Precise Higgs mass calculations in (non-)minimal supersymmetry at both high and low scales

We present FlexibleEFTHiggs, a method for calculating the SM-like Higgs pole mass in SUSY (and even non-SUSY) models, which combines an effective field theory approach with a diagrammatic calculation. It thus achieves an all order resummation of leading logarithms together with the inclusion of all non-logarithmic 1-loop contributions. We implement this method into FlexibleSUSY and study its properties in the MSSM, NMSSM, E6SSM and MRSSM. In the MSSM, it correctly interpolates between the known results of effective field theory calculations in the literature for a high SUSY scale and fixed-order calculations in the full theory for a sub-TeV SUSY scale. We compare our MSSM results to those from public codes and identify the origin of the most significant deviations between the DR-bar programs. We then perform a similar comparison in the remaining three non-minimal models. For all four models we estimate the theoretical uncertainty of FlexibleEFTHiggs and the fixed-order DR-bar programs thereby finding that the former becomes more precise than the latter for a SUSY scale above a few TeV. Even for sub-TeV SUSY scales, FlexibleEFTHiggs maintains the uncertainty estimate around 2-3 GeV, remaining a competitive alternative to existing fixed-order computations.


Introduction
A hallmark of renormalizable supersymmetric (SUSY) theories is that quartic scalar couplings are not free parameters, but fixed in terms of gauge and (in some models) Yukawa couplings. As a result, predictions of the Standard Model (SM)-like Higgs mass are restricted to a limited range and precise calculations are very important for testing SUSY models. Since the discovery of a 125 GeV Higgs boson at the LHC [1,2], the need for precise predictions within SUSY models has increased in several ways. First, the measurement is already far more precise than existing theory predictions, motivating significant improvements in both theory predictions and their associated uncertainty estimates. Second, the non-observation of new physics at the LHC, may imply heavier masses of new particles, so predictions should be reliable both for light or heavy SUSY masses. Third, the heavy Higgs boson mass provokes naturalness questions that motivate non-minimal SUSY models and improving precision Higgs mass calculations there.
Here we present FlexibleEFTHiggs, a new method for calculating the Higgs mass that can improve the precision of the Higgs mass prediction in minimal and non-minimal SUSY models. This method uses effective field theory (EFT) techniques, which improve the precision when the SUSY masses are much heavier than the electroweak (EW) scale. However, FlexibleEFTHiggs also includes terms which are important at low SUSY scales, previously only included in fixed-order calculations. This hybrid approach combines the virtues of both worlds to give precise predictions at both high and low SUSY scales. We also present an extensive analysis of the remaining theory uncertainty and discuss in detail the differences to other calculations, shedding light on the theory uncertainties of existing calculations. The method and uncertainty estimates are applied to the MSSM and three non-minimal models, the NMSSM, the E 6 SSM, and the MRSSM.
Fixed-order calculations are particularly reliable when the new particle masses are around the EW scale. If the new physics scale is too high, large logarithms appear at each order in perturbation theory, and the result can suffer from a large truncation error. Recently, therefore, Refs. [30,33,34] combined fixed-order calculations with the resummation of the leading and next-to-leading logarithms without double counting, reducing the theory uncertainty at high SUSY masses.
As discussed e.g. in Ref. [45], the disadvantage of pure EFT-type calculations is that they miss non-logarithmic contributions that are suppressed by powers of the SUSY mass scale already at the tree-level and 1-loop level. Hence, the theory uncertainty increases strongly if the SUSY masses are close to the EW scale.
FlexibleEFTHiggs is an EFT calculation with specially chosen matching conditions, such that the Higgs mass calculation is exact at the tree-level and 1-loop level. This ensures that the theory uncertainty remains bounded at both high and low SUSY scales, as we will show. We have implemented FlexibleEFTHiggs into FlexibleSUSY [26], a spectrum generator generator for BSM models based on SARAH [53][54][55][56][57][58] and SOFTSUSY [23,25] so that this method can be used in a huge variety of models. The level of precision currently implemented is 1-loop mass matching and 3-loop running in the SM. Currently a limiting assumption is that the SM is the correct low-energy EFT at the EW scale and all non-SM particles are integrated out at a heavy scale.
This paper is structured as follows: In Section 2 we give an overview of the pure EFT and the fixed-order approaches and describe the FlexibleEFTHiggs method in more detail. In Section 3 we apply FlexibleEFTHiggs to the MSSM and compare the results with those from publicly available MSSM spectrum generators. In addition, we analyse the origin of the most significant deviations between the DR fixed-order calculations in FlexibleSUSY, SOFTSUSY and SPheno. We then present several possibilities to estimate the theoretical uncertainty of the Higgs mass prediction in the DR fixed-order approaches and in FlexibleEFTHiggs. In Section 4 we summarize and combine the uncertainty estimates and give an order of magnitude for the SUSY scale above which we expect FlexibleEFTHiggs to lead to a more precise prediction than the DR fixed-order programs. In Sections 5-7 we apply FlexibleEFTHiggs to the NMSSM, E 6 SSM and the MRSSM and perform an uncertainty estimation. We conclude in Section 8.

Procedure of the calculation
The new FlexibleEFTHiggs approach presented here is an EFT-type calculation of the SM-like Higgs mass in the MSSM or any other non-minimal SUSY or BSM model, where we assume the Standard Model is a valid EFT. FlexibleEFTHiggs is implemented into FlexibleSUSY [26], a C++ and Mathematica framework to create modular spectrum generators for SUSY and non-SUSY models. Before introducing FlexibleEFTHiggs, we describe the SM and MSSM to fix our notation and then describe fixed-order and "pure EFT" calculations implemented in several public programs all of which use the DR scheme. There are also very accurate calculations in the on-shell renormalization scheme, for example FeynHiggs [6,19,22,33,[59][60][61][62] and NMSSMCALC [29,63,64], but we will not go into the details of the on-shell calculations. In the following we use the programs

The Standard Model and its minimal supersymmetric extension
The Standard Model is invariant under local gauge transformations of the group, where the gauge couplings associated with SU (3) C , SU (2) L and U (1) Y are g 3 , g 2 and g 1 , respectively, with g 1 under the SU (5) GUT normalization. Sometimes it is more convenient to write expressions in terms of the U (1) Y gauge coupling, which we denote g Y = 3/5 g 1 . As usual, we also use e 2 = g 2 Y g 2 2 /(g 2 Y + g 2 2 ), α e.m. = e 2 /4π and α s = g 2 3 /4π. The spontaneous breakdown of electroweak symmetry SU (2) L ×U (1) Y → U (1) e occurs when the coefficient of the bilinear term in the Higgs potential, is negative. This causes the neutral component of the Higgs field, Φ, which is a SU (2) L doublet, to develop a vacuum expectation value (VEV), v = −µ 2 /(2λ). The Standard Model fermions are the left handed SU (2) L quark and lepton doublets Q i and L i , and the right handed SU (2) L singlets for up-type and down-type quarks and charged leptons, u Ri , d Ri , e Ri . They obtain mass through their Yukawa interactions with the Higgs field, when the neutral Higgs field develops a VEV. Here i, j denote generation indices, and we define the SU (2) L dot product, A · B := A 1 B 2 − A 2 B 1 . To simplify the notation we denote the third generation Yukawa couplings as y t , y b , y τ which are the largest singular values of Y u , Y d , Y e , respectively. The minimal supersymmetric extension of the Standard Model (MSSM) has the superpotential, where all superfields appear with a hat. The chiral superfields have the G SM quantum numbersQ where the first and second symbol in the parentheses denotes the representation of the corresponding superfield with respect to SU (3) C and SU (2) L and the third component is the hypercharge in standard normalization. The neutral components of the up-type Higgs field, H u , and down-type Higgs field, H d , develop the VEVs, v u / √ 2 and v d / √ 2, respectively. As usual we define, where v ≈ 246 GeV. The soft breaking Lagrangian is given by In the following, we trade the soft-breaking trilinear couplings for the customary X f parameters as with the appearing matrices given in the super-CKM basis [65,66]. For the third generation fermions we define X t := (X u ) 33 , X b := (X d ) 33 , X τ := (X e ) 33 . The gauginos have the following G SM quantum numbers: In the scenarios studied in the following we set the dimensionful running DR superpotential and soft-breaking parameters to the common value of the SUSY scale, M SUSY , if not stated otherwise: Sometimes we will go beyond the last equation and keep X t as a free parameter and set it to a non-zero value. In our numerical evaluations we will choose the numerical values α

Fixed-order calculations in FlexibleSUSY, SOFTSUSY and SARAH/SPheno
We now discuss the fixed-order approach for calculating the Higgs mass that is implemented in FlexibleSUSY, SOFTSUSY and SARAH/SPheno. There are two major steps in this calculation: 1. Find all DR parameters at the SUSY scale.
2. Calculate the Higgs pole mass from the DR parameters.
The first step is rather complicated and involves an iteration. One complication is that some parameters may be set at a higher-scale and the values at the SUSY scale only obtained through the RG running, though here we will simply set all non-SM parameters at the SUSY scale. Nonetheless this is still non-trivial because some of the DR parameters must be chosen to fulfill the EWSB equations or are determined by experimental data. For the EWSB conditions we will fix the soft Higgs masses in this work, which is an option available in all of the codes we use. The VEV, v is fixed from the DR running m Z , leaving tan β as a free parameter at the SUSY scale. The gauge couplings, g 1 , g 2 and g 3 and Yukawa couplings, Y u , Y d and Y e can be extracted from data. This can be done using the measured values of the running MS electromagnetic and strong couplings, the Weinberg angle or an equivalent quantity, and the quark and lepton masses. Specifically FlexibleSUSY, SOFTSUSY and SPheno all use the following, where M Z is the Z-boson pole mass, Π T ZZ (p 2 ) is the transverse part of the 1-loop Z self energy and α MS,SM(5) s (M Z ) is the MS strong coupling in the SM with 5-flavours. Using these and further similar relations, all DR gauge couplings and EWSB parameters of the fundamental SUSY theory can be determined at the low scale Q = M Z . The Yukawa couplings are determined similarly from the running vacuum expectation values and fermion masses, but specific corrections beyond the 1-loop level are taken into account. Most importantly, the running top quark mass in FlexibleSUSY and SOFTSUSY is given by where M t denotes the top pole mass, Σ (1),R t (p) denote the scalar, left-handed and right-handed part of the 1-loop top self energy without SM-QCD contributions, evaluated at p = M t , and Σ (1,2),qcd t (m DR t ) denote SM-QCD self energy contributions, with a factor / p removed, evaluated at p = m DR t [67,68]: Eq. (13) is evaluated at the scale M Z and yields the running top mass m DR t (M Z ). SPheno treats the top quark mass differently and requires We will later comment on this difference between Eqs. (13) and (16). Both these equations determine the running top mass implicitly and are solved by an iteration, resulting in slightly different solutions.
In the second step, the Higgs boson mass is computed numerically by solving where m 2 φ denotes the CP-even Higgs tree-level mass matrix, Σ φ and t φ are the DRrenormalized CP-even Higgs self energy and tadpole, respectively, and v In the MSSM, FlexibleSUSY, SOFTSUSY and SPheno use the full 1-loop self energy and 2-loop corrections of the order O( [9,[11][12][13][14]. For non-minimal SUSY models, FlexibleSUSY uses the full 1-loop self energy (optionally extended by the 2-loop MSSM or NMSSM contributions). SARAH/SPheno uses the 2loop self energy in the gauge-less limit and at zero momentum in any given model [31,32].

Pure EFT calculation in SusyHD and FlexibleSUSY/HSSUSY
EFT calculations have the virtue of resumming potentially large logarithms of the generic heavy SUSY mass scale beyond any finite loop level. The calculation is based on the approximation that all non-SM particles, i.e. all SUSY particles and the extra Higgs states, have a common heavy mass of order M SUSY , and that the SM is the correct lowenergy EFT below M SUSY .
The determination of DR parameters and the computation of the Higgs mass is then done in three steps, carried out iteratively, until convergence is reached: 1. Integrate out all SUSY particles at the SUSY scale, and determine the SM parameter λ at M SUSY by a matching of the SUSY theory to the SM.
2. Use the SM renormalization group equations to run the SM parameters down to the EW scale.
3. Match the SM parameters to experiment at the EW scale, and compute the Higgs pole mass.
In the pure EFT approach, the threshold corrections at the SUSY scale are expressed as perturbative functions of the SM parameters at M SUSY , dimensionless (combinations of) SUSY parameters and at most logarithms of SUSY masses. No terms suppressed by powers of M SUSY appear. The known 1-and 2-loop threshold correction to λ from the MSSM read [44,45] (2) can be found in [44,45]. Since λ is directly expressed in terms of running SM parameters and fundamental SUSY input parameters, no other threshold corrections are needed. This pure EFT approach to calculate the Higgs pole mass is implemented in SusyHD (M Z ) and G F and quark and lepton masses using the approach described in [26], thereby taking into account 1-loop and leading 2-loop corrections. For the extraction of the top Yukawa coupling also the known 2-loop and 3-loop QCD corrections are taken into account [71,72]. SusyHD includes several further subleading corrections, e.g. fit formulas for 2-loop threshold corrections to the EW gauge couplings [69]. Finally, the Higgs pole mass is calculated at the scale M t . HSSUSY employs full 1-loop and leading 2-loop corrections of O(α t α s + α 2 t ); SusyHD uses a numerical fit formula approximating the full 2-loop corrections.

EFT calculation in FlexibleEFTHiggs
The calculation of FlexibleEFTHiggs follows the same logic as the EFT calculation of SusyHD and HSSUSY. The difference lies in the choice of the matching conditions. In 1 The FlexibleSUSY/HSSUSY model file has been written by Emanuele Bagnaschi, Georg Weiglein and Alexander Voigt and will be presented and studied in more detail by these authors in an upcoming publication.
2 In HSSUSY we used analytical Mathematica expressions for the 2-loop threshold corrections ∆λ (2) of O(α t α s ) provided by the authors of [44] and O(α 2 t ) provided by the authors of SusyHD.
FlexibleEFTHiggs, λ(M SUSY ) is determined implicitly by the condition i.e. by the condition that the lightest CP-even Higgs pole masses, computed in the effective and the full theory at the SUSY scale in fixed-order perturbation theory in the MS/DR schemes, agree. The Standard Model Higgs pole mass is calculated at the scale M SUSY as is the corresponding tadpole. Using this, the quartic Higgs coupling in the SM reads In the current implementation, only 1-loop self energies and tadpoles are used in this matching condition; in the future it is planned to take into account 2-loop corrections. Likewise, the gauge couplings and the Z-boson and top quark mass, are implicitly fixed by the conditions at the SUSY scale, where Σ Here quantities with the superscript SM are SM quantities, renormalized in the MS scheme. Three-loop RGEs are used to run the SM parameters to the EW scale, as is done in SusyHD and FlexibleSUSY/HSSUSY. The matching to experimental quantities is done at Q = M Z in exactly the same way as for FlexibleSUSY/HSSUSY described in the previous subsection, except that only 2-loop SM-QCD corrections are used to extract y SM t (M Z ). Finally, the Higgs pole mass is calculated in the Standard Model at the scale M t using the full MS-renormalized 1-loop self energy. The crucial advantage of this choice of matching conditions is that the resulting Higgs boson mass is exact at the 1-loop level and contains resummed leading logarithms to all orders. In particular, FlexibleEFTHiggs does not neglect terms of O(v 2 /M 2 SUSY ). This is in contrast to the pure EFT approach, where already at the tree-level terms suppressed by powers of M SUSY originating from the mixing of the light with the heavy Higgs are missing. Thus, FlexibleEFTHiggs has no "EFT uncertainty" [45], which is present in SusyHD and HSSUSY.
For completeness and illustration, the equivalence of the two choices of matching conditions, Eqs. (19) and (22), up to power-suppressed terms is proven analytically at the 1-loop level in Appendix A.
3 Numerical results in the MSSM and differences between calculations In the present section we discuss numerical results for the lightest, SM-like Higgs boson in the MSSM. The results of FlexibleEFTHiggs are compared to the results of SOFTSUSY, SARAH/SPheno and SusyHD and variants of the original FlexibleSUSY. We focus mainly on analysing the differences between the calculations and their origins as well as on discussing theory uncertainties.
3.1 MSSM for X t = 0

Results of FlexibleEFTHiggs and fixed-order calculations
We begin with the special case of zero DR stop mixing, X t (M SUSY ) = 0, and a common value M SUSY for all DR SUSY mass parameters, as defined in Eqs. (10). In this special case it is known that the 2-loop threshold corrections ∆λ (2) are numerically very small, and the leading 2-loop contributions of O(α s α t ) even vanish [44]. As a result, FlexibleEFTHiggs happens to be essentially as accurate as if 2-loop instead of 1-loop threshold corrections for λ were implemented. Our comparisons to other calculations will therefore be sensitive to differences which do not originate from missing 2-loop threshold corrections but from other, more subtle effects.    (2) , y t NNNLO SUSYHD λ (1) , y t NNNLO SUSYHD λ (1) , y t NNLO FlexibleEFTHiggs/MSSM λ (2) , y t NNNLO FlexibleEFTHiggs/MSSM λ (1) , y t NNNLO FlexibleEFTHiggs/MSSM w/ y t (0) , y t NNNLO FlexibleEFTHiggs/MSSM w/ y t (0) , y t NNLO FlexibleEFTHiggs/MSSM w/ y t (1) , y t NNLO -2.5 Terms of this order are ignored in SusyHD, but correctly taken into account in FlexibleEFTHiggs, so the difference between the two matching procedures is a measure of part of the theory uncertainty of SusyHD. In Ref. [45] this theory uncertainty was labelled "EFT uncertainty", and the numerical result of Figure 1 is compatible with the uncertainty estimate given in Ref. [45]: For the scenario shown in Figure 1 and M SUSY > 1 TeV the difference is smaller than 200 MeV. For M SUSY < 500 GeV the difference can reach up to 3 GeV.
• In the blue dashed-double-dotted line the low-scale computation of the running SM top Yukawa coupling has been changed, and the leading 3-loop QCD terms included so far have been switched off. Even though the impact on the Higgs mass is formally of 4-loop order, the resulting numerical difference is rather sizeable, around 600 MeV. The importance of the 3-loop corrections to the top Yukawa coupling was already stressed in Refs. [45,69,73]. The black circles represent the equivalent change in SusyHD, where the 3-loop QCD corrections to the SM top Yukawa coupling are switched off. In SusyHD the omission of this 3-loop correction leads to a change of the same size.    shows in addition that FlexibleEFTHiggs agrees at low M SUSY with all fixedorder calculations. This is the consequence of the choice of the pole-mass matching condition Eq. (20), and it reflects the fact that FlexibleEFTHiggs corresponds to an exact 1-loop calculation plus resummation of higher-loop logarithms.

Theory uncertainty estimations
The comparisons shown in the previous figures allow us to make several observations about various ways to estimate theory uncertainties. Ref. [45] has divided the theory uncertainty of SusyHD into three parts, one of which is the "EFT uncertainty" due to truncating the low-energy EFT at the dimension-4 level (i.e. taking the renormalizable SM as the EFT). This EFT uncertainty arises from missing power-suppressed terms of ; hence it becomes large for M SUSY 500 GeV. As mentioned in the context of Figure 1, the choice of the Higgs pole mass matching condition in FlexibleEFTHiggs avoids this uncertainty by construction. As a consequence, the difference between SusyHD and FlexibleEFTHiggs at low M SUSY can be regarded as a measure of the EFT uncertainty of SusyHD.
In FlexibleEFTHiggs the Higgs mass prediction is exact at the 1-loop level due to the 1-loop Higgs pole mass matching condition. At the 2-loop level, power-suppressed as well as non-power-suppressed (but non-logarithmic) terms are missing; these will be discussed in the next subsection. Now we turn to an extensive discussion of the differences between EFT and fixedorder calculations at high M SUSY , and on the resulting theory uncertainty of the fixedorder calculations. Figure 2 shows that at high M SUSY , the two fixed-order calculations of SPheno and FlexibleSUSY/SOFTSUSY deviate significantly from each other, and that FlexibleSUSY/SOFTSUSY agrees well with the EFT calculations. These differences originate from ≥ 3-loop terms, which are taken into account differently. For a deeper understanding and illustration, we derive the leading 3-loop logarithms for all these approaches: • The all-order leading-log part of the EFT results of FlexibleEFTHiggs and SusyHD can be obtained analytically by integrating 1-loop RGEs and using tree-level matching at the high and low scales. The leading logarithms in α s and α t up to 3-loop level obtained in these ways can be written as where X denotes the calculational approach (X = EFT denotes FlexibleEFTHiggs or SusyHD) and We have worked in the large-tan β limit, and the details of this calculation are shown in Appendix B; for the EFT-case similar analytical results including subleading logarithms are presented in Refs. [42,43]. By construction, all codes agree at the 2-loop level, and the EFT calculations contain the correct 3-loop leading log. However, the implicit 3-loop leading logs of SPheno and FlexibleSUSY/SOFTSUSY in (28) are both incorrect, and different. 4 The analytical results show why FlexibleSUSY/SOFTSUSY and SPheno deviate from each other at high M SUSY . They also make it clear that the difference between FlexibleSUSY/SOFTSUSY and SPheno should be regarded as part of the theory uncertainty of both programs. In fact, inspection of the coefficients of the 3-loop leading logs in Eqs. (28) indicates that the theory uncertainty of both FlexibleSUSY/SOFTSUSY and SPheno could be significantly larger than their difference. In this sense it is surprising that the EFT results are actually close to FlexibleSUSY/SOFTSUSY but far away from SPheno in Figure 2. The reason for this is an accidental cancellation between the O(α 2 s α t ) terms in Eqs. (28) and formally subleading terms. This cancellation can be made more obvious, if one expresses M h in terms of the Standard Model MS parameters at M SUSY : where In the EFT result there is an accidental, numerical cancellation between the different 3-loop terms which has been observed and discussed in Refs. [42,43]. In spite of different numerical coefficients, a similar cancellation happens in FlexibleSUSY/SOFTSUSY and (to a smaller extent) in SPheno. As a consequence, the EFT results are closer to the fixed-order ones than what could be expected.
To highlight this accidentality we show  After these considerations we turn to the question of estimating the theory uncertainty of the DR fixed-order calculations. As noted above, the difference between the fixedorder MSSM calculations in FlexibleSUSY/SOFTSUSY and SPheno is due to a different treatment of the 3-loop leading logarithms, so it can be regarded as an estimate for part of the theory uncertainty of the two calculations. On a more general level, we therefore discuss two ways to estimate the theory uncertainty of these fixed-order calculations: 1. Using known MSSM higher-order results: In the MSSM, we know that the leading 1-loop contributions are governed by the running top mass m t . On the other hand the full 2-loop MSSM SUSY-QCD contributions to m t are known [68]. Evaluating Eq. (13) at the 2-loop leading log level taking into account the full 2-loop SUSY-QCD contributions of [68] would shift the running top mass by Thus, to estimate the theory uncertainty, we can add the term to the r.h.s. of Eq. (13) and vary the coefficient C 1 within the interval [−184/9, 184/9]. This changes the 3-loop leading logarithms in the Higgs boson mass prediction by a motivated amount. The resulting uncertainty band is shown in red in Figure 4. We find that this uncertainty band contains both the FlexibleSUSY curve (green dash-dotted line) and the SPheno curve (turquoise solid line).  Figure 4 we show the four corresponding Higgs mass predictions. The two new ones are shown as the black dashed line and the brown dotted line, respectively. We find that the four approaches to calculate y MSSM t are distributed within the red uncertainty band. Their differences thus represent an alternative way to estimate the theory uncertainty from the missing 3-loop leading logarithms in the fixed-order calculations. We therefore define

Generating higher-order terms
where y (Q) refers to Eq. (16) evaluated at the scale Q. The advantage of this second way is that it can be applied also in nonminimal models, where the 2-loop contributions to y SUSY t are unknown. Another frequently used way to estimate the theory uncertainty is to vary the renormalization scale Q at which M h is calculated and the loop-corrected EWSB conditions are solved. The variation interval is usually chosen to be [Q 0 /2, 2Q 0 ], where Q 0 is the default renormalization scale to be used to calculate the Higgs pole mass in the chosen approach. In the fixed-order programs Q 0 = M SUSY is used, while in FlexibleEFTHiggs we use Q 0 = M t . Figure 5 shows M h as a function of Q in the MSSM calculated for tan β = 5, X t = 0, M SUSY = 2 TeV with FlexibleSUSY in the fixed-order approach (left panel) and with FlexibleEFTHiggs (right panel). The renormalization scale has been varied within the interval [Q 0 /2, 2Q 0 ]. In each approach one can see that the sizes of the resulting upwards and downwards variations of the Higgs pole mass is not equal and might even be highly non-linear. Due to this effect, we define the uncertainty ∆M 3.2 MSSM for X t = 0

Results of FlexibleEFTHiggs and fixed-order calculations
Now we turn to the MSSM with X t = 0. The main new aspect is that the 2-loop threshold correction for λ(M SUSY ), which is not implemented in FlexibleEFTHiggs, is now nonnegligible. Hence, we can in particular discuss the theory uncertainty of FlexibleEFTHiggs from these missing non-logarithmic 2-loop contributions. However, our analysis is intended to be more general. It aims to be applicable also to the case of the non-minimal SUSY models discussed in the subsequent sections, as well as in the future when the 2-loop threshold correction is implemented in FlexibleEFTHiggs. It might also shed light on the theory uncertainty of existing programs such as SusyHD.
In Figure 6 we green dash-dotted line) and SusyHD (brown dashed line).
The large difference between SPheno and FlexibleSUSY/SOFTSUSY in these two figures is again due to the different, incorrect ≥ 3-loop leading logs. As Figure 6 shows, the difference is increasing with M SUSY , and thus should be regarded as an estimate of part of the theory uncertainty of SPheno and FlexibleSUSY/SOFTSUSY. For X t = 0, SusyHD and FlexibleEFTHiggs differ by around 1 GeV, which corresponds mainly to the inclusion of higher-order terms in the matching of y t at M SUSY , to 1-loop terms suppressed by v 2 /M 2 SUSY , which are missing in SusyHD, and to the different definition of the running top mass at the low scale. As can be seen in Figure 6, for X t = 0 the difference between SusyHD and FlexibleEFTHiggs can become larger, due to the missing 2-loop M h matching in FlexibleEFTHiggs. Still, in accordance with the nonlogarithmic nature of the 1-loop threshold corrections, the difference between SusyHD and FlexibleEFTHiggs does not significantly increase with M SUSY .

Theory uncertainty estimations
Now we turn to estimating the theory uncertainty of FlexibleEFTHiggs. As explained in Section 3.1.2 it has no "EFT uncertainty", because power-suppressed terms are automatically taken into account up to the 1-loop level. But FlexibleEFTHiggs is missing the 2-loop threshold corrections in its current version, leading to a theory uncertainty. We propose several methods to estimate the theory uncertainty of M h in FlexibleEFTHiggs originating from these missing 2-loop threshold corrections:   They have the form where the coefficients C 2 and C 3 depend on Q match , X The uncertainty estimated in this way is shown as the dashed area in the panels on the r.h.s. of Figure 7. As can also be seen from Figure 6, the variation of C 2 and C 3 does not reflect the fact that the 2-loop threshold corrections are negligible for X t = 0 and are large for X t ≈ √ 6M SUSY . Therefore, the variation of C 2 and C 3 certainly leads to an overestimation of the theory uncertainty of FlexibleEFTHiggs for small values of X t . However, one can expect that the theory uncertainty estimated in this way is reasonable for maximal mixing scenarios. The turquoise uncertainty band in the panels on the r.h.s. of Figure 7 shows the variation of M h by ±∆M (yt 0L vs. 1L) h , i.e. the variation from using either the tree-level or 1-loop top Yukawa coupling. The figures show that this estimated uncertainty is of the same order as the uncertainty obtained from variation of C 2 and C 3 for most SUSY scales. Furthermore, we find that the uncertainty estimate vanishes for X t ≈ 0 and is maximal for maximal mixing (X t ≈ ± √ 6M SUSY ). Thus, this estimated uncertainty reflects the expectation that the missing 2-loop threshold corrections for λ(M SUSY ) are small for vanishing X t and can be sizable for maximal mixing. The effect of the matching scale variation is shown by the red band on the r.h.s. of Figure 7. We find that the uncertainty is nearly independent of X t , which is in agreement with the expectation: As can be seen from Eq. (71), the renormalization scale-dependent part of the 1-loop threshold correction, ∆λ (1) , is not X t -dependent. Furthermore, the β functions of the MSSM parameters g MSSM Y , g MSSM 2 , tan β, v u and v d , which determine λ at the tree-level, do not depend on X t either [74,75]. For this reason, the variation of Q match is not directly sensitive to X t -dependent terms. Therefore, one can expect that the variation of Q match alone is not sufficient to estimate the theory uncertainty from missing 2-loop threshold corrections. 5 Another source of uncertainty in FlexibleEFTHiggs comes from the missing 2-loop contributions to M h in the SM. As done above, one way to estimate the leading logarithmic 2-loop M h contributions is to vary the renormalization scale Q, at which M h is calculated, within the interval [M t /2, 2M t ]. This uncertainty estimate is shown in form of the yellow band on the r.h.s. of Figure 7. Comparing all uncertainty bands for FlexibleEFTHiggs, we find that for this scenario the Higgs mass theory uncertainty is dominated by the missing 2-loop contributions to λ(M SUSY ).

Combined MSSM uncertainty estimation
In the previous section we discussed many different ways to estimate contributions to theory uncertainties, relevant for existing fixed-order calculations as well as for SusyHD and FlexibleEFTHiggs. In this section we summarize and combine these various uncertainty estimates, focusing on FlexibleEFTHiggs and the fixed-order codes (the discussion equally applies to FlexibleSUSY, SOFTSUSY and SPheno). Figure 8 shows the Higgs pole mass calculated with FlexibleEFTHiggs and the fixed-order FlexibleSUSY, including estimates of theory uncertainties. The plots demonstrate that the new approach always has an uncertainty of around 2-3 GeV and becomes more accurate for M SUSY in the few-TeV range. We now provide the details of the uncertainty estimates.
FlexibleEFTHiggs calculation: Following the classification of the theory uncertainties in Ref. [45], FlexibleEFTHiggs has two basic sources of theory uncertainty: from missing higher-order corrections in the matching procedure at the high scale ("high-scale uncertainty"), and from missing higher-order corrections in the Higgs pole mass computation in the EFT at the low scale ("low-scale uncertainty").
An important property of FlexibleEFTHiggs is the inclusion of all non-logarithmic 1loop contributions to M h due to the special choice of the matching procedure. As discussed in Section 3, the resulting "EFT uncertainty" discussed in Ref. [45] due to missing powersuppressed tree-level or 1-loop terms is therefore not present in FlexibleEFTHiggs by construction.
The high-scale uncertainty of FlexibleEFTHiggs is estimated in two ways, introduced and discussed in detail in Section 3.1.2: 1. Use of y MSSM t (M SUSY ), which has been obtained from the top pole mass matching at the SUSY scale either at the tree-level or at the 1-loop level. We denote the corresponding shift in the Higgs pole mass as ∆M The low-scale uncertainty is estimated as follows: is sensitive to logarithmic higher-order Standard Model corrections, which is why we add it in quadrature to the former: In Figure 8 we find that for small values of X t this combined uncertainty estimate is of the order 2 GeV for FlexibleEFTHiggs. The uncertainty grows up to around 3 GeV for maximal mixing. Since FlexibleEFTHiggs is an EFT calculation, its uncertainty does not depend on the SUSY scale: Even for large M SUSY ≈ 30 TeV the uncertainty is of the order or below 3 GeV. Likewise, because there is no "EFT uncertainty", the uncertainty does not grow significantly for low M SUSY .
Fixed-order calculation: The theory uncertainty of the fixed-order calculations arises from missing higher-order corrections. As described in Section 3.1.2 we propose two measures of leading missing contributions: 1. Using the four different definitions of y MSSM t , as described in Section 3.1.2. We denote the maximum difference between the Higgs masses obtained using these four definitions as ∆M h , see Eq. (33). This is particularly sensitive to subleading logarithms governed by all couplings of the MSSM.
We have combined these two uncertainty estimates as In Figure 8, ∆M FlexibleSUSY h grows logarithmically with M SUSY as expected. For small values of X t and SUSY scales below 1 TeV, the combined uncertainty estimate is below 1 GeV. For larger SUSY scales and larger X t values, the uncertainty can grow up to 9 GeV.
We remark that further subleading effects, such as finite, non-logarithmic corrections arising e.g. from going beyond the O((α t + α b ) 2 + (α t + α b )α s + α 2 τ ) approximation at the 2-loop level, are not necessarily captured by the estimate (37); hence particularly at low M SUSY , the true uncertainty of the fixed-order calculations might be larger than this estimate.
We assume that there is a Z 3 symmetry, which forbids the µ-term so that when the new scalar singlet, S, develops a VEV and generates an effective µ-term, it solves the µ problem of the MSSM. The superpotential is then, The soft breaking Lagrangian density is, The Higgs fields develop VEVs, Here we implement our new method for predicting the Higgs mass, and our uncertainty estimates for this and the FlexibleSUSY fixed-order calculation, to the NMSSM. We then compare our FlexibleEFTHiggs calculation to the predictions using some of the publicly available software. To keep our analysis simple we set the soft-breaking squared sfermion mass parameters and gaugino masses to M SUSY , as defined in Eqs. (10), tan β(M SUSY ) = 5, and require that the two additional Yukawa couplings in the NMSSM are equal, We also require that v s is fixed so that .
The new trilinears are fixed to, where all DR quantities on the right hand side are evaluated at M SUSY . The complicated expression for A λ ensures the mass of the MSSM-like CP-odd state, which appears in the CP-odd mass matrix, is equal to M SUSY . The soft-breaking squared Higgs mass parameters m 2 Hu , m 2 H d , m 2 S are fixed by the EWSB minimization conditions. In Figure 9 we compare the NMSSM predictions for the Higgs mass using the FlexibleSUSY fixed-order calculation and the new FlexibleEFTHiggs calculation. In the top panels one can see that as in the MSSM the FlexibleSUSY prediction is remarkably close to FlexibleEFTHiggs. However the uncertainty bands for fixed-order calculation in the left panels of Figure 9 show that nonetheless this is a coincidence and the true uncertainty of the fixed-order calculation is much larger. Two different values of the new singlet Yukawa couplings, λ and κ, are shown and one can see in this case increasing these couplings reduces the Higgs mass due to increased singlet mixing, but has little impact on the comparison between the two approaches.
The panels on the right of Figure 9 show the uncertainty estimation bands for the FlexibleEFTHiggs calculation. By comparing the plots in the middle panel one can see that as M SUSY is increased, the fixed-order uncertainty rises rapidly while the FlexibleEFTHiggs If we combine these uncertainties in the manner described in Section 4, we find that also in the NMSSM FlexibleEFTHiggs becomes more precise than the fixed-order calculation for values of M SUSY in the few-TeV region. However, by comparing the cases λ = κ = 0.01 and λ = κ = 0.4 (see Figure 10) we find that the precise value of M SUSY at which this happens depends on the singlet Yukawa couplings.
We now turn to a comparison with the results of FlexibleEFTHiggs and various public NMSSM codes: the fixed-order FlexibleSUSY calculation, NMSPEC [28] in the NMSSMTools 4.8.2 package, the next-to-minimal extensions of SOFTSUSY 3.6.2 [25] and an NMSSM module generated with SARAH 4.9.0 and compiled and run in SPheno 3.3.8.   Here we see significant deviation from the MSSM pattern with the SPheno fixed-order calculation, which is due to the same infra-red divergences behind the known Goldstone boson catastrophe [91][92][93].
We also include results from a modified version of SARAH/SPheno, which calculates the top quark Yukawa coupling y NMSSM t (M Z ) using Eq. (13) as is done in FlexibleSUSY and SOFTSUSY, labeled as SARAH/SPheno FS-like. Here we omit the calculation of NMSSMCALC [29], but one may see comparisons between NMSSMCALC calculating in the DR scheme and the other fixed-order codes in Ref. [94]. Note that the SARAH/SPheno calculations take into account the full 2-loop corrections in the gaugeless limit and effective potential approximation, while the other fixed-order codes include 2-loop NMSSM corrections of O((α t +α b )α s ) from Ref. [27] but include only MSSM-like 2-loop corrections of O((α t + α b ) 2 + α 2 τ ). In Figure 10  TeV. This effect has not been discussed in Ref. [94], where only smaller M SUSY were considered. The discrepancy can be traced back to singularities in the 2-loop effective potential calculation used in SPheno, 6 briefly described in Section 2.3 of Ref. [31]. As also mentioned in this reference, these singularities are not present in the corresponding MSSM calculations, and also not present in the other NMSSM codes, since these codes do not take into account NMSSMspecific 2-loop corrections involving α λ and α κ . These singularities are similar to the ones related to Goldstone bosons and discussed in Refs. [91,92], but are related to the smallness of the physical Higgs boson mass compared to the renormalization scale. As explained in the mentioned references, such singularities are spurious and appear only due to the truncation of the perturbation series at fixed order. For this reason we regard the parameter region with large λ, κ and large M SUSY as outside the range of validity of the SPheno calculation. 7 The left panel of Figure 11 confirms that FlexibleEFTHiggs and the fixed-order calculations in FlexibleSUSY and SOFTSUSY agree remarkably well for all values of λ and κ, indicating that these couplings do not disrupt this remarkable numerical cancellation amongst the 3-loop logs, which yield results close to the correct ones calculated using effective field theory techniques. The same cancellation does not take place in the NMSSMTools calculation and the deviation between this result and fixed-order FlexibleSUSY gives an indication of the large uncertainty in these approaches. By contrast the right panel of Figure 11 shows that as with the MSSM, this cancellation depends on the value of X t . The results here are very similar to those of the MSSM, since we are in the MSSM limit. Interestingly the fixed order calculation of NMSSMTools agrees very well with FlexibleEFTHiggs when X t ≈ − √ 6M SUSY .

Numerical results in the E SSM
We now make a much bigger departure from minimality and consider an E 6 inspired model, with an extra U (1) gauge symmetry and matter filling complete multiplets of the fundamental 27 representation of E 6 . Specifically we consider the exceptional supersymmetric standard model (E 6 SSM) [95][96][97] which has previously been shown to have very heavy sfermions [97,98] making effective field theory techniques very important for accurately predicting the Higgs mass. In the past 2-loop expressions for the Higgs mass were obtained [95] by generalising MSSM [40] and NMSSM [82] results from effective field theory calculations that had been expanded to fixed 2-loop order. This was used in determining the spectrum [97,98] and showing consistency with a 125 GeV Higgs [99], though the accuracy was strictly limited due to the very heavy spectra. A first attempt at improving precision of calculations in the model was made when full 1-loop threshold corrections for the gauge and Yukawa couplings were calculated [100], with the top Yukawa threshold corrections having a significant impact on the Higgs mass. With SARAH the full 1-loop self energy can be calculated for the first time in FlexibleSUSY and SPheno and both include the option to use NMSSM and MSSM 2-loop corrections, though these will not be accurate when the exotic couplings are large and must be used with care. SARAH/SPheno can now calculate full fixed-order 2-loop order corrections. 8 Finally, recently when studying the phenomenology of an E 6 inspired model [101], SusyHD was used to resum the logs and obtain the Higgs mass after matching to the MSSM at tree level.
Here we investigate the impact of FlexibleEFTHiggs on the Higgs mass. We will compare our results to the fixed-order calculations of FlexibleSUSY and also compare with SARAH/SPheno. The E 6 SSM extends the matter content of the MSSM with the following superfields: where we include generation indices i = 1, 2, 3 and α = 1, 2 and we specify the G E 6 SSM = G SM × U (1) N gauge group quantum numbers with the quantities in brackets specifically showing the SU (3) representation, the SU (2) representation, the U (1) Y charge without GUT normalization and the U (1) N charge also without GUT normalization. 9 The full E 6 superpotential is rather complicated, but with some simplifying assumptions including a Z H 2 symmetry to forbid flavour changing neutral currents and a Z B 2 symmetry to forbid proton decay, the superpotential can be written as [97], The soft breaking Lagrangian then contains, Calculated in the gaugeless limit with the effective potential approximation, where p 2 = 0. 9 The E 6 GUT normalization for the U (1) N charges is 1 √ 40 , while the E 6 GUT normalization for the U (1) Y charges is the same as the usual SU (5) one, 3/5.
whereB is the gaugino superpartner of the B gauge field, associated with the U (1) N gauge symmetry, and we have defined H d 3 := H d and H u 3 := H u to write the soft trilinear couplings more compactly. The third generation singlet S 3 and the neutral components of H u and H d doublets are the Higgs fields which develop the VEVs, v s / √ 2, v u / √ 2, and v d / √ 2, respectively. In our analysis here we set the soft-breaking scalar and gaugino mass parameters and µ eff = λv s / √ 2 to M SUSY , as defined in Eqs. (10). In addition, we fix To ensure that the exotic quarks, the inert Higgsinos and the Z boson all get DR masses equal to M SUSY we set We also require that v s is fixed so that .
The E 6 SSM-specific trilinear couplings are set to and the soft scalar Higgs masses m 2 Hu , m 2 H d , (m 2 S ) 3,3 are fixed by the EWSB conditions. For the scans we use tan β(M SUSY ) = 5 and λ(M SUSY ) = 0.1.
In Figure 12 one can see that the fixed-order FlexibleSUSY result is quite different from the FlexibleEFTHiggs result. In this case it seems that the cancellation between the logarithms is spoiled, due to the substantially altered RGE running between the EW scale and M SUSY caused by the additional colored matter which dramatically affect the RGE trajectory of α s and then indirectly α t through the gauge coupling contributions to the RGEs. The fact that we are shifted so far away from the cancellation is also reflected in the enhancement of the fixed-order uncertainty estimate from extracting y t in different ways, shown in red in the left panels. Figure 12 also shows our uncertainty estimates for FlexibleEFTHiggs in the right panels. As with the MSSM and NMSSM we can see that our estimation of the theory uncertainty (shown on the right) is not increasing significantly with M SUSY , which is to be expected from the construction of this approach. In Figure 13 we show the SARAH/SPheno prediction, along with combined uncertainty estimations for the fixed-order FlexibleSUSY and FlexibleEFTHiggs results. Here the SPheno prediction is close to the FlexibleSUSY fixed-order prediction, particularly when the top Yukawa is extracted in the same way. This should be expected since the exotic couplings are all quite small in this scenario, making the 2-loop corrections that are only in SPheno rather small. Therefore the main difference between the SPheno and FlexibleEFTHiggs results appears to be due to the resummed logs, which in this case become important at much lower M SUSY values. Notably at M SUSY = 1 TeV there is already a 3 GeV gap between the FlexibleEFTHiggs prediction and the fixed-order predictions, though the results are compatible within estimated uncertainties. It is noteworthy that the estimated uncertainty of FlexibleEFTHiggs is in the range 2-3 GeV, like in the MSSM, while the one of the fixed order results has significantly increased. To improve the precision further, adding 2-loop matching to the FlexibleEFTHiggs calculation will be very important. It is also worth noting that we see no evidence of the problems due to infra-red divergences in the E 6 SSM-specific SARAH/SPheno 2-loop correction here, which can be understood due to the small values of the exotic couplings. We do not investigate varying the exotic couplings here due to large dimensionality of the parameter space, but leave this for dedicated studies of this model. We fix X t = 0, tan β = 5 and λ = 0.1.

Numerical results in the MRSSM
As another example for a non-minimal model, we study the properties of FlexibleEFTHiggs in the MRSSM, a minimal supersymmetric model with unbroken continuous R-symmetry [102]. The model is motivated in a number of ways, and particularly the mass of the SM-like Higgs boson has been shown to be compatible with experiment in a variety of parameter scenarios in Refs. [103][104][105]. In the following we employ the conventions of these references. The MRSSM has the same field content as the MSSM, with the following additional superfields: The superpotential of the MRSSM is given by As in the E 6 SSM and Z 3 -symmetric NMSSM, the µ term is forbidden in the MRSSM. New µ u,d terms and Yukawa-like interactions between theR Higgs fields andĤ u,d are allowed in general. The soft-breaking trilinear couplings as well as the Majorana mass terms for the gauginos are forbidden by the R-symmetry. The Lagrangian of the soft breaking scalar mass terms reads The fermionic components of theŜ,T andÔ fields mix with the gauginosB,W andg into Dirac fermions. The Dirac mass terms can be interpreted as being generated by the soft breaking of a supersymmetric hidden sector model via spurions. The resulting part of the soft-breaking MRSSM Langrangian reads where the auxiliary D fields can be eliminated by their equations of motion, giving rise to triple scalar interactions governed by the Dirac mass parameters. The Higgs fields H u , H d and S develop VEVs as in Eqs. (40). In addition, the electrically neutral linear combination of the Higgs triplet develops a VEV as For the MRSSM study presented in this section, we impose the boundary conditions of Eqs. (10) as well as at the scale M SUSY , which is inspired by BMP3 from Ref. [104]. The parameters m 2 Hu , In the left panel of Figure 14 we show the two uncertainty estimate bands for the fixed-order 2-loop calculation with SARAH/SPheno as described in Section 3.1.2: • The yellow uncertainty band shows ∆M (Q) h , i.e. the variation of M h when the renormalization scale is varied, at which M h is calculated. We find that for SPheno this estimation of part of the uncertainty is of the order 0.5-2 GeV for most of the displayed M SUSY range, which is relatively small, because of the 2-loop Higgs mass computation. In contrast, this uncertainty is between 1-3 GeV for FlexibleSUSY due to the missing 2-loop Higgs mass contributions.
• The red band shows ∆M (M Z ) and the Fermi constant G F are used as input to calculate the DR gauge couplings g 1 (M Z ) and g 2 (M Z ). This approach is a generalization of the one presented in [106] for the MSSM. FlexibleSUSY, in contrast, uses the W and Z pole masses as input to calculate g 1 (M Z ) and g 2 (M Z ) in the MRSSM at the 1-loop level. In the right panel of Figure 14, the three uncertainty bands introduced in Section 3.2 for FlexibleEFTHiggs are shown: • The turquoise uncertainty band shows ∆M . We find that this uncertainty is between 0.5-2.5 GeV and thus dominates in the scenario considered here.
• The yellow band shows ∆M (Q) h in FlexibleEFTHiggs. This estimated uncertainty is below 2 GeV for all SUSY scales above 1 TeV, similarly to the results in the other non-minimal SUSY models.
Based on these estimated theoretical uncertainties, we conclude that for the scenario studied here FlexibleEFTHiggs leads to a more precise prediction of M h than the fixedorder calculation for SUSY scales above a few TeV.
In Figure 15, the lightest CP-even Higgs pole mass in the MRSSM is shown as a function of M SUSY for the parameter point (56), together with the combined uncertainty estimates. The green dash-dotted line shows the fixed-order 1-loop calculation with FlexibleSUSY. The difference between FlexibleSUSY's and SARAH/SPheno's 1loop calculations originates again from the different definition of the running top mass in the MRSSM at M Z . The turquoise solid line shows SPheno's fixed-order 2-loop Higgs pole mass calculation. We find that the pure 2-loop corrections enhance the Higgs mass significantly, up to 12 GeV for M SUSY = 10 TeV, compared to the 1-loop result. 11 Such large 2-loop corrections in the DR scheme have been found and studied already in [104] and have been compared to the on-shell scheme in [107]. The turquoise dashed line again shows SPheno's fixed-order 2-loop calculation, but using the definition (13) for the running top Yukawa coupling. We find that in this scenario all fixed-order curves become linear for M SUSY 1 TeV, which indicates that in this scenario M h is dominated by the leading logarithm for large SUSY scales. The red solid line shows M h as calculated with FlexibleEFTHiggs in the MRSSM. Since in FlexibleEFTHiggs a 1-loop Higgs mass matching and 3-loop renormalization group running is performed, FlexibleEFTHiggs resums the leading logarithmic contributions to all orders. Since FlexibleEFTHiggs resums large logarithms, and since these logarithmic contributions dominate for SUSY scales above 1 TeV in this scenario, we again expect FlexibleEFTHiggs to give a more precise Higgs mass prediction for SUSY scales above a few TeV. In addition, we show in Figure 15 the combined uncertainty estimates introduced in Section 4. In the MRSSM, SPheno is the only publicly available program which can perform a (partial)

Conclusions
We have presented FlexibleEFTHiggs, an EFT calculation of the SM-like Higgs mass in any SUSY or non-SUSY model, that can make precise predictions for both high and low new physics scales. A judicious choice of matching conditions, equating pole masses, ensures that terms of O(v 2 /M 2 SUSY ) are included, which are missed by "pure EFT" calculations such as SusyHD and FlexibleSUSY/HSSUSY. Thus large logarithms can be resummed, while ensuring that the Higgs pole mass calculation is exact at the 1-loop level. Since this choice of matching requires only self energies and tadpoles, it is also very easy to automate and apply to any SUSY (or even non-SUSY) model, where the Standard Model is the valid low energy effective field theory. This method has been implemented in FlexibleSUSY, and we have used this to obtain results in the MSSM, NMSSM, E 6 SSM and MRSSM.
We discussed several ways to estimate the theoretical uncertainty of FlexibleEFTHiggs and the fixed-order approaches of FlexibleSUSY/SOFTSUSY and SPheno. These estimates show the expected behaviour, i.e. the fixed-order uncertainty rises with M SUSY while our FlexibleEFTHiggs estimate does not. For example in the MSSM when M SUSY is larger than a few TeV our combined uncertainty estimate for FlexibleEFTHiggs is smaller than our combination of various fixed-order uncertainty estimates, and similar results are obtained in the NMSSM, E 6 SSM and MRSSM. Moreover, even for SUSY scales close to the EW scale we observe that the uncertainty of FlexibleEFTHiggs is around 2-3 GeV and is thus not much larger than the fixed-order calculations, even for cases where the 2-loop contributions to the Higgs mass are large.
We also compared FlexibleEFTHiggs to other spectrum generators in all four of these models. In the MSSM we demonstrated that we understand all the differences between FlexibleEFTHiggs and SusyHD and showed that the two codes agree well at high M SUSY in scenarios where the 2-loop threshold corrections are negligible. In regions where the 2-loop threshold corrections are non-negligible, the codes disagree by non-logarithmic 2loop terms, which do not increase with M SUSY . However, in these regions the deviation of the codes lies within our uncertainty estimate for FlexibleEFTHiggs. We also found that the fixed-order calculations of FlexibleSUSY and SOFTSUSY agree surprisingly well with the EFT results of FlexibleEFTHiggs, SusyHD and FlexibleSUSY/HSSUSY even at very large M SUSY , owing to an accidental cancellation among the 3-loop leading logarithms. This cancellation however depends on which partial 3-loop corrections are included and SPheno does not have the same tendency despite being accurate to the same formal 2-loop order as FlexibleSUSY and SOFTSUSY. This cancellation also occurs in the NMSSM, even for rather large values of the new singlet Yukawa couplings. There we see on the other hand that the full fixed-order 2-loop calculation in SARAH/SPheno is not reliable when both M SUSY and the exotic couplings are large, due to infrared divergences which appear in the 2-loop functions, which are not present in the other fixed order codes as they neglect these contributions.
In the E 6 SSM we studied cases where all exotic couplings are rather small. Nonetheless, there is already a large impact of the exotic couplings on the fixed-order calculations at large M SUSY and thus we find that there is no longer good agreement between the available fixed-order calculations. The very different renormalization group flow in this model, where the β-function of α s vanishes at 1-loop level, spoils the accidental cancellation between the 3-loop logarithms observed in the MSSM and NMSSM. We see that some of the sources of uncertainty in the fixed-order calculation, specifically the uncertainty that is estimated from different definitions of the Yukawa couplings, rises with M SUSY much more rapidly than in the MSSM or NMSSM. Therefore, an effective field theory calculation in this model is important presumably at lighter values of M SUSY than in the MSSM and NMSSM.
Finally, we applied FlexibleEFTHiggs to the MRSSM and compared the results with the available 1-and 2-loop fixed-order calculations in a scenario with sizable triplet couplings as well as with benchmark points from the literature. Similar to the E 6 SSM, we find that the fixed-order programs no longer agree well with FlexibleEFTHiggs for SUSY scales above a few TeV. One of the reasons is the different running of α s in the MRSSM, which again spoils the accidental cancellation of higher-order logarithms. We also find that the uncertainty of the fixed-order calculations, estimated by the different definitions of the top Yukawa coupling, increases more rapidly with M SUSY than in the MSSM or NMSSM. In contrast, the combined uncertainty estimate for FlexibleEFTHiggs is independent of M SUSY , making the FlexibleEFTHiggs calculation more reliable already above a few TeV.
There are limitations of FlexibleEFTHiggs motivating further developments. The most obvious is the use of 1-loop matching at the SUSY scale instead of 2-loop matching. As a result, the implementation still misses 2-loop power-suppressed as well as non-powersuppressed (but non-logarithmic) terms in the Higgs pole mass. This is complementary to SARAH/SPheno, the only publicly available code that can include these 2-loop terms for all models, which can however become unreliable for large SUSY scales due to the lack of large higher-order logarithms. It is planned to extend FlexibleEFTHiggs by using the Higgs pole mass matching condition at the 2-loop level for a higher accuracy. Another possible extension of FlexibleEFTHiggs is to allow for more diverse mass hierarchies leading to additional intermediate scales at which subsets of particles are sequentially integrated out. In this way further types of potentially large logarithms can be resummed. of the studied MRSSM scenarios. PA thanks Roman Nevzorov and Dylan Harries for helpful comments regarding the E 6 SSM corrections. This work has been supported by the German Research Foundation DFG through grant No. STO876/2-1. JP acknowledges support from the MEC and FEDER (EC) Grants FPA2011-23596 and the Generalitat Valenciana under grant PROMETEOII/2013/017. The work of PA has in part been supported by the ARC Centre of Excellence for Particle Physics at the Tera-scale.
A Equivalence of M h -matching to ∆λ (1) We show the equivalence of FlexibleEFTHiggs' matching procedure, set forth in Section 2.4, to the 1-loop threshold corrections to λ at O(α t ) from the MSSM presented in the literature such as Eq. (10) in [44]. For this, we apply the matching condition (20) at the 1-loop level which requires that the Higgs pole mass calculated in the SM be equal to the lightest Higgs pole mass in the MSSM. In this appendix, the matching scale Q match shall be abbreviated to Q. In the Standard Model Higgs pole mass in Eq.
t DR,MSSM which agrees at the 1-loop level with the EFT result but disagrees at the 2-loop level. The fixed-order calculations of the MSSM Higgs boson mass then plug the SUSY-scale MSSM parameters into the DR-Higgs self energy. At the leading logarithmic level, this gives where m h is the running tree-level Higgs mass and where the values of the coefficients c follow from the agreement of the 2-loop leading logarithms with the correct EFT result. Hence, plugging in number, we obtain the leading logarithms up to the 3-loop level, contained in the fixed-order calculations, So far, all results are expressed in terms of low-scale SM couplings, which are connected to low-energy observables without large logarithms. It is useful to record here the equivalent results, in which these low-scale SM couplings are replaced by SUSY-scale running SM parameters, which are connected to the fundamental high-scale SUSY parameters without large logarithms. For this purpose, Eqs.