarios. That is, if we consider the type I 2HDM, none of the Higgs couplings to the remaining SM particles changes sign relative to the SM one (modulo a global sign change in the heavy scenario). Nevertheless, the shift α → −α still changes the value of κ V , which as previously seen is given by and we recall that in the heavy scenario there is a global minus sign change in the Higgs coupling when cos(β + α) = 1. We call this limit Symmetric Limit to distinguish it from the wrong sign limit because there is no sign change in the couplings even though the shift α → −α occurs. For the case of type LS the only coupling that changes sign relative to the SM is κ L which plays no role in the discussion given the predicted accuracy of future rate measurements at the LHC. For the remainder of this section we will focus on the heavy case (the discussion for the light case is similar).
Since we are taking tan β > 1, it is true for all i, j = F, V that κ H i κ H j > 0 (i, j represent either a fermion or a massive gauge boson). Hence, for type I, and tan β > 1 not only there are no sign changes but we recover exactly the alignment limit when tan β → +∞. In figure 13 we show the predicted allowed parameter space with all rates measured at 5 (red), 10 (blue) and 20% (black) where the regions centred around 1 correspond to the alignment limit and the ones around −1 correspond to the symmetric limit. Type I is shown on the left panel while type LS is in the right panel. There are two points worth discussing. First it is clear that, as the precision increases, the lower bound on tan β grows from about 4 at 20%, to 6 at 10% and finally to 8 at 5%. In type I this behaviour is also present but it is not so striking. is that even for the alignment limit there seems to appear again some kind of non-decoupling effect that excludes the low tan β region. Type I, α < 0 -Heavy 10% 5% u u FIG. 14: Predicted allowed parameter space with all rates measured at 10% (blue) and 5% (red). Left: alignment limit; right: symmetric limit.
Let us start with the first point. The symmetric limit is clearly seen in type LS for low tan β. Now one may ask if the symmetric limit could be distinguished from the alignment limit, for finite tan β, given enough precision in type I. In figure 14 we show the predicted allowed parameter space with all rates within 10% (blue) and 5% (red) of the SM predictions for the alignment limit -all points with α > 0 (left panel) and for the symmetric limit -all points with α < 0 (right panel). Noting that µ V V ≈ cos 2 (β − α) it is going to be extremely hard to distinguish the two limits except for the very low tan β region where they could both be excluded. In fact, this bring us to the second point. Why are values of low tan β excluded in the alignment limit when all rates are measured at 5%? The answer again lies in the behaviour of the H coupling to the charged Higgs bosons together with the remaining theoretical and experimental constraints. In the left panel of figure 15 we present µ γγ as a function of tan β with µ V V measured at 5%. The region where Γ(h → γγ) is below 0.95 is easily identifiable in the plot for the low tan β region. In the right panel we present the plot g Alignment HH ± H ∓ v/m 2 H ± as a function of tan β. The low values of tan β correspond to negative values of the couplings and therefore to a decrease in the two photons width. As suggested by the shape of this plot, where one sees a sharp line cutting the low tan β region, this is mainly due to a combination of the theoretical constraints imposed on the model. Finally, we stress that κ g is the same in the symmetric and in the alignment limits while κ γ shows a negligible difference in the two limits. HH ± H ∓ v/m 2 H ± as a function of tan β.

VII. CONCLUSIONS
We have discussed all the different possibilities for having a wrong sign limit in 2HDMs. A wrong sign limit is defined as a scenario where: (a) one or more Higgs couplings to SM particles change sign relative to the corresponding SM couplings; (b) this difference has physical meaning, that is, it could in principle be measured experimentally. Hence, each scenario is defined by a condition κ i κ j < 0. After listing all possible wrong sign scenario cases when the lightest Higgs is the alignment one, we have also discussed the case where it is the heaviest CP-even Higgs.
We have shown that with all the 7/8 TeV data analysed so far, the wrong sign scenarios for both type II and type F are still allowed even at 1σ. This is true not only for the lightest Higgs case but also for the heaviest Higgs scenario in the regime tan β > 1. The light/heavy Higgs scenarios are very similar except in the range of the parameter scan and in the non-decoupling nature of the heavy scenario. Although we have concluded that each of the wrong sign scenarios can be distinguished from the respective alignment limit, we have also concluded that it is hard to differentiate between the two wrong sign scenarios (light or heavy).
A possibility not previously discussed was the wrong sign scenarios for the case where tan β < 1 which is possible for all Yukawa types. Taking into account all constraints except the ones from B-B mixing data, we have shown that the LHC does allow this particular wrong sign limit for types II and F at 2σ. We have also shown that B-B mixing data at 95 % C.L. completely excludes this region. It is however possible that B-physics constraints taken at 3σ or 4σ would not exclude this scenario. In the end, the next LHC run at 13 TeV will be able to definitely exclude this region.
We have then discussed the non-decoupling nature of the heavy Higgs case. In fact, we have shown that due to a nondecoupling effect in the charged Higgs coupling to the heavy CP-even Higgs boson, there is a reduction in Γ(h → γγ) in the alignment limit, similarly to what happens for the wrong sign limit. We conclude that a measurement of µ γγ with a 5% precision would exclude the alignment limit scenario in the heavy type II case.
Finally, we have also discussed the symmetric limit in the context of the heavy Higgs scenario. This is a limit that occurs in type I (also type LS if we disregard κ L which plays no major role in the LHC results) and corresponds to the flip α → −α, and consequently to κ H U = κ H D = κ H L = −1 while κ H V = (1 − tan 2 β)/(1 + tan 2 β). Although κ H V → −1 when tan β → ∞ this case could be in principle distinguishable from −1 for finite values of tan β (we again recall that in this case −1 corresponds to the SM κ V because there is a global sign change in the limit cos(β + α) = 1). We have shown that although possible, it will be extremely hard to differentiate between the symmetric and the alignment limit.