Leading two-loop corrections to the Higgs boson masses in SUSY models with Dirac gauginos

We compute the two-loop O(as*at) corrections to the Higgs boson masses in supersymmetric extensions of the Standard Model with Dirac gaugino masses. We rely on the effective-potential technique, allow for both Dirac and Majorana mass terms for the gluinos, and compute the corrections in both the DRbar and on-shell renormalisation schemes. We give detailed results for the MDGSSM and the MRSSM, and simple approximate formulae valid in the decoupling limit for all currently-studied variants of supersymmetric models with Dirac gluinos. These results represent the first explicit two-loop calculation of Higgs boson masses in supersymmetric models beyond the MSSM and the NMSSM.


Introduction
In anticipation of new results from the run II of the LHC, supersymmetry (SUSY) as a framework remains the leading candidate for physics beyond the Standard Model (SM). However, the discovery of a SM-like Higgs boson with relatively large mass and the lack of observation of coloured superparticles have spurred considerable interest in SUSY realisations beyond the Minimal Supersymmetric Standard Model (MSSM). A notable extension beyond the minimal case is to allow Dirac masses for the gauginos [1][2][3][4][5][6], in particular instead of -but possibly in addition to -Majorana ones. Among the reasons for the growing interest in this scenario are that Dirac gaugino masses relax constraints on squark masses (through suppressing production) [7][8][9] and flavour constraints [10][11][12], and that they increase the naturalness of the model (because the operators are supersoft [4] and the SM-like Higgs boson mass is enhanced at tree level [13,14]). two-loop corrections [43][44][45] as well as the dominant three-loop corrections [46][47][48] have also been obtained. 1 For the NMSSM, beyond the one-loop level only the two-loop corrections involving the strong gauge coupling together with the top or bottom Yukawa couplings, usually denoted as O(α t α s ) and O(α b α s ), have been computed [49,50]. In contrast, in other supersymmetric extensions of the SM there have been no explicit calculations of the Higgs masses beyond one-loop results.
On the other hand, the public tool SARAH [51][52][53][54][55][56] can, for a generic supersymmetric model, automatically compute the full one-loop corrections to all particle masses, as well as the two-loop corrections to the neutral-scalar masses in the limit of vanishing electroweak gauge couplings and external momenta [57,58], implementing and extending the general two-loop results of refs. [59,60]. Recently, SARAH has made it possible to analyse at the two-loop level the Higgs sector of several non-minimal extensions of the MSSM, see refs. [61][62][63][64][65]. Of particular relevance for this work, it has allowed for Dirac-gaugino masses since version 3.2 [54], incorporating also the results of ref. [66]. Indeed, SARAH has been used for detailed phenomenological analyses of the MDGSSM at one loop in ref. [25] and at two loops in refs. [15,16]; and also for the MRSSM at one loop in ref. [67] and two loops in refs. [68,69].
However, while such a numerical tool for generic models fulfils a significant need of the community, it is also important to have explicit results for specific models, and not just as a cross-check. In this work we shall compute the leading O(α t α s ) corrections to the neutral Higgs boson masses in both the MDGSSM and MRSSM, relying on the effective-potential techniques developed in ref. [36] for the MSSM and in ref. [49] for the NMSSM. This has the following advantages: • We compute the O(α t α s ) corrections in both the DR and on-shell (OS) renormalisation schemes.
The latter turns out to be particularly useful in scenarios with heavy gluinos -a feature of many Dirac-gaugino models in the literature -where the use of DR formulae for the two-loop Higgs-mass corrections can lead to large theoretical uncertainties.
• We have written a simple and fast stand-alone code implementing our results, which we make available upon request (indeed, a version of the code is already included in SARAH).
• We use our results to derive simple approximate expressions for the most important two-loop corrections, applicable in any Dirac gaugino model.
The outline of the paper is as follows. In section 2 we define the important parameters of our theory. In section 3 we present our results for the general case, the MDGSSM and the MRSSM, show how we compute the shift to the OS scheme, and give simplified formulae for the SM-like Higgs boson mass either for a common SUSY-breaking scale or for a heavy Dirac gluino. In section 4 we give numerical examples of our results, illustrating the advantages of our approach and also discussing the inherent theoretical uncertainties. We conclude in section 5. Explicit expressions for the derivatives of the effective potential are given in an appendix.
2 Definition of the theory 2

.1 Adjoint multiplets and the supersoft operator
In order to give gauginos a Dirac mass it is necessary to pair each Weyl fermion of the vector multiplets with another Weyl fermion χ Σ in the adjoint representation of the gauge group. These adjoint fermions sit inside chiral superfields, which we shall denote collectively Σ a = Σ a + √ 2 θχ a Σ + . . . , where the lowest-order component Σ a is a complex scalar. In models with softly-broken supersymmetry 2 , the Dirac gaugino mass arises only in the supersoft operator: where W a α = λ a α + θ α D a + . . . is the field-strength superfield. Integrating out the auxiliary field D a leads to mass terms for the adjoint scalars, as well as to trilinear interactions between the adjoint scalars and the MSSM-like scalars, which we collectively denote as φ: where t a are the generators of the gauge group in the representation appropriate to φ, and a sum over the gauge indices of φ is understood. Considering all sources of mass terms for the adjoint scalars, where m 2 Σ includes in general contributions from both the superpotential and the soft SUSY-breaking Lagrangian, and B Σ is a soft SUSY-breaking bilinear term. In addition, mixing with the MSSM-like Higgs scalars may be induced, upon EWSB, by the D-term interactions in eq. (2.2), as well as by superpotential interactions.
We shall denote the adjoint multiplet for U (1) Y as a singlet S = S + √ 2 θχ S + . . . , the one for SU (2) L as a triplet T a = T a + √ 2 θχ a T +. . . , and the one for SU (3) as an octet O a = O a + √ 2 θχ a O +. . . . In this paper we shall be interested only in the two-loop corrections to the Higgs masses involving the strong gauge coupling g s , thus the relevant trilinear couplings in eq. (2.2) will be the ones involving the octet scalar and the squarks.
We shall make the additional restriction that the octet scalar only interacts via the strong gauge coupling and the above trilinear terms, equivalent to the assumption that it has no superpotential couplings or soft trilinear couplings other than with itself. This shall simplify the computations, and it is true for almost all variants of Dirac gaugino models studied so far. To have renormalisable Yukawa couplings between the octet and the MSSM fields we would need to add new coloured states (such as a vector-like top). However, in the most general version of the MDGSSM there could also be terms that violate the above assumption -which have only recently attracted attention [17,71] -namely couplings between the singlet and the octet of the form The coupling λ SO is typically neglected because it violates R-symmetry and leads to Majorana gaugino masses: for example, in the restricted version of the MDGSSM or the / µSSM the R-symmetry violation is assumed to only occur in the Higgs sector and possibly only via gravitational effects. On the other hand, there is no symmetry preventing the generation of T SO , but it is typically difficult for it to obtain a phenomenologically significant magnitude, hence it has been neglected -see [17] for a full discussion (and for cases when it could be large). Furthermore, T SO is irrelevant in the decoupling limit (when the singlet S is heavy) that we shall employ later in our simplified formulae.
With the above assumptions, we can make a rotation of the superfield O a such that we can take m D to be real without loss of generality, but we cannot simultaneously require that the soft SUSYbreaking bilinear B O be real without additionally imposing CP invariance. The octet mass terms are then If B O is not real, the real and imaginary parts of the octet scalar mix with each other. Their mass matrix can be diagonalised with a rotation by an angle φ O , to obtain the two mass eigenvalues Then the trilinear couplings of the octet mass eigenstates O a 1,2 to squarksq L andq R read where t a are the generators of the fundamental representation of SU (3). These couplings lead to new (compared to MSSM and NMSSM) contributions to the two-loop effective potential involving the octet scalars which will affect the Higgs masses. We remark that, since in eq. (2.5) the superpotential mass term m 2 D affects only the real part of the octet scalar, the mixing angle φ O is suppressed by m 2 D in the limit where the latter is much larger than the soft SUSY-breaking mass terms. In particular, For the remainder of this paper, we shall restrict our attention to the CP-conserving case. This is motivated by clarity and simplicity in the calculations, and also physically in that there are strong constraints upon CP violation, even in the Higgs sector [72][73][74][75]. However, we shall make an exception in allowing a non-zero angle φ O , because it is particularly simple to do so, and its effects are only felt at an order beyond that considered here: it generates CP-violating phases in the stop mass matrix at two loops, and in the Higgs mass at three. This is because the couplings in eq. (2.8) are real, and phases only appear in the octet scalar-gluino-gluino vertex.

Gluino masses and couplings
In the case of Dirac gauginos, there is mixing between the Weyl fermion of the gauge multiplet λ a and its Dirac partner χ a Σ . We shall allow in general both Majorana and Dirac masses which, in two-component notation, we write as As mentioned in the previous section, we can define m D to be real. In general we cannot remove the phases from both M λ and M Σ ; however, as also mentioned above, we shall not consider CP violation in the gluino sector, and thus take all three masses to be real. We then rotate λ a and χ a Σ to mass eigenstates λ a 1 and λ a 2 via a mixing matrix R ij , so that (2.11) In four-component notation, this leads in general to two Majorana gauginos with different masses. In case of a pure Dirac mass, however, we obtain two Majorana gauginos with degenerate masses |m λ 1 | = |m λ 2 | = |m D | , which can also be combined in a single Dirac gaugino. We recall that in the models of interest in this paper there are no Yukawa couplings of the additional octet superfield, therefore the two gluino mass-eigenstates only couple to quarks and squarks via their gaugino component λ a . In particular, the couplings of each (four-component) gluinog a i are simply related to the couplings of the usual (N)MSSM gluino by an insertion of the mixing matrix: where a sum over the SU (3) indices of quarks and squarks is again understood. Consequently, as we shall see below, the gluino contribution to the two-loop effective potential in Dirac-gaugino models can be trivially recovered from the known results valid in the MSSM and in the NMSSM.

Higgs sector
We now consider the Higgs sector of the theory. Dirac gaugino models extend the (N)MSSM, so we shall assume that we have at least the usual two Higgs doublets H u and H d . To these we must add the adjoint scalars S and T a mentioned above, which mix with the Higgs fields. The couplings of the adjoint scalars, as well as the presence of any additional fields in the Higgs sector, will, however, depend on the model under consideration. In the following we shall focus on the minimal Dirac-gaugino extension of the MSSM, the MDGSSM, and on the minimal R-symmetric extension, the MRSSM.
In the MDGSSM there are no additional superfields apart from the adjoint ones, and the superpotential reads 14) where σ a are Pauli matrices, and the dot-product denotes the antisymmetric contraction of the SU (2) L indices. In addition to the terms explicitly shown in eqs. (2.14) and (2.15), the most general renormalisable superpotential contains terms involving only the adjoint superfields -namely, mass terms for each of them, all trilinear terms allowed by the gauge symmetries, and a linear term for the singlet -which we denote collectively as W Σ . The most general soft SUSY-breaking Lagrangian for the MDGSSM contains non-holomorphic mass terms for all of the scalars, as well as Majorana mass terms for the gauginos, plus A-type (i.e., trilinear), B-type (i.e., bilinear) and tadpole (i.e., linear) holomorphic terms for the scalars with the same structure as the terms in the superpotential. With the assumption, discussed in section 2.1, that we neglect the couplings λ SO and T SO defined in eq. (2.4), the superpotential W Σ and the soft SUSY-breaking terms that involve only the adjoint fields are not relevant to the calculation of the two-loop O(α t α s ) corrections to the Higgs masses presented in this paper, apart from contributing to the masses and mixing of the adjoint fields as discussed in sections 2.1 and 2.2 above.
In the case of the MRSSM, we must add two superfields R u and R d with the same gauge quantum numbers as H d and H u , respectively, but with different charges under a conserved R-symmetry. The superpotential reads while all terms involving only the MSSM-like Higgs superfields and/or the adjoint superfields, such as those in eq. (2.15), are forbidden by the R-symmetry. The most general soft SUSY-breaking Lagrangian for the MRSSM contains non-holomorphic mass terms for all of the scalars, plus all of the holomorphic terms involving only the MSSM-like Higgs scalars and/or the adjoint scalars (which, as mentioned above, have no equivalent in the superpotential). In contrast, the R-symmetry forbids Majorana mass terms for the gauginos, and holomorphic terms for the scalars with the same structure as the terms in the MRSSM superpotential. The requirement that the R-symmetry is conserved also means that the scalar doublets R u and R d do not develop a vacuum expectation value (vev), and do not mix with either the MSSM-like Higgs scalars or the adjoint scalars.

Two-loop corrections in the effective potential approach
In this section we adapt to the calculation of two-loop corrections to the neutral Higgs masses in Dirac-gaugino models the effective-potential techniques developed in ref. [36] for the MSSM and in ref. [49] for the NMSSM. We start by deriving general results valid for all variants of Dirac-gaugino extensions of the MSSM, then we provide explicit formulae for the MDGSSM and MRSSM models discussed in section 2.

General results
The effective potential for the neutral Higgs sector can be decomposed as V eff = V 0 + ∆V , where ∆V incorporates the radiative corrections. We denote collectively as Φ 0 i the complex neutral scalars whose masses we want to calculate, and split them into vacuum expectation values v i , real scalars S i and pseudoscalars P i as Then the mass matrices for the scalar and pseudoscalar fields can be decomposed as and the radiative corrections to the mass matrices are where v i , which we assume to be real, denote the vevs of the full radiatively-corrected potential V eff , and the derivatives are in turn evaluated at the minimum of the potential. The single-derivative terms in eqs. (3.3) and (3.4) arise when the minimum conditions of the potential, are used to remove the soft SUSY-breaking mass for a given field Φ 0 i from the tree-level parts of the mass matrices. It is understood that those terms should be omitted for fields that do not develop a vev (such as, e.g., the fields R u,d in the MRSSM).
With a straightforward application of the chain rule for the derivatives of the effective potential, the mass-matrix corrections in eqs. (3.3) and (3.4) and the minimum conditions in eq. (3.5) can be computed by exploiting the Higgs-field dependence of the parameters appearing in ∆V . We restrict for simplicity our calculation to the so-called "gaugeless limit", i.e. we neglect all corrections controlled by the electroweak gauge couplings g and g . At the two-loop level, we focus on the contributions to ∆V from top/stop loops that involve the strong interactions. In Dirac-gaugino models, this results in corrections to mass matrices and minimum conditions that are proportional to α s times various combinations of the top Yukawa coupling y t with the superpotential couplings of the singlet and triplet fields. It is therefore with a slight abuse of notation that we maintain the MSSM-inspired habit of denoting collectively those corrections as O(α t α s ).
As detailed in refs. [36,49], if we neglect the electroweak contributions to the stop mass matrix the parameters in the top/stop sector depend on the neutral Higgs fields only through two combinations, which we denote as X ≡ |X| e iϕ and X ≡ | X| e iφ . They enter the stop mass matrix as where m 2 Q and m 2 U are the soft SUSY-breaking mass terms for the stops. While X = y t H 0 u both in the (N)MSSM and in Dirac-gaugino models, the precise form of X depends on the model under consideration and will be discussed later. For the time being, we only assume that X is real at the minimum of the potential, to prevent CP-violating contributions to the Higgs mass matrices. The top/stop O(α s ) contribution to ∆V can then be expressed in terms of five field-dependent parameters, which can be chosen as follows. The squared top and stop masses a mixing angleθt, with 0 ≤θt ≤ π/2, which diagonalises the stop mass matrix after the stop fields have been redefined to make it real and symmetric and a combination of the phases of X and X that we can choose as Finally, the gluino masses mg i and the octet masses m 2 O i do not depend on the Higgs background, since we neglect the singlet-octet couplings λ SO and T SO . In the following we will also refer to θ t , with −π/2 < θ t < π/2, i.e. the usual field-independent mixing angle that diagonalises the stop mass matrix at the minimum of the scalar potential.
We find general expressions for the top/stop contributions to the minimum conditions of the effective potential and to the corrections to the scalar and pseudoscalar mass matrices: where all quantities are understood as evaluated at the minimum of the potential, no summation is implied over repeated indices, the fields are ordered as The angle β is defined as in the MSSM by tan β = v 2 /v 1 . Here and thereafter we also adopt the shortcuts c φ ≡ cos φ and s φ ≡ sin φ for a generic angle φ. The functions F, G, F 1 , F 2 , F 3 and F ϕ entering eqs. (3.10)-(3.12) are combinations of the derivatives of ∆V . Explicit expressions for most of those functions can be found e.g. in ref. [49], but we display all of them here for completeness: (3.14) (3.15) where we defined z t ≡ sign( X| min ).
O ĩ t j t k Figure 1: Novel two-loop contribution to the effective potential involving stops and octet scalars.

Two-loop top/stop contributions to the effective potential
For the computation of the two-loop O(α t α s ) corrections to the Higgs mass matrices in models with Dirac gauginos we need the explicit expression for the top/stop O(α s ) contribution to ∆V , expressed in terms of the field-dependent parameters defined in the previous section. In addition to the contributions of diagrams involving gluons, gluinos or the D-term-induced quartic stop couplings, which are in common with the (N)MSSM and can be found in ref. [36], ∆V receives a contribution from the diagram shown in figure 1, involving stops and octet scalars. We assume that the gaugino masses are real so that the diagonalising matrix R ij is real and R 2 1i is positive, but allow mg i to be negative. Since R 2 11 + R 2 12 = 1, we can simply write the top/stop O(α s ) contribution to the two-loop effective potential (in units of α s C F N c /(4π) 3 , where C F = 4/3 and N c = 3 are colour factors) as where ∆V αs MSSM is the analogous contribution in the (N)MSSM, with The two-loop integrals J(x, y), I(x, y, z) and L(x, y, z) entering eqs. (3.20) and (3.22) are defined, e.g., in eqs. (D1)-(D3) of ref. [49], and were first introduced in ref. [76]. Explicit expressions for the derivatives of ∆V αs , valid for all Dirac-gaugino models considered in this paper, are provided in appendix A. We remark that, by using the "minimally subtracted" two-loop integrals of ref. [76], we are implicitly assuming a DR renormalisation for the parameters entering the tree-level and one-loop parts of the effective potential. Consequently, our results for the two-loop top/stop contributions to mass matrices and minimum conditions also assume that the corresponding tree-level and one-loop parts are expressed in terms of DR-renormalised parameters. We will describe in section 3.5 how our two-loop formulae should be modified if the top/stop parameters entering the one-loop part of the corrections are expressed in a different renormalisation scheme. For what concerns the parameters entering the tree-level mass matrices for scalars and pseudoscalars -whose specific form depends on the Diracgaugino model under consideration -they can be taken directly as DR-renormalised inputs at some reference scale Q, at least in the absence of any experimental information on an extended Higgs sector. Exceptions are given by the electroweak gauge couplings and by the combination of doublet vevs v ≡ (v 2 1 + v 2 2 ) 1/2 , which in general should be extracted from experimentally known observables such as, e.g., the muon decay constant and the gauge-boson masses. As was pointed out for the NMSSM in ref. [50], the extraction of the DR parameter v(Q) involves two-loop corrections whose effects on the scalar and pseudoscalar mass matrices are formally of the same order as some of the O(α t α s ) corrections computed in this paper 3 . However, a two-loop determination of v(Q) goes beyond the scope of our calculation, as it requires two-loop contributions to the gauge-boson self-energies which cannot be obtained with effective-potential methods. Besides, ref. [50] showed that, at least in the NMSSM scenarios considered in that paper, the O(α t α s ) effects on the scalar masses arising from the two-loop corrections to v are quite small, typically of the order of a hundred MeV.

Mass corrections in the MDGSSM
The MDGSSM contains a singlet S and an SU (2) triplet T a which mix with the usual Higgs fields H d and H u . In this model, the stop mixing term X defined in eq. (3.6) reads where A t is the soft SUSY-breaking trilinear interaction term for Higgs and stops. We order the neutral components of the fields as Φ 0 i = (H 0 d , H 0 u , S, T 0 ) and expand them as in eq. (3.1). For the minimum conditions of the effective potential, eq. (3.10) gives For the corrections to the mass matrices of scalars and pseudoscalars, eqs. (3.11) and (3.12) give

Mass corrections in the MRSSM
The MRSSM is defined to be R-symmetric, and has fields R u , R d which pair with the Higgs fields without themselves developing vevs. In this model the gluino mass terms are purely Dirac, therefore, in our conventions, R 2 11 = R 2 12 = 1/2 and mg 1 = −mg 2 = m D . The trilinear Higgs-stop coupling A t is forbidden, and the term X defined in eq. (3.6) reads 48) and vanishes at the minimum of the scalar potential, hence the stops do not mix. Moreover, the term proportional to c ϕ−φ in the second line of eq. (3.20) cancels out in the sum over the gluino masses. As a consequence, the radiative corrections induced by top/stop loops are remarkably simple. Ordering the neutral components of the fields as Φ 0 , we find that the only non-vanishing contributions to the minimum conditions of the potential and to the Higgs mass matrices are where we definedμ u ≡ µ u + λ Su v 3 + λ Tu v 4 .

On-shell parameters in the top/stop sector
The results presented so far for the two-loop corrections to the neutral Higgs masses in models with Dirac gauginos were obtained under the assumption that the parameters entering the tree-level and one-loop parts of the mass matrices are renormalised in the DR scheme. While this choice allows for a straightforward implementation of our results in automated calculations such as the one of SARAH, it is well known that, in the DR scheme, the Higgs-mass calculation can be plagued by unphysically large contributions if there is a hierarchy between the masses of the particles running in the loops [36].
In particular, the contributions of two-loop diagrams involving stops and gluinos include terms proportional to m 2 g i /m 2 t j , which can become very large in scenarios with gluinos much heavier than the stops. Since this kind of hierarchy can occur naturally (i.e., without excessive fine tuning in the squark masses) in scenarios with Dirac gluino masses [4], it is useful to re-express the one-loop part of the corrections to the Higgs masses in terms of OS-renormalised top/stop parameters. In that case, the terms proportional to m 2 g i in the two-loop part of the corrections cancel out against analogous contributions induced by the OS counterterms, leaving only a milder logarithmic dependence of the Higgs masses on the gluino masses.
Since we are focusing on the O(α t α s ) corrections to the Higgs masses, we need to provide an OS prescription only for parameters in the top/stop sector that are subject to O(α s ) corrections, i.e. m t , m 2 ) sin 2θ t = 2 X| min (in general, the stop mixing X| min contains other terms in addition to m t A t , but they are exempt from O(α s ) corrections). Finally, since the vevs v i are not renormalised at O(α s ), the top Yukawa coupling y t receives the same relative correction as the top mass. Defining , θ t , A t ), the DR -OS shifts of top and stop masses and mixing are given in terms of the finite parts (here denoted by a hat) of the top and stop self-energies 52) and the shift for the trilinear coupling reads As in the case of the two-loop effective potential in eq. (3.19), the DR -OS shifts δx k can be cast as where (δx MSSM k ) i are obtained, with the trivial replacement mg → mg i , from the MSSM shifts given in appendix B of ref. [36], whereas δx octet k are novel contributions involving the octet scalar. In particular, δm octet t = 0, and the remaining shifts can be obtained by combining as in eqs. (3.52) and (3.53) the octet contributions to the finite parts of the stop self-energies: where Q is the renormalisation scale at which the parameters entering the tree-level and one-loop parts of the mass matrices are expressed. As mentioned above, the DR -OS shifts derived in eq. (with k > 2) and tan β ↔ cot β must be performed in the formulae of sections 3.3 and 3.4. In the case of the bottom/sbottom corrections, however, passing from the DR scheme to the OS scheme would involve additional complications, as explained in ref. [38].

Simplified formulae
Having computed the general expressions for the two-loop corrections to the neutral Higgs masses in models with Dirac gauginos, it is now interesting to provide some approximate results for the dominant corrections to the mass of a SM-like Higgs. We focus on the case of a purely-Dirac mass term for the gluinos, which -as mentioned earlier -implies that we can set R 2 11 = R 2 12 = 1/2 and mg 1 = −mg 2 = mg , with mg ≡ m D . We also restrict ourselves to the decoupling limit in which all neutral states except a combination of H 0 d and H 0 u are heavy, so that where v ≈ 174 GeV, and all other fields have negligible mixing with the lightest scalar h, which is SM-like. We can then approximate the correction to the squared mass m 2 h as ∆m 2 h ≈ cos 2 β ∆M 2 S 11 + sin 2 β ∆M 2 S 22 + sin 2β ∆M 2 S 12 . (3.61) Finally, we assume that the superpotential couplings of the adjoint fields (e.g., the couplings λ S and λ T in the MDGSSM) are subdominant with respect to the top Yukawa coupling, so that we can focus on the two-loop corrections proportional to α s m 4 t /v 2 . With these restrictions, we shall give useful formulae valid for a phenomenologically interesting subspace of all extant Dirac gaugino models; while in the following we refer to simplified MDGSSM and MRSSM scenarios, this merely reflects whether stop mixing is allowed.

Common SUSY-breaking scale
We first consider a simplified MDGSSM scenario in which the soft SUSY-breaking masses for the two stops and the Dirac mass of the gluinos are large and degenerate, i.e. m Q = m U = mg = M S with M S m t . Expanding our result 4 for the top/stop contributions to ∆m 2 h at the leading order in m t /M S , we can decompose it as , (3.62) 4 We have verified that, for MS = 1 TeV and for |Xt| up to the "maximal mixing" value of √ 6, the predictions for m h obtained with the simplified formulae of this section agree at the per-mil level with the unexpanded result. For larger MS the accuracy of our approximation improves, and for |Xt| > √ 6 it degrades.
whereX t ≡ X t /M S , in which X t = A t −μ cot β is the left-right mixing term in the stop mass matrix withμ defined as in section 3.3. The first term in ∆m 2 h is the dominant 1-loop contribution from diagrams with top quarks or stop squarks, which is the same as in the MSSM. The second term is the O(α t α s ) contribution from two-loop, MSSM-like diagrams involving gluons, gluinos or a four-stop interaction. Under the assumption that the parameters m t , M S and A t entering the one-loop part of the correction are renormalised in the DR scheme at the scale Q, it reads (3.63) We remark that this correction differs from the usual one in the MSSM, see e.g. eq. (21) of ref. [34], due to the absence of terms involving odd powers ofX t . Indeed, those terms are actually proportional to the gluino masses, and in the considered scenario they cancel out of the sum over the gluino mass eigenstates, because mg 1 = −mg 2 . If the parameters m t , M S and A t are renormalised in the OS scheme as described in section 3.5, the correction reads instead Note that the explicit dependence on the renormalisation scale Q drops out. Again, this correction differs from the usual one in the MSSM, see e.g. the first line in eq. (20) of ref. [35], due to the absence of a term linear inX t . Finally, the last two terms on the right-hand side of eq. (3.62) represent the O(α t α s ) contributions of two-loop diagrams with stops and octet scalars, which are specific to models with Dirac gluinos. In the DR scheme they read φ(z) being the function defined in eq. (45) of ref. [37]. Special limits of the function in eq.
Again, it can be easily checked that the explicit dependence on Q cancels out in the sum of eqs. (3.65) and (3.67).

MRSSM with heavy Dirac gluino
The second simplified scenario we consider is the R-symmetric model of section 3.4, in the limit of heavy Dirac gluino, i.e. mg mt i . This is a phenomenologically interesting limit because Dirac gaugino masses are "supersoft", i.e. they can be substantially larger than the squark masses without spoiling the naturalness of the model [4].
In the MRSSM the left and right stops do not mix, hence we set θ t = 0 in our formulae, but we allow for the possibility of different stop masses mt 1 and mt 2 . In the decoupling limit of the Higgs sector, where we neglect the mixing with the heavy neutral states, the correction to the SM-like Higgs mass reduces to ∆m 2 h ≈ sin 2 β ∆M 2 S 22 . In analogy to eq. (3.62), the correction can in turn be decomposed in a dominant one-loop part, a two-loop, MSSM-like O(α t α s ) contribution and two-loop octet-scalar contributions: Assuming that the top and stop masses in the one-loop part of the correction are DR-renormalised parameters at the scale Q, and expanding our results in inverse powers of m 2 g , the contribution of two-loop, MSSM-like diagrams involving gluons, gluinos or a four-stop coupling reads . This non-decoupling behaviour of the corrections to the Higgs mass in the DR scheme has already been discussed in the context of the MSSM in ref. [36]. Indeed, the correction in eq. (3.70) corresponds to the one obtained by setting µ = A t = 0 in the MSSM result. The terms enhanced by m 2 g /m 2 t i can be removed by expressing the top and stop masses in the one-loop part of the correction as OS parameters. After including the resulting shifts in the two-loop correction, we find where g(x) is the function defined in eq. (3.68). It would appear from eqs. (3.72) and (3.73) above that, independently of the renormalisation scheme adopted for the stop masses, the octet-scalar contributions to ∆m 2 h are enhanced by a factor m 2 g . This is due to the fact that the trilinear squark-octet interaction, see eq. (2.8), is proportional to the Dirac mass term m D -i.e., to mg . However, as discussed in section 2.1, one of the mass eigenvalues for the octet scalars -to fix the notation, let us assume it is m 2 O 1 -does in turn grow with the gluino mass, namely m 2 O 1 ≈ 4 m 2 D when m 2 D becomes much larger than the soft SUSY-breaking mass terms for the octet scalars. Expanding the corresponding contribution to ∆m 2 h in inverse powers of m 2 O 1 we find, in the DR scheme, which does indeed contain potentially large terms enhanced by the ratio m 2 g /m 2 t i . Note that those terms cancel only partially the corresponding terms in the MSSM-like contribution -see the first term in the curly brackets of eq. (3.70) -leaving residues proportional to m 2 g /m 2 On the other hand, in the OS scheme we find Thus, we see that in the OS scheme the contribution to ∆m 2 h from two-loop diagrams involving the heaviest octet scalar O 1 does not grow unphysically large when m 2 g increases, because the ratio m 2 g /m 2 O 1 tends to 1/4. In contrast, for the contribution of the lightest octet scalar O 2 , whose squared mass does not grow with m 2 g , the unexpanded formulae in eqs. 2 is compensated for by the factor s 2 φ O , which, as discussed in section 2.1, is in fact suppressed by m −4 g in the heavy-gluino limit. In summary, we find that, in the MRSSM with heavy Dirac gluino, neither of the octet scalars can induce unphysically large contributions to ∆m 2 h , as long as the stop masses in the one-loop part of the correction are renormalised in the OS scheme.

Numerical examples
In this section we discuss the numerical impact of the two-loop O(α t α s ) corrections to the Higgs boson masses whose computation was described in the previous section. As we did for the simplified formulae of section 3.7, we focus on "decoupling" scenarios in which the lightest neutral scalar is SM-like and the superpotential couplings λ S,T are subdominant with respect to the top Yukawa coupling. Our purpose here is to elucidate the dependence of the corrections to the SM-like Higgs boson mass m h on relevant parameters such as the stop masses and mixing and the gluino masses, rather than provide accurate predictions for all Higgs boson masses in realistic scenarios. We therefore approximate the one-loop part of the corrections with the dominant top/stop contributions at vanishing external momentum, obtained by combining the formulae for the Higgs mass matrices given for MDGSSM and MRSSM in sections 3.3 and 3.4, respectively, with the one-loop functions given in eq. (3.59). We recall that a computation of the Higgs boson masses in models with Dirac gauginos could also be obtained in an automated way by means of the package SARAH [51][52][53][54][55][56]. That would include the full one-loop corrections [54] and the two-loop corrections computed in the gaugeless limit at vanishing external momentum [57,58]. However, the computation implemented in SARAH employs the DR renormalisation scheme, and does not easily lend itself to an adaptation to the OS scheme which, as discussed in section 3.7.2, can be more appropriate in scenarios with heavy gluinos.
The SM parameters entering our computation of the Higgs boson masses, which we take from ref. [77], are the Z boson mass m Z = 91.1876 GeV, the Fermi constant G F = 1.16637 × 10 −5 GeV −2 (from which we extract v = (2 √ 2 G F ) −1/2 ≈ 174 GeV), the pole top-quark mass m t = 173.21 GeV and the strong gauge coupling of the SM in the MS renormalisation scheme, α s (m Z ) = 0.1185. Concerning the SUSY parameters entering the scalar mass matrix at tree-level, we set λ S = λ T = 0 and push the parameters that determine the heavy-scalar masses to multi-TeV values, so that (m 2 h ) tree ≈ m 2 Z cos 2 2β. We also set tan β = 10, so that the tree-level mass of the SM-like Higgs boson is almost maximal but the corrections from diagrams involving sbottom squarks, which we neglect, are not particularly enhanced. For the parameters in the stop mass matrices we take degenerate soft SUSY-breaking masses m Q = m U = M S , we neglect D-term-induced electroweak contributions and we treat the whole leftright mixing term X t = A t − µ cot β as a single input. Finally, for what concerns the parameters that determine the gluino and octet-scalar masses we focus again on the case of purely-Dirac gluinos, with mg 1 = −mg 2 = mg and R 2 11 = R 2 12 = 1/2 . We also take a vanishing soft SUSY-breaking bilinear B O , so that φ O = 0 and only the CP-even octet scalar O 1 , with mass m 2 O 1 = m 2 O + 4 m 2 g , participates in the O(α t α s ) corrections to the Higgs masses.

An example in the MDGSSM
In figure 2 we illustrate some differences between the O(α t α s ) corrections to the SM-like Higgs boson mass in the MDGSSM and in the MSSM. We plot m h as a function of the ratio X t /M S , setting M S = 1.5 TeV and mg = m O = 2 TeV and adopting the OS renormalisation scheme for the parameters m t , M S and X t . We employ the renormalisation-group equations of the SM to evolve the coupling α s from the input scale m Z to the scale M S , then we convert it to the DR-renormalised coupling of the considered SUSY model, which we denote asα s (M S ), by including the appropriate threshold corrections (in this step, we assume that all soft SUSY-breaking squark masses are equal to M S ). The solid (black) and dashed (red) curves in figure 2 represent the SM-like Higgs boson mass in the MDGSSM and in the MSSM, respectively. The comparison between the two curves highlights the fact that, in contrast with the case of the MSSM, in the MDGSSM with purely-Dirac gluinos the O(α t α s ) corrections to m h are symmetric with respect to a change of sign in X t . As mentioned in section 3.7.1, this stems from cancellations between terms proportional to odd powers of the gluino masses. In the points where m h is maximal, which in the OS calculation happens for |X t /M S | ≈ 2, the difference between the MDGSSM and MSSM predictions for m h is about 1 or 2 GeV, depending on the sign of X t . Finally, the dotted (blue) curve in figure    Besides the top mass and the stop masses and mixing, there are a few more parameters entering the O(α t α s ) corrections to the Higgs boson masses whose O(α s ) definition amounts to a three-loop O(α t α 2 s ) effect, namely the gluino and octet-scalar masses and the strong gauge coupling itself. Concerning the masses, in an OS calculation it seems natural to interpret them as pole ones. For α s , on the other hand, there is no obvious "on-shell" definition available, and different choices of scheme, scale and even underlying theory -while all formally equivalent at O(α t α s ) for the Higgs-mass calculation -can lead to significant variations in the numerical results. As mentioned earlier, the solid curve in figure 3 was obtained with top/stop parameters in the OS scheme, but with α s defined as the DR-renormalised coupling of the MDGSSM at the stop-mass scale, i.e.α s (M S ). However, since both stop squarks and top quarks enter the relevant two-loop diagrams, it would not seem unreasonable to evaluate the strong gauge coupling at the top-mass scale either. The dashed (red) and dot-dashed (green) curves in figure 3 represent the predictions for m h obtained with top/stop parameters still in the OS scheme, but with α s defined as the DR-renormalised coupling of the MDGSSM at the top-mass scale,α s (m t ), and as the MS-renormalised coupling of the SM at the same scale, α s (m t ), respectively. The comparison of these two curves with the solid curve shows that a variation in the definition of the coupling α s entering the two-loop corrections provides a less-optimistic estimate of the uncertainty associated to the O(α t α 2 s ) corrections compared with the scheme variation of the top/stop parameters. In particular, for the considered scenario the use of α s (m t ) would induce a negative variation with respect to the results obtained withα s (M S ) of about 4 GeV for X t ≈ 0 and about 7 GeV for |X t /M S | ≈ 2. In contrast, the use ofα s (m t ) would induce a positive variation of about 1 GeV for X t ≈ 0 and about 2 GeV for |X t /M S | ≈ 2, i.e. more modest than the previous one but still larger than the one induced by a scheme change in the top/stop parameters. While remaining agnostic about the true size (and sign) of the three-loop O(α t α 2 s ) corrections, we take this as a cautionary tale against putting too much stock in any single estimate of the theoretical uncertainty of a fixed-order calculation of m h in scenarios with TeV-scale superparticles.

An example in the MRSSM
In our second numerical example we consider the MRSSM, and illustrate the dependence of the SMlike Higgs boson mass on the gluino mass. In ref. [68] it was pointed out that, for multi-TeV values of mg, the contribution of two-loop diagrams involving octet scalars can increase the prediction for m h by more than 10 GeV. We will show that such large effects are related to the non-decoupling behaviour of the DR calculation of m h that we discussed in section 3.7.2, and that the octet-scalar contributions are much more modest in an OS calculation.
The upper (blue) and lower (red) solid curves in figure 4 represent the SM-like Higgs boson mass obtained from the DR calculation as a function of mg, with and without the octet-scalar contributions, respectively. We set m O = 2 TeV and M S = 1 TeV. The latter is interpreted as a DR-renormalised soft SUSY-breaking parameter evaluated at a scale equal to M S itself, which means that each point in the solid curves corresponds to a different value of the physical stop masses. Both curves show a from the OS calculation with and without octet-scalar contributions, respectively. In this case the input M S = 1 TeV is interpreted as an OS-renormalised parameter, meaning that the physical stop masses correspond to (M 2 S + m 2 t ) 1/2 ≈ 1015 GeV for all points in the curves. We stress that direct comparisons between these two curves and the solid (and dashed) ones would not be appropriate, because they refer to different points of the MRSSM parameter space. However, the dotted and dotdashed curves show that, when the physical stop masses are taken as input, the prediction for m h in the MRSSM depends only mildly on the value of mg, and the effect of the octet-scalar contributions is below one GeV. This is explained by the fact that, as discussed in section 3.7.2, in the OS scheme there are no terms enhanced by m 2 g /M 2 S in either the gluino or the octet-scalar contributions to the O(α t α s ) corrections.
Before concluding, we note that there are extreme situations in which a DR calculation of m h is not workable at all, and a conversion to the OS scheme such as the one represented by the dashed lines in figure 4 is necessary. In the so-called supersoft scenario, all soft SUSY-breaking masses vanish, and sizeable sfermion masses -proportional to the Dirac-gaugino masses -are induced only by radiative corrections. Such a scenario can be realised e.g. in the MRSSM by setting m O = 0 and M S = 0, where the latter is interpreted as a DR-renormalised parameter. At the scale where this condition is imposed, the DR stop masses coincide with the top mass, with the result that, in the DR calculation, the oneloop correction in the first term of eq. (3.69) vanishes, while the two-loop corrections in eqs. (3.70) and (3.74) contain terms enhanced by m 2 g /m 2 t (concerning the octet-scalar contributions, we recall that m O 1 = 2 mg in this scenario). Since the Dirac-gluino mass needs to be in the multi-TeV range to generate realistic values for the physical stop masses, the non-decoupling terms in the two-loop corrections can become unphysically large. This is illustrated by the solid (red) curve in figure 5, which represents the SM-like Higgs boson mass obtained with the DR calculation as a function of the gluino mass (here we fix the renormalisation scale as Q = m t and use α s (m t ) in the two-loop corrections). It appears that the DR prediction for m h becomes essentially proportional to mg, and quickly grows to nonsensical values as the latter increases. In contrast, the dashed (blue) curve is obtained with the same procedure as the dashed curves in figure 4, i.e. by computing the physical stop masses at O(α s ) as a function of mg and using them in conjunction with the appropriate OS formulae for the O(α t α s ) corrections to m h . In our example the stop masses range between 302 GeV and 1272 GeV, while the SM-like Higgs boson mass shows only a mild dependence on mg and remains confined to values well below the observed one.

Conclusions
Supersymmetric models with Dirac gaugino masses have attracted considerable attention in the past few years, because they are subject to looser experimental constraints and require less fine-tuning than the MSSM. Besides the extended gaugino sector, such models feature additional colourless scalars which mix with the usual Higgs doublets of the MSSM, as well as additional coloured scalars in the octet representation of SU (3) which contribute to the Higgs boson masses at the two-loop level. In this paper we presented a computation of the dominant two-loop corrections to the Higgs boson masses in Dirac-gaugino models, relying on effective-potential techniques that had previously been applied to the MSSM [36] and to the NMSSM [49]. We obtained analytic formulae for the O(α t α s ) corrections to the scalar and pseudoscalar Higgs mass matrices valid for arbitrary choices of parameters in the squark and gaugino sectors, both in the DR and in the OS renormalisation schemes, which we make available upon request as a fortran code. We also presented compact approximate formulae for the dominant corrections to the mass of the SM-like Higgs boson, valid under a number of simplifying assumptions for the SUSY parameters. Finally, we studied the numerical impact of the newly-computed corrections on the predictions for the SM-like Higgs boson mass in some representative scenarios. In particular, we elucidated the differences between the predictions for m h in the MSSM and those in its Dirac-gaugino extensions; we discussed the theoretical uncertainty of our predictions stemming from uncomputed higher-order corrections; we stressed that a judicious choice of renormalisation scheme is required to obtain reliable predictions in scenarios where the gluinos are much heavier than the squarks, which can occur naturally in Dirac-gaugino models. If our community's hopes are fulfilled and the run II of the LHC brings on a wealth of new discoveries, our results will contribute to their accurate interpretation in the framework of a well-motivated SUSY extension of the SM.