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

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

Peter Athron, Jae-hyeon Park, Tom Steudtner,
Dominik Stöckinger, Alexander Voigt

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 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 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 , remaining a competitive alternative to existing fixed-order computations.

ARC Centre of Excellence for Particle Physics at the Terascale,

School of Physics and Astronomy, Monash University, Victoria 3800

[0.5em] Quantum Universe Center, Korea Institute for Advanced Study,

85 Hoegiro Dongdaemungu, Seoul 02455, Republic of Korea

[0.5em] Institut für Kern- und Teilchenphysik,

TU Dresden, Zellescher Weg 19, 01069 Dresden, Germany

[0.5em] Deutsches Elektronen-Synchrotron DESY,

Notkestraße 85, 22607 Hamburg, Germany

1 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 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 E6SSM, and the MRSSM.

The fixed-order and EFT approaches have both been used extensively in the literature, for a complete picture, see e.g. the recent review [3]. In a fixed-order, or Feynman diagrammatic computation, a perturbative expansion is performed to a specified order in the gauge or Yukawa couplings. In the MSSM, the dominant 2-loop corrections were added long ago [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Recent progress for the MSSM includes incorporating electroweak gauge couplings [16], a genuine calculation of leading 3-loop effects [17, 18], and momentum-dependent 2-loop contributions [19, 20, 21]. Many public codes for MSSM Higgs mass calculations are available, see e.g. [22, 23, 24, 25, 26]. There are also dedicated calculations and public codes for the NMSSM, see e.g. [27, 28, 29, 25, 26, 30]. For any user-defined model, SARAH/SPheno performs an automatic 2-loop calculation at zero momentum in the gauge-less limit [31, 32].

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. [33, 30, 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.

EFT calculations use a matching-and-running procedure. In the simplest case, all non-SM particles are integrated out at some high SUSY scale. The running SM parameters at the high scale are then determined by matching, run down to the EW scale using renormalization group methods, and the Higgs mass is computed at the weak scale in the SM. Since the early works in this approach [35, 36, 37, 38, 39, 40, 41], developments include the analytical evaluation of 3-loop terms [42], next-to-next-to-leading logarithm (NNLL) accuracy [43, 44, 45, 34], non-SM EFTs potentially for additional thresholds [46, 47, 44, 48, 49, 50]. The RGEs can be solved either numerically as in this work or perturbatively as done to two loops [38, 39, 40, 41] and three loops [42] (see also Appendix B). Public programs implementing this EFT-type calculation (for MSSM only) are SusyHD [45], FlexibleSUSY/HSSUSY [51] and the MhEFT [52].

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 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 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 fixed-order programs. In Sections 57 we apply FlexibleEFTHiggs to the NMSSM, E6SSM and the MRSSM and perform an uncertainty estimation. We conclude in Section 8.

2 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  scheme. There are also very accurate calculations in the on-shell renormalization scheme, for example FeynHiggs [33, 59, 60, 6, 22, 19, 61, 62] and NMSSMCALC [63, 64, 29], but we will not go into the details of the on-shell calculations. In the following we use the programs FlexibleSUSY 1.5.1, SOFTSUSY 3.6.2, SARAH 4.9.0, SPheno 3.3.8, FeynHiggs 2.12.0, SusyHD 1.0.2 and NMSSMTools 4.8.2, if not stated otherwise.

2.1 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 , and are , and , respectively, with under the GUT normalization. Sometimes it is more convenient to write expressions in terms of the gauge coupling, which we denote . As usual, we also use , and .

The spontaneous breakdown of electroweak symmetry occurs when the coefficient of the bilinear term in the Higgs potential,


is negative. This causes the neutral component of the Higgs field, , which is a doublet, to develop a vacuum expectation value (VEV), . The Standard Model fermions are the left handed quark and lepton doublets and , and the right handed singlets for up-type and down-type quarks and charged leptons, , , . They obtain mass through their Yukawa interactions with the Higgs field,


when the neutral Higgs field develops a VEV. Here denote generation indices, and we define the dot product, . To simplify the notation we denote the third generation Yukawa couplings as which are the largest singular values of , 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 quantum numbers


where the first and second symbol in the parentheses denotes the representation of the corresponding superfield with respect to and and the third component is the hypercharge in standard normalization. The neutral components of the up-type Higgs field, , and down-type Higgs field, , develop the VEVs, and , respectively. As usual we define,


where . The soft breaking Lagrangian is given by


In the following, we trade the soft-breaking trilinear couplings for the customary parameters as


with the appearing matrices given in the super-CKM basis [65, 66]. For the third generation fermions we define , , . The gauginos have the following quantum numbers:


In the scenarios studied in the following we set the dimensionful running superpotential and soft-breaking parameters to the common value of the SUSY scale, , if not stated otherwise:


Sometimes we will go beyond the last equation and keep as a free parameter and set it to a non-zero value.

In our numerical evaluations we will choose the numerical values for the running fine structure constant, , , for the top quark, lepton and boson pole masses, and for the running -quark mass, if not stated otherwise.

2.2 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 parameters at the SUSY scale.

  2. Calculate the Higgs pole mass from the 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 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, is fixed from the running , leaving as a free parameter at the SUSY scale. The gauge couplings, , and and Yukawa couplings, , and can be extracted from data. This can be done using the measured values of the running 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 is the -boson pole mass, is the transverse part of the 1-loop self energy and is the strong coupling in the SM with 5-flavours. Using these and further similar relations, all gauge couplings and EWSB parameters of the fundamental SUSY theory can be determined at the low scale . 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 denotes the top pole mass, , and denote the scalar, left-handed and right-handed part of the 1-loop top self energy without SM-QCD contributions, evaluated at , and denote SM-QCD self energy contributions, with a factor removed, evaluated at [67, 68]:


Eq. (13) is evaluated at the scale and yields the running top mass . 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 denotes the CP-even Higgs tree-level mass matrix, and are the -renormalized CP-even Higgs self energy and tadpole, respectively, and , . In the MSSM, FlexibleSUSY, SOFTSUSY and SPheno use the full 1-loop self energy and 2-loop corrections of the order from [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 2-loop self energy in the gauge-less limit and at zero momentum in any given model [31, 32].

2.3 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 , and that the SM is the correct low-energy EFT below .

The determination of 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 at 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 , dimensionless (combinations of) SUSY parameters and at most logarithms of SUSY masses. No terms suppressed by powers of appear. The known 1- and 2-loop threshold correction to from the MSSM read [44, 45]


where and . In Eqs. (18)–(19) , and denote the Standard Model electroweak gauge and top Yukawa couplings at the SUSY scale, respectively, all defined in the scheme. With we denote the matching scale, which we identify with , if not stated otherwise. The loop functions as well as can be found in [44, 45]. Since 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 [45] and FlexibleSUSY/HSSUSY [51].111The 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. HSSUSY is now part of the public FlexibleSUSY distribution and has the same essential features and method of SusyHD within a C++ framework. Both programs use the same definition222In HSSUSY we used analytical Mathematica expressions for the 2-loop threshold corrections of provided by the authors of [44] and provided by the authors of SusyHD. (18) for and 3-loop RGEs to evolve to the scale [69, 70]. At the scale, both programs determine the SM gauge and Yukawa couplings by matching to experiment. HSSUSY extracts the SM gauge and Yukawa couplings at the 1-loop level from , and 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 . HSSUSY employs full 1-loop and leading 2-loop corrections of ; SusyHD uses a numerical fit formula approximating the full 2-loop corrections.

2.4 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 FlexibleEFTHiggs, 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 / schemes, agree. The Standard Model Higgs pole mass is calculated at the scale as


where is the running Higgs mass in the Standard Model, is the -renormalized Standard Model Higgs self energy and 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 -boson and top quark mass, are implicitly fixed by the conditions


at the SUSY scale, where again denote the 1-loop top self-energy contributions without the SM QCD part, and denote SM QCD self energy contributions in the scheme, with a factor removed, evaluated at [71]:


Here quantities with the superscript are SM quantities, renormalized in the 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 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 . Finally, the Higgs pole mass is calculated in the Standard Model at the scale using the full -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 . This is in contrast to the pure EFT approach, where already at the tree-level terms suppressed by powers of 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

3.1.1 Results of FlexibleEFTHiggs and fixed-order calculations

We begin with the special case of zero stop mixing, , and a common value for all  SUSY mass parameters, as defined in Eqs. (10). In this special case it is known that the 2-loop threshold corrections are numerically very small, and the leading 2-loop contributions of even vanish [44]. As a result, FlexibleEFTHiggs happens to be essentially as accurate as if 2-loop instead of 1-loop threshold corrections for 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.

Figure 1: Influence of switching on or off different contributions to the lightest CP-even Higgs pole mass, , in SusyHD and FlexibleEFTHiggs for , .

Figure 1 compares FlexibleEFTHiggs to SusyHD. It demonstrates the validity of FlexibleEFTHiggs and shows the numerical impact of the various different design choices made in FlexibleEFTHiggs and SusyHD. The red solid line shows as a function of for FlexibleEFTHiggs with maximum precision, i.e. with 1-loop mass matching conditions Eqs. (20)–(25) at the scale , 3-loop running in the Standard Model, 1-loop matching to the known low-energy parameters, including 2-loop QCD corrections to . The other lines correspond to SusyHD and variants of FlexibleEFTHiggs, where the SusyHD-like calculation is transformed step by step into the FlexibleEFTHiggs-like one. We will now explain each step in detail.

  • The brown dashed line corresponds to SusyHD with maximum precision. The brown pluses correspond to FlexibleEFTHiggs, where the calculation of all Standard Model parameters is performed using the same expressions as in SusyHD. This means in particular that the fit formulas of Ref. [69] are used to obtain the running gauge and Yukawa couplings at the scale. Thus, both programs use 2-loop threshold corrections to from Eq. (18), 3-loop running in the Standard Model, calculation of using 4-loop QCD and 2-loop electroweak running from to plus 3-loop matching, and calculation of at NNNLO [69]. The two programs agree exactly with each other.333For this reason the FlexibleEFTHiggs version modified in this way might be viewed as a replica of SusyHD within the C++ framework of FlexibleSUSY.

  • The green crosses and the green dash-dotted line correspond to SusyHD and the modified FlexibleEFTHiggs respectively with only 1-loop matching of at the high scale . The numerical difference from what would result from 2-loop matching is very small for large , namely below for . This confirms the statement that the 2-loop threshold correction is negligible for and a common SUSY mass scale.

  • The black dotted line corresponds to replacing the -matching, Eq. (18), by the matching procedure of FlexibleEFTHiggs, Eqs. (20)–(25), except that the equality of the top pole masses at has been required at the tree-level only. The Higgs pole mass matching is the essential design choice of FlexibleEFTHiggs. The line converges to the -matching curves for large , confirming that the two matching procedures become equivalent for . For the SusyHD-approach becomes unreliable. The difference between the two matching procedures is formally of . 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 the difference is smaller than . For the difference can reach up to .

  • 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 . The importance of the 3-loop corrections to the top Yukawa coupling was already stressed in Refs. [73, 69, 45]. 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.

  • The red line shows the calculation in FlexibleEFTHiggs. It differs from the blue dashed-double-dotted line in the following ways: (i) is calculated by matching the top pole mass at the 1-loop level (including 2-loop SM-QCD corrections) at using Eq. (25), (ii) is calculated at the scale by numerically solving Eq. (17) using the full momentum-dependent 1-loop Higgs self-energy, instead of setting the momentum to the Higgs mass, , as done in SusyHD. The inclusion of both changes leads to an approximately constant decrease of of about .

Figure 2: Comparison of predictions for in the MSSM using the EFT with pole mass matching method (FlexibleEFTHiggs/MSSM), the pure EFT calculation (FlexibleSUSY/HSSUSY and SusyHD) and the diagrammatic calculations (FlexibleSUSY/MSSM, SARAH/SPheno, SOFTSUSY and FeynHiggs) for and . The green and brown bands show the theory uncertainty as estimated by FeynHiggs and SusyHD, respectively.

Figure 2 compares the results of FlexibleEFTHiggs and SusyHD/HSSUSY to the fixed-order results of SOFTSUSY, SARAH/SPheno, and the original FlexibleSUSY. For comparison, also the results of FeynHiggs are shown; the differences between the recent versions of FeynHiggs and other calculations have been discussed e.g. in Refs. [20, 45, 3]. In line with the discussion of Figure 1, SusyHD and FlexibleEFTHiggs agree up to at high , but SusyHD deviates more strongly at low due to the missing terms of .

Figure 2 shows in addition that FlexibleEFTHiggs agrees at low with all fixed-order 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.

3.1.2 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 . 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 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 fixed-order calculations at high , and on the resulting theory uncertainty of the fixed-order calculations. Figure 2 shows that at high , 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 -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.

  • SPheno, SOFTSUSY and FlexibleSUSY do a fixed-order 2-loop computation of in the -scheme at the scale . Once the running parameters at the scale are replaced by their low-energy counterparts via the definitions of Section 2.2, implicit terms of -loop order are generated. These implicit higher-order terms are different in FlexibleSUSY/SOFTSUSY and SPheno, because of the different definitions of the top Yukawa coupling in Eqs. (13), (16), respectively.

The leading logarithms in and up to 3-loop level obtained in these ways can be written as


where denotes the calculational approach ( denotes FlexibleEFTHiggs or SusyHD) and , , , , . We have worked in the large- 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.444In Ref. [43], the EFT calculation was compared to “fixed-order calculations”. In that reference, “fixed-order” was simulated via perturbative truncation of the full EFT result. Hence, even at the 3-loop order, the “fixed-order” calculations of Ref. [43] always agree with the EFT result. This is different from the concrete fixed-order calculations implemented in SPheno, SOFTSUSY and FlexibleSUSY, which are 2-loop codes but nonetheless include partial corrections at the -loop level.

Figure 3: Accidentality of the agreement between EFT and () fixed-order results: in the MSSM as a function of for , , , , , .

The analytical results show why FlexibleSUSY/SOFTSUSY and SPheno deviate from each other at high . 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 terms in Eqs. (28) and formally subleading terms. This cancellation can be made more obvious, if one expresses in terms of the Standard Model parameters at :


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 Figure 3, which displays in the three approaches for different values of . is set to