Probing Unified Theories with Reduced Couplings at Future Hadron Colliders

The search for renormalization group invariant relations among parameters to all orders in perturbation theory constitutes the basis of the reduction of couplings concept. Reduction of couplings can be achieved in certain $N=1$ supersymmetric Grand Unified Theories and few of them can become even finite at all loops. We review the basic idea, the tools that have been developed as well as the resulting theories in which successful reduction of couplings has been achieved so far. These include: (i) a reduced version of the minimal $N = 1$ $SU(5)$ model, (ii) an all-loop finite $N = 1$ $SU(5)$ model, (iii) a two-loop finite $N = 1$ $SU(3)^3$ model and finally (vi) a reduced version of the Minimal Supersymmetric Standard Model. In this paper we present a number of benchmark scenarios for each model and investigate their observability at existing and future hadron colliders. The heavy supersymmetric spectra featured by each of the above models are found to be beyond the reach of the 14 TeV HL-LHC. It is also found that the reduced version of the MSSM is already ruled out by the LHC searches for heavy neutral MSSM Higgs bosons. In turn the discovery potential of the 100 TeV FCC-hh is investigated and found that large parts of the predicted spectrum of these models can be tested, but the higher mass regions are beyond the reach even of the FCC-hh.


Introduction
which should satisfy the following partial differential equation (PDE) where βa is the β-functions of ga. The above PDE is equivalent to the following set of ordinary differential equations (ODEs), which are called Reduction Equations (REs) [2][3][4], where now g and βg are the primary coupling and its corresponding β-function. There are obviously A − 1 relations in the form of Φ(g1, · · · , gA) = const. in order to express all other couplings in term of the primary one.
The crucial demand is that the above REs admit power series solutions which preserve perturbative renormalizability. Without this requirement, we just trade each "dependent" coupling for an integration constant. The power series, which are a set of special solutions, fix that constant. It is very important to point out that the uniqueness of such a solution can be already decided at the one-loop level [2][3][4]. In supersymmetric theories, where the asymptotic behaviour of several parameters are similar, the use of power series as solutions of the REs are justified. But, usually, the reduction is not "complete", which means that not all of the couplings can be reduced in favor of the primary one, leading to the so called "partial reduction" [38,39]. We proceed to the reduction scheme for massive parameters, which is far from being straightforward. A number of conditions is required (see for example [40]). Nevertheless, progress has been achieved, starting from [41], and finally we can introduce mass parameters and couplings carrying mass dimension [42,43] in the same way as dimensionless couplings.
Consider the superpotential and the SSB sector Lagrangian where φi's are the scalar fields of the corresponding superfields Φi's and λi are the gauginos. Let us write down some well known relations: (i) The β-function of the gauge coupling at one-loop level is given by [44][45][46][47][48] β (1) where T (Ri) is the Dynkin index of the rep Ri where the matter fields belong and C2(G) is the quadratic Casimir operator of the adjoint rep G.
(ii) The anomalous dimension γ (1) i j , at a one-loop level, of a chiral superfield is (iii) The β-functions of C ijk 's, at one-loop level, following the N = 1 non-renormalization theorem [49][50][51], are expressed in terms of the anomalous dimensions of the fields involved We proceed by assuming that the REs admit power series solutions: Trying to obtain all-loop results we turn to relations among β-functions. The spurion technique [51][52][53][54][55] gives all-loop relations among SSB β-functions [56][57][58][59][60][61][62]. Then, assuming that the reduction of C ijk is possible to all orders as well as for h ijk it can be proven [63,64] that the following relations are all-loop RGI where M0 is an arbitrary reference mass scale to be specified and Eq. (12) is the Hisano-Shifman relation [59] (note that in both assumptions we do not rely on specific solutions of these equations).
As a next step we substitute the last equation, Eq. (15), by a more general RGI sum rule that holds to all orders [65] m 2 i + m 2 j + m 2 k = |M | 2 which leads to the following one-loop relation Finally, note that in the case of product gauge groups, Eq. (12) takes the form where i denotes the group of the product. This will be used in the Reduced MSSM case. Consider an N = 1 globally supersymmetric gauge theory, which is chiral and anomaly free, where G is the gauge group and g the associated gauge coupling. The theory has the superpotential of Eq. (4), while the one-loop gauge and C ijk s β-functions are given by Eq. (6) and Eq. (8) respectively and the oneloop anomalous dimensions of the chiral superfields by Eq. (7). Demanding the vanishing of all one-loop β-functions, Eqs. (6,7) lead to the relations The finiteness conditions for an N = 1 supersymmetric theory with SU (N ) associated group is found in [66] while discussion of the no-charge renormalization and anomaly free requirements can be found in [67]. It should be noted that conditions (19) and (20) are necessary and sufficient to ensure finiteness at the two-loop level [44][45][46][47][48].
The requirement of finiteness, at the one-loop level, in softly broken SUSY theories demands additional constraints among the soft terms of the SSB sector [68], while, once more, these one-loop requirements assure two-loop finiteness, too [69]. These conditions impose restrictions on the irreducible representations Ri of the gauge group G as well as on the Yukawa couplings. For example, since U (1)s are not compatible with condition (19), the MSSM is excluded. Therefore, a GUT is initially required with the MSSM being its low energy theory. Also, since condition (20) forbids the appearance of gauge singlets (C2(1) = 0), F-type spontaneous symmetry breaking [70] are not compatible with finiteness. Finally, D-type spontaneous breaking [71] is also incompatible since it requires a U (1) group.
The nontrivial point is that the relations among couplings (gauge and Yukawa) which are imposed by the conditions (19) and (20) should hold at any energy scale. The necessary and sufficient condition is to require that such relations are solutions to the REs (see Eq. (10)) holding at all orders. We note, once more, that the existence of one-loop level power series solution guarantees the all-order series. There exist the following theorem [72,73] which points down which are the necessary and sufficient conditions in order for an N = 1 SUSY theory to be all-loop finite. In refs [72][73][74][75][76][77][78] it was shown that for an N = 1 SUSY Yang-Mills theory, based on a simple gauge group, if the following four conditions are fulfilled: (i) No gauge anomaly is present. (ii) The β-function of the gauge coupling is zero at one-loop level (iii) The condition of vanishing for the one-loop anomalous dimensions of matter fields, admits solution of the form (iv) When considered as solutions of vanishing Yukawa β-functions (at one-loop order), i.e. β ijk = 0, the above solutions are isolated and non-degenerate; then, each of the solutions in Eq. (24) can be extended uniquely to a formal power series in g, and the associated super Yang-Mills models depend on the single coupling constant g with a vanishing, at all orders, β-function. While the validity of the above cannot be extended to non-SUSY theories, it should be noted that reduction of couplings and finiteness are intimately related.

Phenomenological Constraints
In this section we briefly review several experimental constraints that were applied in our phenomenological analysis. The used values do not correspond to the latest experimental results, which, however, has a negligible impact on our analysis.
In our models we evaluate the pole mass of the top quark while the bottom quark mass is evaluated at the MZ scale (to avoid uncertainties to its pole mass). The experimental values, taken from ref. [79] are: We interpret the Higgs-like particle discovered in July 2012 by ATLAS and CMS [30,31] as the light CP-even Higgs boson of the MSSM [80][81][82]. The Higgs boson experimental average mass is [79] a M exp h = 125.10 ± 0.14 GeV .
The theoretical uncertainty [33,34], however, for the prediction of M h in the MSSM dominates the total uncertainty, since it is much larger than the experimental one. In our following analyses we shall use the new FeynHiggs code [33][34][35] [36]. The theoretical uncertainty calculated is added linearly to the experimental error in Eq. (26). Furthermore, recent results from the ATLAS experiment [84] set limits to the mass of the pseudoscalar Higgs boson, MA, in comparison with tan β. For models with tan β ∼ 45 − 55, as the ones examined here, the lowest limit for the physical pseudoscalar Higgs mass is We also consider the following four flavor observables where SUSY has non-negligible impact. For the branching ratio BR(b → sγ) we take a value from [85,86], while for the branching ratio BR(Bs → µ + µ − ) we use a combination of [87][88][89][90][91]: For the Bu decay to τ ν we use [86,92,93] and for ∆MB s we use [94,95]: In the following sections we will apply these constraints to each model and discuss the corresponding collider phenomenology.

Computational setup
The setup for our phenomenological analysis is as follows. Starting from an appropriate set of MSSM boundary conditions at the GUT scale, parameters are run down to the SUSY scale using a private code. Two-loop RGEs are used throughout, with the exception of the soft sector, in which one-loop RGEs are used.   [97,98]. It should be noted that FeynHiggs requires the m b (m b ) scale, the physical top quark mass mt as well as the physical pseudoscalar boson mass MA as input. The first two values are calculated by the private code while MA is calculated only in DR scheme. This single value is obtained from the SPheno output where it is calculated at the two-loop level in the gaugeless limit [99,100]. The flow of information between codes in our analysis is summarised in Fig. 1. At this point both codes contain a consistent set of all required parameters. SM-like Higgs boson mass as well as low energy observables mentioned in Sec. 3 are evaluated using FeynHiggs. To obtain collider predictions we use SARAH to generate UFO [101,102] model for MadGraph event generator. Based on SLHA spectrum files generated by SPheno, we use MadGraph5 aMC@NLO [103] to calculate cross sections for Higgs boson and SUSY particle production at the HL-LHC and a 100 TeV FCC-hh. Processes are generated at the leading order, using NNPDF31 lo as 0130 [104] structure functions interfaced through LHAPDF6 [105].  We start with the partial reduction of the N = 1 SUSY SU (5) model [16,41]. Our notation is as follows: Ψ I (10) and Φ I (5) refer to the three generations of leptons and quarks (I = 1, 2, 3), Σ(24) is the adjoint which breaks SU (5) to SU (3)C × SU (2)L × U (1)Y and H(5) represent the two Higgs superfields for the electroweak symmetry breaking (ESB) [106,107]. The choice of using only one set of (5 +5) for the ESB renders the model asymptotically free (i.e. βg < 0 ). The superpotential of the model is described by where only the third generation Yukawa couplings are taken into account. The indices α, β, γ, δ, τ are SU (5) ones. A detailed presentation of the model can be found in [108] as well as in [109,110]. Our primary coupling is the gauge coupling g. In this model the gauge-Yukawa unification can be achieved through two sets of solutions which are asymptotically free [108]: where the higher order terms denote uniquely computable power series in g. Let us note that the reduction of the dimensionless sector is independent of the dimensionful one. These solutions describe the boundaries of a RGI surface in the parameter space which is AF and where g f and g λ could be different from zero. Therefore, a partial reduction is possible where g λ and g f are independent (non-vanishing) parameters without endangering asymptotic freedom (AF). The proton decay constraints favor solution a, therefore we choose this one for our discussion. c The SSB Lagrangian is where the hat denotes the scalar components of the chiral superfields. The parameters M , µΣ and µH are treated as independent ones, since they cannot be reduced in a suitable form. The lowest-order reduction for the parameters of the SSB Lagrangian are given by: We choose the gaugino mass M for characterizing the SUSY breaking scale. Finally, we note that (i) BΣ and BH are treated as independent parameters without spoiling the one-loop reduction solution of Eq. (34) and (ii) the soft scalar mass sum rule still holds despite the specific relations among the gaugino mass and the soft scalar masses. We analyze the particle spectrum predicted for µ < 0 as the only phenomenologically acceptable choice (in the µ > 0 the quark masses do not match the experimental measurements). Below MGUT all couplings and masses of the theory run according to the RGEs of the MSSM. Thus we examine the evolution of these parameters according to their RGEs up to two-loops for dimensionless parameters and at one-loop for dimensionful ones imposing the corresponding boundary conditions. As presented in [37], the pole top mass mt is predicted within 2σ of Eq. (25). Concerning the m b (MZ ) prediction (also in [37]), we take into account a theoretical uncertainty of ∼ 3%. But even taking theoretical and experimental uncertainties into account in combination, we find agreement with the experimental value only at the 4σ level. However, since there additional uncertainties of a few percent on the quark Yukawa couplings at the SUSY-breaking scale, that were not fully included (see [20]) into the evaluation of the bottom mass, we still consider the model as viable and proceed with its analysis.
The prediction for M h as a function of the unified gaugino mass M with µ < 0 is given in Fig. 2 (left). The ∆MB s channel is responsible for the gap at the B-physics allowed points (green points). The scattered points come from the fact that for each M we vary the free parameters µΣ and µH . Fig. 2 (right) gives the theoretical uncertainty of the Higgs mass for each point, calculated with FeynHiggs 2.16.0 [36]. There is substantial improvement to the Higgs mass uncertainty compared to past analyses, since it has dropped by more than 1 GeV. c g λ = 0 is inconsistent, but g λ < ∼ 0.005 is necessary in order for the proton decay constraint [20] to be satisfied. A small g λ is expected to not affect the prediction of unification of SSB parameters. Large parts of the predicted particle spectrum are in agreement with the B-physics observables and the lightest Higgs boson mass measurement and its theoretical uncertainty. We choose three benchmarks in the low-mass region, marking the points with the lightest SUSY particle (LSP) above 1200 GeV (MINI-1), 1500 GeV (MINI-2) and 2200 GeV (MINI-3), respectively. The mass of the LSP can go as high as ∼ 3800 GeV, but the cross sections calculated below will then be negligible and we restrict ourselves here to the low-mass region. The values presented in Table 1 were used as input to get the full supersymmetric spectrum from SPheno 4.0.4 [97,98]. Mi are the gaugino masses and the rest are squared soft sfermion masses which are diagonal (m 2 = diag(m 2 1 , m 2 2 , m 2 3 )), and soft trilinear couplings (also diagonal A i = 13×3Ai).   The resulting masses of all the particles that will be relevant for our analysis can be found in   Table 3 shows the expected production cross section for selected channels at the 100 TeV future FCC-hh collider. We do not show any cross sections for √ s = 14 TeV, since the prospects for discovery of MINI scenarios at the HL-LHC are very dim. SUSY particles are too heavy to be produced with cross sections greater that 0.01 fb. Concerning the heavy Higgs bosons, the main search channels will be H/A → τ + τ − . Our heavy Higgs-boson mass scale shows values > ∼ 2500 GeV with tan β ∼ 50. The corresponding reach of the HL-LHC has been estimated in [142]. In comparison with our benchmark points we conclude that they will not be accessible at the HL-LHC. d The situation changes for the FCC-hh. Theory analyses [143,144] have shown that for large tan β heavy Higgs-boson mass scales up to ∼ 8 TeV may be accessible, both for neutral as well as for charged Higgs bosons. The relevant decay channels are H/A → τ + τ − and H ± → τ ν τ , tb. This places our three benchmark points well within the covered region (MINI-1 and MINI-2) or at the border of the parameter space that can be probed (MINI-3).
The energy of 100 TeV is big enough to produce SUSY particles in pairs. However, the cross sections remain relatively small. Only for the MINI-1 scenario the squark pair and squark-gluino (summed over all squarks) production cross sections can reach tens of fb. For MINI-2 and MINI-3 scenarios the cross sections are significantly smaller. In these scenarios squarks decay preferentially into a quark+LSP (with BR ∼ 0.95), gluino intott andbb +h.c with BR ∼ 0.33 each.
The SUSY discovery reach at the FCC-hh with 3 ab −1 was evaluated in [145] for a certain set of simplified models. In the following we will compare these simplified model limits with our benchmark points to get an idea, which part of the spectrum can be covered at the FCC-hh. A more detailed evaluation with the future limits implemented into proper recasting tools would be necessary to obtain a firmer statement. However, such a detailed analysis goes beyond the scope of our paper and we restrict ourselves to the simpler direct comparison of the simplified model limits with our benchmark predictions.
Concerning the scalar tops, the mass predictions of MINI-1 and MINI-2 are well within the anticipated reach of the FCC-hh, while MINI-3 predicts a too heavy stop mass. On the other hand, even for MINI-1 and MINI-2 no 5 σ discovery can be expected. The situation looks more favorable for the first and second generation squarks. All the predicted masses can be excluded at the FCC-hh, whereas a 5 σ discovery will be difficult, but potentially possible (see Fig. 19 in [145]). Even more favorable appear the prospects for gluino searches at the FCC-hh. All three benchmark points may lead to a 5 σ discovery (see Fig. 13 in [145]). On the other hand, chances for chargino/neutralino d The analysis presented in [142] only reaches M A ≤ 2000 GeV, where an exclusion down to tan β ∼ 30 is expected. An extrapolation to tan β ∼ 50 reaches Higgs-boson mass scales of ∼ 2500 GeV.
searches are slim at the FCC-hh. The Next-to LSP (NLSP) can only be accessed for Mχ0 1 < ∼ 1 TeV (see Fig. 21 in [145]), where all our benchmark points have Mχ0 1 > 1 TeV. Taking into account that our three benchmark points represent only the lower part of the possible mass spectrum (with LSP masses of up to ∼ 1.5 TeV higher), we conclude that even at the FCC-hh large parts of the possible SUSY spectrum will remain elusive.  We proceed now to the finite to all-orders SU (5) gauge theory, where the reduction of couplings is restricted to the third generation. An older examination of this specific Finite Unified Theory (FUT) was shown to be in agreement with the experimental constraints at the time [29] and has predicted, almost five years before its discovery, the light Higgs mass in the correct range. As discussed below, improved Higgs calculations predict a somewhat different interval that is still in agreement with current experimental data. The particle content of the model has three (5 + 10) supermultiplets for the three generations of leptons and quarks, while the Higgs sector consists of four supermultiplets (5 + 5) and one 24. The finite SU (5) group is broken to the MSSM, which of course in no longer a finite theory [14-17, 21, 24].
In order for this finite to all-orders SU (5) model to achieve Gauge Yukawa Unification (GYU), it should have the following characteristics: The superpotential of the model, with an enhanced symmetry due to the reduction of couplings, is given by [25,27]: Discussion of the model with a more detailed description can be found in [14][15][16]. The nondegenerate and isolated solutions to the vanishing of γ (1) i are: We have also the relation h = −M C, while the sum rules lead to: Therefore, we only have two free parameters, namely m 10 and M in the dimensionful sector. When SU (5) breaks down to the MSSM, a suitable rotation in the Higgs sector [14,15,[111][112][113][114], permits only a pair of Higgs doublets (coupled mostly to the third family) to remain light and acquire vev's. Avoiding fast proton decay is achieved with the usual doublet-triplet splitting, although different from the one applied to the minimal SU (5) due to the extended Higgs sector of the finite model. Therefore, below the GUT scale we get the MSSM where the third generation is given by the finiteness conditions while the first two remain unrestricted.
Conditions set by finiteness do not restrict the renormalization properties at low energies, so we are left with boundary conditions on the gauge and Yukawa couplings (36), the h = −M C relation and the soft scalar-mass sum rule at M GUT . The quark masses m b (M Z ) and m t are predicted within 2σ and 3σ uncertainty, respectively of their experimental values (see [37] for details). The only phenomenologically viable option is to consider µ < 0, as shown in [37,[115][116][117][118][119][120][121].
The scatter plot of the light Higgs boson mass is given in Fig. 3 (left), while its theory uncertainty [36] is given in Fig. 3 (right), with the same color coding as in Fig. 2. This point-by-point uncertainty (calculated with FeynHiggs) drops significantly (w.r.t. past analyses) to 0.65−0.70 GeV. The scattered points come from the free parameter m 10 .
Compared to our previous analyses [37,[115][116][117][118][119][120][121][122][123][124], the improved evaluation of M h and its uncertainty prefer a heavier (Higgs) spectrum and thus allows only a heavy supersymmetric spectrum    (which is in agreement with all existing experimental constraints). In particular, very heavy colored SUSY particles are favored (nearly independent of the M h uncertainty), in agreement with the non-observation of those particles at the LHC [126]. We choose three benchmarks, each featuring the LSP above 2100 GeV, 2400 GeV and 2900 GeV respectively. Again, they are chosen from the low-mass region. Although the LSP can be as heavy as ∼ 4000 GeV, but in such cases the production cross sections even at the FCC-hh would be too small. The input and output of SPheno 4.0.4 [97,98] can be found in Table 4 and Table 5 (with the notation as in Sect. 5).
The expected production cross sections for various final states are listed in Table 6. At 14 TeV HL-LHC none of the Finite N = 1 SU (5) scenarios listed in Table 4 has a SUSY production cross section above 0.01 fb, and thus will (likely) remain unobservable. All superpartners are too heavy to be produced in pairs. Also the heavy Higgs bosons are far outside the reach of the HL-LHC [142].
At the FCC-hh the discovery prospects for the heavy Higgs-boson spectrum is significantly better. With tan β ∼ 50 the first two benchmark points, FUTSU5-1 and FUTSU5-2, are well within the reach of the FCC-hh. The third point, FUTSU5-3, however, with M A ∼ 16 TeV will be far outside  the reach of the FCC-hh. Prospects for detecting production of squark pairs and squark-gluino pairs are also very dim since their production cross section is also at the level of a few fb. This is as a result of a heavy spectrum in this class of models (see [145] with the same Figures as discussed in Sec. 5).
Concerning the stops, the lighter one might be accessible in FUTSU5-1. For the squarks of the first two generations the prospects of testing the model are somewhat better. All three benchmark models could possibly be excluded at the 2 σ level, but no discovery at the 5 σ can be expected. The same holds for the gluino. Charginos and neutralinos will remain unobservable due to the heavy LSP. As in the previous section, since only the lower part of the possible mass spectrum has been considered (with LSP masses higher by up to ∼ 1 TeV), we have to conclude that again large parts of the possible mass spectra will not be observable at the FCC-hh.

The Finite SU (N ) 3 Model
We proceed now to a FUT based on a product gauge group. Consider an N = 1 SUSY theory with SU (N ) 1 × SU (N ) 2 × · · · × SU (N ) k having n f families transforming as (N, N  *  , 1, . . . , 1) + (1, N, N * , . . . , 1) + · · · + (N * , 1, 1, . . . , N ). Then, the first order coefficient of the β-function, for each SU (N ) group is: Demanding the vanishing of the gauge one-loop β-function, i.e. b = 0, we are led to the choice n f = 3. Phenomenological reasons lead to the choice of the SU [128], while a detailed discussion of the general well known example can be found in [129][130][131][132]. The leptons and quarks transform as: where D are down-type quarks acquiring masses close to M GUT . A cyclic Z 3 symmetry is imposed on the multiplets to achieve equal gauge couplings at the GUT scale and in that case the vanishing of the first-order β-function is satisfied. Continuing to the vanishing of the anomalous dimension of all the fields (see Eq. (20)), we note that there are two trilinear invariant terms in the superpotential, namely: with f and f the corresponding Yukawa couplings. The superfields (Ñ ,Ñ c ) obtain vev's and provide masses to leptons and quarks Having three families, 11 f couplings and 10 f couplings are present in the most general superpotential. Demanding the vanishing of all superfield anomalous dimensions, 9 conditions are imposed where The masses of leptons and quarks are acquired from the vev's of the scalar parts of the superfields N 1,2,3 andÑ c 1,2,3 . At M GUT the SU (3) 3 FUT breaks e to the MSSM, where as was already mentioned, both Higgs doublets couple mostly to the third generation. The FUT breaking leaves its mark in the form of Eq. (42), i.e. boundary conditions on the gauge and Yukawa couplings, the relation among the soft trilinear coupling, the corresponding Yukawa coupling and the unified gaugino mass and finally the soft scalar mass sum rule at M GUT . In this specific model the sum rule takes the form: The model is finite to all-orders if the solution of Eq. (42) is both isolated and unique. Then, f = 0 and we have the relations Since all f vanish, at one-loop order, the lepton masses vanish. Since these masses, even radiatively, cannot be produced because of the finiteness conditions, we are faced with a problem which needs further study. If the solution of Eq. (42) is unique but not isolated (i.e. parametric), we can have non zero f leading to non-vanishing lepton masses and at the same time achieving two-loop finiteness.
In that case the set of conditions restricting the Yukawa couplings read: where r parametrises the different solutions and as such is a free parameter. It should be noted that we use the sum rule as boundary condition for the soft scalar masses.
In our analysis we consider the two-loop finite version of the model, where again below M GUT we get the MSSM. We take into account two new thresholds for the masses of the new particles at ∼ 10 13 GeV and ∼ 10 14 GeV resulting in a wider phenomenologically viable parameter space [125].
Looking for the values of the parameter r which comply with the experimental limits, we find that both the top and bottom masses are in the experimental range (within 2σ) for the same value of r between 0.65 and 0.80 (we singled out the µ < 0 case as the most promising). The inclusion of the above-mentioned thresholds gives an important improvement on the top mass from past versions of the model [128,[135][136][137].   Fig. 4 (right). The scattered points are due to the fact that we vary five parameters, namely r and four of the parameters that form the sum rule. The uncertainty is found in the range between 0.6 GeV and 1.0 GeV. All constraints regarding quark masses, the light Higgs boson mass and B-physics are   satisfied, rendering the model very successful. The prediction of the SUSY spectrum results in relatively heavy particles, in full agreement with the current experimental searches. Again, we choose three benchmarks, each featuring the LSP above 1500 GeV, 2000 GeV and 2400 GeV respectively (but the LSP can go as high as ∼ 4100 GeV, again with too small cross sections). The input and output of SPheno 4.0.4 [97,98] can be found in Table 7 and Table 8 respectively (with the notation as in Sect. 5).

MH
MA M H ± Mg Mχ0  It should be noted that in this model the scale of the heavy Higgs bosons does not vary monotonously with Mχ0 1 , as in the previously considered models. This can be understood as follows. The Higgs bosons masses are determined by a combination of the sum rule at the unification scale, and the requirement of successful electroweak symmetry breaking at the low scale. Like in the finite scenario of the previous section, there are no direct relations between the soft scalar masses and the unified gaugino mass, but they are related through the corresponding sum rule and thus vary correlatedly, a fact that makes the dependence on the boundary values more restrictive. Furthermore (and even more importantly), the fact that we took into account the two thresholds at ∼ 10 13 GeV and ∼ 10 14 GeV (as mentioned above), allows the new particles, mainly the Higgsinos of the two other families (that were considered decoupled at the unification scale in previous analyses) and the down-like exotic quarks (in a lower degree), to affect the running of the (soft) RGEs in a nonnegligible way. Thus, since at low energies the heavy Higgs masses depend mainly on the values of m 2 Hu , m 2 H d , |µ| and tan β, they are substantially less connected to Mχ0 1 than in the other models, leading to a different exclusion potential, as will be discussed in the following.
Scenarios of Finite SU (3) 3 are beyond the reach of the HL-LHC. Not only superpartners are too heavy, but also heavy Higgs bosons with a mass scale of ∼ 7 TeV cannot be detected at the HL-LHC. At 100 TeV collider (see Table 9), on the other hand, all three benchmark points are well within the reach of the H/A → τ + τ − as well as the H ± → τ ν τ , tb searches [143,144], despite the slightly smaller values of tan β ∼ 45. This is particularly because of the different dependence of the heavy Higgs-boson mass scale on Mχ0 1 , as discussed above. However, we have checked that M A can go up to to ∼ 11 TeV, and thus the heaviest part of the possible spectrum would escape the heavy Higgs-boson searches at the FCC-hh.
Interesting are also the prospects for production of squark pairs and squark-gluino, which can reach ∼ 20 fb for the FSU33-1 case, going down to a few fb for FSU33-2 and FSU33-3 scenarios. The lightest squarks decay almost exclusively to the third generation quark and chargino/neutralino, while gluino enjoys many possible decay channels to quark-squark pairs each one with branching fraction of the order of a percent, with the biggest one ∼ 20% to tt 1 + h.c..
We briefly discuss the SUSY discovery potential at the FCC-hh, referring agian to [145] with the same Figures as discussed in Sec. 5. Stops in FSU33-1 and FSU33-2 can be tested at the FCC-hh, while the masses turn out to be too heavy in FSU33-3. The situation is better for scalar quarks, where all three scenarios can be tested, but will not allow for a 5 σ discovery. Even more favorable are the prospects for gluino. Possibly all three scenarios can be tested at the 5 σ level. As in the previous scenario, the charginos and neutralinos will not be accessible, due to the too heavy LSP. Keeping in mind that only the lower part of possible mass spectrum is represented by the three benchmarks (with the LSP up to ∼ 1.5 TeV heavier), we conclude that as before large parts of the parameter space will not be testable at the FCC-hh. The only partial exception here is the Higgsboson sector, where only the the part with the highest possible Higgs-boson mass spectra would escape the FCC-hh searches.  Table 9: Expected production cross sections (in fb) for SUSY particles in the FSU33 scenarios.

The Reduced MSSM
We finish our phenomenological analyses with the application of the method of coupling reduction to a version of the MSSM, where a covering GUT is assumed. The original partial reduction can be found in refs. [138,139] where only the third fermionic generation is considered. Following this restriction, the superpotential reads: where Y t,b,τ refer only to the third family, and the SSB Lagrangian is given by by (with the trilinear couplings h t,b,τ for the third family) We start with the dimensionless sector and consider initially the top and bottom Yuakwa couplings and the strong gauge coupling. The rest of the couplings will be treated as corrections.
, the REs and the Yukawa RGEs give If the tau Yukawa is included in the reduction, the corresponding G 2 coefficient for tau turns negative [140], explaining why this coupling is treated also as a correction (i.e. it cannot be reduced). We assume that the ratios of the top and bottom Yukawa to the strong coupling are constant at the GUT scale, i.e. they have negligible scale dependence, Then, including the corrections from the SU (2), U (1) and tau couplings, at the GUT scale, the coefficients G 2 t,b become: where We shall treat Eqs. (49) as boundary conditions at the GUT scale.
Going to the two-loop level, we assume that the corrections take the following form: Then, the two-loop coefficients, J i , including the corrections from the gauge and the tau Yukawa couplings, are: where D, N t and N b are known quantities which can be found in ref. [141].
Proceeding to the the SSB Lagrangian, Eq. (48), and the dimension-one parameters, i.e the trilinear couplings h t,b,τ , we first reduce h t,b and we get where M 3 is the gluino mass. Adding the corrections from the gauge and the tau couplings we have in Fig. 5 (right) has dropped below 1 GeV. The Higgs mass predicted by the model lies perfectly in the experimentally measured range. The M h limits set a limit on the low-energy supersymmetric masses, which we briefly discuss. The three selected benchmarks correspond to DR pseudoscalar Higgs boson masses above 1900 GeV, 1950 GeV and 2000 GeV respectively. The input of SPheno 4.0.4 [97,98] can be found in Table 10 (notation as in Sect. 5).    Table 11 shows the resulting masses of Higgs bosons and some of the lightest SUSY particles. In particular, we find M A < ∼ 1.5 TeV (for large values of tan β as in the other models), values substantially lower than in the previously considered models. This can be understood as follows.
In this model, we have direct relations between the soft scalar masses and the unified gaugino mass, which receive corrections from the two gauge couplings g 1 and g 2 and the Yukawa coupling of the τ lepton. As mentioned above, in the absence of these corrections the relations obey the soft scalar mass sum rule. However, unlike all the previous models, these corrections make the sum rule only approximate. Thus, these unique boundary conditions result in very low values for the masses of the heavy Higgs bosons (even compared to the minimal SU (5) case presented above, which also exhibits direct relations which however obey the sum rule). A relatively light spectrum is also favored by the prediction for the light CP-even Higgs boson mass, which turns out to be relatively high in this model and does not allow us to consider heavier spectra. Thus, in this model, contrary to the models analyzed before, because of the large tan β ∼ 45 found here, the physical mass of the pseudoscalar Higgs boson, M A , is excluded by the searches H/A → τ τ at ATLAS with 139/fb [84] for all three benchmarks. One could try considering a heavier spectrum, in which we would have M A 1900 GeV, but in that case the light Higgs mass would be well above its acceptable region. Particularly, it would be above 128 GeV, a value that is clearly excluded, especially given the improved (much smaller) uncertainty calculated by the new FeynHiggs code). Thus, the current version of this model has been ruled out experimentally. Consequently, we do not show any SUSY or Higgs production cross sections.

MH
MA M H ± Mg Mχ0

Conclusions
The reduction of couplings scheme consists in searching for RGE relations among parameters of a renormalizable theory that hold to all orders in perturbation theory. In certain N = 1 theories such a reduction of couplings indeed appears to be theoretically realised and therefore it developed to a powerful tool able to reduce the parameters and increase the predictivity of these theories.
In the present paper first we briefly reviewed the ideas concerning the reduction of couplings of renormalizable theories and the theoretical methods which have been developed to confront the problem. Then we turned to the question of testing experimentally the idea of reduction of couplings. Four specific models, namely the Reduced Minimal N = 1 SU (5), the all-loop Finite N = 1 SU (5), the two-loop Finite N = 1 SU (3) 3 and the Reduced MSSM, have been considered for which new results have been obtained using the updated Higgs-boson mass calculation of FeynHiggs. In each case benchmark points in the low-mass regions have been chosen for which the SPheno code has been used to calculate the spectrum of SUSY particles and their decay modes. Finally the MadGraph event generator was used to compute the production cross sections of relevant final states at the 14 TeV (HL-)LHC and 100 TeV FCC-hh colliders. The first three (unified) models were found to be in comfortable agreement with LHC measurements and searches, with the exception of the bottom quark mass in the Reduced Minimal SU (5), for which agreement with measurements can be achieved only at the 4σ level. In addition it was found that all models predict relatively heavy spectra, which evade largely the detection in the HL-LHC. We found one noticeable exception. The reduced MSSM features a relatively light heavy Higgs-boson mass spectrum. Together with the relatively high value of tan β this spectrum is excluded already by current searches at ATLAS and CMS for in the pp → H/A → τ + τ − mode. We also analyzed the accessibility of the SUSY and heavy Higgs spectrum at the FCC-hh with √ s = 100 TeV. We found that the lower parts of the parameter space will be testable at the 2 σ level, with only an even smaller part discoverable at the 5 σ level. However, the heavier parts of the possible SUSY spectra will remain elusive even at the FCC-hh. One exception here is the heavy Higgs-boson sector of the two-loop finite N = 1 SU (3) 3 model, which exhibits a spectrum where only the highest possible mass values could escape the searches at the FCC-hh.