The light CP-even MSSM Higgs mass including N$^3$LO+N$^3$LL QCD corrections

We present a calculation of the light neutral CP-even Higgs boson pole mass in the real MSSM which combines state-of-the-art EFT and fixed-order results, including the three-loop fixed-order QCD corrections as well as the resummation of logarithmic terms in the ratio of the weak to the SUSY scale up to fourth logarithmic order. This hybrid calculation should be valid for arbitrary SUSY scales above the weak scale. Comparison to the pure fixed-order and EFT results provides an estimate of their individual validity range.


Introduction
With the discovery of the Higgs boson with a mass of M h = (125.10 ± 0.14) GeV [1][2][3][4], the Standard Model (SM) of particle physics is complete and appears to be a good description of nature around and below the electroweak scale. However, the SM does not describe gravity and cannot account for phenomena typically associated with dark matter, for example, or for CP-violation at the level required to explain the observed baryon anti-baryon asymmetry. Supersymmetry (SUSY) has been an attractive proposal to address some of the deficits of the SM. One particular feature of the Minimal Supersymmetric Standard Model (MSSM) is its constrained Higgs sector which, for a given set of SUSY parameters, results in a theoretical value of the lightest Higgs boson mass. Comparison to the measured mass of the observed Higgs boson as quoted above provides a stringent constraint of the MSSM. It is well known that this theory value of the lightest Higgs boson mass receives large radiative corrections, so that higher-order calculations are required in order to achieve a precision which is competitive with the experimental accuracy.
There are different methods to calculate the Higgs pole mass in the MSSM, which can be divided into fixed-order (FO), effective field theory (EFT) and hybrid approaches. In the fixed-order calculation, loop corrections to the Higgs mass are calculated in the full MSSM, and the perturbation series is truncated at a fixed order of the coupling constants. If the SUSY particles have masses not too far above the electroweak scale, the FO calculation typically leads to a reliable value. However, if (some of) the SUSY particles are very heavy, then the perturbative coefficients receive large logarithmic contributions, which spoil the perturbative series. Currently, loop corrections up to the two-loop level are known in the on-shell scheme [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and up to three-loop level in the DR scheme [10-13, 16, 22-37]. The corresponding FO Higgs pole mass results are available through implementations into publicly available spectrum generators [8,[38][39][40][41][42][43][44][45][46][47].
An EFT calculation, on the other hand, is based on the assumption that the SUSY particles are very heavy compared to the electro-weak scale. Integrating them out leaves the SM as an EFT. The latter retains the SUSY constraints through the matching conditions between the MSSM and the SM parameters, which are imposed at some large mass scale. The Higgs pole mass is then calculated from the SM MS parameters after evolving them down to the electro-weak scale through SM renormalization group equations (RGEs), thereby resumming contributions which are logarithmic in the ratio of the SUSY and the electro-weak scale ("large logarithms"). This procedure has been implemented through third logarithmic order (NNLL) in several publicly available pure-EFT spectrum generators [47][48][49]. Resummation through fourth logarithmic order (N 3 LL) has recently been achieved through the evaluation of the three-loop matching coefficient for the quartic coupling [50], which complemented the available two-loop matching relations [48,49,[51][52][53][54][55].
It turns out that, in order for the theoretical value of the light MSSM Higgs mass to be compatible with the observed Higgs mass of M h ≈ 125 GeV, the SUSY spectrum requires TeV-scale stops (see Refs. [48,52,[56][57][58], for example). It is not clear a priori whether a FO or an EFT approach provides the best value for the Higgs mass at these mass scales. For this reason, so-called hybrid approaches have been devised [41,47,56,57,[59][60][61]. They combine the virtues of a FO and an EFT calculation, and lead to a reliable value for the Higgs pole mass at all SUSY scales. So far, they rely on two-loop FO results with a resummation of the large logarithms at NNLL level at most. Comparison to the highest available FO result shows good agreement up to remarkably large SUSY scales of the order of 5-10 TeV [62], in accordance with earlier comparisons of FO and EFT results [35].
In this paper, we adopt a hybrid approach for the real MSSM by including the next perturbative order in the strong coupling. More precisely, we combine the FO and the EFT results at order y 4 t g 4 3 , resulting in a value at fourth perturbative order (N 3 LO) with N 3 LL resummation. 1 By comparing our three-loop hybrid result with the individual FO and EFT approximations, we infer the size of the terms of O(v 2 /M 2 S ) which are usually neglected in a pure EFT approach. This allows us to derive an estimate for the SUSY scale above which a pure EFT calculation is sufficient (see also Ref. [58]).
The remaining part of this paper is structured as follows: In Sect. 2 we describe our procedure to combine the three-loop FO and EFT results. The numerical implications of the resulting hybrid result are discussed in Sect. 3. Sect. 4 contains our conclusions.
2 Matching procedure 2.1 General outline So far, two approaches to combine FO and EFT results in the context of the SUSY Higgs mass have been pursued in the literature: • Subtraction approach: Here one writes the squared Higgs pole mass as where (M FO h ) 2 denotes the FO result, (M logs h ) 2 are the large logarithmic FO corrections, and (M res h ) 2 are the resummed logarithmic corrections. An advantage of this approach is that existing fixed-order results can be used and different effective theories can be considered in a straightforward way. The generalization of this approach to models beyond the MSSM is non-trivial, because it requires model-specific FO and EFT loop calculations.
This approach is implemented in FeynHiggs at the two-loop level, for example [41,56,57].
• FlexibleEFTHiggs approach [47,59,60]: Here one employs the identity where M SM h denotes the Higgs pole mass expressed through SM MS parameters, and M MSSM h is the Higgs pole mass calculated in the MSSM in the DR scheme. The MS and DR parameters appearing in Eq. (2) depend on the renormalization scale Q S , which is set close to the SUSY scale. This determines the SM quartic Higgs coupling in the MS scheme at the scale Q S , which is then evolved down to the electro-weak scale using SM RGEs in order to evaluate the Higgs pole mass from it.
Due to the simplicity of the matching condition (2), this approach can be generalized to other models in a rather straightforward way. However, the extension of the approach to the two-loop level is non-trivial, because care must be taken to cancel potential large logarithmic corrections in the matching.
The FlexibleEFTHiggs approach is implemented at one-loop level into FlexibleSUSY [47,59], and at two-loop level into SARAH/SPheno [60]. 2 In this paper, we adopt a hybrid scheme which is similar to the subtraction approach of Eq. (1). However, we work in the DR scheme and go one loop level higher, combining the FO and EFT approximations which include three-loop level QCD corrections. In our scheme, we calculate the (squared) Higgs pole mass as where M EFT h denotes the three-loop EFT result of FlexibleSUSY/HSSUSY+Himalaya [50]; it resums large logarithms of order y 4 t g 6 3 to N 3 LL, while others are resummed to NNLL. 3 Its fixed-order expansion would reproduce the full fixed-order result in the limit v 2 /M 2 S → 0, including the known two-loop corrections in the gaugeless limit and the three-loop terms of order y 4 t g 4 3 from Himalaya [31,32,35], including non-logarithmic terms. ∆ v supplies the terms that are suppressed by powers of v 2 /M 2 S as M S v at fixed order up to the two-loop level. We separate ∆ v into a tree-level+one-loop and a two-loop part, These terms are extracted from the FlexibleEFTHiggs result implemented in Flexible-SUSY, and from the two-loop contributions included in the Himalaya library, as described in what follows. The tree-level and one-loop contribution ∆ 0 +1 v is obtained by taking the difference between the one-loop FlexibleEFTHiggs result M FEFT h and the one-loop pure EFT result obtained from HSSUSY as Due to the structure of the FlexibleEFTHiggs calculation, this difference contains all treelevel and one-loop SUSY contributions of higher order in v 2 /M 2 S , and formally two-loop non-logarithmic electroweak SUSY terms (see below). In particular, large logarithmic corrections as well as two-loop non-electroweak SUSY contributions are absent. The two-loop contribution ∆ 2 v is obtained as The terms on the r.h.s. of Eq. (6) represent the difference between the two-loop fixed-order contribution O(y 4 t g 2 3 + y 6 t ) calculated with Himalaya, and the same two-loop fixed-order contribution where all O(v 2 /M 2 S ) terms are neglected. This difference thus contains all . Large logarithmic as well as non-electroweak three-loop corrections of order (v 2 /M 2 S ) 0 are absent. 2 Note that in the implementation of the FlexibleEFTHiggs approach in SARAH/SPheno, large higher-order logarithmic corrections are induced at the matching scale. As a result, SARAH/SPheno resums large logarithms only up to (including) the leading-log level. 3 Our identification of the logarithmic order refers to the required order of the β function of the SM Higgs self coupling λ. Specifically, our N n LL terms involve the β function to O(y 4 t g 2n 3 ).

Explicit result in the degenerate-mass case
for a general SUSY spectrum, but we find that the full analytic expression is too long to be displayed in this paper. For reference, however, we include the results for the case of degenerate DR SUSY mass parameters at the SUSY scale M S , i.e., Here mq 3 and mũ 3 denote the left-and right-handed third generation squark mass parameters, mg the gluino mass, m A the CP-odd Higgs boson mass, and µ the superpotential µ-parameter. We parameterize our expressions in terms of the DR stop mixing parameter where A t is the trilinear Higgs-stop-stop coupling, and tan β = v u /v d with v u and v d being the running vacuum expectation values of the MSSM up-and down-type Higgs doublets, respectively. Furthermore we define the short-hand notation Here, Q S denotes the renormalization scale at which the matching is performed, m t = y t s β v/ √ 2 is the running top quark mass, y t denotes the top Yukawa coupling, g 3 is the strong gauge coupling, and v = (v 2 u + v 2 d ) 1/2 is the SM-like Higgs vacuum expectation value, all defined in the MSSM in the DR scheme.
Following the procedure described in Ref. [50], the two-loop subtraction term on the r.h.s. of Eq. (6) can be expressed in terms of threshold corrections as (y 2 and the one-and two-loop threshold factors [30,52] The two-loop correction ∆ 2 (y 6 t ) λ is given by Eq. (21) of Ref. [48] (without the prefactor). Inserting the threshold corrections, the subtraction terms become In this section, we study the effect of the O(v 2 /M 2 S ) terms ∆ v on the Higgs pole mass as a function of the SUSY scale. Even though our approach is applicable to a general SUSY mass spectrum, we focus on the degenerate mass case in our numerical examples. For convenience we define the (non-squared) contribution of these terms as Setting tan β = 20, we find that the O(v 2 /M 2 S ) terms can be sizable below M S 0.5 TeV, while they are small as long as M S 1 TeV, see Fig. 1. Specifically, we find for tan β = 20 and M S 1 TeV: Other values of tan β lead to similar observations. The sign and the order of magnitude of these results are in agreement with the contribution due to higher-dimensional operators as presented in Ref. [53]. Since the remaining uncertainty on the Higgs pole mass is dominated by the uncertainty induced by the extraction of the running top Yukawa coupling, which has been estimated to be between 0.

Convergence for high SUSY scales
In Fig. 2 [50] (black dashdotted line), which resums large logarithms through N 3 LO. The red band is our uncertainty estimate on the hybrid result (see Sect. 3.2.3 below for details). Since ∆ v → 0 for M S → ∞, the hybrid curve converges towards the EFT curve in this limit. Note that in the scenario with x t = − √ 6 for values of M S below ∼ 600 GeV, no suitable mass hierarchy is available in Himalaya. The three-loop fixed-order contribution is set to zero in this case, which means that the EFT curve and the hybrid calculation is formally consistent only at the two-loop level for lower scales. On the other hand, for M S → M Z one may expect the hybrid curve to converge towards the three-loop fixed-order curve. However, we find a finite offset at low  Figure 1: energies of up to ∼ 0.5 GeV for x t = 0 and ∼ 1.5 GeV for x t = − √ 6. This offset results from higher order O(v 2 /M 2 S ) terms, which are not suppressed in the low M S region. The origin of these will be investigated in the following sub-section.
In Fig. 3 a comparison of the hybrid results with the three-loop FO and EFT ones is shown as a function of x t for the degenerate scenario with tan β = 20 and M S = 3 TeV, where the MSSM value of the Higgs pole mass can be in agreement with the experimentally measured value. As our derivation of the ∆ v terms from above suggests, we find agreement of the hybrid result with the EFT within 0.5 GeV for such a large SUSY scale. The largest deviations of 0.5 GeV occur in the region |x t | > 3, while in the region |x t | < 3 the deviation is smaller than 0.1 GeV. However, this region suffers from a problematic feature of the fixed-order calculation, which is the occurrence of tachyonic DR masses of the heavy CP-even, the CP-  odd, and the charged Higgs bosons at the electroweak scale for x t > 0; this will be discussed in more detail in Sect. 3.3.

Convergence for low SUSY scales
As described in Refs. [47,59], the (hybrid) FlexibleEFTHiggs calculation implemented in the one-loop fixed-order DR result of FlexibleSUSY, one finds very good agreement in the limit M S → M Z if tan β → 1 and x t = 0. However, for larger values of tan β or x t the FlexibleEFTHiggs calculation does not converge well towards the fixed-order calculation for M S → M Z due to larger incomplete higher-order terms picked up by both calculations, as can be seen in Fig. 2. In the following we give examples of sources of such incomplete higherorder terms. We start from a scenario with small tan β and x t = 0, where the incomplete higher-order terms are small. We then step-wise increase tan β and M S and discuss the occurring deviations between the two calculations. Note, that the incomplete higher-order terms are also sensitive to the value of x t . However, their x t dependence at the electroweak scale cannot be properly studied for large values of x t with the DR calculation implemented in FlexibleSUSY due to the occurrence of a tachyonic DR stop mass at M S ∼ M Z .
The first row in Tab. 1 shows the scenario with tan β = 3, M S = M Z , and x t = 0, where both results agree within 5 MeV (0.01%). When increasing tan β, the two-loop differences between the two Higgs mass values become more sizable, increasing to 0.089 GeV (0.1%) for tan β = 20, see the second row of Tab. 1. There are multiple sources of such tan β-dependent higher-order terms in both calculations: In the fixed-order calculation, for example, an iteration over the squared momentum p 2 is used to find the solution of the equation where Σ h (p 2 ) is the momentum-dependent CP-even Higgs self-energy matrix and t h the tadpole vector (see Ref. [64], for example). This iteration leads to higher-order SUSY contributions of O(y n t y m b v 2 /M 2 S ) (n + m ≥ 6) which increase with tan β, for example due to the increasing bottom Yukawa coupling y b . In the FlexibleEFTHiggs approach such terms Since the tree-level MSSM DR Higgs mass (m MSSM h ) 2 initially depends on g 2 1 , g 2 2 , and c 2β ≡ cos 2β, the insertion of the threshold corrections generates two-loop terms, which are of electroweak order O(g n 1 g m 2 c 2k 2β v 2 /m 2 A ) and depend on tan β. Note that these are just two of several possible sources for incomplete higher-order tan β-dependent terms by which the two formally one-loop approximations differ.
When the SUSY scale is increased to M S = M t (third row in Tab. 1), renormalization group running effects come into play, because the scale at which the running couplings are extracted (Q = M Z ) is no longer identical to the scale where the Higgs pole mass is calculated at (Q = M S = M t ). While in FlexibleEFTHiggs the SM RGEs are used to evolve the running couplings from M Z → M t , the fixed-order calculation uses MSSM RGEs. This raises the difference between the two results to −0.387 GeV (−0.4%) in our example. For larger SUSY scales, this difference increases further, as shown in 4 th and 5 th rows of Tab. 1 for M S = 200 GeV and M S = 500 GeV, respectively. For these scales, logarithmic corrections of the form log(M S /M t ) occur, which get resummed in the Flex-ibleEFTHiggs calculation, but not in the fixed-order one. In the latter, the inclusion of two-loop corrections must account for this difference. In fact, when two-loop corrections are included in the fixed-order calculation, see the bottom row of Tab. 1, the difference is reduced again to −0.078 GeV (−0.07%).
This analysis shows that one cannot expect perfect agreement between the FlexibleEFT-Higgs and the fixed-order results at low SUSY scales M S 200 GeV, even though both calculations are formally consistent at their respective accuracy level. Since the Flexible-EFTHiggs result is part of our hybrid scheme (3)-(6), the described deviation translates into a non-convergence of M hyb h towards the three-loop fixed-order result at low SUSY scales in Fig. 2.

Uncertainty estimate
We estimate the uncertainty of the hybrid result by taking the minimum uncertainty of the FO and EFT results for each parameter point, The uncertainty of the three-loop fixed-order calculation, ∆M FO h , is estimated by (a) varying the renormalization scale Q S at which the Higgs pole mass is calculated and (b) by in-/excluding the two-loop threshold correction for the strong gauge coupling in the MSSM [65][66][67]: with Even though this uncertainty estimate implicitly assumes a common SUSY mass M S , in accordance with our numerical examples in this paper, its application is not restricted to the exactly degenerate mass case, of course. A general SUSY spectrum may require more sophisticated estimates of the FO and the EFT uncertainties though, but Eq. (17)   with is approximately independent of the SUSY scale and amounts to about 0.2 GeV. The matching scale uncertainty ∆ (Q S ) M EFT h has been estimated in Refs. [48,52,58]. It was found that for scenarios as those considered here, the uncertainty does not exceed 0.5 GeV for M S 1 TeV. A generalization of this uncertainty estimate to our result would require the extension of the underlying procedure to N 3 LL, which would involve the logarithmic terms at N 4 LO and their implementation into HSSUSY. As long as this is not available, we content ourselves to conservatively associate the maximal value of 0.5 GeV (see above) with the matching scale uncertainty, independent of M S . For the uncertainty ∆ (y SM t ) M EFT h induced by the SM top Yukawa coupling we follow the prescription of Refs. [48,58], but apply it at the next order in perturbation theory as required by our results. It amounts to approximately 0.1 GeV and increases slightly with the SUSY scale. For SUSY scales above 1-2 TeV, the total uncertainty of the EFT calculation ∆M EFT h is dominated by these three contributions and amounts to slightly less than 1 GeV, while is negligible. This is in agreement with the results from Sect. 3.1, where it was found that the O(v 2 /M 2 S ) terms are below 0.25 GeV for M S 1 TeV. Finally, we find that the uncertainty |δ xt + δ exp | of the three-loop calculation of λ given in Ref. [50] is below 2 MeV for the degenerate-mass scenarios considered here with M S 1 TeV, and is thus negligible.
Our combined (hybrid) uncertainty (17)  Very rarely it happens that the central value of the approach (EFT or FO) that determines the hybrid uncertainty through Eq. (17) is not itself contained in the resulting uncertainty band. In this case, we widen the band correspondingly.
Quite generally, we find that the SUSY scale M equal S , where both the FO and the EFT calculation have the same uncertainty, is between M S ∼ 0.7-0.8 TeV. This region is slightly lower than our estimate from Sect. 3.1. Due to our different uncertainty estimate of the fixedorder calculation, this region is also below the region of M equal S = 1.0-1.3 TeV estimated in Ref. [58]. For larger values of M S the EFT calculation deviates not more than 0.5 GeV from the hybrid calculation. The behavior of the curves and the associated uncertainty also justifies our estimate with hindsight. For example, the FO result visibly impacts the hybrid result only for rather low values of M S 1 TeV, but there the interval [M t , M S ] for the renormalization scale variation is sufficiently close to the weak scale to provide a reasonable estimate of the theory uncertainty. As described in the previous section, in the fixed-order calculation the DR masses of the heavy CP-even, the CP-odd, and the charged Higgs bosons are tachyonic at the scale Q = M Z for x t 0. The reason for this is the Bµ parameter, which is negative at that scale due to the renormalization group running, see Fig. 4a. In our scenario, the value of Bµ is fixed at the SUSY scale by the DR CP-odd Higgs mass m A (M S ) as

Tachyonic Higgs bosons at the electroweak scale
where we have set tan β(M S ) = 20 and m 2 A (M S ) = M 2 S in the last step. For such a large value of tan β, the one-loop β-function of the Bµ parameter is approximately given by For x t < −0.025 the β-function is negative, which means that Bµ increases during the renormalization group running from M S down to M Z , see the green dashed line in Fig. 4a. However, if x t > −0.025 the β-function is positive so that Bµ decreases when running down and changes sign at some low scale Q tach (green dotted line). The value of the scale Q tach can be larger than M Z if x t and/or M S are large enough, for example for x t > 0 and M S 3 TeV. When this happens, the DR masses of the heavy CP-even, the CP-odd, and the charged Higgs bosons are tachyonic at Q = M Z , because In Fig. 4b  In general, the occurrence of these tachyonic states due to higher order effects appears to make the approach [64] of matching SM and MSSM parameters at the scale M Z questionable. For SUSY scales above the TeV scale it might thus be advisable to perform the matching at a larger scale to avoid tachyonic states. To our knowledge, this program has not been pursued in all generality up to now (see Ref. [69], however). For very large SUSY scales, the FO approach is bound to fail anyway due to the large logarithms as discussed in the introduction.

Conclusions
We presented a hybrid calculation of the light CP-even Higgs boson pole mass in the real MSSM by combining FO and EFT results. Our procedure is based on the DR scheme. Beyond the relevant two-loop FO corrections and the corresponding resummation of large logarithms through NNLL, our result includes the three-loop FO corrections and the resummation through N 3 LL w.r.t. the strong coupling.
The estimated uncertainty of our hybrid result is below 1 GeV in most of the relevant parameter space. An exception is the transition region M S = 0.7-2 TeV, where both the fixed-order and the EFT calculation are less precise and the uncertainty can be up to ∼ 2 GeV.
By comparing the hybrid calculation with the pure EFT calculation, we can estimate the size of the terms of O(v 2 /M 2 S ) which are typically neglected in a pure EFT approach. For degenerate SUSY mass parameters we find that these terms are smaller than 0.25 GeV as long as M S 1 TeV, which is the region where the degenerate scenarios can be compatible with the experimental value for the Higgs mass [52]. Combining this with the fact that for M S 0.7-0.8 TeV the pure EFT calculation has a smaller uncertainty than the FO calculation (see also Ref. [58]), we conclude that a pure EFT calculation provides an excellent approximation in the MSSM for the degenerate SUSY mass parameter scenarios.