Doubling Up on Supersymmetry in the Higgs Sector

We explore the possibility that physics at the TeV scale possesses approximate $N = 2$ supersymmetry, which is reduced to the $N=1$ minimal supersymmetric extension of the Standard Model (MSSM) at the electroweak scale. This doubling of supersymmetry modifies the Higgs sector of the theory, with consequences for the masses, mixings and couplings of the MSSM Higgs bosons, whose phenomenological consequences we explore in this paper. The mass of the lightest neutral Higgs boson $h$ is independent of $\tan \beta$ at the tree level, and the decoupling limit is realized whatever the values of the heavy Higgs boson masses. Radiative corrections to the top quark and stop squarks dominate over those due to particles in $N=2$ gauge multiplets. We assume that these radiative corrections fix $m_h \simeq 125$ GeV, whatever the masses of the other neutral Higgs bosons $H, A$, a scenario that we term the $h$2MSSM. Since the $H, A$ bosons decouple from the $W$ and $Z$ bosons in the $h$2MSSM at tree level, only the LHC constraints on $H, A$ and $H^\pm$ couplings to fermions are applicable. These and the indirect constraints from LHC measurements of $h$ couplings are consistent with $m_A \gtrsim 200$ GeV for $\tan \beta \in (2, 8)$ in the $h$2MSSM.


Introduction
Since the Standard Model is chiral, it can accommodate only N = 1 supersymmetry, as in the minimal supersymmetric extension of the Standard Model (MSSM). On the other hand, any new physics beyond the Standard Model would contain vector-like representations of the SU(2)×U(1) gauge group of the Standard Model. As such, it could accommodate N = 2 supersymmetry. One could even argue that it should possess the maximum possible degree of supersymmetry, namely N = 2. Indeed, there are plenty of theoretical set-ups that lead naturally to a chiral N = 1 supersymmetry model at the electroweak scale with a vector-like N = 2 extension at the TeV scale, including models invoking extra dimensions and superstring model constructions [1][2][3][4].
Studies of possible N = 2 extensions of the Standard Model have a long history, with considerable attention paid to the gauge and matter sectors of such models. An N = 2 vector multiplet would contain more degrees of freedom than in the MSSM. In particular, gauginos would no longer be Majorana particles, but Dirac. Moreover, additional adjoint scalar fields would appear, namely a new singlet S, triplet T and octet O. The phenomenology of the Dirac gauginos has been explored in a number of papers [5,6], and attention also been paid to the Higgs sector of an N = 2 extension of the Standard Model, which has interesting differences from the Higgs sector of the MSSM [1]. This is a natural entry point into phenomenological studies of N = 2 models, since the Higgs sector of the MSSM is necessarily vector-like, and hence readily modified to realize N = 2 supersymmetry. Moreover, the exploration of Higgs phenomenology is well underway, with important experimental constraints coming from measurements of the h(125) Higgs boson [7] and searches for the heavier MSSM Higgs bosons.
As has been pointed out in previous studies, the N = 2 version of the tree-level supersymmetric Higgs potential (2.3) contains an extra term 1 2 (g 2 1 + g 2 2 )|H 1 H 2 | 2 , which has important phenomenological consequences [1]. In particular, the masses of the Higgs bosons are independent of tan β at the tree level, and the rotation from the doublet basis H 1 , H 2 to the mass eigenstate basis h, H is trivial, so that at the tree level the N = 2 model realizes automatically the decoupling limit of the MSSM. Hence the treelevel couplings of the lighter neutral scalar Higgs boson h are necessarily identical to those of a Standard Model Higgs, and the heavier neutral scalar boson H plays no role in electroweak symmetry breaking.
These observations are modified by the radiative corrections to the Higgs sector, of which the most important are those due to the top-stop sector, as in the MSSM 1 . As in the MSSM, a practical way to analyze Higgs phenomenology in the model with N = 2 supersymmetry is to use the measured mass of the observed Standard Model-like Higgs boson m h 125 GeV as a constraint on the other parameters of the model. In the MSSM case, this has been called the hMSSM scenario: the analogous scenario we propose here is termed the h2MSSM scenario.
As we show, an important difference between the hMSSM and h2MSSM scenarios is that the latter can be realized with smaller stop masses than the former for any value of m A m h , and for smaller m A for any fixed values of the stop masses and tan β. This observation then raises the question how light the heavier Higgs bosons H, A can be in the h2MSSM, for what range of tan β.
The LHC constraints on H → W + W − , Z 0 Z 0 and A → Zh decays are not relevant for the h2MSSM, since it realizes automatically the decoupling limit at the tree level, and the HW + W − and HZ 0 Z 0 couplings induced at the loop level are relatively small. On the other hand, LHC constraints on decays of the heavy Higgs bosons into fermions are in principle relevant. Specifically, the constraint from the search for H ± → τ ± ν decays is the same as in the hMSSM. Before saying the same for the LHC constraint on A/H → τ + τ − , one must check the near-degeneracy of the H and A, as assumed in the experimental analyses. As we show, in the h2MSSM m H − m A is actually typically significantly smaller in magnitude than in the hMSSM. Consequently, the LHC constraints on A/H → τ + τ − are directly applicable to the h2MSSM.
Also, measurements at LHC Run 1 of the couplings of the h(125) to fermions impose important indirect constraints on the h2MSSM in the (m A , tan β) plane, though they are weaker than in the hMSSM. As we show, the principal constraints are those on the ratios of h couplings to up-type quarks, down-type quarks and massive vector bosons, and that on the hγγ coupling. We find that the direct searches for heavy Higgs bosons exclude ranges of m A when tan β 7, and the h coupling measurements require m A 185 GeV in the h2MSSM, compared with 350 GeV in the hMSSM.
This paper is organized as follows. In Section 2, we show the differences between the MSSM and the N = 2 Higgs sector, at the tree level in Section 2.2 and including radiative corrections in Section 2.3, and we use the dominant loop corrections from the stop sector in both the hMSSM and the h2MSSM to evaluate possible stop masses in Section 2.4. Constraints from the LHC are studied in Section 3, where we discuss the current direct constraints from searches for H, A and H ± in Section 3.1, bounds on the N = 2 Higgs sector from hff , hW + W − and hZ 0 Z 0 couplings in Section 3.2 and those from the hγγ and hgg couplings in Section 3.3. We also discuss the sizes of anomalous couplings of h(125) that could be constrained by future measurements in Section 3.4. We conclude in Sec. 4. 2 The N = 2 Supersymmetric Higgs Sector

Model Framework
The Lagrangian for an N = 2 extension of the Standard Model possesses an R symmetry, and its SU(2) R ×U(1) N =2 R -invariant form can be written in the N = 1 language as [3,4]: where Φ V ≡ Φ a V T a and V ≡ V a T a , where the T a are the gauge group generators. The second F -term in the upper line of (2.1) is the superpotential, whose only free parameter is the gauge coupling constant g. The coupling constant of the Yukawa term in the superpotential is determined by the gauge coupling due to the SU(2) R global symmetry. The SU(2) R symmetry forbids any chiral Yukawa terms, so that fermion mass generation in the N = 2 sector is linked to supersymmetry breaking. We note also that the U(1) N =2 R symmetry forbids any mass terms of the form W 2 µ XY , and specifically that the usual N = 1 µ term W ∼ µH 1 H 2 is forbidden by the full R-symmetry. A theory with no µ-term would lead to unacceptably light charginos [9], but couplings of the Higgs multiplet to the adjoint scalars of an N=2 gauge sector provide mechanisms to lift the chargino masses and additional µ-like contributions to the scalar potential [6]. Note that, unlike the SU(2) R global symmetry, the U (1) N =2 R symmetry can survive supersymmetry breaking.
Finally, the N = 2 Higgs sector belongs to a hypermultiplet H = (H c , H) whose interactions with the gauge sector are given by the Lagrangian In the following we analyze the phenomenology of this N = 2 framework for the Higgs sector of the MSSM.

Tree-Level Analysis
We can write the tree-level N = 2 Higgs potential in the usual MSSM notation where H 1,2 are the lowest components of the chiral superfields H and H c respectively. The H 2 field gives masses to up-type quarks and the H 1 field gives masses to down-type fermions. The potential for these neutral components of the Higgs doublets is where m 2 i = m 2 H i + µ 2 are the effective low-energy mass parameters including the soft supersymmetry-breaking and µ terms. In the last line of (2.3), the first quartic term is the usual D-term of the N = 1 MSSM, whereas the second is a specific N = 2 effect. This extra quartic term in the potential has interesting consequences for the minimization of the potential and the Higgs spectrum, as we now review.
The conditions to have a vacuum that breaks electroweak symmetry with the correct value of m Z for a specific value of tan β are: We note the difference between (2.5) and the corresponding MSSM minimization condition m 2 A = m 2 H 1 + m 2 H 2 + 2µ 2 , which has the consequence that the value of m A in the N = 2 model is larger than that in the MSSM for the same input mass parameters.
In the (H 1 , H 2 ) basis for the two Higgs doublet fields, the CP-even h/H mass matrix can be written in terms of the Z and A boson masses and the angle β. In the MSSM, the tree-level mass-squared matrix is On the other hand, if the Higgs sector has N = 2 supersymmetry, the tree-level masssquared matrix is [1]: where we note the crucial change: Z in the off-diagonal terms from the MSSM case (2.6) 2 . The eigenvalues of the matrices (2.6, 2.7) correspond to the physical masses-squared of the neutral CP-even Higgs bosons. In the MSSM case they are (2.8) and the mass of the charged Higgs boson is at the tree level 3 , whereas in the N = 2 case they are 10) and the charged Higgs boson mass is We see that, as in the MSSM, the spectrum of the N = 2 Higgs sector is controlled by m A . However, in contrast to the MSSM, it has no dependence on tan β at the tree level. The left panel of Fig. 1 shows the tree-level N = 1 MSSM CP-even neutral Higgs boson masses as functions of m A for different values of tan β, and we see that m h increases with tan β, its upper limit being m Z . The right panel of Fig. 1 shows the corresponding N = 2 CP-even neutral Higgs boson masses at the tree level, where we see that m h = m Z independently of m A and tan β, and that m H crosses m h without the 'level repulsion' effect seen in the left panel.
The physical CP-even Higgs bosons are obtained from the Higgs doublet fields (H 1 , H 2 ) by rotation through an angle α: MSSM@ Tree-Level The MSSM mass-squared matrix (2.6) is diagonalized by the following mixing angle: which satisfies the relation −π/2 ≤ α ≤ 0. On the other hand, the N = 2 mass matrix (2.7) is diagonalized by the following mixing angle: 14) which also satisfies the relation −π/2 ≤ α ≤ 0. This implies that at the tree level the N = 2 theory realizes automatically the decoupling limit, in which the lighter CP-even neutral Higgs boson h has Standard Model-like couplings and the heavier one, H, does not couple to gauge bosons.

Radiative Corrections
In our approach, the Higgs sector is described in terms of just the parameters entering the tree-level expressions for the masses and mixing, supplemented by the experimentallyknown value of m h . In this sense, the hMSSM and h2MSSM approaches can be considered as 'model-independent', as the predictions for the properties of the Higgs bosons do not depend on the details of the unobserved supersymmetric sector. We write the mass matrix for the neutral CP-even states as where the tree-level matrix M 2 tree is given in (2.6) and (2.7) for the MSSM and its N = 2 extension, respectively, and the ∆M 2 ij are the radiative corrections. The importance of radiative corrections is manifested by the experimental measurement m h = 125 GeV. The most important quantum corrections to the CP-even neutral Higgs masses come from top and stop loops, which alter only the ∆M 2 22 element of the mass-squared matrix. In the MSSM we have: where M SSM depends on the top quark mass, the stop masses through the combination M SU SY ≡ √ mt 1 mt 2 , and the mixing parameter in the stop mass matrix, X t . A useful approximate expression for M SSM is: ( 2.17) In general MSSM models, the value of m h is a complicated function on the model parameters, particularly if one takes into account two-and more-loop effects.
Other radiative corrections to the Higgs mass matrix have been studied in [10,11]. Direct analysis of the dominant one-loop contributions from top-stop loops shows that the corrections to the ∆M 2 11 and ∆M 2 12 elements of the CP-even Higgs mass matrix are proportional to powers of the quantity µX t /M 2 SU SY . Consequently, they are negligible to the extent that µX t /M 2

1.
In MSSM-like scenarios with M SU SY up to a few TeV, the consideration of the full one-loop contributions or of the known two-loop contributions does not alter this simple picture 4 . When the SUSY scale is very large, additional checks on the value of m h are required at low tan β, for which a comparison with an effective field theory calculation is necessary. Results of such an analysis [12] indicate that, even in such heavy-M SU SY scenarios, the predictions of the hMSSM agree within a few percent with the exact results for m H , α and λ Hhh , as long as the condition µX t /M 2 SU SY 1 is satisfied. For the purposes of our N = 2 study here, which is restricted to the Higgs sector, we follow the philosophy proposed in [10,11], in which the hMSSM scenario was introduced to discuss the N = 1 MSSM Higgs sector. The idea is again to use the known output m h instead of the unknown input , adjusting so as to obtain m h = 125 GeV. Here we extend this idea to the N = 2 case, in a scenario we call the h2MSSM.
In the N = 1 case, diagonalizing the one-loop corrected mass-squared matrix (2.16) and requiring that one of the eigenvalues of the mass matrix be m h = 125 GeV yields the following simple analytical formula for : In this hMSSM approach the mass of the heavier neutral CP-even H boson and the mixing 4 For more details about this particular point, the reader should consult references in [11].
7 angle α that diagonalises the h, H states are given by the following simple expressions: 19) in terms of the inputs m A , tan β and the mass of the lighter CP-even eigenstate m h = 125 GeV. Turning now to the N = 2 Higgs sector, we can perform the same analysis as before, starting with the mass matrix (2.20) Requiring m N 2 h = 125 GeV, we then obtain ( 2.21) The heavier CP-even mass-squared eigenvalue and the rotation angle of the mass matrix are then found to be We note that in both the hMSSM and the h2MSSM scenarios there is the same minimal value for m A : The general form of the one-loop stop/top contribution to the ∆M 2 22 element of the CP-even Higgs mass matrix, M SSM , is the same as in the N = 1 MSSM, see (2.17), and one can apply the same arguments about the relative unimportance of other MSSM loop contributions.
However, in the N = 2 Higgs sector, there are additional loop contributions to the CP-even mass matrix from singlet and triplet adjoint scalars. We use the estimate of their contribution from [8,13], where more details about the assumptions behind this estimate can be found: where m S , m T are the masses of the adjoint singlet and triplet scalars, respectively. In the last line of (2.24) we show the limiting value when these additional scalars are degenerate in mass. In our approximation, the total radiative correction to the mass matrix is then N 2 = M SSM + ∆ N 2 . The relative orders of magnitude of these two pieces can be estimated from their ratio when the adjoint singlet and triplet are mass degenerate: . (2.25) This shows that ∆˜ N 2 is relatively unimportant for our current purposes: in our subsequent numerical estimates we use m S = m T = 1 TeV as a default. Fig. 2 displays the differences between the hMSSM scenario in the N = 1 case and the h2MSSM scenario in the N = 2 case. The left panel of Fig. 2 compares the values of the mass of the heavier CP-even Higgs boson H in the h2MSSM (red curve) and the hMSSM (green curve) as functions of m A for tan β = 1. We see that the H boson has quite a different mass in the h2MSSM as compared to the hMSSM. An interesting point is that, in both scenarios, m H diverges for some specific value of m A slightly above 125 GeV, the exact value depending on tan β as shown in (2.23). This corresponds to the fact that there is no value of that satisfies the requirement m h = 125 GeV for a region of the (m A , tan β) parameter plane. However, in the N = 2 h2MSSM scenario, the divergence in the required value of m H is less severe.
The eagle-eyed reader will notice that the red curve for m H in the left panel of  it is a general feature of the h2MSSM that m H − m A is smaller than in the MSSM. In the middle panel of Fig. 2, we plot the mass splitting m H − m A in the h2MSSM as a function of m A for tan β = 3 (red curve). The right panel of Fig. 2 shows the corresponding calculation of the mass splitting m H − m A in the h2MSSM as a function of tan β for m A = 300 GeV (red curve). The similar feature of a smaller magnitude is again apparent. The fact that m H − m A is small is relevant to the LHC experimental searches for H/A → τ + τ − that we discuss later, since they assume that this mass difference is smaller than their experimental resolution. Fig. 3 displays contours of cos 2 (β−α) in the (m A , tan β) plane for the hMSSM scenario (left panel) and the N = 2 h2MSSM scenario (right panel). This quantity determines the coupling of the heavier CP-even Higgs boson H to the electroweak gauge sector. We can see that this coupling is significantly reduced in the h2MSSM, compared to the hMSSM, reducing the impact of the experimental constraints, as we also discuss later.

The Stop Sector in the hMSSM and the h2MSSM
Thus far, we have simply assumed that the stop sector is such that m h = 125 GeV. Now we study what properties the stop sector must have in order for this to be possible. We recall from (2.17) that the two relevant parameters in M SSM are M SU SY and X t . As can be seen there, the radiative correction increases monotonically with M SU SY , but depends in a nontrivial and nonlinear way on X t . This means that any statement about the required size of M SU SY is dependent on the assumed value of X t , and more than one value of X t may yield m h = 125 GeV with the same value of M SU SY . These remarks apply to both the hMSSM and the h2MSSM. Looking at Fig. 1, however, we recall that the tree-level value of m h is larger in the N = 2 extension of the MSSM than in its N = 1 version. This implies that the required magnitude of M SSM is smaller in the h2MSSM than in the hMSSM and hence that, for any fixed value of X t , the required value of M SU SY is also smaller, as we now discuss in more detail. We display in Fig. 4 the values of M SU SY that are required in the hMSSM (green dotted lines) and the h2MSSM (red full lines) to yield m h = 125 GeV, as functions of X t /M SU SY . The first point visible in these plots is that the required value of M SU SY is very sensitive to X t , in both scenarios. It is occasionally said that m h = 125 GeV requires, within the MSSM, values of M SU SY in the multi-TeV range. We see that this is true in the hMSSM for X t = 0 and tan β = 1 (left panel), but is not true in general. For example, as seen in the middle panel, for most values of X t , M SU SY < 1000 GeV is sufficient in the hMSSM if tan β = 3, and even M SU SY < 600 GeV for a suitable choice of X t . The trend to lower M SU SY continues for tan β = 10 (right panel) and larger.
However, the key new point of our analysis is that the required values of M SU SY are indeed significantly lower in the h2MSSM than in the hMSSM. Some caveats are in order. As discussed earlier, in this analysis we consider only the stop contributions to the ∆M 2 22 element in the CP-even Higgs mass matrix. However, as argued previously, the contributions to other entries in this mass matrix are subdominant, at least for small µ. Secondly, we have neglected two-and multi-loop effects, but these should not change our qualitative results. Finally, as also argued previously, the specifically N = 2 one-loop corrections due to the adjoint scalar fields are also expected not to affect significantly our results: for definiteness, we have chosen m S =m T = 1 TeV The left panel is for X t = 0, and the right panel is for the maximal-mixing scenario with X t = √ 6M SU SY . The grey areas correspond to the region disallowed in our scenarios, cf, (2.23).
in the h2MSSM plots in the right panels of Fig. 4.
A different way of visualizing our results for the hMSSM and h2MSSM is shown in Fig. 5. Comparing the two panels, we see that much lower values of M SU SY are required for the maximal-mixing scenario X t = √ 6M SU SY (right panel) than for X t = 0 (left panel). However, the most striking and novel feature is that, as remarked above, the h2MSSM requires much smaller values of M SU SY . When X t = 0 (left panel), for tan β ∼ 3 in the hMSSM values of M SU SY ∼ 2000 GeV are required, whereas M SU SY > 1000 GeV are sufficient in the h2MSSM. In the maximal-mixing scenario these values are reduced to M SU SY ∼ 900 GeV in the hMSSM and M SU SY ∼ 250 GeV in the h2MSSM.

Constraints from LHC Measurements
In light of these differences between the masses and couplings of the Higgs bosons in the h2MSSM and hMSSM, we now examine the impacts of LHC constraints in the (m A , tan β) plane.

Constraints from H/A/H ± searches
Since the mixing angle of the tree-level scalar mass matrix is exactly α = β − π/2 in the h2MSSM, the heavy Higgs bosons decouple from pairs of gauge bosons at this level, and the loop-induced HW + W − , HZ 0 Z 0 and AZh couplings are relatively small. The limits in the (m A , tan β) plane of the N = 1 hMSSM coming from H decays to W + W − and Z 0 Z 0 and A decay to Zh [10,14] are therefore not applicable to the h2MSSM. Only the constraints from H, A and H ± couplings to Standard Model fermions are applicable to the h2MSSM. As we have seen, the H − A mass difference is smaller in the h2MSSM than in the hMSSM, so the LHC constraints on A/H → τ + τ − are applicable without modification. This is shown in Fig. 6 as a grey excluded region excluding a range of m A for tan β 7. We do not display the constraint from H ± → τ ± ν searches, which exclude a small region at small m A and large tan β that is contained within the grey area [10].

Constraints from h(125) Coupling Measurements
The couplings of the Standard Model-like Higgs boson h(125) [7] can be analysed using the following effective field theory (EFT): where y t,c,b,τ = m t,c,b,τ /v are the Standard Model Yukawa couplings in the mass eigenbasis, the subscripts L/R label the left and right chirality states of the fermions, and we consider only the fermions with the largest couplings to the Higgs boson. The quantities g hW W = 2m 2 W /v and g hZZ = m 2 Z /v are the couplings of h to the electroweak gauge bosons, and v is the vacuum expectation value of the Higgs field. The parameters κ X are the free parameters of this EFT.
These parameters can be constrained using the Higgs signal strengths in various channels, denoted by XX: as measured in all the Higgs production/decay channels available from the LHC Run 1. A full analysis requires performing an appropriate three-parameter fit in the threedimensional (κ V , κ t , κ b ) space, where we assume that κ c = κ t , κ τ = κ b , which is consistent with the current experimental accuracies, and κ V = κ W = κ Z , the custodial symmetry relations that should hold to a good approximation in the supersymmetric models of interest. In our two supersymmetric models, the N = 1 MSSM and the N = 2 h2MSSM scenario, the κ parameters take the following similar forms: where α is the rotation angle that diagonalizes the Higgs mass-squared matrix in the hMSSM or h2MSSM, respectively, after including the dominant one-loop radiative corrections as discussed above. The expressions (3.3) do not include the effects of subdominant loop corrections, which may not be negligible if the supersymmetric particles are not very heavy, in which case there are direct radiative corrections to the Higgs couplings that are not contained in the expression of the mass matrix. We neglect such possible effects in the present study. At tree level, α only depends on two unknown quantities, namely tan β and m A . Moreover, only two of the three quantities κ V , κ t and κ b are independent. This is still the case when we include the dominant one-loop radiative corrections and fix m h = 125 GeV as discussed above. In both the hMSSM and the h2MSSM we can derive κ V (tan β, m A ), κ t (tan β, m A ) and κ b (tan β, m A ) for any pair of values of (tan β, m A ).
The values may be derived by plugging the explicit expressions for α M SSM in (2.19) and α N 2 in (2.22) into (3.3). Alternatively, one can proceed directly from the MSSM or N = 2 mass-squared matrix, associating the mass eigenvalue m h with the normalized eigenvector V h = (V h1 , V h2 ) such that the physical field is h = V hi H i with i = 1, 2 and the mass eigenvalue m H with the normalized eigenvector V H = (V H1 , V H2 ) such that the physical field is H = V Hi H i with i = 1, 2. We then have 3.4) In terms of tan β we find 3.5) where in the case of the hMSSM: and in the case of the N = 2 h2MSSM: These results can be used to apply the constraints on Higgs couplings derived from a combination of CMS and ATLAS data at Run1 [15]. In particular, the analysis relevant to constraining the hMSSM and h2MSSM scenarios tests for deviations from the Standard Model in couplings to up-and down-type quarks and to vector bosons via the ratios λ du and λ V u : The results of this fit are shown in Fig. 6, where the excluded region in the hMSSM lies to the left of the green line, whereas in the N = 2 case the bounds (in red) are very much weakened.
We conclude from Fig. 6 that m A 200 GeV is allowed in the h2MSSM for tan β ∈ (2,8), whereas m A 350 GeV would be required in the hMSSM.

Constraints from Γ(h → gg, γγ)
We now analyze the corrections to the couplings of the SM-like Higgs boson to gluons and photons that arise at the loop level, and the corresponding constraints on the hMSSM and h2MSSM.
The decay width of the Standard Model-like h(125) into pairs of gluons and photons can be expressed as [16,17]: 3.11) where the variable τ i ≡ m 2 h /4m 2 i , m i being the mass of the particle propagating in the loop. In the case of the loops for the hgg coupling, whereas one has only contributions from quarks in the Standard Model, in the MSSM additional contributions are provided by the scalar partners of those quarks. The normalized amplitudes of these two contributions are 3.12) In the case of the loop for the hγγ coupling, in the Standard Model the W boson and charged fermions are the only contributors, whereas in the MSSM there are additional contributions from the two chargino fermionic fields, the scalar partners of the fermions and the charged Higgs boson. The normalized amplitudes of these contributions are 13) where N c is the color factor and Q f the electric charge of the fermion or sfermion in units of the proton charge. The spin 1, 1/2 and 0 amplitudes are [16] with the function f (τ ) defined as The amplitudes are real when m h < 2m i , but are complex above that threshold. In the regime τ 1, i.e., heavy masses in the loop, the amplitudes reach asymptotic values (3.16) Standard Model particle loops give finite contributions in the heavy-mass limit, whereas the new supersymmetric contributions decouple in the limit of large mass, since their amplitudes A i are divided by their masses.
As we have discussed in the previous Section, the top quark superpartners are responsible for a substantial shift in the tree-level Higgs mass of ∼ 34 GeV in the h2MSSM (and more in the hMSSM). We will focus in the following on the loop-level correction to the hgg and hγγ couplings due to the stops, neglecting other potential supersymmetric contributions. generic common adjoint scalar mass m S = m T = 1 TeV. The constraints on κ g are less severe than those on κ γ , so we do not display them in Fig. 7.
The Higgs mass requirement has, in general, zero, one or two solutions for X 2 t , and it is possible that one or more of them might be in conflict with the constraint coming from the soft masses: from which we can derive the maximum allowed value for X t , |X sof t t |, which is given by ( 3.24) When scanning the (mt 1 , mt 2 ) plane, we must ensure that our solutions in X t are below this maximal value. The grey regions in Fig.7 with dotted (full) border contours are forbidden by this consideration in the case of the hMSSM (h2MSSM). There are no values of X t able to accommodate m h = 125 GeV in the hMSSM (h2MSSM) in the regions at low mt 1 and/or mt 2 that are shaded yellow (blue). The left panels of Fig. 7 consider the maximal value of X t allowing m h = 125 GeV, including the case where there is only one possible choice for X t . The right panels of Fig. 7 consider the minimal value of X t allowing m h = 125 GeV, including the case where there is only one possible choice for X t . This explains the particular shape of the grey region for relatively high stop masses.
The current constraints on κ g,γ in the hMSSM and the h2MSSM are outlined in green (red) in Fig. 7. We see that they are generally weak. Indeed, for m A = 500 GeV and tan β = 1.5 (top two panels) there is no constraint at all. However, for higher values of tan β (middle and bottom panels) these constraints do exclude some scenarios with low supersymmetry-breaking scales.

Anomalous h(125) Couplings
In addition to these modifications of the h couplings measured in Higgs production and decay, integrating out the heavy scalars can also induce anomalous couplings of the Higgs to vector bosons with non-standard momentum dependence. One can parametrize these effects in the coupling of the Higgs to two W bosons as follows [20]: hW W W ν ∂ µ W † µν h + h.c. + g hW W W µ W † µ h . (3.25) We note that the coupling g (3) causes a shift in the usual Standard Model coupling structure. Indeed, the interpretation of the Higgs data described by the Lagrangian (3.2) corresponds to g (3) = (κ V − 1)g hW W and setting g (1,2) to zero. However, with more precise measurements of differential distributions in Run 2 one may be able to disentangle different Lorentz structures, which could give a handle for discriminating between an anomaly due to the MSSM and an underlying N = 2 supersymmetric structure. Generic expressions for the effects of one-loop scalar contributions to Higgs anomalous couplings can be found in [21]. These correspond to integrating out the heavy MSSM Figure 7: Compilation of the constraints in (mt 1 , mt 2 ) planes fixing X t so as to obtain Another interesting feature of the N = 2 extension of the MSSM is that the heavy Higgs bosons H, A, H ± decouple from the massive vector bosons W ± , Z 0 at the tree level. This observation is subject to radiative corrections, but the decoupling limit is a sufficiently good approximation that current searches for H → W + W − , Z 0 Z 0 and A → Zh do not constrain the h2MSSM significantly. On the other hand, the constraints from the decays of the heavy Higgs bosons to fermions are the same in the h2MSSM as in the hMSSM.
The most stringent constraints on the h2MSSM come from LHC Run 1 measurements of the h(125) couplings, including those to fermions, massive and massless gauge bosons. However, these constraints are considerably weaker than in the hMSSM. We find that m A 185 GeV is possible in the h2MSSM, whereas m A 350 GeV is required in the hMSSM.
Looking to the future, we have also calculated the possible N = 2 Higgs sector contributions to anomalous couplings of the h(125) boson. Current limits on these couplings do not constrain the N = 2 model, but this may be an interesting window for future measurements at the LHC and elsewhere.
Doubling up supersymmetry opens up the possibility that supersymmetric Higgs bosons and stop squarks could be significantly lighter than in the MSSM. Maybe Run 2 of the LHC will discover not just one supersymmetry, but two?