The light CP-even MSSM Higgs mass resummed to fourth logarithmic order

We present the calculation of the light neutral CP-even Higgs mass in the MSSM for a heavy SUSY spectrum by resumming enhanced terms through fourth logarithmic order (N3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{3}$$\end{document}LL), keeping terms of leading order in the top Yukawa coupling αt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _t$$\end{document}, and NNLO in the strong coupling αs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _s$$\end{document}. To this goal, the three-loop matching coefficient for the quartic Higgs coupling of the SM to the MSSM is derived to order αt2αs2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _t^2\alpha _s^2$$\end{document} by comparing the perturbative EFT to the fixed-order expression for the Higgs mass. The new matching coefficient is made available through an updated version of the program Himalaya. Numerical effects of the higher-order resummation are studied using specific examples, and sources of theoretical uncertainty on this result are discussed.


Introduction
In the MSSM (the minimal supersymmetric (SUSY) extension of the Standard Model (SM)), the mass of the lightest CP-even Higgs boson is predicted to be of the order of the electroweak scale. More precisely, at the tree-level, the Higgs boson mass is restricted to be smaller than or equal to the mass of the Z boson, M h ≤ M Z . In viable parameter regions of the MSSM, the loop corrections to the mass of the light CP-even Higgs boson must therefore be large in order for the MSSM to accommodate for the measured Higgs mass value of [3], M h = (125.09 ± 0.32) GeV. (1) It has been known for a long time that these loop corrections are indeed large, predominantly due to contributions from top quarks and their super-partners, the "stops" [4][5][6][7][8][9][10]. To be specific, in the limit where the superpartners are much heavier than the electroweak scale, the pole mass of the light CP-even Higgs boson, including the dominant one-loop contribution, reads [11], where m t is the top-quark mass, m 2 t = mt 1 mt 2 is the average of the two stop masses mt i (i = 1, 2), g t is the SM top Yukawa coupling, v ∼ 246 GeV is the vacuum expectation value of the SM, X t = A t − μ/ tan β is the stop mixing parameter, A t is the trilinear Higgs-stop coupling, μ is an MSSM superpotential parameter and tan β = v u /v d is the ratio of the up-and down-type MSSM Higgs boson VEVs. Eq. (2) illustrates that a heavy SUSY spectrum logarithmically enhances the corrections to the Higgs mass, and that the effect of the stop mixing parameter maximally enhances the Higgs mass at |X t /mt | = √ 6. Including higher order effects, it turns out that the stop masses must be larger than mt i 1 TeV in order to predict the physical Higgs mass of Eq. (1) in scenarios with degenerate SUSY mass parameters and arbitrary stop mixing [12][13][14][15][16].
For stop masses larger than about 1 TeV, logarithmic corrections like the ln(m 2 t /m 2 t ) term in Eq. (2) may spoil the precision of the perturbative fixed-order result. However, using an effective field theory (EFT) approach, the leading (nextto-leading, etc.) powers of these logarithmic terms can be resummed to all orders in the coupling constants. Terms of order v 2 /M 2 S , where M S is the typical SUSY particle mass, are usually neglected in an EFT calculation, which is justified at M S 1 TeV [15]. Their inclusion can be achieved by taking into account higher-dimensional operators [17], or through so-called "hybrid" approaches [2,14,15,[18][19][20][21].
The resummation of the logarithmic terms through an EFT calculation is achieved by integrating out the SUSY partners at a high scale μ S ∼ M S . This means that the MS parameters of the effective theory (the SM), in particular the quartic Higgs couplingλ, which itself is not a free MSSM parameter, are expressed in terms of the MSSM parameters at that scale. The SM parameters are then evolved down to a low scale μ t ∼ v through numerical SM renormalization group running, which implicitly resums all logarithms of ratios of the high and the low scale, μ S /μ t . This allows to evaluate the Higgs pole mass within the SM in terms of SM parameters: wherev is the vacuum expectation value of the Higgs field in the MS scheme, and the ellipsis denotes terms of higher order in the SM couplings. The crucial ingredients in the EFT approach are therefore the running MSSM parameters, which can be obtained from spectrum generators such as FlexibleSUSY [18,22], SARAH/SPheno [20,[23][24][25][26][27][28], SOFTSUSY [29,30], or SuSpect [31], the β functions of the SM parameters, and the matching relations of the SM to the MSSM parameters. In order to consistently resum through first (leading), second (next-to-leading), . . . , kth logarithmic order (LL, NLL, . . ., N k−1 LL), one needs to take into account the β function of the quartic Higgs coupling, β λ , through k-loop order, and the corresponding matching coefficient λ through (k − 1)-loop order, while for the other parameters, the corresponding functions are required only at lower orders. While β λ is known through four loops [39,61], however, the matching coefficient λ has been available only through two loops [1,12,13,17]. The logarithmic order for the resummed expression of the Higgs mass has thus been limited to the third logarithmic order (NNLL) up to now.
In this paper, we show how the three-loop matching coefficient for the quartic Higgs coupling can be extracted from the three-loop fixed-order expression [33,34] for the Higgs pole mass in the MSSM. The latter has recently been implemented into the Himalaya library [35]. We make the three-loop threshold correction to the quartic Higgs coupling available in Himalaya 2.0.1, which can be downloaded from https://github.com/Himalaya-Library This result allows us to study the impact of the resummation to fourth logarithmic order on the numerical prediction of the Higgs boson mass in the decoupling limit of the MSSM by implementing the three-loop correction into HSSUSY, an EFT spectrum generator from the FlexibleSUSY package.

Formalism
As briefly described in the introduction, there are different approximation schemes commonly used to calculate the light CP-even Higgs boson mass in the MSSM: The fixed-order, the EFT, and the hybrid calculation. The fixed-order calculation includes the SUSY effects through an expansion in terms of couplings up to a fixed order. In this expansion, logarithmic corrections appear, which may be large if there is a large split between the SUSY and the electroweak scale, M S v. The fixed-order calculation is therefore a suitable approximation as long as M S ∼ v. In an EFT calculation, an expansion in powers ofv 2 /M 2 S is performed, and the leading (sub-leading, …) powers of such logarithms are resummed to all orders in the couplings. An EFT calculation is therefore a suitable approximation if M S v, but becomes invalid when M S ∼ v.
In the following sections, we describe both the fixed-order and the EFT calculation in more detail, in order to prepare for the extraction of the three-loop correction to the quartic Higgs coupling of the Standard Model later in Sect. 3.
The set of SM MS parameters relevant to our calculation will be denoted as wherē λ denotes the quartic Higgs coupling,ḡ t the SM top Yukawa coupling,ḡ 3 the strong gauge coupling, andv the vacuum expectation value of the Higgs field in the SM. Furthermore, we use the following set of MSSM parameters, renormalized in the DR scheme [36], whereas y t denotes the MSSM top Yukawa coupling, g 3 the strong gauge coupling, v u and v d the vacuum expectation values of the neutral up-and down-type Higgs bosons, X t = A t −μ/ tan β the stop mixing parameter, mg the gluino mass, and mq the average mass of all squarks but the stops. The running stop masses mt 1 ≤ mt 2 are the eigenvalues of the stop mass matrix: with the SUSY breaking parameters m Q,3 and m U, 3 . Note that, due to the SUSY constraints, Y does not contain a separate parameter for the quartic Higgs coupling.

Fixed-order calculation
In the Standard Model, the pole mass of the Higgs boson can be expressed as a series expansion in terms of the SM couplings and logarithms. The dominant terms in the expansion are those which involve the strong and the top Yukawa coupling. In the following, we consider only corrections to the tree-level Higgs mass of the form O(ᾱ 2 tā n s ) with n ≥ 0, in which case the pole mass of the Higgs boson can be expressed in terms of MS parameters as wherē and μ t is the renormalization scale. The auxiliary parameter κ = 1 has been introduced to label the orders of perturbation theory. The c (n, p) SM are pure numbers; through three-loop order (n = 2), the non-logarithmic coefficients read [37][38][39] where ζ 2 = π 2 6 = 1.64493 . . . , ζ 3 = 1.20206 . . . , The logarithmic coefficients ( p = 0) can be easily obtained from the renormalization-group (RG) invariance of M 2 h and the RG-equations (RGEs) of the parameters [37], withx i ∈X . The terms in the SM β functions that are relevant for our discussion read In the MSSM one can write an analogous expression for the light CP-even Higgs boson mass in terms of the MSSM parameters. Neglecting sub-leading terms of v 2 /M 2 S , one obtains the expansion in the decoupling limit, which reads with The coefficients c (n, p) MSSM have been calculated analytically through n = 1 and can be extracted from Refs. [40][41][42][43]. The result for n = 2 was obtained in Refs. [33,34] in terms of "hierarchies", i.e., expansions in various limits of the MSSM particle spectrum. 1 The c (n, p) MSSM contain logarithmic terms of the form ln(m t /M S ) which spoil the convergence properties of the purely fixed-order result of Eq. (15) if M S m t . To make this more explicit, let us introduce a second scale μ S = μ t by perturbatively evolving the running MSSM parameters in Eq. (15) from μ t to μ S , using the corresponding β functions defined in analogy to Eq. (13). This means that we apply the replacement to Eq. (15) for all MSSM parameters y i ∈ Y , where the d (n, p) i are determined by the perturbative coefficients of the respective β functions. After re-expanding in κ, this results in a relation of the form In a fixed-order calculation, the perturbative expansion is truncated at finite order in κ. Keeping terms through order κ N , we will denote this result as For m t M S , any choice of μ t and μ S will result in large logarithms in Eq. (19). This is avoided in the EFT approach which allows to resum the (leading, sub-leading, etc. powers of) logarithms l t S to all orders in perturbation theory. This will be the subject of the next section. Of course, a reexpansion of the EFT result must take the fixed-order form of Eq. (19) again. Comparison of this re-expanded result to the fixed-order three-loop result will allow us to derive the three-loop matching coefficient forλ in Sect. 3.

EFT calculation
The idea behind the EFT calculation is to resum the logarithms of the form l t S in Eq. (18) ("large logarithms") by integrating out the heavy (i.e., SUSY) particles. As a result, 1 As has been shown recently, the three-loop calculation of the Higgs mass in the MSSM in the DR scheme is consistent with supersymmetry [44][45][46]; see also Refs. [47,48] concerning the consistency of dimensional reduction [49] and perturbative calculations in SUSY. one obtains a relation between the parameters of the effective theory (the SM) and the full theory (the MSSM) of the form In particular, one obtains a relation betweenλ and the MSSM parameters, which means that the Higgs mass in the SM, given by Eq. (9), is fixed in terms of the parameters Y . The f i in Eq. (20) are known in terms of perturbative expansions, neglecting terms of the order v 2 /M 2 S . They depend explicitly on the renormalization scale μ in the form of ln(μ/M S ). Therefore, if Eq. (20) is employed at the scale μ ∼ M S , no large logarithms appear in the matching. For our purpose, the relevant matching relations of Eq. (20) take the form where the perturbative coefficients ( x i ) can be found in Refs. [12,50,51], except for ( λ) α 2 t a 2 s , which will be one of the central results of this paper. Explicit expressions for the degenerate-mass case will be given in Sect. 3.3. The dependence on the renormalization scale μ, indicated in Eq. (20), has been suppressed here.
Assuming that the numerical values for the y i (μ S ∼ M S ) are known, 2 Eq. (20) provides numerical values for the MS SM parametersx i (μ S ). Then one may use the numerical solution of the SM MS RGEs of Eq. (13) to evolve thē x i (μ S ) down to μ t ∼ M t . In solving the RGEs numerically, one effectively resums large logarithms of the form l t S = ln(μ t /μ S ). This is in contrast to the fixed-order calculation, where these large logarithms appear explicitly in M 2 h up to a fixed order, see Eq. (19). Thex i (μ t ) are then inserted into Eq. (9) in order to calculate M 2 h up to terms of order v 2 /M 2 S . We denote this result as The only fixed-order logarithms involved in this result are of the form ln(μ S /M S ) from Eq. (20), and ln(μ t /m t ) from Eq. (9). They can be made small by choosing μ S ∼ M S and μ t ∼m t , respectively.

Re-expanding the EFT result
The perturbative version of the approach described in the previous section would be to first evolve thex i (μ) perturbatively from μ = μ t to μ S , i.e., to solve Eq. (13) in the form Eq. (17), which explicitly introduces large logarithms of the form l t S : Subsequently, one expresses thex i (μ S ) by the y i (μ S ) through Eq. (20). This last step only introduces small logarithms of the form ln(μ S /M S ). Re-expanding in κ, one thus arrives at a result which coincides with Eq. (18). If we keep terms through order κ N , this result will be denoted as Obviously, the following formal relation applies: if the same order in the perturbative expansions of the βfunctions, the matching relations, and the SM expression for M 2 h is used in deriving the results on both sides of this equation. Since the perturbative expression for M 2 h is unique, we also have with the fixed-order result of Eq. (19). These relations will be used in the next section to extract the three-loop matching relation for the quartic Higgs couplingλ(μ S ). The goal of this paper is to calculate the light CP-even Higgs pole mass of the MSSM in the decoupling limit including the fixed-order through O(α 2 t a 2 s ) (N 3 LO), as well as resummation in α 2 t α n s through fourth logarithmic order (N 3 LL). This calculation requires to include Currently, all of the necessary expressions are known, except for the three-loop matching relation forλ to orderᾱ 2 tā 2 s . In the next section, we will derive this quantity from the H3m result, i.e., the known fixed-order corrections of O(α 2 t a 2 s ) for M 2 h from Refs. [33,34].

Extraction of the three-loop matching coefficient
3.1 General procedure Using Eqs. (9), (11), (14) and (21), and setting μ t = μ S , the three-loop SUSY QCD result for M 2 h,EFT,3 (μ S , μ S ) can be written in the following form: where l St = ln(μ 2 S /m 2 t ) and, as before, the μ S dependence of α t , a s , α t , a s and λ is suppressed. The only unknown term on the r.h.s. of Eq. (27) is the three-loop matching coefficient for the quartic Higgs coupling ( λ) α 2 t a 2 s . Assuming that the three-loop fixed-order result M 2 h,FO,3 (μ S , μ S ) is known, we could insert Eq. (26) into (27) and solve for the unknown matching coefficient: Note that all large logarithms l St cancel on the l.h.s. of Eq. (28). Thus, we may write Eq. (28) as where, The matching coefficient ( λ) α 2 t a 2 s obtained in this way is defined in the MS scheme and expressed in terms of the MSSM DR parameters α t and a s , in accordance with Eq. (21). 3 Inverting the matching relations for α t and a s , it can also be expressed in terms of SM MS parameters according to, where, and

Tree-loop fixed-order result
Equation (28) shows how the three-loop matching coefficient for the quartic Higgs coupling can be extracted from the three-loop fixed-order result for the MSSM Higgs mass. The latter has been calculated in Refs. [33,34] in the form of a set of expansions around various limiting cases for the SUSY masses ("hierarchies"). Since the explicit formulae for this result are available in the Mathematica package H3m [54], we will refer to it as the "H3m result" in what follows. In all of the different expansions, terms of O(v 2 /M 2 S ) have been neglected. The calculation was performed in the DR scheme with an on-shell renormalization condition for the -scalars were m 2 = 0. 4 We refer to this renormalization scheme as the "H3m scheme".

Transformation to DR
In order to be able to seamlessly combine the three-loop result in the H3m scheme with existing lower-order calculations, it is necessary to convert it to the more commonly used DR scheme, where m completely decouples from the model. To do that, we need to reconstruct the m -terms in the H3m result. This can be done by noting that, up to two-loop O(α 2 t a s ), the analytic form of the corrections to the Higgs mass are identical in the DR, the DR , and the H3m scheme for m = 0. Since the DR result is independent of m to all orders in perturbation theory, we can convert the known two-loop O(α 2 t a s ) DR expression to the DR scheme by shifting the stop masses according to Refs. [36,41,55]. Expanding the resulting expression to O(α 2 t a 2 s ) generates all m -dependent terms up this order in the DR scheme. From there, we can convert the stop masses and m to the H3m scheme, using the formulae of Ref. [34]. This generates a non-vanishing term at O(α 2 t a 2 s ), which is non-zero even when the on-shell condition m = 0 is applied. For m = 0, this term reads 5 with l Sx = ln μ 2 S /m 2 x and 12 = m 2 Adding these terms to the H3m result provides the three-loop Higgs mass corrections in the DR scheme: We checked that the resulting DR expression is renormalization scale independent by using the corresponding stop mass β functions in the DR scheme. Furthermore, we explicitly verified the cancellation of the l St terms in Eq. (28) up to higher orders in the hierarchy expansions of the H3m result.

Reconstruction of the logarithmic terms
After transforming the H3m result into the DR scheme according to Eq. (36), it can be inserted into Eq. (28). This results in the three-loop matching coefficient for the quartic Higgs coupling, expressed in terms of the H3m-hierarchies defined in Ref. [34]. We denote this result as ( λ H3m ) α 2 t a 2 s in what follows.
Due to renormalization group invariance of the MSSM Higgs mass, we can actually derive the logarithmic terms of the form ln(μ 2 /M 2 S ) in λ for general MSSM particle masses by requiring that with M 2 h,3 from Eq. (30), and using the three-loop MSSM β functions. We refer to the corresponding matching coefficient which includes the exact mass dependence of the logarithmic terms reconstructed in this way as ( λ EFT ) α 2 t a 2 s . Note that only the non-logarithmic term of the fixed-order three-loop result of Ref. [34] enters this result. Of course, expanding ( λ EFT ) α 2 t a 2 s in terms of the H3m hierarchies up to the appropriate orders, we recover ( λ H3m ) α 2 t a 2 s as defined above.

Example: degenerate-mass case
In this paper, we refer to the limit m U,3 = m Q,3 = mg = mq = M S as the "degenerate-mass case", where m Q,3 and m U,3 are soft-breaking parameters of the Lagrangian introduced in Eq. (8). Since we have made the x t dependence explicit in our result and we neglect all but the leading terms in α t ∝ m 2 t , we can set mt 1 = mt 2 = M S in our expressions. In the degenerate-mass limit, the expression for ( λ) α 2 t a 2 s is simple enough to be quoted here. In this case, the matching coefficients for the top Yukawa coupling, defined by Eq. (21), are given by, where L S = ln(μ 2 S /M 2 S ). This leads to a subtraction term (see Eq. (30)) with c (2,0) SM from Eq. (11). Using the "h3 hierarchy" of H3m, where all SUSY masses are assumed to be of comparable size and the expansion is performed in the mass differences, the H3m result for the degenerate-mass case reads where we set μ t = μ S . Note that higher orders in x t are not included in the H3m result. The corresponding shift from the H3m to the DR scheme is (see Eq. (35)) Combining Eqs. (40), (41), and (42) according to Eq. (28), we obtain for the matching coefficient in terms of DR parameters If one re-expresses the one-and two-loop corrections in terms of SM MS parameters the following shift must be added to Eq. (43) in the degenerate-mass case,

Implementation into Himalaya
Recently, the original Mathematica [56] implementation H3m of the three-loop fixed-order results of Ref. [34] was translated into the C++ library Himalaya 1.0 [35] in order to facilitate the combination of these terms with lowerorder codes such as FlexibleSUSY, SARAH/SPheno, SOFTSUSY or SuSpect, which typically work in the DR scheme. Himalaya 2.0.1 extends the functionality of Himalaya 1.0 to provide the three-loop matching coefficient ( λ) α 2 t a 2 s by implementing Eq. (28), including the conversion from the H3m to the DR scheme. In addition, we implemented the shift of Eq. (34) which converts the parameters in the matching coefficient from the DR to the MS scheme. This allows to directly use the result in existing EFT codes such as HSSUSY [18] or SusyHD [13], where the oneand two-loop corrections are expressed in terms of SM MS parameters.
Since the H3m result is given as an expansion in mass hierarchies, it is important to provide uncertainty estimates due to missing higher order terms in these expansions. We employ two largely complementary ways to estimate this uncertainty, referring to the logarithmic and the non-logarithmic terms, respectively.
Concerning the logarithmic terms, we proceed as follows. As described in Sect. 3.2.2, within the DR scheme, there are two possible extractions of the matching relation for the quartic Higgs coupling. Both of them use the hierarchy expansions of H3m for the non-logarithmic terms. However, while ( λ H3m ) α 2 t a 2 s uses these expansions also for the logarithmic terms, ( λ EFT ) α 2 t a 2 s contains their exact mass dependence, derived from RG invariance (see Sect. 3.2.2). We thus use the difference of ( λ EFT ) α 2 t a 2 s to ( λ H3m ) α 2 t a 2 s at the scale μ S as an uncertainty estimate: For the non-logarithmic terms, on the other hand, we consider the conversion term (δλ) α 2 t a 2 s defined in Eq. (34), whose mass dependence is known exactly. Since the main source of uncertainty in these expansions occurs for large mixing, we determine the highest power n max of x t taken into account in the specific H3m hierarchy, and use the size of the terms of order x n t with n max < n ≤ 4 in the non-logarithmic part of (δλ) α 2 t a 2 s as uncertainty estimate, named δ x t . Note that powers higher than x 4 t cannot appear in ( λ) α 2 t a 2 s when the result is expressed in terms of the MSSM top Yukawa coupling. The reason is that the one-loop correction ( λ) α 2 t contains no terms with x n>4 t , and additional loops involving only (s)quarks, gluons, and gluinos do not introduce any additional X t -dependence. To be specific, let us again consider the limit of degenerate MSSM mass parameters. In this case, H3m uses the h3 hierarchy described in Sect. 3.3, which includes only terms through x 3 t though. The uncertainty is thus estimated with the help of the non-logarithmic terms of order x 4 t in (δλ) α 2 t a 2 s , given by, We combine these two uncertainties linearly and define the total uncertainty due to the hierarchy expansions as, Technical details on how to calculate the three-loop corrections and the combined uncertainties with Himalaya 2.0.1 can be found in Appendix.

Numerical study and comparison with other calculations
To study the numerical impact of the three-loop matching coefficient ( λ)ᾱ2  [12,17] on the pure EFT calculation of HSSUSY is shown as a function of the SUSY scale M S for degenerate soft-breaking mass parameters, all set equal to M S . Furthermore, we set μ(μ S ) = m A (μ S ) = μ S , tan β(μ S ) = 10, A t = X t + μ/ tan β, while all other trilinear couplings are set to zero. The upper row shows a scenario with vanishing stop mixing, X t (μ S ) = 0, the lower row shows one with maximal stop mixing, X t (μ S ) = − √ 6M S .   . For comparison, the yellow horizontal band shows the current experimental value for the Higgs mass, see Eq. (1). As was already observed for example in Refs. [16,18,19], we find that in the range M S ≥ 1 TeV the fixed-order and the EFT calculations deviate by several GeV. This is to be expected, because the EFT calculation resums the large logarithmic corrections (in contrast to the fixed-order calculation) and above M S 1 TeV the neglected terms of O(v 2 /M 2 S ) are negligible [15,18,20]. As the black dashed and solid red line are hardly distinguishable in these plots, we show the shift relative to the oneand two-loop calculations of HSSUSY in the right column of Fig. 1. The gray band in Fig. 1d corresponds to the theoretical  For X t = 0, this uncertainty is zero, see Eq. (46), because we also set μ S = M S . This is consistent with the fact that in this case, the degenerate-mass limit of the H3m result is exact. The red band shows the "EFT uncertainty" as defined in Refs. [12,13,16], estimating effects from missing terms of O(v 2 /M 2 S ). We see that the impact of ( λ EFT )ᾱ2 tā 2 s is largely negative with respect to the two-loop threshold correction, λ 2L , and may reduce the Higgs mass by up to 0.6 GeV for maximal mixing when considering all values in the grey uncertainty band. For zero stop mixing, the shift is significantly smaller ( 20 MeV).
In Fig. 2, the Higgs mass prediction is shown as a function of the relative stop mixing parameter x t = X t /M S for a scenario with tan β = 10 and M S = 5 TeV, where both the fixed-order and the EFT approach can accommodate for the experimentally observed value of M h , Eq. (1), as long as |x t | is sufficiently large. The right panel shows again the difference of the three-loop calculation of HSSUSY with respect to the one-and two-loop calculations. In accordance with Fig. 1, we find that the shift induced by including ( λ EFT )ᾱ2 tā 2 s is negative by trend, and below about 200 MeV for x t > −2. Below that value, the effects could be of order 1 GeV, but the uncertainty of our approximation grows to about 100% in this case, because are not included in the hierarchy expansions of the H3m result.
To get an idea of the maximal effect that ( λ EFT )ᾱ2 ). 6 The hatched region marks the range of SUSY scales where the lightest running stop mass is below 1 TeV for at least one of the scanned points; in this case, the EFT may not be applicable. For zero stop mixing (left panel), we find that ( λ EFT )ᾱ2 can have an effect up to ≈ −150 MeV for M S ≥ 1 TeV. In the region where mt 1 > 1 TeV, the correction reduces to −130 MeV at most. The three-loop correction decreases for larger SUSY scales, mainly due to the fact that the SM couplings become smaller. For maximal stop mixing, x t = − √ 6, the effect of the three-loop correction is significantly larger, and can reach −1.25 GeV for mt 1 1 TeV. The correction becomes particularly large when the soft-breaking stop-mass parameters m Q,3 and m U,3 become small.

Conclusions
We have calculated the light CP-even Higgs mass of the MSSM by including all known fixed-order radiative corrections through O(α 2 t α 2 s ), and resumming the logarithmically enhanced terms for a heavy SUSY spectrum through fourth logarithmic order in SUSY QCD. The only ingredient entering this result that was unavailable in the literature up to now was the three-loop matching coefficient at O(α 2 t α 2 s ) for the quartic Higgs coupling from the SM to the MSSM. We . In the scanned parameter region, the most frequently chosen hierarchy is h3 or one of its sub-hierarchies.
) is included, with respect to the two-loop calculation. In the hatched region there is mt 1 (M S ) ≤ 1 TeV for at least one of the scanned parameter points derived it from the known three-loop corrections to the light CP-even Higgs boson mass of Refs. [33,34]. The coefficient is provided both in terms of DR and MS parameters through its implementation into the public Himalaya library, version 2.0.1. This should facilitate its inclusion into spectrum generators which implement the EFT approach. An uncertainty estimate is provided to account for missing higher order terms in the mass hierarchy expansions.
Implementing ( λ) α 2 t a 2 s through Himalaya 2.0.1 into HSSUSY, our numerical analysis shows that the three-loop correction tends to be negative and may decrease the predicted Higgs boson pole mass by up to 0.6 GeV for maximal stop mixing. In scenarios with zero stop mixing, the shift is significantly smaller, dropping to about −25 MeV for SUSY mass parameters of around 1 TeV. For non-degenerate spectra with mt 1 1 TeV, the three-loop correction can be of the same size and reach up to −1.25 GeV for low stop masses in scenarios where a suitable mass hierarchy exists. In scenarios where no such hierarchy exists the correction may be significantly larger, accompanied by a large expansion uncertainty. ) in the hierarchy expansions of H3m, we found disagreement with the logarith-mic terms of the EFT approach. We therefore discarded these orders completely (also the non-logarithmic terms) in Himalaya.
Input parameters With Himalaya 2.0.1 we extend the input parameters struct to a more general form. Its new form is summarized in the following listing: The parameters initialized to NaN are optional and will be calculated internally if not set to a finite value by the user. Note that all input parameters are interpreted as running MSSM parameters in the DR scheme at the renormalization scale scale.

Calling at the C++ level
Since the input parameters and the output of The HierarchyCalculator class takes the parameter point as the only mandatory argument. To calculate the three-loop corrections to the CP-even Higgs mass matrix or to the quartic Higgs coupling λ, one needs to call the calculateDMh3L member function of the created HierarchyCalculator object. The calculate DMh3L function takes a boolean argument to calculate the corrections of O(α 2 t a 2 s ) (argument is false) or O(α 2 b a 2 s ) (argument is true) to the CP-even Higgs mass matrix. The function returns a HierarchyObject which contains the calculated three-loop results.
To convert the three-loop results to other renormalization schemes, the HierarchyObject class provides new member functions which return additive shifts from the DR to any other scheme. The new member functions are listed in the following sub-section.
The following source code listing represents a complete example which illustrates how the three-loop correction of O(α 2 t a 2 s ) to the CP-even Higgs mass matrix and to the quartic Higgs coupling can be calculated with Himalaya 2.0.1.