]>PLB34757S03702693(19)30461710.1016/j.physletb.2019.06.065The Author(s)PhenomenologyFig. 1Our approach to determine the value at low energies of the U(1)′ coupling strength α4, the unification scale MGUT and the unification value of the gauge couplings αGUT. We consider ΛZ′=2 TeV, the three cases ΛSUSY=ΛZ′,1012 GeV, 1017 GeV and Mstr ≈ 1017 GeV. We illustrate the case ΛZ′<ΛSUSY<MGUT.Fig. 1Fig. 2Vacua with different values of MGUT, α4−1(ΛZ′) and αGUT−1 for three choices of ΛSUSY and partial unification, αGUT ≡ α1 = α2 ≠ α3. In the left panels, the bubbles in different colors and sizes indicate the number of vacua with the given values of α4(ΛZ′) and MGUT at their center. Analogously, the right panels count vacua with different values of αGUT and MGUT.Fig. 2Fig. 3Vacua with different values of MGUT, α4−1(ΛZ′) and αGUT−1 for three choices of ΛSUSY and partial unification, αGUT ≡ α1 = α2 ≈ α3. In the left panels, the bubbles in different colors and sizes indicate the number of vacua with the given values of α4(ΛZ′) and MGUT at their center. Analogously, the right panels count vacua with different values of αGUT and MGUT.Fig. 3Fig. 4RGE running of the quartic Higgs coupling and the gauge couplings of U(1)Y,SU(2)L,SU(3)c and U(1)′ in a Z8–II (2,1) sample vacuum. The gauge couplings meet at MGUT ≈ 1012 GeV with a value αGUT ≈ 1/32. We see that at ΛZ′=2 TeV, the U(1)′ coupling has the value g4 ≈ 0.49. From this plot, we observe that the quartic Higgs coupling remains perturbative and positive, yielding a plausible solution of the vacuum metastability problem of the SM.Fig. 4Table 1Orders and relations of the WL of Z8 orbifolds, depending on the geometry of the compactification. A ≈ B indicates that A = B up to translations in the root lattice of E8 × E8.Table 1OrbifoldConditions on the Wilson lines
Z8–I (1,1)2W1 ≈ 2W5 ≈ 0; W1 ≈ W2 ≈ W3 ≈ W4; W5 ≈ W6
Z8–I (2,1)2W1 ≈ 2W5 ≈ 0; W1 ≈ W2 ≈ W3 ≈ W4; W5 ≈ W6
Z8–I (3,1)4W1 ≈ 0; W1 ≈ W2 ≈ W3 ≈ W4 ≈ W5 ≈ W6

Z8–II (1,1)2W1 ≈ 2W5 ≈ 2W6 ≈ 0; W1 ≈ W2 ≈ W3 ≈ W4
Z8–II (2,1)4W1 ≈ 2W6 ≈ 0; W1 ≈ W2 ≈ W3 ≈ W4 ≈ W5
Table 2Number of inequivalent heterotic orbifold models with MSSMlike properties found in ref. [32] for each Z8 orbifold geometry. We further provide in the third and sixth columns the number of vacuum configurations with MSSMlike properties and gauge group Geff=GSM×U(1)′ in each case.Table 2Orbifold# MSSMlike modelsEffective vacuaOrbifold# MSSMlike modelsEffective vacua
Z8–I(1,1)2681,362Z8–II(1,1)2,02310,023
Z8–I(2,1)2461,097Z8–II(2,1)5052,813
Z8–I(3,1)3891,989
Table 3Number of effective MSSMlike vacua arising from Z8 heterotic orbifolds with U(1)′ symmetries and gauge coupling unification. For each orbifold geometry, the second column shows the number of vacua with the exact dynamics of the MSSM, i.e. such that bi=biMSSM∈{33/5,1,−3,0} in their RGE. Excluding these and other inconsistent models, in the remaining (pairs of) columns we show the number of vacua satisfying our constraints and with partial unification (α1 = α2) or total gauge coupling unification (αGUT = αi), corresponding to three choices of ΛSUSY.Table 3OrbifoldMSSM vacuaEffective vacua of interest
ΛSUSY=ΛZ′ΛSUSY=1012 GeVΛSUSY=1017 GeV
α1 = α2unifiedα1 = α2unifiedα1 = α2unified
Z8–I (1,1)5868121,21881,25320
Z8–I (2,1)116421094029583
Z8–I (3,1)761,10181,79271,84420
Z8–II (1,1)2456,476608,9701819,245114
Z8–II (2,1)1111,92972,567712,63151

totals60610,6087715,48726915,931208
Table 4Massless spectrum for a vacuum with gauge group Geff. Representations with respect to SU(3)c × SU(2)L are given in bold face, the hypercharge and the U(1)′ charge are indicated as subscript. The frame on the left corresponds to the SM fermions, the middle frame to fermionic exotics, and the right frame shows scalars including the Higgs fields.Table 4#Fermionic irrepLabel
2(1,2)(−1/2,−7/122)ℓ1,2
1(1,2)(−1/2,1/32)ℓ3
2(1,1)(1,−1/62)e¯1,2
1(1,1)(1,1/122)e¯3
2(3,2)(1/6,−1/62)q1,2
1(3,2)(1/6,1/42)q3
2(3‾,1)(−2/3,−1/62)u¯1,2
1(3‾,1)(−2/3,1/122)u¯3
2(3‾,1)(1/3,−7/122)d¯1,2
1(3‾,1)(1/3,3/42)d¯3
#Fermionic irrepLabel
1(3‾,1)(1/3,3/42)x¯i
1(3,1)(−1/3,1/122)xi
8(1,2)(0,1/62)ηi
8(1,1)(1/2,1/62)ζi
8(1,1)(−1/2,7/122)ζ¯i
8(1,1)(−1/2,−1/122)κ¯i
8(1,1)(1/2,−1/62)κi

11(1,1)(0,1/32)Nia
10(1,1)(0,−2/32)Nib
8(1,1)(0,−1/122)Nic
6(1,1)(0,−5/122)Nid
4(1,1)(0,7/122)Nie
2(1,1)(0,−1/42)Nif
#Scalar irrepLabel
1(1,2)(1/2,−1/32)Hu
1(1,2)(−1/2,1/122)Hd
1(1,1)(0,−1/42)s1
1(1,1)(0,1/32)s2
U(1)′ coupling constant at low energies from heterotic orbifoldsYesseniaOlguínTrejoayes.olt@ciencias.unam.mxOmarPérezFigueroaaomar_perfig@ciencias.unam.mxRicardoPérezMartínezabricardoperezm@estudiantes.fisica.unam.mxSaúlRamosSáncheza⁎ramos@fisica.unam.mxaInstituto de Física, Universidad Nacional Autónoma de México, POB 20364, Cd.Mx. 01000, MexicoInstituto de FísicaUniversidad Nacional Autónoma de MéxicoPOB 20364Cd.Mx. 01000MexicoInstituto de Física, Universidad Nacional Autónoma de México, POB 20364, Cd.Mx. 01000, MéxicobFacultad de Ciencias FísicoMatemáticas, Universidad Autónoma de Coahuila, Edificio A, Unidad Camporredondo, 25000, Saltillo, Coahuila, MexicoFacultad de Ciencias FísicoMatemáticasUniversidad Autónoma de CoahuilaEdificio AUnidad CamporredondoSaltilloCoahuila25000MexicoFacultad de Ciencias FísicoMatemáticas, Universidad Autónoma de Coahuila, Edificio A, Unidad Camporredondo, 25000, Saltillo, Coahuila, México⁎Corresponding author.Editor: A. RingwaldAbstractAdditional Abelian gauge interactions are generic to string compactifications. In heterotic string models, gauge coupling unification of such forces and other gauge interactions is natural due to their common origin. In this letter we study systematically the 1loop running of the coupling constants in effective vacua emerging from Z8 heterotic orbifold compactifications that provide the matter spectrum of the MSSM plus some vectorlike exotics, restricting to vacua that yield a nonanomalous U(1)′ symmetry, gauge coupling unification and the observed values of known gauge couplings. We determine the lowenergy value of the U(1)′ coupling constant for different scales of supersymmetry breakdown. We find that the U(1)′ coupling constant is quite restricted in string models to lie in the range 0.46–0.7 for lowscale supersymmetry or 0.44–0.6 in other cases. We argue that the phenomenology of these string vacua should be further explored to solve some extant issues, such as the stability of the Higgs vacuum.1IntroductionSome open questions in the standard model (SM) and cosmology have led to conjecture the existence of additional U(1)′ gauge symmetries, under which different SM particles may be charged. These symmetries lead to a rich phenomenology (for details, see ref. [1] and references therein). To mention a few of their qualities, they may shed some light on neutrino physics and dark matter simultaneously [2–4], or the gμ−2 anomaly [5], or the metastability of the Higgs vacuum [6]. They could also yield interesting signals at colliders [7–9] and alleviate some issues of models with supersymmetry (SUSY) [10,11]. Although they must be broken at low energies and mZ′ is very constrained [12–15], the bounds can be avoided (e.g. via a leptophobic U(1)′) and Z′ signals could be soon confirmed at colliders.The origin of U(1)′ symmetries is frequently related to grand unified theories (GUT) beyond SU(5). For example, in E6 GUTs, U(1)′ symmetries have been classified and studied phenomenologically [16–19]. It is also known that U(1)′s are natural to models resulting from different string compactifications [20–24]. In particular, in heterotic orbifolds in the fermionic formulation, plausible scenarios with a light Z′ and rich phenomenology have been identified [23,25,26]. Also, orbifolds in the bosonic formulation have shown that in models resembling the minimal extension of the SM (the MSSM), matter parity [27,28] or even a Z4R symmetry [29] for proton stability can arise from U(1)′s (and other symmetries) of the model.Motivated by these findings, in this letter we aim at characterizing the couplings of nonanomalous U(1)′s natural to some string compactifications. We focus on Z8 toroidal orbifold compactifications of the E8×E8 heterotic string with MSSMlike features. This kind of models has been investigated before [30], but with a different purpose and only in a small subset (about 20%) of all promising models due to some far too restrictive priors which do not improve the phenomenology of the models. We avoid such restrictions to obtain a richer variety of models and a more general analysis.We study special effective vacua with gauge group SU(3)c×SU(2)L×U(1)Y×U(1)′, where the kinetic mixing of Abelian symmetries is unimportant [31] and the few exotics in their spectra are vectorlike w.r.t. the SM. Assuming that a Higgslike mechanism breaks the U(1)′ at a scale ΛZ′=2 TeV, where also some exotics acquire masses, and a SUSY breaking scale ΛSUSY≥ΛZ′, we determine systematically the running of all coupling constants of our effective vacua by using the renormalization group equations (RGE) at 1loop, which suffice at this level because of further small corrections that we neglect, such as threshold effects. Restricting to vacua with gauge coupling unification, we observe that besides the usual family nonuniversality of stringy U(1)′s, heterotic orbifolds limit the values of the U(1)′ coupling at the TeV scale as well as the unification scale and the unified coupling.In what follows, we discuss the main features of Z8 heterotic orbifold compactifications and their effective vacua with U(1)′ and analyze how to arrive at limits for the couplings of such symmetries. Our findings are reported in section 4, followed by a sample model with the potential to solve the metastability of the Higgs vacuum thanks to a U(1)′.2MSSMlike Z8 toroidal orbifold modelsOur starting point is the N=1 E8×E8 heterotic string theory in the bosonic formulation, and we compactify the six extra dimensions on a toroidal Z8 orbifold that preserves N=1 in four dimensions. This choice is taken because Z8 has shown to be the symmetry that yields the largest fraction of ZN MSSMlike heterotic orbifold models available so far [32], so that we can be sure to be focused on a representative patch of promising string compactifications.In general, a toroidal ZN orbifold is defined as the quotient T6/P of a sixtorus over a point group, which is generated by a single twist ϑ of order N, i.e. so that ϑN=1. ϑ must be chosen to act as an isometry on T6. ϑ can always be diagonalized on three twodimensional actions, so that ϑ=diag(e2πiv1,e2πiv2,e2πiv3), where v=(v1,v2,v3) is called the twist vector. It is possible to combine together the point group with the lattice Γ of the torus to build the space group S=P⋉Γ, such that the orbifold can be analogously defined as R6/S.A complete classification of the T6 geometries and point groups for Abelian toroidal orbifolds [33] reveals that there are only two Z8 point groups, denoted Z8–I and Z8–II and defined by the twist vectors vZ8–I=18(1,2,−3) and vZ8–II=18(1,3,−4), respectively. There are five inequivalent T6 geometries (see [30] for details) acceptable for these twists. Following the notation of [33], we shall label them as Z8–I (i,1), i=1,2,3, and Z8–II (j,1), j=1,2.A consistent heterotic orbifold requires to embed the action of the sixdimensional orbifold into the E8×E8 gauge degrees of freedom. The ZN twist ϑ can be embedded as a 16dimensional shift vector V of order N (i.e. such that NV is in the E8×E8 root lattice), whereas the six independent directions of the torus can be embedded as 16dimensional discrete Wilson lines (WL) Wa, a=1,…,6. Given a ZN twist and a toroidal geometry, there are several admissible choices of shifts and WL, as long as they fulfill the modular invariance conditions [34],(1)N(V2−v2)=0mod2,Na(V⋅Wa)=0mod2,a=1,…,6,NaWa2=0mod2,gcd(Na,Nb)(Wa⋅Wb)=0mod2,a≠b, which ensure that the fourdimensional emergent field theory be nonanomalous and compatible with string theory. The space group of the orbifold constrains the order11The smallest integer Na, such that NaWa (with no summation over a) is contained in the root lattice of E8×E8, is defined as the order of the WL Wa. Na of a WL Wa and its relations with other WL. Interestingly, these restrictions can be understood in terms of the Abelianization of the space group as requirements for the WL to be compatible with the embedding of the socalled space group flavor symmetry into the gauge degrees of freedom [35].In Z8 orbifolds, the order and relations of the admissible WL are given in Table 1. Two Z8–I geometries admit two independent WL of order two, whereas the third case admits only one independent order4 WL. Further, Z8–II (1,1) allows for three order2 WL, and Z8–II (2,1), one WL of order two and one of order four.Applying standard techniques (see e.g. [36–39]), one can use the modular invariant solutions to eqs. (1) (with N=8) complying with the WLconstraints of Table 1 to compute the spectrum of massless string states. These techniques have been implemented in the orbifolder [40] to automatize the search of admissible models, the computation of the massless spectrum, and the identification of phenomenologically viable models. By using this tool, we have previously [32] found 3,431 Z8 orbifold compactifications of the E8×E8 heterotic string with the following properties:• the gauge group at the compactification scale is(2)G4D=GSM×[U(1)′]n×Ghidden, where GSM=SU(3)c×SU(2)L×U(1)Y with nonanomalous hypercharge (satisfying sin2θw=3/8), Ghidden is a nonAbelian gauge factor (typically a product of SU(M) subgroups), at most one U(1)′ is (pseudo)anomalous and n≤10, depending on the model; and• the massless spectrum includes the MSSM superfields plus only vectorlike exotic matter w.r.t. GSM.Ghidden is commonly considered a hidden gauge group because the MSSM fields are mostly uncharged under that group. The number of models found in each orbifold geometry is presented in Table 2. The defining shifts and WL of the models can be found in [41].Aiming at the study of the U(1)′ symmetries of these models, we must point out some of their properties. Most of the models present an anomalous U(1)′ [42], whose anomaly can be canceled through the GreenSchwarz mechanism [43]. Besides, in this type of models, the gauge fields of the U(1)α′ symmetries can be decomposed as(3)Tα=∑I=116tαIHI,α=1,…,n, in terms of the Cartan generators HI of the original E8×E8, such that the corresponding U(1)α′ charges for fields of the spectrum with gauge momentum p∈E8×E8 are given by qα=tα⋅p. This is why tα is frequently called the generator of U(1)α′. It is known that, if we adopt22Despite this U(1)′ normalization, we allow the GUTcompatible hypercharge normalization t12=5/6. the U(1)′ normalization ktα2=1 and consider all algebras associated with the gauge group to have KačMoody level k=2, the treelevel gauge kinetic function is universally given by fα=S, where S corresponds to the bosonic component of the (axio)dilaton. Consequently, the treelevel gauge coupling satisfies gs−2=〈ReS〉 at the (heterotic) string scale, Mstr≈1017 GeV. Further, there is some kinetic mixing between different U(1)′ gauge symmetries which one might believe relevant for phenomenology; however, it has been found to be typically of order 10−4–10−2 in semirealistic heterotic orbifolds [31], unimportant for our purposes.Let us make a couple of additional remarks about the models we explore. First, as in the MiniLandscape [28,44,45] of Z6–II heterotic orbifolds, in all the Z8 models discussed here, there exists a large number of SMsinglets, which naturally develop O(0.1) VEVs in order to cancel the FayetIliopoulos term, ξ=gs2trTanom/192π2 (in Planck units), appearing in models with a (pseudo)anomalous U(1)′. As a consequence, the allowed couplings of such singlets among themselves and with vectorlike exotics yield large masses for the additional matter, decoupling it from the lowenergy effective field theory. Simultaneously, since the singlets are charged under the [U(1)′]n gauge sector, those symmetries can be broken in the vacuum.Secondly, it is always possible to find SMsinglet VEV configurations, such that, SUSY is retained while only the effective gauge group,(4)Geff=GSM×U(1)′⊂G4D, remains after the spontaneous breakdown triggered by the singlet VEVs, where we ignore the hidden group Ghidden. The surviving (nonanomalous) U(1)′ can be any of the original U(1)α′, α=1,…,n, symmetries or a linear combination of them, depending on the details of the model. In this work, for practical purposes, we shall study only the former case, i.e. the effective vacua with the effective gauge group Geff, where the U(1)′ corresponds to each of the nonanomalous U(1)α′ of the Z8 orbifold models. We find that there is a total of 17,284 such effective vacua, distributed in all admissible Z8 orbifold geometries, as shown in Table 2.Third, in the most general case, (at least) some of the MSSM superfields and the exotics exhibit some U(1)′ charges. This is true in most of the vacua where one of the U(1)′ remains unbroken. Thus, only the exotics that are vectorlike w.r.t. Geff, and not just GSM, decouple at low energies, allowing interactions among SM fields charged under U(1)′ and the extra matter which can yield interesting new phenomenology. These interactions affect in particular the RGE running of the gauge couplings, as we discuss in the following section.3Searching effective vacua with U(1)′ and unificationWe shall study the value of the U(1)′ coupling constants of the effective field theories emerging from Z8 heterotic orbifold compactifications at currently reachable energies, restricting ourselves to those that are consistent with unification, in the sense that the gauge couplings meet (perhaps accidentally) at a given scale, and where SM gauge couplings are compatible with the observed values at low energies.Selecting the vacua with these features requires some additional knowledge of the details of the effective spectrum, and some reasonable priors. The first hurdle is that, even though gs−2=〈ReS〉 at Mstr, below this scale, the coupling constants get different contributions from all other moduli too, whose stabilization represents a challenge by itself [46,47], hindering to know the exact values of the gauge couplings in the effective field theory. Thus, in order to figure out which models yield the measured values of the gauge couplings, one could start with an ad hoc value of all SM gauge couplings at high energies and retain only models where the RGE lead to the observed values at MZ. This approach seems rather arbitrary.Instead, we assume that the SM coupling strengths αi of our effective vacua have the observed values at MZ [48], (with i=1 for U(1)Y, i=2 for SU(2)L and i=3 for SU(3)c)(5)α1−1(MZ)=59.01±0.01,α2−1(MZ)=29.59±0.01,α3(MZ)=0.1182±0.0012, and then let the RGE define the value of the gauge couplings at all scales up to Mstr, using the spectrum of the effective vacua we find. These effective vacua exhibit N=1 SUSY and an additional Z′ boson, but no SUSY partner (in some models, the lightest neutralino with masses lighter than few hundred GeV has been excluded [49,50]) nor extra vector boson has been detected at the LHC so far (with a lower limit for mZ′ around 2 TeV [12–15]). Hence, we suppose that SUSY is broken at a scale ΛSUSY>MZ and the U(1)′ breakdown scale is ΛZ′=2 TeV, as a benchmark value.33We have verified that our results are qualitatively the same for other scales near this value. We further assume that ΛSUSY≥ΛZ′.Given these remarks, we consider different matter spectra depending on the energy scale μ. As sketched in Fig. 1, above the SUSY scale ΛSUSY the spectrum of the effective vacua includes the MSSM superfields and a few vectorlike exotics w.r.t. GSM with nontrivial U(1)′ charges. Below ΛSUSY, if it is larger than ΛZ′, the gauge group is still Geff, eq. (4), but we assume that all SM superpartners and the bosonic superpartners of the exotics decouple. Only one SM singlet with U(1)′ charge is taken as scalar below ΛSUSY, so that its VEV can trigger the breakdown of U(1)′ and provide masses around ΛZ′ for the remaining exotics. Consequenly, below ΛZ′ only the SM particles and gauge group are left.The next step is to choose only models that are consistent with gauge coupling unification, so that we recover at some level the unification provided at Mstr by string theory, and justify why we have restricted ourselves to hypercharges with GUT normalization. To do so, we let the RGE determine the value of the SM couplings and retain vacua with unification at some scale MGUT, which varies from vacuum to vacuum due to their matter content.44Note that, even though MGUT defines the scale at which all gauge couplings meet, no true unification is implied in the sense of usual GUTs. At MGUT, one naturally can assume that all coupling strengths of Geff have the same value αGUT. Thus, the U(1)′ coupling strength α4=αGUT at that scale and its RGE running down to ΛZ′ in a vacuum is a lowenergy consequence of such a vacuum. This approach is depicted in Fig. 1. We display an intermediate ΛSUSY, but, in this work, we consider three wellmotivated cases: ΛSUSY=ΛZ′, ΛSUSY=1012 GeV and ΛSUSY=1017 GeV. The first one arises from the common expectation that SUSY may show up at reachable energies, the second one from constraints on the metastability of the Higgs potential [51] and the last one from considering that SUSY may be broken at the string scale, as in nonSUSY string compactifications [52–54]. It is known that gaugino condensates can render various ΛSUSY in these models [55].As stated before, all vectorlike exotics w.r.t. Geff naturally acquire masses just below Mstr while G4D breaks down to Geff, but it is easy to conceive that this process happens gradually at various scales between MGUT and Mstr. Furthermore, we expect that at those scales string threshold corrections, effects of the GreenSchwarz mechanism and moduli stabilization take place. We shall assume that all these effects do not alter the unification, even though they set deviations of αGUT from αs=gs2/4π that differ for each effective vacuum.In this work, we use the 1loop RGE for the gauge factors of the effective gauge group, Gi∈Geff. The running of the coupling strengths αi=gi2/4π is given by(6)∂αi−1∂lnμ=−bi2π,i=1,…,4, where the βfunction coefficients for nonAbelian groups (i=2,3) are given at 1loop by(7)bi={−113C2(Gi)+23∑fmfC(Rf)+13∑bmbC(Rb),nonSUSY,−3C2(Gi)+∑SmSC(RS),SUSY. Here C2(Gi) is the quadratic Casimir of the group Gi, C(Rb,f,S) denotes respectively the quadratic index of the Gi representations Rb,f,S of the bosons, fermions and superfields included in the spectrum, over which the sums run, and mb,f,S denotes their multiplicities. Conventionally, we take C(Rb,f,S)=1/2 if Rb,f,S corresponds to the fundamental representation of SU(N). For Abelian gauge symmetries (i=1,4), eq. (7) reduces to(8)bi={23∑fmfqi(f)2+13∑bmbqi(b)2,nonSUSY,∑SmStrqi(S)2,SUSY, in terms of the matter U(1)′ charges qi(b,f,S), defined around eq. (3).The solutions to (6) have the general form αi−1(μ)=αi−1(μ0)−bi/2πln(μ/μ0), in terms of a reference scale μ0. For the SM couplings, we take their observed values, eq. (5), with μ0=MZ. Since below ΛZ′=2 TeV we assume that only the SM particles are present, we determine readily that α1−1=56.99, α2−1=31.15 and α3−1=11.9 at ΛZ′.4U(1)′ couplings in Z8 heterotic orbifold vacuaAs a first step, we compute systematically the βfunction coefficients, according to eqs. (7)(8), of all effective vacua counted in Table 2. We observe that a fraction (3.5%) of all vacua yield the properties of the MSSM above ΛSUSY. That is, their matter spectra match the MSSM spectrum and, consequently, (b1,b2,b3,b4)=(33/5,1,−3,0). In these cases, b4=0 arises because the MSSM fields have no U(1)′ charges. In the second column of Table 3 we show the number of these MSSM vacua. There is also a smaller fraction (1.3%) of vacua with b4=0, but bi≠biMSSM. Since in all these cases the coupling of Z′ with observable matter is very suppressed, we shall not consider these models here.The running of the couplings described by the RGE reveals that there is a significant number of vacua that are inadmissible for our study. First, computing the scale at which couplings meet leads in some cases to MGUT<MZ or MGUT>Mstr, which are either excluded (the former) or meaningless since our effective models apply only below Mstr. Second, vacua in which any of the coupling strengths αi=gi2/4π reaches negative values in its running are not acceptable. Finally, since our work is based on weakly coupled string theory, nonperturbative couplings, αi>1, are equally undesirable.The number of vacua with these weaknesses varies depending on the choice of ΛSUSY and the Z8 orbifold geometry. For instance, in the case of Z8I (1,1) with ΛSUSY=ΛZ′=2 TeV, we find that out of the 1,362 effective vacua, 205 lead to unification at a scale MGUT<ΛZ′ or MGUT>Mstr. Further, 321 vacua produce negative values of some αi, and in 89 we find nonperturbative values for some couplings (αi>1). Disregarding these (and those with b4=0), we arrive at the 681 vacua of Table 2 for this case.To obtain the Z8 orbifold vacua of interest, with unification of all coupling strengths, we proceed in two steps. We analyze first the qualities of those vacua with SU(2)L−U(1)Y unification, i.e. those with α1=α2≠α3, and then select among them those with αGUT≡α1=α2≈α3 at a scale MGUT, allowing for a small deviation αGUT−1−α3−1(MGUT)<0.26, corresponding to the 3σ interval of the measured value of α3−1(MZ). These observations are considered in the third through eighth columns of Table 3, where we display the number of vacua with partial and full unification for each choice of ΛSUSY. We realize that, from the huge number of possible effective vacua with Geff, only a quite small set of Z8 orbifold vacua of order hundred in each case satisfies all of our constraints. The details of these vacua are available in [56].In Fig. 2 we present our results for all Z8 orbifold vacua with only SU(2)L−U(1)Y unification. The left panels correspond to frequency plots for three different choices of ΛSUSY of (the inverse of) the U(1)′ coupling strength α4−1 at the lowenergy scale ΛZ′=2 TeV against the scale at which α1=α2, denoted MGUT. The central values of the largest (red) bubble corresponds to the most frequent combination of α4−1(ΛZ′) and MGUT. The small (purple) bubbles correspond to at most six vacua with the combination of values at their center. The right panels are also frequency plots of the values of αGUT−1 and MGUT achieved by our vacua, where αGUT≡α1(MGUT)=α2(MGUT). In these plots, small purple bubbles correspond to up to 50 models with the central value of the circles.Since this is only an intermediate result, we content ourselves with some semiqualitative remarks. The first observation is that, independently of whether SUSY is broken at low, intermediate or high energies, Z8 orbifold vacua with MSSMlike properties do not allow any arbitrary values of U(1)′ couplings constants or unification scale. We find that roughly only 20<α4−1(2 TeV)<80, corresponding to 0.4<g4(2 TeV)<0.8, is allowed in our string constructions. We expect this to hold for any heterotic orbifold model with semirealistic properties. The most common value of α4−1(2 TeV) depends on ΛSUSY: for lowscale SUSY α4−1∼30, whereas α4−1∼60 for other cases. We observe also other rough limits: MGUT>108 GeV, and αGUT−1<30 for lowscale SUSY and αGUT−1<45 for other SUSY scales.On the other hand, taking averages over all models, we find MGUT‾≈1016 GeV and αGUT−1‾≈13 for lowscale SUSY, and MGUT‾≈1015 GeV and αGUT−1‾≈30 otherwise. Unfortunately, most of these vacua are far from our ideal scenario, with full unification, which can also be measured by the average difference of Δ≡αGUT−α3(MGUT), which is as large as Δ‾≈αGUT for lowscale SUSY, and Δ‾≈13αGUT for higher ΛSUSY.Let us comment on the vertical (diagonal) alignment pattern of the points in the left (right) plots of Fig. 2. Consider e.g. the top plots, with ΛSUSY=ΛZ′, where one can find that the RGE lead tolnMGUTΛZ′=2π(α1−1(ΛZ′)−α2−1(ΛZ′))1b1−b2. Since only MGUT and b1−b2 are modeldependent, all vacua with the same difference b1−b2 lead to the same MGUT, yielding a point on the same vertical line in the left panels of the figure. Further, as not any arbitrary bi can appear in our vacua (but only rational numbers), this difference does not build a continuous, producing separate lines. The origin of the diagonal lines in the right panels is simpler. The RGE lead to αGUT−1=α2−1(MGUT)=α2−1(ΛZ′)−b22πlnMGUTΛZ′, as a result of the running of α2; each diagonal line describes this running for a given b2, populated by all vacua with the same b2.In Fig. 3 we present our main results: the values of α4(ΛZ′), αGUT and MGUT in the Z8 orbifold vacua with gauge coupling unification. As before, we present how often we find in our effective vacua the few admissible values of the U(1)′ coupling strength at reachable energies, α4(ΛZ′), the scale at which all coupling strengths meet, MGUT, and the value of the coupling strengths when they meet, αGUT=αi(MGUT),i=1,2,3. From the top to the bottom plots, we display these results for the SUSY scales, ΛSUSY=2 TeV, 1012 GeV and 1017 GeV.For lowSUSY scale, we observe that the U(1)′ coupling strength is quite restricted by 25≤α4−1(2 TeV)≤60, or equivalently 0.46≤g4(2 TeV)≤0.7; the only allowed unification scales are MGUT∈{1012 GeV,6.6×1013 GeV,4.1×1016 GeV}; and the coupling at unification takes only a few values, restricted by 5.6≤αGUT−1≤21.4. We note that g4(2 TeV)≈0.6 is the most commonly present value, just below the observed value of the SU(2)L coupling. Additionally, most of the vacua (62 out of 77) find unification at the largest MGUT. At that scale, the preferred value of the GUT coupling corresponds to αGUT≈1/21, very close to the value taken traditionally in GUTs, αGUT≈1/25. As a side remark, although we have considered αi=1 as our perturbativity limit, a stricter bound is achieved if one demands gi<1, which would imply that the values αGUT−1≲11 should be disregarded and, in turn, so should the vacua with the lowest GUT scale MGUT=1012 GeV in this case.For intermediate scale SUSY breaking, the variety of Z8 orbifold vacua with unification is richer. However, once more, there are strong restrictions on the possibly observable values of the U(1)′ couplings, set by 38<α4−1(2 TeV)≤64, or equivalently 0.44≤g4(2 TeV)<0.6. Since the distribution of U(1)′ coupling values at low energies is quite uniform, its average value can also be of some interest: g4(2 TeV)‾≈0.5. Concerning the unification scale and the coupling at those energies, we find a very compact distribution of values with 4.3×1011 GeV≤MGUT≤1016 GeV and 17<αGUT−1<36. Most of the vacua yield MGUT≈4.3×1011 GeV and αGUT≈1/33. It is interesting that the higher the SUSY breaking scale, the lower the unification scale.An intriguing observation is that models without SUSY below Mstr emerging from heterotic orbifold compactifications produce very similar results (roughly identical in our approximations) to those of intermediate SUSY scale. In particular, inspecting the bottomleft panel, we see that the range of values for α4−1(2 TeV) coincides with the previous case, except for an isolated vacuum, which we might ignore. As a consequence, again, g4‾≈0.5 at low energies. In fact, most of the vacua of this type render exactly the same MGUT and αGUT as in the previous case.5Sample model with potentially stable Higgs vacuumTo illustrate the features of our promising vacua with U(1)′, let us examine in one Z8 orbifold sample vacuum the potential of a U(1)′ as a tool to solve the metastability problem of the Higgs potential. According to ref. [6], SM fermions and some extra singlets with U(1)′ charges, subject to a series of constraints, can ameliorate the RGE running of the relevant couplings, yielding a positive Higgs selfcoupling at all scales.The shift vector V and WL Wa (satisfying the constraints in Table 1) that define the gauge embedding of a particular Z8–II (2,1) orbifold are(9)V=14(−7/2,0,0,0,1/2,1/2,5/2,3)×(−4,−1,0,0,0,1/2,1/2,3),W1=14(1,−7,−7,−5,2,2,1,−3)(−3,3,−6,−4,1,−3,3,5),W6=0. The resulting gauge group reads G4D=GSM×[U(1)′]6×SU(2)6, where one U(1)′ is (pseudo)anomalous. We choose a vacuum of SM singlet VEVs, such that G4D→Geff (see eq. (4)) spontaneously and the (correctly normalized) hypercharge and U(1)′ generators are given by(10)t1=14(1,5/3,5/3,−5/3,1,1,1,1)(0,0,0,0,0,0,0,0),t4=1122(−3,0,0,0,1,1,1,−2)(0,0,0,0,0,8,8,0). The spectrum of the chosen vacuum, after the decoupling of vectorlike exotics w.r.t. Geff, is displayed in Table 4, considering scales below ΛSUSY=1012 GeV. It contains the SM particles, an extra Higgs boson, few fermionic exotics and some SM singlets, which are mostly fermions. We choose as scalars two (instead of one) SM singlets, s1 and s2, to trigger the spontaneous breakdown of U(1)′ and facilitate the decoupling of SM exotics.We compute the RGE running of the couplings in this model by using SARAH [57]. First, by applying our approach, we find g4(2 TeV)≈0.49 and a unification scale MGUT∼ΛSUSY=1012 GeV. Supposing that the scalar fields s1 and s2 develop VEVs, we note that the fermionic exotics exhibit couplings that allow them to be decoupled below ΛZ′=2 TeV while the U(1)′ is spontaneously broken, so that we can consider only the SM spectrum below ΛZ′. Taking g1=0.3587, g2=0.6482, g3=1.1645, the top Yukawa Y33u=0.9356 and the quartic Higgs selfcoupling λ=0.127 at the topmass scale mt=173.1 GeV (see e.g. [58]), we let the SM couplings evolve below ΛZ′. For ΛZ′<μ<ΛSUSY, we include all exotics of Table 4 and further suppose that Hu dominates the quartic Higgs selfcoupling in order to carefully study the evolution of that coupling. Our findings are shown in Fig. 4, where we have extended our description of λ above ΛSUSY to make sure that perturbativity is not lost. In order to test the strength of our study, we have also allowed for nontrivial values of other quartic couplings (those of Hd, s1 and s2) and found that our result is not altered as long as those couplings are taken close to the value of λ at mt. Thus, it is possible to state that, although our model differs from those of ref. [6], our charges and U(1)′ coupling constant combine together to yield a stable Higgs vacuum, as in their cases.This model admits further interesting phenomenology. Let us roughly explore here some aspects concerning the fermion masses in this model. Based on the compactification scheme, the dominant contributions to the mass terms in the effective Lagrangian are given byL⊃−Y33uu¯3Hu†q3−Yu11,22u¯1,2Hu†q1,2s22−Y33dd¯3Hd†q3s17s22−Y11,22dd¯1,2Hd†q1,2s22−Y33ℓe¯3Hd†ℓ3s12−Y11,22ℓe¯1,2Hd†ℓ1,2s22−YiiνNibHu†ℓ3s22−kijNiaNjcs1+h.c., where we suppose that the singlet VEVs can be chosen allowing some tuning. For example, we find, at this level, that the top quark has the largest mass as the corresponding Yukawa coupling appears unsupressed, allowed by all symmetries of the string construction. Other Yukawas are suppressed by the singlet VEVs. For example, if U(1)′ is spontaneously broken such that 〈s2〉2∼O(10−5), 〈s1〉∼O(10), and 〈Hd〉∼O(10−4)〈Hu〉, one arrives at the correct relations mt/mu≈105, mt/mb≈102 and mt/mτ≈102, where all coefficients Yu,d,ℓ are (unsupressed) of order unity because untwisted fields appear in each coupling. Additionally, we observe that neutrino masses of the right order are generated through a typeI seesaw mechanism, where Ni are heavy righthanded neutrinos. Further, while U(1)′ breaks down, the exotics of the middle frame of Table 4 also develop masses of order 〈s1,2〉 and could also be detected as a signal of this kind of models. On the less bright side, in our model the electron and down quark are very light, mt/me=mt/md≈109, the chosen VEVs require large finetuning because the effective theory is defined at ΛSUSY, and there is a residual flavor symmetry between the first and second generation. We expect that a more careful analysis of additional details of the model, such as the SUSY and flavor breakdown, shall provide solutions to these issues, but this analysis is beyond the scope of this letter.6Final remarks and outlookBy means of the 1loop RGE, we have systematically studied the TeVscale value of the U(1)′ coupling constant in vacua arising from Z8 heterotic orbifold compactifications whose matter content exhibits the MSSM spectrum plus vectorlike exotics at the string scale. We have restricted ourselves to vacua with only one nonanomalous U(1)′ gauge symmetry, and whose SM gauge couplings have the observed values and unify at a modeldependent GUT scale, below the string scale. Only between 0.5% and 1.5% of all possible vacua satisfy these conditions.Supposing that the U(1)′ breakdown scale is of order of few TeV, reachable at colliders, we find that for TeV SUSY the U(1)′ coupling constant is restricted in our constructions to lie in the small range 0.46<g4<0.7. This range is further reduced to 0.44<g4<0.6 if one allows SUSY to be broken at a scale larger than 1012 GeV. Models with such couplings exhibit exotic fermions, in addition to a multiTeV Z′, that may be detected soon.We have found that also the unification scale is restricted in Z8 orbifold vacua to be roughly either 1014 GeV or 1016 GeV for lowscale SUSY, or preferably about 1012 GeV for intermediate SUSY breaking scale or higher.We have also studied the properties of a sample model, finding that, if intermediate scale SUSY is realized, there are Z8 orbifold vacua that may be furnished with the ingredients to stabilize the Higgs vacuum. The details of such vacua and mechanism are left for future work.In our scheme, the dynamics of the spontaneous breaking of U(1)′ requires large finetuning to establish the hierarchies ΛZ′≪ΛSUSY≪Mstr. In a modeldependent basis, it could however be possible that the potential of SM singlets and gaugino condensates conspire to yield such hierarchies. One may also wonder whether this simplifies in nonsupersymmetric heterotic orbifolds. Another issue is the details of the RGE at the SUSY breaking scale, including the decoupling of superpartners, which may require a treatment such as in [59]. These important questions shall be the goal of future projects.AcknowledgementsIt is a pleasure to thank Jens Erler for very useful discussions and motivation to pursue this work. SRS would like to thank Rafael AlapiscoArámbula, who participated in an early stage of this project. SRS is grateful to the Bethe Center for Theoretical Physics and the Mainz Institute for Theoretical Physics for the hospitality. This work was partly supported by DGAPAPAPIIT grant IN100217 and CONACYT grants F252167 and 278017.References[1]P.LangackerRev. Mod. Phys.8120091199arXiv:0801.1345 [hepph]P. Langacker, Rev. Mod. Phys. 81 (2009), 1199, arXiv:0801.1345 [hepph].[2]O.LebedevY.MambriniPhys. Lett. B7342014350arXiv:1403.4837 [hepph]O. Lebedev and Y. Mambrini, Phys. 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