U(1)' coupling constant at low energies from heterotic orbifolds

TL;DR

This paper analyzes the 1-loop running of U(1)' gauge coupling constants in heterotic orbifold models, finding a restricted range at low energies that could impact phenomenological studies.

Contribution

It systematically computes the low-energy U(1)' coupling in heterotic orbifold vacua with MSSM-like spectra, considering various supersymmetry breaking scales.

Findings

U(1)' coupling constant ranges from 0.46 to 0.7 at low-scale SUSY

Coupling values are similarly constrained in other SUSY breaking scenarios

Results support further phenomenological exploration of string vacua

Abstract

Additional 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 1-loop 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 non-anomalous U(1)' symmetry, gauge coupling unification and the observed values of known gauge couplings. We determine the low-energy 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 low-scale 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.

Tables

Table 1. Table 1: Orders and relations of the WL of ℤ 8 subscript ℤ 8 \mathbbm{Z}_{8} orbifolds, depending on the geometry of the compactification. A ≈ B 𝐴 𝐵 A\approx B indicates that A = B 𝐴 𝐵 A=B up to translations in the root lattice of E 8 subscript E 8 \mathrm{E}_{8} × \times E 8 subscript E 8 \mathrm{E}_{8} .

Orbifold	Conditions on the Wilson lines
${\mathbb{Z}}_8$–I (1,1)	$2W_1\approx 2W_5\approx 0;W_1\approx W_2\approx W_3\approx W_4;W_5\approx W_6$
${\mathbb{Z}}_8$–I (2,1)	$2W_1\approx 2W_5\approx 0;W_1\approx W_2\approx W_3\approx W_4;W_5\approx W_6$
${\mathbb{Z}}_8$–I (3,1)	$4W_1\approx 0;W_1\approx W_2\approx W_3\approx W_4\approx W_5\approx W_6$
${\mathbb{Z}}_8$–II (1,1)	$2W_1\approx 2W_5\approx 2W_6\approx 0;W_1\approx W_2\approx W_3\approx W_4$
${\mathbb{Z}}_8$–II (2,1)	$4W_1\approx 2W_6\approx 0;W_1\approx W_2\approx W_3\approx W_4\approx W_5$

Table 2. Table 2: Number of inequivalent heterotic orbifold models with MSSM-like properties found in ref.  [ 32 ] for each ℤ 8 subscript ℤ 8 \mathbbm{Z}_{8} orbifold geometry. We further provide in the third and sixth columns the number of vacuum configurations with MSSM-like properties and gauge group 𝒢 eff = 𝒢 S ​ M × U ​ ( 1 ) ′ subscript 𝒢 eff subscript 𝒢 𝑆 𝑀 U superscript 1 ′ \mathcal{G}_{\text{eff}}=\mathcal{G}_{SM}\times\mathrm{U}(1)^{\prime} in each case.

Orbifold	# MSSM-like	effective	Orbifold	# MSSM-like	effective
	models	vacua		models	vacua
${\mathbb{Z}}_8\text{–I}(1,1)$	268	1,362	${\mathbb{Z}}_8\text{–II}(1,1)$	2,023	10,023
${\mathbb{Z}}_8\text{–I}(2,1)$	246	1,097	${\mathbb{Z}}_8\text{–II}(2,1)$	505	2,813
${\mathbb{Z}}_8\text{–I}(3,1)$	389	1,989

Table 3. Table 3: Number of effective MSSM-like vacua arising from ℤ 8 subscript ℤ 8 \mathbbm{Z}_{8} heterotic orbifolds with U ​ ( 1 ) ′ U superscript 1 ′ \mathrm{U}(1)^{\prime} 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 b i = b i M ​ S ​ S ​ M ∈ { 33 / 5 , 1 , − 3 , 0 } subscript 𝑏 𝑖 superscript subscript 𝑏 𝑖 𝑀 𝑆 𝑆 𝑀 33 5 1 3 0 b_{i}=b_{i}^{MSSM}\in\{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 ( α 1 = α 2 subscript 𝛼 1 subscript 𝛼 2 \alpha_{1}=\alpha_{2} ) or total gauge coupling unification ( α G ​ U ​ T = α i subscript 𝛼 𝐺 𝑈 𝑇 subscript 𝛼 𝑖 \alpha_{GUT}=\alpha_{i} ), corresponding to three choices of Λ S ​ U ​ S ​ Y subscript Λ 𝑆 𝑈 𝑆 𝑌 \Lambda_{SUSY} .

orbifold	$\begin{array}{c}\text{MSSM}\\ \text{vacua}\end{array}$	effective vacua of interest
		${\Lambda }_{SUSY}={\Lambda }_{Z^{\prime }}$		${\Lambda }_{SUSY}={10}^{12}$ GeV		${\Lambda }_{SUSY}={10}^{17}$ GeV
		${\alpha }_1={\alpha }_2$	unified	${\alpha }_1={\alpha }_2$	unified	${\alpha }_1={\alpha }_2$	unified
${\mathbb{Z}}_8$–I (1,1)	58	681	2	1,218	8	1,253	20
${\mathbb{Z}}_8$–I (2,1)	116	421	0	940	2	958	3
${\mathbb{Z}}_8$–I (3,1)	76	1,101	8	1,792	7	1,844	20
${\mathbb{Z}}_8$–II (1,1)	245	6,476	60	8,970	181	9,245	114
${\mathbb{Z}}_8$–II (2,1)	111	1,929	7	2,567	71	2,631	51
totals	606	10,608	77	15,487	269	15,931	208

Table 4. Table 4: Massless spectrum for a vacuum with gauge group 𝒢 eff subscript 𝒢 eff \mathcal{G}_{\text{eff}} . Representations with respect to SU ​ ( 3 ) c × SU ​ ( 2 ) L SU subscript 3 𝑐 SU subscript 2 𝐿 \mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L} are given in bold face, the hypercharge and the U ​ ( 1 ) ′ U superscript 1 ′ \mathrm{U}(1)^{\prime} 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.

#	Fermionic irrep	Label		#	Fermionic irrep	Label		#	Scalar irrep	Label
2	${(\mathrm{𝟏},\mathrm{𝟐})}_{(-1/2,-7/12\sqrt{2})}$	${\ell }_{1,2}$		1	${(\overline{\mathrm{𝟑}},\mathrm{𝟏})}_{(1/3,\mathrm{\hspace{0.17em}3}/4\sqrt{2})}$	${\overline{x}}_i$		1	${(\mathrm{𝟏},\mathrm{𝟐})}_{(1/2,-1/3\sqrt{2})}$	$H_u$
1	${(\mathrm{𝟏},\mathrm{𝟐})}_{(-1/2,\mathrm{\hspace{0.17em}1}/3\sqrt{2})}$	${\ell }_3$		1	${(\mathrm{𝟑},\mathrm{𝟏})}_{(-1/3,\mathrm{\hspace{0.17em}1}/12\sqrt{2})}$	$x_i$		1	${(\mathrm{𝟏},\mathrm{𝟐})}_{(-1/2,\mathrm{\hspace{0.17em}1}/12\sqrt{2})}$	$H_d$
2	${(\mathrm{𝟏},\mathrm{𝟏})}_{(1,-1/6\sqrt{2})}$	${\overline{e}}_{1,2}$		8	${(\mathrm{𝟏},\mathrm{𝟐})}_{(0,\mathrm{\hspace{0.17em}1}/6\sqrt{2})}$	${\eta }_i$		1	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,-1/4\sqrt{2})}$	$s_1$
1	${(\mathrm{𝟏},\mathrm{𝟏})}_{(1,\mathrm{\hspace{0.17em}1}/12\sqrt{2})}$	${\overline{e}}_3$		8	${(\mathrm{𝟏},\mathrm{𝟏})}_{(1/2,\mathrm{\hspace{0.17em}1}/6\sqrt{2})}$	${\zeta }_i$		1	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,\mathrm{\hspace{0.17em}1}/3\sqrt{2})}$	$s_2$
2	${(\mathrm{𝟑},\mathrm{𝟐})}_{(1/6,-1/6\sqrt{2})}$	$q_{1,2}$		8	${(\mathrm{𝟏},\mathrm{𝟏})}_{(-1/2,\mathrm{\hspace{0.17em}7}/12\sqrt{2})}$	${\overline{\zeta }}_i$
1	${(\mathrm{𝟑},\mathrm{𝟐})}_{(1/6,\mathrm{\hspace{0.17em}1}/4\sqrt{2})}$	$q_3$		8	${(\mathrm{𝟏},\mathrm{𝟏})}_{(-1/2,-1/12\sqrt{2})}$	${\overline{\kappa }}_i$
2	${(\overline{\mathrm{𝟑}},\mathrm{𝟏})}_{(-2/3,-1/6\sqrt{2})}$	${\overline{u}}_{1,2}$		8	${(\mathrm{𝟏},\mathrm{𝟏})}_{(1/2,-1/6\sqrt{2})}$	${\kappa }_i$
1	${(\overline{\mathrm{𝟑}},\mathrm{𝟏})}_{(-2/3,\mathrm{\hspace{0.17em}1}/12\sqrt{2})}$	${\overline{u}}_3$		11	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,\mathrm{\hspace{0.17em}1}/3\sqrt{2})}$	$N_i^a$
2	${(\overline{\mathrm{𝟑}},\mathrm{𝟏})}_{(1/3,-7/12\sqrt{2})}$	${\overline{d}}_{1,2}$		10	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,-2/3\sqrt{2})}$	$N_i^b$
1	${(\overline{\mathrm{𝟑}},\mathrm{𝟏})}_{(1/3,\mathrm{\hspace{0.17em}3}/4\sqrt{2})}$	${\overline{d}}_3$		8	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,-1/12\sqrt{2})}$	$N_i^c$
				6	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,-5/12\sqrt{2})}$	$N_i^d$
				4	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,\mathrm{\hspace{0.17em}7}/12\sqrt{2})}$	$N_i^e$
				2	${(\mathrm{𝟏},\mathrm{𝟏})}_{(0,-1/4\sqrt{2})}$	$N_i^f$

Equations

$$N(V^2-v^2)=0\mathrm{mod}2,$$

$$N_a{W}_a^2=0\mathrm{mod}2,$$

$${\mathcal{G}}_{4D}={\mathcal{G}}_{SM}\times [\mathrm{U}(1{)}^{\prime }{]}^n\times {\mathcal{G}}_{\text{hidden}},$$

$$T_{\alpha }=\sum _{I=1}^{16}{t}_{\alpha }^I{H}_I,\alpha =1,\dots ,n,$$

$${\mathcal{G}}_{\text{eff}}={\mathcal{G}}_{SM}\times \mathrm{U}(1{)}^{\prime }\subset {\mathcal{G}}_{4D},$$

$${\alpha }_1^{-1}(M_Z)=59.01\pm 0.01,{\alpha }_2^{-1}(M_Z)=29.59\pm 0.01,{\alpha }_3(M_Z)=0.1182\pm 0.0012,$$

$$\frac{\partial {\alpha }_i^{-1}}{\partial \ln \mu }=-\frac{b_i}{2\pi },i=1,\dots ,4,$$

$$b_{i}=\left\{\begin{array}[]{lcl}-\frac{11}{3}C_{2}(G_{i})+\frac{2}{3}\sum_{f}m_{f}C(\boldsymbol{R}_{f})+\frac{1}{3}\sum_{b}m_{b}C(\boldsymbol{R}_{b})\,,&&\text{non-SUSY}\,,\\ -3\,C_{2}(G_{i})+\sum_{S}m_{S}\,C(\boldsymbol{R}_{S})\,,&&\text{SUSY}\,,\\ \end{array}\right.$$

$$b_{i}=\left\{\begin{array}[]{lcl}\frac{2}{3}\sum_{f}m_{f}|q_{i}^{(f)}|^{2}+\frac{1}{3}\sum_{b}m_{b}|q_{i}^{(b)}|^{2}\,,&&\text{non-SUSY}\,,\\ \sum_{S}m_{S}\,\operatorname{tr}|q_{i}^{(S)}|^{2}\,,&&\text{SUSY}\,.\\ \end{array}\right.$$

$$\ln \frac{M_{GUT}}{{\Lambda }_{Z^{\prime }}}=2\pi \left({\alpha }_1^{-1}({\Lambda }_{Z^{\prime }})-{\alpha }_2^{-1}({\Lambda }_{Z^{\prime }})\right)\frac{1}{b_1-b_2}.$$

$$V$$

$$W_1$$

$$t_1$$

$$t_4$$

$$\begin{array}{r}\mathcal{L}\supset -Y_{33}^u{\bar{u}}_3{H}_u^{†}{q}_3-Y_{11,22}^u{\bar{u}}_{1,2}{H}_u^{†}{q}_{1,2}{s}_2^2-Y_{33}^d{\bar{d}}_3{H}_d^{†}{q}_3{s}_1^7{s}_2^2-Y_{11,22}^d{\bar{d}}_{1,2}{H}_d^{†}{q}_{1,2}{s}_2^2\\ -Y_{33}^{\ell }{\bar{e}}_3{H}_d^{†}{\ell }_3{s}_1^2-Y_{11,22}^{\ell }{\bar{e}}_{1,2}{H}_d^{†}{\ell }_{1,2}{s}_2^2-Y_{ii}^{\nu }{N}_i^b{H}_u^{†}{\ell }_3{s}_2^2-k_{ij}{N}_i^a{N}_j^c{s}_1+h.c.,\end{array}$$

Full text

1 Introduction

Some open questions in the standard model (SM) and cosmology have led to conjecture the existence of additional $\mathrm{U}(1)^{\prime}$ 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, 3, 4], or the $g_{\mu}-2$ anomaly [5], or the metastability of the Higgs vacuum [6]. They could also yield interesting signals at colliders [7, 8, 9] and alleviate some issues of models with supersymmetry (SUSY) [10, 11]. Although they must be broken at low energies and $m_{Z^{\prime}}$ is very constrained [12, 13, 14, 15], the bounds can be avoided (e.g. via a leptophobic $\mathrm{U}(1)^{\prime}$) and $Z^{\prime}$ signals could be soon confirmed at colliders.

The origin of $\mathrm{U}(1)^{\prime}$ symmetries is frequently related to grand unified theories (GUT) beyond $\mathrm{SU}(5)$. For example, in $\mathrm{E}_{6}$ GUTs, $\mathrm{U}(1)^{\prime}$ symmetries have been classified and studied phenomenologically [16, 17, 18, 19]. It is also known that $\mathrm{U}(1)^{\prime}$s are natural to models resulting from different string compactifications [20, 21, 22, 23, 24]. In particular, in heterotic orbifolds in the fermionic formulation, plausible scenarios with a light $Z^{\prime}$ 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 $\mathbbm{Z}_{4}^{R}$ symmetry [29] for proton stability can arise from $\mathrm{U}(1)^{\prime}$s (and other symmetries) of the model.

Motivated by these findings, in this letter we aim at characterizing the couplings of non-anomalous $\mathrm{U}(1)^{\prime}$s natural to some string compactifications. We focus on $\mathbbm{Z}_{8}$ toroidal orbifold compactifications of the \mathrm{E}_{8}$$\times$$\mathrm{E}_{8} heterotic string with MSSM-like 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 $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}\times\mathrm{U}(1)^{\prime}$, 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 Higgs-like mechanism breaks the $\mathrm{U}(1)^{\prime}$ at a scale $\Lambda_{Z^{\prime}}=2$ TeV, where also some exotics acquire masses, and a SUSY breaking scale $\Lambda_{SUSY}\geq\Lambda_{Z^{\prime}}$, we determine systematically the running of all coupling constants of our effective vacua by using the renormalization group equations (RGE) at 1-loop, 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 non-universality of stringy $\mathrm{U}(1)^{\prime}$s, heterotic orbifolds limit the values of the $\mathrm{U}(1)^{\prime}$ coupling at the TeV scale as well as the unification scale and the unified coupling.

In what follows, we discuss the main features of $\mathbbm{Z}_{8}$ heterotic orbifold compactifications and their effective vacua with $\mathrm{U}(1)^{\prime}$ 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 $\mathrm{U}(1)^{\prime}$.

2 MSSM-like $\boldsymbol{\mathbbm{Z}_{8}}$ toroidal orbifold models

Our starting point is the $\mathcal{N}=1$ \mathrm{E}_{8}$$\times$$\mathrm{E}_{8} heterotic string theory in the bosonic formulation, and we compactify the six extra dimensions on a toroidal $\mathbbm{Z}_{8}$ orbifold that preserves $\mathcal{N}=1$ in four dimensions. This choice is taken because $\mathbbm{Z}_{8}$ has shown to be the symmetry that yields the largest fraction of $\mathbbm{Z}_{N}$ MSSM-like 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 $\mathbbm{Z}_{N}$ orbifold is defined as the quotient $\mathbb{T}^{6}/P$ of a six-torus over a point group, which is generated by a single twist $\vartheta$ of order $N$, i.e. so that $\vartheta^{N}=\mathbbm{1}$. $\vartheta$ must be chosen to act as an isometry on $\mathbb{T}^{6}$. $\vartheta$ can always be diagonalized on three two-dimensional actions, so that $\vartheta=\text{diag}(e^{2\pi iv_{1}},e^{2\pi iv_{2}},e^{2\pi iv_{3}})$, where $v=(v_{1},v_{2},v_{3})$ is called the twist vector. It is possible to combine together the point group with the lattice $\Gamma$ of the torus to build the space group $S=P\ltimes\Gamma$, such that the orbifold can be analogously defined as $\mathbbm{R}^{6}/S$.

A complete classification of the $\mathbb{T}^{6}$ geometries and point groups for Abelian toroidal orbifolds [33] reveals that there are only two $\mathbbm{Z}_{8}$ point groups, denoted $\mathbbm{Z}_{8}$–I and $\mathbbm{Z}_{8}$–II and defined by the twist vectors $v_{\mathbbm{Z}_{8}\text{--I}}=\frac{1}{8}(1,2,-3)$ and $v_{\mathbbm{Z}_{8}\text{--II}}=\frac{1}{8}(1,3,-4)$, respectively. There are five inequivalent $\mathbb{T}^{6}$ geometries (see [30] for details) acceptable for these twists. Following the notation of [33], we shall label them as $\mathbbm{Z}_{8}$–I ($i$,1), $i=1,2,3$, and $\mathbbm{Z}_{8}$–II ($j$,1), $j=1,2$.

A consistent heterotic orbifold requires to embed the action of the six-dimensional orbifold into the \mathrm{E}_{8}$$\times$$\mathrm{E}_{8} gauge degrees of freedom. The $\mathbbm{Z}_{N}$ twist $\vartheta$ can be embedded as a 16-dimensional shift vector $V$ of order $N$ (i.e. such that $NV$ is in the \mathrm{E}_{8}$$\times$$\mathrm{E}_{8} root lattice), whereas the six independent directions of the torus can be embedded as 16-dimensional discrete Wilson lines (WL) $W_{a}$, $a=1,\ldots,6$. Given a $\mathbbm{Z}_{N}$ twist and a toroidal geometry, there are several admissible choices of shifts and WL, as long as they fulfill the modular invariance conditions [34],

[TABLE]

which ensure that the four-dimensional emergent field theory be non-anomalous and compatible with string theory. The space group of the orbifold constrains the order555The smallest integer $N_{a}$, such that $N_{a}W_{a}$ (with no summation over $a$) is contained in the root lattice of \mathrm{E}_{8}$$\times$$\mathrm{E}_{8}, is defined as the order of the WL $W_{a}$. $N_{a}$ of a WL $W_{a}$ 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 so-called space group flavor symmetry into the gauge degrees of freedom [35].

In $\mathbbm{Z}_{8}$ orbifolds, the order and relations of the admissible WL are given in table 1. Two $\mathbbm{Z}_{8}$–I geometries admit two independent WL of order two, whereas the third case admits only one independent order-4 WL. Further, $\mathbbm{Z}_{8}$–II (1,1) allows for three order-2 WL, and $\mathbbm{Z}_{8}$–II (2,1), one WL of order two and one of order four.

Applying standard techniques (see e.g. [36, 37, 38, 39]), one can use the modular invariant solutions to eqs. (1) (with $N=8$) complying with the WL-constraints 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 $\mathbbm{Z}_{8}$ orbifold compactifications of the \mathrm{E}_{8}$$\times$$\mathrm{E}_{8} heterotic string with the following properties:

$\bullet$ the gauge group at the compactification scale is

[TABLE]

where $\mathcal{G}_{SM}=\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{U}(1)_{Y}$ with non-anomalous hypercharge (satisfying $\sin^{2}\theta_{w}=3/8$), $\mathcal{G}_{\text{hidden}}$ is a non-Abelian gauge factor (typically a product of $\mathrm{SU}(M)$ subgroups), at most one $\mathrm{U}(1)^{\prime}$ is (pseudo-)anomalous and $n\leq 10$, depending on the model; and

$\bullet$ the massless spectrum includes the MSSM superfields plus only vectorlike exotic matter w.r.t. $\mathcal{G}_{SM}$.

$\mathcal{G}_{\text{hidden}}$ 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 $\mathrm{U}(1)^{\prime}$ symmetries of these models, we must point out some of their properties. Most of the models present an anomalous $\mathrm{U}(1)^{\prime}$ [42], whose anomaly can be canceled through the Green-Schwarz mechanism [43]. Besides, in this type of models, the gauge fields of the $\mathrm{U}(1)^{\prime}_{\alpha}$ symmetries can be decomposed as

[TABLE]

in terms of the Cartan generators $H_{I}$ of the original \mathrm{E}_{8}$$\times$$\mathrm{E}_{8}, such that the corresponding $\mathrm{U}(1)^{\prime}_{\alpha}$ charges for fields of the spectrum with gauge momentum $p\in\mathrm{E}_{8}\times\mathrm{E}_{8}$ are given by $q_{\alpha}=t_{\alpha}\cdot p$. This is why $t_{\alpha}$ is frequently called the generator of $\mathrm{U}(1)^{\prime}_{\alpha}$. It is known that, if we adopt666Despite this $\mathrm{U}(1)^{\prime}$ normalization, we allow the GUT-compatible hypercharge normalization $|t_{1}|^{2}=5/6$. the $\mathrm{U}(1)^{\prime}$ normalization $k|t_{\alpha}|^{2}=1$ and consider all algebras associated with the gauge group to have Kač-Moody level $k=2$, the tree-level gauge kinetic function is universally given by $f_{\alpha}=S$, where $S$ corresponds to the bosonic component of the (axio-)dilaton. Consequently, the tree-level gauge coupling satisfies $g_{s}^{-2}=\langle\text{Re}S\rangle$ at the (heterotic) string scale, $M_{str}\approx 10^{17}$ GeV. Further, there is some kinetic mixing between different $\mathrm{U}(1)^{\prime}$ gauge symmetries which one might believe relevant for phenomenology; however, it has been found to be typically of order $10^{-4}$–$10^{-2}$ in semi-realistic 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 [44, 28, 45] of $\mathbbm{Z}_{6}$–II heterotic orbifolds, in all the $\mathbbm{Z}_{8}$ models discussed here, there exists a large number of SM-singlets, which naturally develop $\mathcal{O}(0.1)$ VEVs in order to cancel the Fayet-Iliopoulos term, $\xi=g_{s}^{2}\operatorname{tr}T_{\text{anom}}/192\pi^{2}$ (in Planck units), appearing in models with a (pseudo-)anomalous $\mathrm{U}(1)^{\prime}$. 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 low-energy effective field theory. Simultaneously, since the singlets are charged under the $[\mathrm{U}(1)^{\prime}]^{n}$ gauge sector, those symmetries can be broken in the vacuum.

Secondly, it is always possible to find SM-singlet VEV configurations, such that, SUSY is retained while only the effective gauge group,

[TABLE]

remains after the spontaneous breakdown triggered by the singlet VEVs, where we ignore the hidden group $\mathcal{G}_{\text{hidden}}$. The surviving (non-anomalous) $\mathrm{U}(1)^{\prime}$ can be any of the original $\mathrm{U}(1)^{\prime}_{\alpha}$, $\alpha=1,\ldots,n$, 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 $\mathcal{G}_{\text{eff}}$, where the $\mathrm{U}(1)^{\prime}$ corresponds to each of the non-anomalous $\mathrm{U}(1)^{\prime}_{\alpha}$ of the $\mathbbm{Z}_{8}$ orbifold models. We find that there is a total of 17,284 such effective vacua, distributed in all admissible $\mathbbm{Z}_{8}$ 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 $\mathrm{U}(1)^{\prime}$ charges. This is true in most of the vacua where one of the $\mathrm{U}(1)^{\prime}$ remains unbroken. Thus, only the exotics that are vectorlike w.r.t. $\mathcal{G}_{\text{eff}}$, and not just $\mathcal{G}_{SM}$, decouple at low energies, allowing interactions among SM fields charged under $\mathrm{U}(1)^{\prime}$ 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.

3 Searching effective vacua with $\boldsymbol{\mathrm{U}(1)^{\prime}}$ and unification

We shall study the value of the $\mathrm{U}(1)^{\prime}$ coupling constants of the effective field theories emerging from $\mathbbm{Z}_{8}$ 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 $g_{s}^{-2}=\langle\text{Re}S\rangle$ at $M_{str}$, 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 $M_{Z}$. This approach seems rather arbitrary.

Instead, we assume that the SM coupling strengths $\alpha_{i}$ of our effective vacua have the observed values at $M_{Z}$ [48], (with $i=1$ for $\mathrm{U}(1)_{Y}$, $i=2$ for $\mathrm{SU}(2)_{L}$ and $i=3$ for $\mathrm{SU}(3)_{c}$)

[TABLE]

and then let the RGE define the value of the gauge couplings at all scales up to $M_{str}$, using the spectrum of the effective vacua we find. These effective vacua exhibit $\mathcal{N}=1$ SUSY and an additional $Z^{\prime}$ 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 $m_{Z^{\prime}}$ around 2 TeV [12, 13, 14, 15]). Hence, we suppose that SUSY is broken at a scale $\Lambda_{SUSY}>M_{Z}$ and the $\mathrm{U}(1)^{\prime}$ breakdown scale is $\Lambda_{Z^{\prime}}=2$ TeV, as a benchmark value.777We have verified that our results are qualitatively the same for other scales near this value. We further assume that $\Lambda_{SUSY}\geq\Lambda_{Z^{\prime}}$.

Given these remarks, we consider different matter spectra depending on the energy scale $\mu$. As sketched in figure 1, above the SUSY scale $\Lambda_{SUSY}$ the spectrum of the effective vacua includes the MSSM superfields and a few vectorlike exotics w.r.t. $\mathcal{G}_{SM}$ with non-trivial $\mathrm{U}(1)^{\prime}$ charges. Below $\Lambda_{SUSY}$, if it is larger than $\Lambda_{Z^{\prime}}$, the gauge group is still $\mathcal{G}_{\text{eff}}$, eq. (4), but we assume that all SM superpartners and the bosonic superpartners of the exotics decouple. Only one SM singlet with $\mathrm{U}(1)^{\prime}$ charge is taken as scalar below $\Lambda_{SUSY}$, so that its VEV can trigger the breakdown of $\mathrm{U}(1)^{\prime}$ and provide masses around $\Lambda_{Z^{\prime}}$ for the remaining exotics. Consequenly, below $\Lambda_{Z^{\prime}}$ 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 $M_{str}$ 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 $M_{GUT}$, which varies from vacuum to vacuum due to their matter content.888Note that, even though $M_{GUT}$ defines the scale at which all gauge couplings meet, no true unification is implied in the sense of usual GUTs. At $M_{GUT}$, one naturally can assume that all coupling strengths of $\mathcal{G}_{\text{eff}}$ have the same value $\alpha_{GUT}$. Thus, the $\mathrm{U}(1)^{\prime}$ coupling strength $\alpha_{4}=\alpha_{GUT}$ at that scale and its RGE running down to $\Lambda_{Z^{\prime}}$ in a vacuum is a low-energy consequence of such a vacuum. This approach is depicted in figure 1. We display an intermediate $\Lambda_{SUSY}$, but, in this work, we consider three well-motivated cases: $\Lambda_{SUSY}=\Lambda_{Z^{\prime}}$, $\Lambda_{SUSY}=10^{12}$ GeV and $\Lambda_{SUSY}=10^{17}$ 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 non-SUSY string compactifications [52, 53, 54]. It is known that gaugino condensates can render various $\Lambda_{SUSY}$ in these models [55].

As stated before, all vectorlike exotics w.r.t. $\mathcal{G}_{\text{eff}}$ naturally acquire masses just below $M_{str}$ while $\mathcal{G}_{4D}$ breaks down to $\mathcal{G}_{\text{eff}}$, but it is easy to conceive that this process happens gradually at various scales between $M_{GUT}$ and $M_{str}$. Furthermore, we expect that at those scales string threshold corrections, effects of the Green-Schwarz mechanism and moduli stabilization take place. We shall assume that all these effects do not alter the unification, even though they set deviations of $\alpha_{GUT}$ from $\alpha_{s}=g_{s}^{2}/4\pi$ that differ for each effective vacuum.

In this work, we use the 1-loop RGE for the gauge factors of the effective gauge group, $G_{i}\in\mathcal{G}_{\text{eff}}$. The running of the coupling strengths $\alpha_{i}=g_{i}^{2}/4\pi$ is given by

[TABLE]

where the $\beta$-function coefficients for non-Abelian groups ($i=2,3$) are given at 1-loop by

[TABLE]

Here $C_{2}(G_{i})$ is the quadratic Casimir of the group $G_{i}$, $C(\boldsymbol{R}_{b,f,S})$ denotes respectively the quadratic index of the $G_{i}$ representations $\boldsymbol{R}_{b,f,S}$ of the bosons, fermions and superfields included in the spectrum, over which the sums run, and $m_{b,f,S}$ denotes their multiplicities. Conventionally, we take $C(\boldsymbol{R}_{b,f,S})=1/2$ if $\boldsymbol{R}_{b,f,S}$ corresponds to the fundamental representation of $\mathrm{SU}(N)$. For Abelian gauge symmetries ($i=1,4$), eq. (7) reduces to

[TABLE]

in terms of the matter $\mathrm{U}(1)^{\prime}$ charges $q_{i}^{(b,f,S)}$, defined around eq. (3).

The solutions to (6) have the general form $\alpha_{i}^{-1}(\mu)=\alpha_{i}^{-1}(\mu_{0})-b_{i}/2\pi\ln(\mu/\mu_{0})$, in terms of a reference scale $\mu_{0}$. For the SM couplings, we take their observed values, eq. (5), with $\mu_{0}=M_{Z}$. Since below $\Lambda_{Z^{\prime}}=2$ TeV we assume that only the SM particles are present, we determine readily that $\alpha^{-1}_{1}=56.99$, $\alpha^{-1}_{2}=31.15$ and $\alpha^{-1}_{3}=11.9$ at $\Lambda_{Z^{\prime}}$.

4 $\boldsymbol{\mathrm{U}(1)^{\prime}}$ couplings in $\boldsymbol{\mathbbm{Z}_{8}}$ heterotic orbifold vacua

As a first step, we compute systematically the $\beta$-function 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 $\Lambda_{SUSY}$. That is, their matter spectra match the MSSM spectrum and, consequently, $(b_{1},b_{2},b_{3},b_{4})=(33/5,1,-3,0)$. In these cases, $b_{4}=0$ arises because the MSSM fields have no $\mathrm{U}(1)^{\prime}$ 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 $b_{4}=0$, but $b_{i}\neq b_{i}^{MSSM}$. Since in all these cases the coupling of $Z^{\prime}$ 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 $M_{GUT}<M_{Z}$ or $M_{GUT}>M_{str}$, which are either excluded (the former) or meaningless since our effective models apply only below $M_{str}$. Second, vacua in which any of the coupling strengths $\alpha_{i}=g_{i}^{2}/4\pi$ reaches negative values in its running are not acceptable. Finally, since our work is based on weakly coupled string theory, non-perturbative couplings, $\alpha_{i}>1$, are equally undesirable.

The number of vacua with these weaknesses varies depending on the choice of $\Lambda_{SUSY}$ and the $\mathbbm{Z}_{8}$ orbifold geometry. For instance, in the case of $\mathbbm{Z}_{8}$-I (1,1) with $\Lambda_{SUSY}=\Lambda_{Z^{\prime}}=2$ TeV, we find that out of the 1,362 effective vacua, 205 lead to unification at a scale $M_{GUT}<\Lambda_{Z^{\prime}}$ or $M_{GUT}>M_{str}$. Further, 321 vacua produce negative values of some $\alpha_{i}$, and in 89 we find non-perturbative values for some couplings ($\alpha_{i}>1$). Disregarding these (and those with $b_{4}=0$), we arrive at the 681 vacua of table 2 for this case.

To obtain the $\mathbbm{Z}_{8}$ orbifold vacua of interest, with unification of all coupling strengths, we proceed in two steps. We analyze first the qualities of those vacua with $\mathrm{SU}(2)_{L}-\mathrm{U}(1)_{Y}$ unification, i.e. those with $\alpha_{1}=\alpha_{2}\neq\alpha_{3}$, and then select among them those with $\alpha_{GUT}\equiv\alpha_{1}=\alpha_{2}\approx\alpha_{3}$ at a scale $M_{GUT}$, allowing for a small deviation $|\alpha_{GUT}^{-1}-\alpha_{3}^{-1}(M_{GUT})|<0.26$, corresponding to the $3\sigma$ interval of the measured value of $\alpha_{3}^{-1}(M_{Z})$. 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 $\Lambda_{SUSY}$. We realize that, from the huge number of possible effective vacua with $\mathcal{G}_{\text{eff}}$, only a quite small set of $\mathbbm{Z}_{8}$ orbifold vacua of order hundred in each case satisfies all of our constraints. The details of these vacua are available in [56].

In figure 2 we present our results for all $\mathbbm{Z}_{8}$ orbifold vacua with only $\mathrm{SU}(2)_{L}-\mathrm{U}(1)_{Y}$ unification. The left panels correspond to frequency plots for three different choices of $\Lambda_{SUSY}$ of (the inverse of) the $\mathrm{U}(1)^{\prime}$ coupling strength $\alpha_{4}^{-1}$ at the low-energy scale $\Lambda_{Z^{\prime}}=2$ TeV against the scale at which $\alpha_{1}=\alpha_{2}$, denoted $M_{GUT}$. The central values of the largest (red) bubble corresponds to the most frequent combination of $\alpha_{4}^{-1}(\Lambda_{Z^{\prime}})$ and $M_{GUT}$. 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 $\alpha_{GUT}^{-1}$ and $M_{GUT}$ achieved by our vacua, where $\alpha_{GUT}\equiv\alpha_{1}(M_{GUT})=\alpha_{2}(M_{GUT})$. 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 semi-qualitative remarks. The first observation is that, independently of whether SUSY is broken at low, intermediate or high energies, $\mathbbm{Z}_{8}$ orbifold vacua with MSSM-like properties do not allow any arbitrary values of $\mathrm{U}(1)^{\prime}$ couplings constants or unification scale. We find that roughly only $20<\alpha_{4}^{-1}(2\text{ TeV})<80$, corresponding to $0.4<g_{4}(2\text{ TeV})<0.8$, is allowed in our string constructions. We expect this to hold for any heterotic orbifold model with semi-realistic properties. The most common value of $\alpha_{4}^{-1}(2\text{ TeV})$ depends on $\Lambda_{SUSY}$: for low-scale SUSY $\alpha_{4}^{-1}\sim 30$, whereas $\alpha_{4}^{-1}\sim 60$ for other cases. We observe also other rough limits: $M_{GUT}>10^{8}\text{ GeV}$, and $\alpha_{GUT}^{-1}<30$ for low-scale SUSY and $\alpha_{GUT}^{-1}<45$ for other SUSY scales.

On the other hand, taking averages over all models, we find $\overline{M_{GUT}}\approx 10^{16}$ GeV and $\overline{\alpha_{GUT}^{-1}}\approx 13$ for low-scale SUSY, and $\overline{M_{GUT}}\approx 10^{15}$ GeV and $\overline{\alpha_{GUT}^{-1}}\approx 30$ 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 $\Delta\equiv|\alpha_{GUT}-\alpha_{3}(M_{GUT})|$, which is as large as $\overline{\Delta}\approx\alpha_{GUT}$ for low-scale SUSY, and $\overline{\Delta}\approx\frac{1}{3}\alpha_{GUT}$ for higher $\Lambda_{SUSY}$.

Let us comment on the vertical (diagonal) alignment pattern of the points in the left (right) plots of figure 2. Consider e.g. the top plots, with $\Lambda_{SUSY}=\Lambda_{Z^{\prime}}$, where one can find that the RGE lead to

[TABLE]

Since only $M_{GUT}$ and $b_{1}-b_{2}$ are model-dependent, all vacua with the same difference $b_{1}-b_{2}$ lead to the same $M_{GUT}$, yielding a point on the same vertical line in the left panels of the figure. Further, as not any arbitrary $b_{i}$ 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 $\alpha_{GUT}^{-1}=\alpha_{2}^{-1}(M_{GUT})=\alpha_{2}^{-1}(\Lambda_{Z^{\prime}})-\frac{b_{2}}{2\pi}\ln\frac{M_{GUT}}{\Lambda_{Z^{\prime}}}$, as a result of the running of $\alpha_{2}$; each diagonal line describes this running for a given $b_{2}$, populated by all vacua with the same $b_{2}$.

In figure 3 we present our main results: the values of $\alpha_{4}(\Lambda_{Z^{\prime}})$, $\alpha_{GUT}$ and $M_{GUT}$ in the $\mathbbm{Z}_{8}$ orbifold vacua with gauge coupling unification. As before, we present how often we find in our effective vacua the few admissible values of the $\mathrm{U}(1)^{\prime}$ coupling strength at reachable energies, $\alpha_{4}(\Lambda_{Z^{\prime}})$, the scale at which all coupling strengths meet, $M_{GUT}$, and the value of the coupling strengths when they meet, $\alpha_{GUT}=\alpha_{i}(M_{GUT}),i=1,2,3$. From the top to the bottom plots, we display these results for the SUSY scales, $\Lambda_{SUSY}=2$ TeV, $10^{12}$ GeV and $10^{17}$ GeV.

For low-SUSY scale, we observe that the $\mathrm{U}(1)^{\prime}$ coupling strength is quite restricted by $25\leq\alpha_{4}^{-1}(2\text{ TeV})\leq 60$, or equivalently $0.46\leq g_{4}(2\text{ TeV})\leq 0.7$; the only allowed unification scales are $M_{GUT}\in\{10^{12}\text{ GeV},6.6\times 10^{13}\text{ GeV},4.1\times 10^{16}\text{ GeV}\}$; and the coupling at unification takes only a few values, restricted by $5.6\leq\alpha_{GUT}^{-1}\leq 21.4$. We note that $g_{4}(2\text{ TeV})\approx 0.6$ is the most commonly present value, just below the observed value of the $\mathrm{SU}(2)_{L}$ coupling. Additionally, most of the vacua (62 out of 77) find unification at the largest $M_{GUT}$. At that scale, the preferred value of the GUT coupling corresponds to $\alpha_{GUT}\approx 1/21$, very close to the value taken traditionally in GUTs, $\alpha_{GUT}\approx 1/25$. As a side remark, although we have considered $\alpha_{i}=1$ as our perturbativity limit, a stricter bound is achieved if one demands $g_{i}<1$, which would imply that the values $\alpha_{GUT}^{-1}\lesssim 11$ should be disregarded and, in turn, so should the vacua with the lowest GUT scale $M_{GUT}=10^{12}$ GeV in this case.

For intermediate scale SUSY breaking, the variety of $\mathbbm{Z}_{8}$ orbifold vacua with unification is richer. However, once more, there are strong restrictions on the possibly observable values of the $\mathrm{U}(1)^{\prime}$ couplings, set by $38<\alpha_{4}^{-1}(2\text{ TeV})\leq 64$, or equivalently $0.44\leq g_{4}(2\text{ TeV})<0.6$. Since the distribution of $\mathrm{U}(1)^{\prime}$ coupling values at low energies is quite uniform, its average value can also be of some interest: $\overline{g_{4}(2\text{ TeV})}\approx 0.5$. Concerning the unification scale and the coupling at those energies, we find a very compact distribution of values with $4.3\times 10^{11}\text{ GeV}\leq M_{GUT}\leq 10^{16}$ GeV and $17<\alpha_{GUT}^{-1}<36$. Most of the vacua yield $M_{GUT}\approx 4.3\times 10^{11}$ GeV and $\alpha_{GUT}\approx 1/33$. It is interesting that the higher the SUSY breaking scale, the lower the unification scale.

An intriguing observation is that models without SUSY below $M_{str}$ emerging from heterotic orbifold compactifications produce very similar results (roughly identical in our approximations) to those of intermediate SUSY scale. In particular, inspecting the bottom-left panel, we see that the range of values for $\alpha_{4}^{-1}(2\text{ TeV})$ coincides with the previous case, except for an isolated vacuum, which we might ignore. As a consequence, again, $\overline{g_{4}}\approx 0.5$ at low energies. In fact, most of the vacua of this type render exactly the same $M_{GUT}$ and $\alpha_{GUT}$ as in the previous case.

5 Sample model with potentially stable Higgs vacuum

To illustrate the features of our promising vacua with $\mathrm{U}(1)^{\prime}$, let us examine in one $\mathbbm{Z}_{8}$ orbifold sample vacuum the potential of a $\mathrm{U}(1)^{\prime}$ as a tool to solve the metastability problem of the Higgs potential. According to ref. [6], SM fermions and some extra singlets with $\mathrm{U}(1)^{\prime}$ charges, subject to a series of constraints, can ameliorate the RGE running of the relevant couplings, yielding a positive Higgs self-coupling at all scales.

The shift vector $V$ and WL $W_{a}$ (satisfying the constraints in table 1) that define the gauge embedding of a particular $\mathbbm{Z}_{8}$–II (2,1) orbifold are

[TABLE]

The resulting gauge group reads $\mathcal{G}_{4D}=\mathcal{G}_{SM}\times[\mathrm{U}(1)^{\prime}]^{6}\times\mathrm{SU}(2)^{6}$, where one $\mathrm{U}(1)^{\prime}$ is (pseudo-)anomalous. We choose a vacuum of SM singlet VEVs, such that $\mathcal{G}_{4D}\to\mathcal{G}_{\text{eff}}$ (see eq. (4)) spontaneously and the (correctly normalized) hypercharge and $\mathrm{U}(1)^{\prime}$ generators are given by

[TABLE]

The spectrum of the chosen vacuum, after the decoupling of vectorlike exotics w.r.t. $\mathcal{G}_{\text{eff}}$, is displayed in table 4, considering scales below $\Lambda_{SUSY}=10^{12}$ 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, $s_{1}$ and $s_{2}$, to trigger the spontaneous breakdown of $\mathrm{U}(1)^{\prime}$ 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 $g_{4}(2\text{ TeV})\approx 0.49$ and a unification scale $M_{GUT}\sim\Lambda_{SUSY}=10^{12}$ GeV. Supposing that the scalar fields $s_{1}$ and $s_{2}$ develop VEVs, we note that the fermionic exotics exhibit couplings that allow them to be decoupled below $\Lambda_{Z^{\prime}}=2$ TeV while the $\mathrm{U}(1)^{\prime}$ is spontaneously broken, so that we can consider only the SM spectrum below $\Lambda_{Z^{\prime}}$. Taking $g_{1}=0.3587$, $g_{2}=0.6482$, $g_{3}=1.1645$, the top Yukawa $Y^{u}_{33}=0.9356$ and the quartic Higgs self-coupling $\lambda=0.127$ at the top-mass scale $m_{t}=173.1$ GeV (see e.g. [58]), we let the SM couplings evolve below $\Lambda_{Z^{\prime}}$. For $\Lambda_{Z^{\prime}}<\mu<\Lambda_{SUSY}$, we include all exotics of table 4 and further suppose that $H_{u}$ dominates the quartic Higgs self-coupling in order to carefully study the evolution of that coupling. Our findings are shown in fig. 4, where we have extended our description of $\lambda$ above $\Lambda_{SUSY}$ to make sure that perturbativity is not lost. In order to test the strength of our study, we have also allowed for non-trivial values of other quartic couplings (those of $H_{d}$, $s_{1}$ and $s_{2}$) and found that our result is not altered as long as those couplings are taken close to the value of $\lambda$ at $m_{t}$. Thus, it is possible to state that, although our model differs from those of ref. [6], our charges and $\mathrm{U}(1)^{\prime}$ 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 by

[TABLE]

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 $\mathrm{U}(1)^{\prime}$ is spontaneously broken such that $\langle s_{2}\rangle^{2}\sim\mathcal{O}(10^{-5})$, $\langle s_{1}\rangle\sim\mathcal{O}(10)$, and $\langle H_{d}\rangle\sim\mathcal{O}(10^{-4})\langle H_{u}\rangle$, one arrives at the correct relations $m_{t}/m_{u}\approx 10^{5}$, $m_{t}/m_{b}\approx 10^{2}$ and $m_{t}/m_{\tau}\approx 10^{2}$, where all coefficients $Y^{u,d,\ell}$ 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 type-I see-saw mechanism, where $N_{i}$ are heavy right-handed neutrinos. Further, while $\mathrm{U}(1)^{\prime}$ breaks down, the exotics of the middle frame of table 4 also develop masses of order $\langle s_{1,2}\rangle$ 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, $m_{t}/m_{e}=m_{t}/m_{d}\approx 10^{9}$, the chosen VEVs require large fine-tuning because the effective theory is defined at $\Lambda_{SUSY}$, 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.

6 Final remarks and outlook

By means of the 1-loop RGE, we have systematically studied the TeV-scale value of the $\mathrm{U}(1)^{\prime}$ coupling constant in vacua arising from $\mathbbm{Z}_{8}$ 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 non-anomalous $\mathrm{U}(1)^{\prime}$ gauge symmetry, and whose SM gauge couplings have the observed values and unify at a model-dependent GUT scale, below the string scale. Only between 0.5% and 1.5% of all possible vacua satisfy these conditions.

Supposing that the $\mathrm{U}(1)^{\prime}$ breakdown scale is of order of few TeV, reachable at colliders, we find that for TeV SUSY the $\mathrm{U}(1)^{\prime}$ coupling constant is restricted in our constructions to lie in the small range $0.46<g_{4}<0.7$. This range is further reduced to $0.44<g_{4}<0.6$ if one allows SUSY to be broken at a scale larger than $10^{12}$ GeV. Models with such couplings exhibit exotic fermions, in addition to a multi-TeV $Z^{\prime}$, that may be detected soon.

We have found that also the unification scale is restricted in $\mathbbm{Z}_{8}$ orbifold vacua to be roughly either $10^{14}$ GeV or $10^{16}$ GeV for low-scale SUSY, or preferably about $10^{12}$ 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 $\mathbbm{Z}_{8}$ 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 $\mathrm{U}(1)^{\prime}$ requires large fine-tuning to establish the hierarchies $\Lambda_{Z^{\prime}}\ll\Lambda_{SUSY}\ll M_{str}$. In a model-dependent 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 non-supersymmetric 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.

Acknowledgments.

It is a pleasure to thank Jens Erler for very useful discussions and motivation to pursue this work. SR-S would like to thank Rafael Alapisco-Arámbula, who participated in an early stage of this project. SR-S 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 DGAPA-PAPIIT grant IN100217 and CONACyT grants F-252167 and 278017.