{'fraction_non_alphanumeric': 0.043022295623451695, 'fraction_numerical': 0.019405450041288193, 'mean_word_length': 4.3612217795484725, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Finite Unified Theories and the Higgs boson

All-loop Finite Unified Theories (FUTs) are very interesting N = 1 supersymmetric Grand Unified Theories (GUTs) realising an old field theory dream, and moreover have a remarkable predictive power due to the required reduction of couplings. Based on this theoretical framework phenomenologically consistent FUTs have been constructed. Here we review two FUT models based on the SU(5) gauge group, which can be seen as special, restricted and thus very predictive versions of the MSSM. We show that from the requirement of correct prediction of quark masses and other experimental constraints a light Higgs-boson mass in the range M_h ~ 121 - 126 GeV is predicted, in striking agreement with recent experimental results from ATLAS and CMS. The model furthermore naturally predicts a relatively heavy spectrum with colored supersymmetric particles above ~ 1.5 TeV in agreement with the non-observation of those particles at the LHC.


Introduction
The success of the Standard Model (SM) of Elementary Particle Physics is seriously limited by the presence of a plethora of free parameters. An even more disturbing fact is that the best bet for Physics beyond the SM namely the minimal supersymmetric extension of the SM (MSSM), which is expected to bring us one step further towards a more fundamental understanding of Nature, introduces around a hundred additional free parameters. To reduce the number of free parameters of a theory, and thus render it more predictive, one is usually led to introduce a symmetry. Grand Unified Theories (GUTs) are very good examples of such a procedure [1][2][3][4][5]. For instance, in the case of minimal SU(5), because of (approximate) gauge coupling unification, it was possible to reduce the gauge couplings by one and give a prediction for one of them. In fact, LEP data [6] seem to suggest that a further symmetry, namely N = 1 global supersymmetry (SUSY) [7,8] should also be required to make the prediction viable. GUTs can also relate the Yukawa couplings among themselves, again SU(5) provided an example of this by predicting the ratio M τ /M b [9] in the SM. Unfortunately, requiring more gauge symmetry does not seem to help, since additional complications are introduced due to new degrees of freedom, in the ways and channels of breaking the symmetry, and so on.
A natural extension of the GUT idea is to find a way to relate the gauge and Yukawa sectors of a theory, that is to achieve Gauge-Yukawa Unification (GYU) [10][11][12]. A symmetry which naturally relates the two sectors is supersymmetry, in particular N = 2 SUSY [13]. It turns out, however, that N = 2 supersymmetric theories have serious phenomenological problems due to light mirror fermions. Also in superstring theories and in composite models there exist relations among the gauge and Yukawa couplings, but both kind of theories have phenomenological problems, which we are not going to address here.
Finite Unified Theories (FUTs) are N = 1 supersymmetric Grand Unified Theories (GUTs) which can be made finite to all-loop orders, including the soft supersymmetry breaking sector. The constructed finite unified N = 1 supersymmetric SU(5) GUTs predicted correctly from the dimensionless sector (Gauge-Yukawa unification), among others, the top quark mass [14,15]. Eventually, the full theories can be made all-loop finite and their predictive power is extended to the Higgs sector and the s-spectrum [16]. For a detailed discussion see [11,17,18]. Here we limit ourselves to a brief review.
Consider a chiral, anomaly free, N = 1 globally supersymmetric gauge theory based on a group G with gauge coupling constant g. The superpotential of the theory is given by where m ij (the mass terms) and C ijk (the Yukawa couplings) are gauge invariant tensors, and the matter field Φ i transforms according to the irreducible representation R i of the gauge group G. All the one-loop β-functions of the theory vanish if the β-function of the gauge coupling β (1) g , and the anomalous dimensions of the Yukawa couplings γ where ℓ(R i ) is the Dynkin index of R i , and C 2 (G) is the quadratic Casimir invariant of the adjoint representation of G. A theorem given in [19][20][21][22] then guarantees the vanishing of the β-functions to all-orders in perturbation theory. This requires that, in addition to the one-loop finiteness conditions (2), the Yukawa couplings are reduced in favour of the gauge coupling [23][24][25][26][27][28].
In the soft breaking sector, the one-and two-loop finiteness for the trilinear terms h ijk can be achieved by [29] It was also found that the soft supersymmetry breaking (SSB) scalar masses in Gauge-Yukawa and finite unified models satisfy a sum rule [30,31] ( for i, j, k, where ∆ (2) is the two-loop correction, which vanishes when all the soft scalar masses are the same at the unification point.

SU (5) Finite Unified Theories
We will examine here all-loop Finite Unified theories with SU(5) gauge group, where the reduction of couplings has been applied to the third generation of quarks and leptons. An extension to three families, and the generation of quark mixing angles and masses in Finite Unified Theories has been addressed in [32], where several examples are given. These extensions are not considered here. The particle content of the models we will study consists of the following supermultiplets: three (5 + 10), needed for each of the three generations of quarks and leptons, four (5 + 5) and one 24 considered as Higgs supermultiplets. When the gauge group of the finite GUT is broken the theory is no longer finite, and we will assume that we are left with the MSSM.
A predictive Gauge-Yukawa unified SU(5) model which is finite to all orders, should have the following properties: 2. Three fermion generations, in the irreducible representations 5 i , 10 i (i = 1, 2, 3), which obviously should not couple to the adjoint 24.
3. The two Higgs doublets of the MSSM should mostly be made out of a pair of Higgs quintet and anti-quintet, which couple to the third generation.
In the following we discuss two versions of the all-order finite model. The model of ref. [14,15], which will be labeled A, and a slight variation of this model (labeled B), which can also be obtained from the class of the models suggested in ref. [33] with a modification to suppress non-diagonal anomalous dimensions.
The superpotential which describes the two models takes the form [14,15,34] where H a and H a (a = 1, . . . , 4) stand for the Higgs quintets and antiquintets.
The main difference between model A and model B is that two pairs of Higgs quintets and anti-quintets couple to the 24 in B, so that it is not necessary to mix them with H 4 and H 4 in order to achieve the triplet-doublet splitting after the symmetry breaking of SU(5) [34]. Thus, although the particle content is the same, the solutions to the finiteness equations and the sum rules are different, which will reflect in the phenomenology.
The non-degenerate and isolated solutions to γ (1) i = 0 for model FUTA, which are the boundary conditions for the Yukawa couplings at the GUT scale, are: In the dimensionful sector, the sum rule gives us the following boundary conditions at the GUT scale for this model [34][35][36]: and thus we are left with only three free parameters, namely m 5 ≡ m 5 3 , m 10 ≡ m 10 3 and M.
For the model FUTB the non-degenerate and isolated solutions to γ (1) i = 0 give us: and from the sum rule we obtain: i.e., in this case we have only two free parameters m 10 ≡ m 10 3 and M.

Predictions of the SU (5) models
We confront now the predictions of the four models, FUTA and FUTB, each with µ < 0 and µ > 0, with the experimental data, starting with the heavy quark masses (see refs. [16,18] for more details). Since the gauge symmetry is spontaneously broken below M GUT , the finiteness conditions do not restrict the renormalization properties at low energies, and all it remains are boundary conditions on the gauge and Yukawa couplings (6), the h = −MC relation eq.(3), and the soft scalar-mass sum rule (4) at M GUT , as applied in each of the models. 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 with the relevant boundary conditions. Below M GUT their evolution is assumed to be governed by the MSSM. We further assume a unique supersymmetry breaking scale M s and therefore below that scale the effective theory is just the SM. As a first step, we compare the predictions of the two models, FUTA and FUTB (both with µ > 0 and µ < 0), with the experimental values of the top and bottom quark masses. We use for the top quark the value for the pole mass where we notice that the theoretical values for M t may suffer from a correction of ∼ 4% [11]. For the bottom quark mass we use the value at M Z [39] m b (M Z ) = (2.83 ± 0.10) GeV (11) to avoid uncertainties that come from the furhter running from the M Z to the m b mass, and where we have taken the ∆ b effects into account [40]. From Fig. 1 it is clear that the model FUTB with µ < 0 is the only one where both top and bottom quark masses lie within experimental limits. In this case the value of tan β is found to be ∼ 48. Thus, we will concentrate now on the results for FUTB, µ < 0.
In the case where all the soft scalar masses are universal at the unfication scale, there is no region of M below O(few TeV) in which mτ > m χ 0 is satisfied (where mτ is the lightestτ mass, and m χ 0 the lightest neutralino mass). However, this problem can be solved naturally, thanks to the sum rule (9), see Refs. [16,18] for details. A related problem concerns the agreement of our predictions with the cold dark matter constraints. Again a detailed discussion can be found in Refs. [16,18].
As a second step, we impose the conditions of successful radiative electroweak symmetry breaking, m 2 τ > 0 and mτ > m χ 0 . We furthermore require ¶ Using the most up-to-date value of M exp t = (173.2 ± 0.9) GeV [38] would have a minor impact on our analysis. agreement at the 95% C.L. with constraints coming from B-physics, namely the experimental bounds on BR(b → sγ) [41] and BR(B s → µ + µ − ) [42] (which we have evaluated with Micromegas [43]). This way, we find a prediction for the lightest Higgs mass and the SUSY spectra. From the analysis we find that the lightest observable particle (LOSP) is either the stau or the second lightest neutralino, with mass starting around ∼ 500 GeV. The prediction of the lightest Higgs boson mass, evaluated with Feyn-Higgs [44,45], as a function of M is shown in Fig. 2. The light (green) points shown are in agreement with all the constraints discussed listed above. The lightest Higgs mass ranges in where the uncertainty comes from variations of the soft scalar masses, and from finite (i.e. not logarithmically divergent) corrections in changing renormalization scheme. To this value one has to add ±2 GeV coming from unknown higher order corrections [44]. We have also included a small variation, due to threshold corrections at the GUT scale, of up to 5% of the FUT boundary conditions. Thus, taking into account the B-physics constraints results naturally in a light Higgs boson that not only fulfills the LEP bounds [46,47], but naturally falls in the range favored by recent ATLAS and CMS measurements [48]. A more detailed discussion can be found in Refs. [16,18]. The full particle spectrum of model FUTB with µ < 0, again compliant with quark mass constraints and the B-physics observables is shown in Fig. 3. The masses of the particles increase with increasing values of the unified gaugino mass M.
One can see that the lighter parts of the spectrum, especially of the colored particles, are in the kinematic reach of the LHC. A numerical example of such a light spectrum is shown in Table 1. However, large parts are beyond this kinematic reach, and one possibility offered by FUTB with µ < 0 would be to observe only a SM-like Higgs boson around 125 GeV, but no additional SUSY particles.