On two-loop corrections to the Higgs trilinear coupling in models with extended scalar sectors

We investigate the possible size of two-loop radiative corrections to the Higgs trilinear coupling $\lambda_{hhh}$ in two types of models with extended Higgs sectors, namely in a Two-Higgs-Doublet Model (2HDM) and in the Inert Doublet Model (IDM). We calculate the leading contributions at two loops arising from the additional (heavy) scalars and the top quark of these theories in the effective-potential approximation. We include all necessary conversion shifts in order to obtain expressions both in the $\overline{\text{MS}}$ and on-shell renormalisation schemes, and in particular, we devise a consistent"on-shell"prescription for the soft-breaking mass of the 2HDM at the two-loop level. We illustrate our analytical results with numerical studies of simple aligned scenarios and show that the two-loop corrections to $\lambda_{hhh}$ remain smaller than their one-loop counterparts, with a typical size being 10-20% of the one-loop corrections, at least while perturbative unitarity conditions are fulfilled. As a consequence, the existence of a large deviation of the Higgs trilinear coupling from the prediction in the Standard Model, which has been discussed in the literature at one loop, is not altered significantly.


I. INTRODUCTION
Although the Standard Model (SM) particle spectrum has been completed by the discovery of a 125-GeV Higgs particle at the CERN LHC [1,2], no sign of any new Physics has been found so far, and direct searches of non-SM particles are currently putting increasingly stringent bounds on parameter spaces of Beyond-the-Standard-Model (BSM) theories.At the moment, the measured properties of the Higgs boson appear to be in close agreement with their SM predictions, which tends to indicate that new Physics is either heavy or made difficult to observe by some mechanism such as alignment [3] -which is defined as the situation where the Higgs vacuum expectation value (VEV) is colinear in field space with one (often the lightest) of the CP-even Higgs mass eigenstates.As a consequence, in aligned scenarios of BSM models, the coupling constants of the 125-GeV Higgs boson are equal at tree level to those in the SM, and deviations can only arise via radiative corrections.However, in models with extended Higgs sectors, some of the couplings of the SM-like Higgs boson can deviate significantly from the SM case because of non-decoupling loop effects involving the additional scalar states of the theory, as was found first in Refs.[4,5].Among these is the Higgs trilinear coupling λ hhh , on which we will focus in this letter.This coupling is especially important because it determines the shape of the Higgs potential, and in turn the type and strength of the electroweak phase transition (EWPT).In particular, it has been shown in Refs.[6,7] that large -O(20 − 30%) or more -deviations in λ hhh from its SM prediction are required for the EWPT to be of strong first order, which is necessary for the scenario of electroweak baryogenesis (EWBG) [8][9][10] to be successful.
Radiative corrections to the Higgs trilinear coupling were first investigated at the one-loop order in the SM and the minimal supersymmetric SM (MSSM) in Refs.[22][23][24].Moreover, one-loop effects have also been investigated for various (non-supersymmetric) BSM theories with extended Higgs sectors -namely with additional singlets [25][26][27][28], doublets [4,5,[28][29][30][31], or triplets [32] -and most of these results are now available in the program H-COUP [33].Since Refs.[4,5] it is known that, in the non-decoupling regime, the dominant one-loop BSM corrections to the Higgs trilinear coupling can cause a deviation of λ hhh by several tens of or even a hundred percent from its SM prediction, while still verifying the criterion of tree-level unitarity [34].After encountering such large effects at one loop, one may at first ask whether perturbativity is still preserved.This is indeed the case because the one-loop expressions are not a perturbation of the tree-level formula, and instead involve new parameters that only enter the calculation at loop level.However, it remains natural to enquire about the situation at two loops: i.e., whether new effects as large as O(100%) may add up with the one-loop ones, or whether contributions at two loops stay smaller than at one loop.
The first study of leading two-loop corrections in a model exhibiting non-decoupling effects was performed in Ref. [35] for the case of the Inert Doublet Model (IDM), and indicated that two-loop corrections enhance the Higgs trilinear coupling by a few percent and slightly weaken the firstorder EWPT.We should also mention anterior works in the context of supersymmetric models, motivated by the need for a consistent theoretical determination of the Higgs mass(es) and trilinear coupling: these are namely Refs.[36] and [37] where the leading O(α t α s ) SUSY-QCD corrections to λ hhh were computed in the MSSM and NMSSM respectively, and their effects were found to be of the order of 10%.
In this work, we continue along this line of research and investigate the possible size of two-loop corrections to λ hhh both in the IDM and in an aligned scenario of a Two-Higgs-Doublet Model (2HDM), using the effective-potential method.For the former, we include new scalar diagrams that were overlooked in Ref. [35], while for the latter the expressions that we obtain constitute the first results in the literature for the 2HDM and in general for two-loop diagrams involving both heavy Higgs scalars and top quarks.Moreover, we find the need for a careful treatment of the renormalisation of the soft-breaking mass M of the 2HDM and we therefore devise a new prescription ensuring explicitly the decoupling of our expressions in terms of on-shell-renormalised parameters.We will restrict our attention here to the two-loop BSM contributions to λ hhh from the additional states in the extended Higgs sectors.

II. MODELS
We here briefly recall our conventions for the 2HDM and the IDM.For more complete reviews of these models, see Refs.[38,39] for the 2HDM and Refs.[40,41] for the IDM.

A. Two-Higgs-Doublet Model
The first type of model that we consider is a CP-conserving 2HDM, defined in terms of two SU (2) L doublets Φ 1 , Φ 2 of hypercharge 1/2.To avoid flavour changing neutral currents that are strongly constrained experimentally, we impose a Z 2 symmetry under which the two doublets of the theory transform as Φ 1 → Φ 1 and Φ 2 → −Φ 2 [42], but which is softly broken by a mass term 3 ) in the potential.We follow the conventions of Ref. [5] and write the tree-level scalar potential as Because we assume that CP is conserved, all mass parameters m 2 i and quartic coupling constants λ i are real.We choose then to expand the doublets Φ 1 and Φ 2 as Φ i = (φ + i , φ 0 i / √ 2).Depending on the parameters of the Lagrangian, the neutral components of the doublets may acquire non-zero (real) vacuum expectation values (VEVs), which are denoted v i ≡ φ 0 i and verify v 2 1 + v 2 2 = v 2 , where v 246 GeV.
Assuming that both v 1 and v 2 are non-zero, two dimensionful parameters -typically m 2 1 and m 2 2 -can be eliminated using the tadpole equations (see e.g.eqs ( 9)- (10) in [5] for their tree-level expressions).Seven free parameters then remain in the scalar sector: m 2 3 , λ i (i = 1 − 5) and the ratio of the two VEVs v 2 /v 1 = tan β.The latter defines an angle β that diagonalises the charged and CP-odd Higgs mass matrices at tree-level, while for the CP-even Higgs mass matrix a second mixing angle α needs to be introduced.We can then obtain the charged and neutral components of the SU (2) L doublets in terms of mass eigenstates as with R x ≡ cos x − sin x sin x cos x , and where h and H are CP-even Higgs bosons, A is a CP-odd Higgs boson, and H + is a charged Higgs boson.In addition, G and G + are respectively the neutral and charged Goldstone bosons associated with electroweak symmetry breaking (EWSB).
It is common to replace the Lagrangian mass parameter m 2  3 by the soft-breaking mass M 2 ≡ 2m 2  3 /s 2β , and to express the five quartic couplings in terms of the four scalar mass eigenvalues and the mixing angle α, using tree-level relations given for example in equations ( 26)-(30) of Ref. [5].
In this respect, it is important to emphasize that the mass eigenvalues in these equations should be interpreted as the tree-level ones -otherwise radiative corrections would need to be taken into account to obtain the relation between quartic couplings and loop-level mass eigenvalues (see for example Refs.[28,29,[43][44][45][46][47]).
To ensure compatibility with experimental constraints, we will throughout this letter consider the so-called alignment limit.This limit is defined by the requirement that one of the CP-even Higgs mass eigenstates is aligned in field space with the full Higgs VEV v [3].Additionally, we want the SM-like state to be the lightest eigenstate h, which in terms of mixing angles implies The tree-level couplings of h to other particles are then equal to their SM values, and in particular λ Furthermore, in the alignment limit and for m h m Φ , we can obtain simple expressions for the field-dependent tree-level masses of the additional scalars Φ = H, A, H ± as Finally, it should be noted that we neglect throughout this letter contributions from quarks other than the top and from leptons, so there is no need to specify the type of fermion couplings in our setting.Indeed, at tree-level, the couplings of the top quark to the scalar sector are the same in all types of 2HDMs, and its field-dependent mass is

B. The Inert Doublet Model
The IDM [40,48] is one of the simplest extensions of the SM, and corresponds to the limit of the 2HDM in which the previously-mentioned Z 2 symmetry is exact after EWSB.This ensures that there is no mixing between the SM-like doublet Φ 1 , and the Z 2 -odd one Φ 2 .Furthermore, it allows the model to accommodate a dark matter candidate -namely the lightest Z 2 -odd scalar.
Under the gauge and Z 2 symmetries, the scalar potential is given by Following Ref. [49], we here decompose the two doublets in terms of mass eigenstates as where we use the same notations as in the 2HDM.Finally, from the above tree-level potential, we can derive field-dependent masses for the inert (i.e.Z 2 -odd) scalars H, A, and , where λ H,A = λ 3 + λ 4 ± λ 5 and λ H ± = λ 3 .

III. EFFECTIVE-POTENTIAL CALCULATION OF λ hhh AT TWO LOOPS
We investigate leading two-loop corrections to the effective Higgs trilinear coupling in the effective-potential approximation, which is equivalent to setting external momenta to zero in a diagrammatic calculation.We define our loop expansion of the effective potential V eff as where κ ≡ 1/(16π 2 ) is the usual loop factor.While they miss potential threshold effects (shown at one loop for example in Ref. [5]), effective-potential computations are considerably simpler than diagrammatic ones and are sufficient for a first study of the magnitude of two-loop corrections.
Furthermore, from past experience with scalar mass calculations we may expect the inclusion of momentum at two loops to give only subleading effects -see e.g.[45,[50][51][52].
Normalising the effective Higgs trilinear coupling as L ⊃ − 1 6 λ hhh h 3 , the radiative corrections that it receives can be computed by taking derivatives of the effective potential as hhh + κδ (1) In the scenarios without mixing in the scalar sector that we consider in section IV, the tree-level result λ (0) , the effectivepotential (or curvature) mass of the lightest Higgs boson, as The above definition of the differential operator D 3 ensures that tadpole conditions are taken into account -i.e. the calculation is performed at the minimum of the loop-corrected potential.
We follow the common choice of performing renormalisation before taking derivatives of the potential, which allows us to make use of existing results for the effective potential [53].The renormalised effective potential is calculated in terms of field-dependent (MS) tree-level masses, and therefore the results we find for λ hhh are also expressed in terms of MS-renormalised parameters.While theoretically consistent and simple, MS-scheme calculations may be plagued by large logarithmic contributions that appear because of the explicit renormalisation scale dependence, and furthermore it requires the inclusion of renormalisation group equations (RGEs) for all running parameters.Therefore, we choose to use an OS scheme instead and express our results in terms of physical parameters.For this purpose, we translate the relevant parameters, i.e. all particle masses and the Higgs VEV, from their MS values , and we also include finite wave-function renormalisation (WFR) as where δZ OS h and δZ MS h are the WFR counterterms in the OS and MS schemes respectively, and is the finite part of the Higgs self-energy evaluated at external momentum equal to p 2 .We recall that the pole and curvature masses of the Higgs boson are related as (see e.g.[54]) As our main concern is the size of the dominant two-loop BSM contributions to λ hhh due to the additional scalar states in the 2HDM and the IDM, we make the further approximation of neglecting contributions from the 125-GeV Higgs and the would-be Goldstone bosons, at both oneand two-loop orders, throughout the following.
Before turning to the two-loop computation and our new results, we briefly review here the effective-potential calculation of one-loop corrections to the Higgs trilinear coupling.The dominant terms in the one-loop effective potential, for both the 2HDM and the IDM, are [55] where m 2 t (h) and m 2 Φ (h) are the field-dependent masses of the top quark and of the extra scalars, respectively, and n Φ is 1 for H and A, and 2 for H ± -as mentioned earlier, we have neglected here the SM-like Higgs and Goldstone boson terms.
One can then derive the leading one-loop contributions to λ hhh by using the operator D 3 [4,35] where μ is either M in the 2HDM or µ 2 in the IDM.
FIG. 1. Topologies of diagrams with scalars and fermions contributing to the effective potential at two loops.
Corrections to the two-loop effective potential are obtained by calculating one-particle-irreducible vacuum bubble diagrams [55].For our study, we expand the two-loop part of V eff as F F S , where each index S or F indicates a scalar or Dirac-fermion propagator -the corresponding diagrams are shown in figure 1. Analytic expressions for each of these terms can be obtained in the Landau gauge and MS scheme for any renormalisable model using 1 the results of Ref. [53] (see also [56] for results with a general gauge fixing).These involve only two loop functions, namely the one-loop function A and the two-loop sunrise integral I, for both of which complete expressions are given e.g. in Refs.[53,[57][58][59], and useful limits of I with one or more mass arguments equal or vanishing can be found in Refs.[58,60].
At two loops, the MS to OS scheme conversion requires adding finite one-loop or two-loop shifts to the parameters that enter at one loop and tree level respectively.However for the latter, i.e.

the Higgs mass [M 2
h ] V eff and VEV v, the two-loop shifts yield corrections to λhhh proportional to the 125-GeV Higgs mass and should hence be neglected in our approximation.Consequently, we only need one-loop scheme translations for the Higgs VEV, the scalar masses, the top quark mass.
Finally, before considering BSM corrections, we should mention also the case of the SM calculation, performed, for example, in Ref. [35].Starting from the two-loop SM effective potential given in Ref. [57], we obtain the same result as equation (11) of [35] in terms of MS parameters.
However, when translating that expression to the OS scheme, both using the results of Ref. [54] as well as with a standalone calculation, we have δ (2)  λhhh = 72M 1 Note that our notation differs slightly from that of Ref. [53] because we work here with Dirac fermions, and not Weyl fermions, therefore our V F F S corresponds to the sum V F F S in [53].
Our results do not agree with equation ( 12) of [35] as we find for the numerical coefficient of the two-loop M 6 t term −936 instead of 336.We have furthermore checked that the numerical values for our OS expression and the MS one evaluated at renormalisation Q = M t are in excellent agreement.

IV. NUMERICAL EXAMPLES
A. An aligned scenario with degenerate heavy scalars in the 2HDM As our first numerical example, we consider a simplified scenario of the 2HDM where the additional scalars are degenerate in mass.This ensures that our calculations contain only three mass scales M , M Φ , and M t and allow relatively compact analytical expressions to be obtained.Furthermore, to avoid complications arising from mixing between h and H, the CP-even mixing angle α is fixed2 as α = β − π/2 to ensure alignment.
In the 2HDM, there are with respect to the SM fifteen new diagrams involving heavy scalars and top quarks that contribute to the effective potential at two loops, which we can write as ttA , and V (2) tbH ± .Applying the operator D 3 to these effective-potential terms, we obtain δ (2) in terms of the MS-renormalised parameters -m Φ being an MS-scheme degenerate mass for the heavy scalars H, A, H ± -and with log x ≡ log(x/Q 2 ), Q being the renormalisation scale.The complete expression of the third derivative of the heavy scalar and top quark sunrise V F F S diagrams is rather long, so for brevity we only write here the leading O(m 4 Φ m 2 t /v 5 ) term (while we use the complete result for our following numerical investigation).We performed some consistency checks of these results by: (i) verifying that the log Q 2 dependence of the total result for λ hhh is eliminated when including the running of all parameters appearing at lower orders; (ii) confirming that each of the terms in this MS expression independently decouples when taking the limit M → ∞.The latter can be understood as each term is proportional to with n = 3 or 4, and where λ denotes some combination of Lagrangian scalar quartic couplings.We should expect to observe decoupling of the BSM corrections when taking M → ∞ while keeping λv 2 finite, so that the additional scalar masses go to infinity without the calculation entering the nondecoupling regime associated with large scalar quartic couplings, which would cause perturbativity to be lost.Here, when taking the limit M → ∞ with λv 2 fixed our expressions do indeed decouple properly, as can be seen from eq. ( 15).
Instead of MS-renormalised parameters, we prefer to work in terms of physical parameters, and therefore we now convert our expressions to the OS scheme.For the masses of the top quark and the heavy scalars, this simply requires shifting the masses in the one-loop corrections given in eq. ( 12) by the corresponding self-energies.At this point, we should emphasise that the scenarios where the heavy scalars have degenerate MS masses or OS masses correspond to different points in the parameter space of the 2HDM, because the radiative corrections that relate MS and OS masses are not the same for the different scalars.Keeping this in mind, we choose however to consider parameter points for which the scalars H, A, and H ± have a common physical mass M Φ , after the conversion of our results to the OS scheme.Then, for tan β, we do not need to perform any conversion, as this parameter only enters the calculation at two loops.
Finally, the treatment of M 2 is more subtle, as we will discuss now.When working at one-loop order, one may find decoupling of the effects of the heavy-scalar loops in the OS scheme result in the limit M → ∞ when using a relation M 2 Φ = M 2 + λv 2 , with M Φ renormalised in the OS scheme and M in the MS scheme [5].However, when going to two-loop order this is not the case any more, and one needs to relate parameters that appear at one-loop order in different schemes with a one-loop equation, so as not to miss two-loop order effects.We have checked that decoupling does occur if we consistently use a one-loop relation between M Φ and M in our results -this is essentially equivalent to using expressions with the heavy scalar masses renormalised in the MS scheme.Nevertheless, this situation motivates devising an "on-shell" renormalisation3 condition for the soft-breaking mass, which we then denote M , that would make decoupling apparent when using a relation of the form M 2 Φ = M 2 + λv 2 phys .Furthermore, we still have the freedom to choose this renormalisation condition in such a way that it ensures the complete cancellation of all log m 2 Φ terms in δ (2) λ hhh .
With this choice, we can derive the finite "OS" counterterm δ OS M 2 for M 2 -defined so that M 2 = M 2 + δ OS M 2 -and we obtain at the one-loop order where B 0 is the (finite part of the) usual Passarino-Veltman function [61].Our final OS scheme result for δ (2)  λhhh is then where the terms on the third line come from finite WF and VEV renormalisation.
These two-loop corrections indeed decouple explicitly when taking the limit M → ∞ with λv 2 phys fixed.An example of this is shown in the left side of figure 2, where we plot the deviation δR of λ hhh calculated in the 2HDM with respect to the SM prediction as a function of the OS-renormalised M , at one-and two-loop orders, for different fixed values of M 2 Φ − M 2 and for tan β = 1.5other physical inputs being taken from the PDG [62].
The other interesting limit to consider with these results is the case of maximal non-decoupling effects that we obtain for M = 0. We illustrate the non-decoupling behaviour of the BSM corrections to λhhh in the right side of figure 2, where we show the same deviation δR as in the left-side plot but now as a function of the heavy scalar (degenerate) pole mass M Φ , in the case of M = 0 (to enhance as much as possible the non-decoupling effects) and tan β = 1.1.From essentially negligible before the one-loop corrections for M Φ v phys , the two-loop contributions become as large as 80% for M Φ = 500 GeV -the one-loop deviation is then 250%.One should note that the value M Φ = 500 GeV is close to the upper limit on M Φ allowed by the criterion of tree-level unitarity -using [63], we find this limit to be M Φ 600 GeV for tan β = 1.1 and M = 0.For M Φ = 400 GeV, well below the bound from perturbative unitarity, the BSM contributions cause a deviation of λ2HDM hhh with respect to its SM prediction of 101% at one loop, and of a further 22% at two loops.Finally, we should mention also the dependence on tan β that appears at two loops in λhhh , even in the alignment limit that we have considered.In particular, the effect of tan β is largest in the scalar contributions to δ (2)  λhhh (the first two terms in eq. ( 17)), because of their   cot 2 2β dependence, and hence these terms are greatly enhanced when tan β increases.However, we observe that the perturbative expansion is not broken -i.e.two-loop corrections to λhhh remain smaller than their one-loop counterparts -at least while perturbative unitarity conditions are not violated.To illustrate this, we consider two example points.For the first one, we take M Φ = 400 GeV and M = 0, and the criterion of tree-level unitarity then implies [63] an upper bound tan β 1.7.With this maximal value of tan β, the one-and two-loop deviations of λhhh from BSM contributions are respectively 101% and 34%.We fix for the second example point M Φ = 250 GeV and M = 0, which gives the bound tan β 2.8, and in turn we obtain for the deviation of λ2HDM hhh from λSM hhh at one-and two-loop orders respectively 15% and 6%.It therefore appears that, under the criterion of tree-level unitarity, the two-loop BSM corrections to the Higgs trilinear coupling can become at most O(30 − 40%) of the one-loop ones.

B. A dark-matter-inspired scenario in the IDM
The second case that we consider is a scenario of the IDM, already studied in Ref. [35], in which the additional inert scalar H is light (M H M h /2 M A,H ± ) and becomes a DM candidate.The leading two-loop corrections to λ hhh are then due to the pseudoscalar and charged Higgs bosons, and in order to maximise the size of the radiative corrections, we set the mass parameter µ 2 to zero throughout this section.
In this scenario, there are eight BSM diagrams that give contributions to V (2) , namely V hAA , V (2) AH ± , and V H ± H ± .Only the first two of these were included in Ref. [35], and we find agreement between our results for these and equation (16) of [35].Taking into account the other of the above diagrams, we present here for the first time the complete O(M 6 Φ /v 5 phys ) and O(λ 2 M 4 Φ /v 3 phys ) contributions to δ (2)  λhhh (here by M Φ we mean M A , M H ± , or some combination of the two).After conversion to the OS scheme, and inclusion of finite WF and VEV renormalisation, these read We emphasise that, as λ 2 only appears in the calculation of λ hhh at two-loop order, we do not need here to specify a choice of renormalisation scheme for it, as opposed to the inert scalar masses, which appear at one-loop.Moreover, we expect from our result for M in the previous section that the tree-level (MS) condition µ 2 = 0 will also hold in the OS scheme (see eq. ( 16)).Interestingly, one may notice that although the inert scalars do not couple directly to the top quark, terms involving both the top and inert-scalar masses do appear at two loops through the interplay of one-loop scalar contributions to the Higgs WFR with the one-loop top quark correction to λ hhh (and vice versa).
We illustrate our numerical results in figure 3, where we plot the deviation δR of the Higgs trilinear coupling λIDM hhh with respect to its SM prediction λSM hhh , as a function of the pole masses of the heavy scalars, which we take to be equal -i.e.M A = M H ± = M Φ -both for convenience and to keep the ρ parameter close to 1.One should note that this implies that the third term in the above equation (18), which corresponds to the V (2) HAG , and V HH ± G ± -is given by the difference between the solid blue (one-loop) and dashed red (two-loop, with λ 2 = 0) curves in figure 3. We also try to evaluate the maximal possible size of the contributions proportional to λ 2 -i.e.coming H ± H ± -under the constraint of perturbative unitarity, and for this purpose evaluate δR at two loops for λ 2 = 6; note that we do not take λ 2 larger because we find the tree-level unitarity conditions to be violated [63,64] for M Φ = 500 GeV, µ 2 = 0 and λ 2 6.5.
For M Φ = 400 GeV, λIDM hhh deviates at one loop by about 76% with respect to its SM prediction, while the sunrise diagrams give a further enhancement of ∼ 18% and the remaining two-loop diagrams (with λ 2 ) another ∼ 21%.On the one hand one may notice that, in relative size, the corrections from the sunrise diagrams grow significantly faster than the one-loop and two-loop λ 2 -dependent ones, which can be understood from their expression proportional to M 6 Φ as opposed to M 4 Φ for the others.On the other hand, the remaining two-loop diagrams can potentially give large contributions for low M Φ , if λ 2 is large, because of their large combinatorial factor, but their relative importance diminishes for increasing M Φ .To summarise, similarly to what we found in the 2HDM, the two-loop corrections in the IDM remain smaller than the one-loop ones, meaning that the perturbative expansion is not breaking down, at least as long as perturbative unitarity is fulfilled.
Before concluding, we point out that while the coupling λ 2 is quite difficult to access experi-mentally as it is the quartic coupling between inert scalars (see the potential in eq. ( 4)), a precise measurement of λ hhh could allow us to obtain some information about the value of λ 2 .We can expect this observation to hold also for couplings of the Higgs boson with other particles, such as for example the hγγ or hZZ couplings, because the quartic coupling λ 2 should appear in internal scalar loops therein as well.

V. SUMMARY
We have discussed the magnitude of the deviation of the Higgs trilinear coupling λ hhh from its SM prediction at two loops in two models with extended Higgs sectors, namely the 2HDM -for which we have obtained for the first time leading two-loop corrections to λ hhh -and the IDMwhere we have improved on the existing results of Ref. [35].We have performed calculations both in the MS and the on-shell schemes using the effective-potential method.In the cases where it was possible, we have compared our expressions with existing works in the literature, explaining the origin of some differences in the SM and IDM.We also devised a new "on-shell" renormalisation prescription for the soft-breaking scale M to maintain explicitly the decoupling of the two-loop corrections for M 2 Φ = M 2 + λv 2 phys with M → ∞ and fixed λv 2 phys .
In the two models we studied, we found new dependences of λ hhh on parameters -respectively tan β in the aligned 2HDM and λ 2 in the IDM -entering the calculation only from two loops, and which may cause large enhancements of δ (2)  λhhh .However, we have shown that, provided one considers parameter points within the region of parameter space allowed under the criterion of tree-level unitarity, the two-loop corrections to λ hhh do not grow out of control and remain smaller than the one-loop effects.When expressed in terms of OS-scheme parameters, our new two-loop contributions (moderately) enhance the non-decoupling effects appearing at one-loop, but we should emphasise that we do not obtain new large -i.e.O(100%) or so -corrections at two loops.The typical size that we find for the two-loop corrections in the OS scheme, up to O(∼ 20%) of the one-loop corrections, implies, on the one hand, that higher-order contributions do not change the existence of the non-decoupling effects observed from one loop.But on the other hand, it also means that in the perspective of precise measurements of the Higgs trilinear coupling, a careful theoretical calculation of λ hhh -including radiative corrections beyond one loop -will be necessary.
Details about the calculations and discussions in this letter will be shown elsewhere [65].
FIG. 2. Illustrations of our results for the deviation δR of λhhh computed in the 2HDM with respect to its SM prediction, with δR ≡ ∆ λ2HDM hhh / λSM hhh = λ2HDM hhh / λSM hhh − 1. (Left side): Decoupling behaviour of the BSM contributions to λhhh , shown by plotting δR at one loop (solid blue curve) and two loops (red dot-dashed curve) as a function of M .The degenerate pole mass of the additional scalars M Φ is taken to be M 2 Φ = M 2 + λv 2 phys , with λv 2 phys = (200 GeV) 2 , (300 GeV) 2 , (400 GeV) 2 , and we fix tan β = 1.5.(Right side): δR computed at one loop (solid blue curve) and two loops (red dot-dashed curve) as a function of M Φ , for the maximal non-decoupling case of M = 0, and with tan β = 1.1.

FIG. 3 .
FIG.3.Deviation δR of the Higgs trilinear coupling calculated in the IDM ( λIDM hhh ) with respect to the SM ( λSM hhh ) -i.e.δR = λIDM hhh / λSM hhh − 1 -as a function of the degenerate pole masses of the pseudoscalar and charged Higgses M Φ = M A = M H ± , at one loop (solid blue curve) and two loops (dashed and dot-dashed red curves).For the two-loop results, the dashed and dot-dashed correspond to different values of the inert doublet quartic coupling λ 2 , respectively, λ 2 = 0 and 6.