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Non-Tachyonic Semi-Realistic Non-Supersymmetric Heterotic-String Vacua

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Non-Tachyonic Semi-Realistic Non-Supersymmetric Heterotic-String Vacua Eur. Phys. J. C (2016) 76:208 DOI 10.1140/epjc/s10052-016-4056-2 Regular Article - Theoretical Physics Non-tachyonic semi-realistic non-supersymmetric heterotic-string vacua Johar M. Ashfaquea, Panos Athanasopoulosb, Alon E. Faraggic, Hasan Sonmezd Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK Received: 1 December 2015 / Accepted: 31 March 2016 / Published online: 15 April 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract The heterotic-string models in the free fermionic 1 Introduction formulation gave rise to some of the most realistic-string models to date, which possess N = 1 spacetime supersym- The discovery of the agent of electroweak symmetry break- metry. Lack of evidence for supersymmetry at the LHC insti- ing at the LHC [1,2] is a pivotal moment in particle physics. gated recent interest in non-supersymmetric heterotic-string While confirmation of this agent as the Standard Model vacua. We explore what may be learned in this context from electroweak doublet representation will require experimental the quasi-realistic free fermionic models. We show that con- scrutiny in the decades to come, the data to date seems to vin- structions with a low number of families give rise to pro- dicate this possibility. Substantiation of this interpretation of liferation of a priori tachyon producing sectors, compared to the data will reinforce the view that the electroweak symme- the non-realistic examples, which typically may contain only try breaking mechanism is intrinsically perturbative, and that one such sector. The reason being that in the realistic cases the SM provides a viable perturbative parameterisation up to the internal six dimensional space is fragmented into smaller the Planck scale. Moreover, the large scale unification sce- units. We present one example of a quasi-realistic, non- nario is further motivated by the embedding of the SM matter supersymmetric, non-tachyonic, heterotic-string vacuum and states in the chiral SO(10) representation; by the logarith- compare the structure of its massless spectrum to the cor- mic evolution of the SM parameters; by proton longevity; and responding supersymmetric vacuum. While in some sec- by the suppression of left-handed neutrino masses. Gaining tors supersymmetry is broken explicitly, i.e. the bosonic and further insight into the fundamental origins of the SM param- fermionic sectors produce massless and massive states, other eters can then only be obtained by incorporating gravity into sectors, and in particular those leading to the chiral families, the picture. continue to exhibit Fermi–Bose degeneracy. In these sectors String theory provides the most developed contemporary the massless spectrum, as compared to the supersymmetric approach to study how the Standard Model parameters may cases, will only differ in some local or global U(1) charges. arise from a unified theory of the gauge and gravitational We discuss the conditions for obtaining nb = n f at the mass- interactions. For this purpose several models that repro- less level in these models. Our example model contains an duce the spectrum of the Minimal Supersymmetric Standard anomalous U(1) symmetry, which generates a tadpole dia- Model have been produced [3–8]. Amongst them the free gram at one-loop order in string perturbation theory. We spec- fermionic models [3–5,9–23] are the most studied exam- ulate that this tadpole diagram may cancel the correspond- ples. The heterotic string in particular provides a compelling ing diagram generated by the one-loop non-vanishing vac- framework to study the gauge–gravity synthesis in the large uum energy and that in this respect the supersymmetric and scale unification scenario, as it reproduces the embedding of non-supersymmetric vacua should be regarded on an equal the SM chiral spectrum in spinorial SO(10) representations. footing. Finally we discuss vacua that contain two supersym- The majority of semi-realistic heterotic-string models metry generating sectors. constructed to date possess N = 1 spacetime supersym- metry, while non-supersymmetric vacua were investigated sporadically [24–31]. In the absence of evidence of super- a e-mail: [email protected] symmetry at the LHC recent interest in non-supersymmetric b e-mail: [email protected] heterotic-string vacua has emerged [32–38]. It is therefore c e-mail: [email protected] prudent to examine what may be learned in that context from d e-mail: [email protected] the quasi-realistic free fermionic models. In this paper this 123 208 Page 2 of 17 Eur. Phys. J. C (2016) 76 :208 question is considered. We discuss the different avenues that tains 44 real worldsheet fermions φa. The worldsheet super- may be used to break supersymmetry directly at the string symmetry is realised non-linearly in the left-moving sector scale and how they compare with the recent analysis [34]. and the worldsheet supercurrent is given by Our paper is organised as follows: in Sect. 2,wereviewthe μ I J K T = ψ ∂ Xμ + f χ χ χ , (2.1) structure of the phenomenological free fermionic heterotic- F IJK string models. In Sect. 3, we discuss the phases that break where fIJK are the structure constants of the 18 dimen- supersymmetry in the string models and the different patterns sional semi-simple Lie group. The 18 left-moving world- that they induce. Further discussion of the existence of sec- sheet fermions χ I transform in the adjoint representation of tors producing tachyons in these models and the relation of the Lie group, which in the case of the fermionic Z2 × Z2 the abundance of these sectors with the number of families is orbifolds with N = 1SUSYisSU(2)6. Such models pro- given. Moreover, in Sect. 4 a non-supersymmetric tachyon- vide our starting point and we will discuss in later sections free model is presented and its relation to the supersymmetric how supersymmetry is broken. The χ I therefore transform in counterpart is inferred. In Sect.5 we discuss the construction the adjoint representation of SU(2)6, and they are denoted of string vacua with split supersymmetry, in which super- by χ I , y I ,ωI with I = 1,...,6. Under parallel transport symmetry is produced by two sectors. Section 6 contains our around a non-contractible loop of the one-loop vacuum to conclusions. vacuum amplitude the worldsheet fermions pick up a phase πα( ) f →−ei f f , (2.2) 2 Phenomenological free fermionic models with α( f ) ∈ (−1, +1]. The phases for all worldsheet ferm- ions constitute the spin structure of the models and are given In this section, we review the structure of the phenomenologi- in the form of 64 dimensional boundary condition basis vec- cal free fermionic models. It should be stressed that these free tors. The partition function, fermionic models correspond to Z × Z toroidal orbifolds 2 2 α α and their phenomenological characteristics are deeply rooted Z(τ) = c , β Tr β (2.3) in the structure of the Z2 × Z2 orbifolds. In this respect, the α,β∈ free fermionic formalism merely provides an accessible set of α tools to extract the spectra of the string vacua and their prop- is a sum over all spin structures, where c β are Gener- α erties. Furthermore, the free fermionic machinery extends to alised GSO (GGSO) projection coefficients and Tr β ≡ πβ πτ the massive string spectrum via the analysis of the relevant Tr(ei Fα ei Hα ) with Hα being the hamiltonian, is the trace partition function. This provides important insight into the over the mode excitations of the worldsheet fields in the sec- symmetries that underly the string landscape and eventually tor α, subject to the GSO projections induced by the sec- may prove instrumental in understanding how the string vac- tor β. Requiring invariance under modular transformations uum is selected. However, one should not tie the cart before results in a set of constraints on the allowed spin struc- the horse. The fermionic and bosonic representations only tures and the GGSO projection coefficients. The Hilbert α provide complementary tools that are formally identical in space of a given sector in the finite abelian additive group two dimensions. The physically relevant properties of these α ∈ = = ,..., − kni bi , where ni 0 gzi 1, is obtained free fermionic models are due to their underlying Z2 × Z2 by acting on the vacuum of the sector α with bosonic, as orbifold structure. well as fermionic oscillators with frequencies ν f , ν f ∗ , and In the free fermionic formulation of the heterotic string in subsequently imposing the GGSO projections four dimensions, all the extra degrees of freedom needed α iπ(b Fα) ∗ to cancel the conformal anomaly are represented as free e i − δαc |sα = 0(2.4) fermions propagating on the string worldsheet. It is important bi to note that the two dimensional fermions are free only at a with special point in the moduli space [39]. However, the models ⎧ ⎫ ⎪ ⎪ can be deformed away from that point by incorporating the ⎨ ⎬ moduli as worldsheet Thirring interactions [40–42]. Since (bi Fα) ≡ − (bi ( f )Fα( f )), (2.5) ⎪ ⎪ the twisted matter spectrum of the Z2 × Z2 orbifolds, which ⎩real+complex real+complex⎭ gives rise to the Standard Model matter states, is independent left right of the moduli, working at the free fermionic point is just a where δα is the spacetime spin statistics index and Fα( f ) is convenient choice. In the light-cone gauge the supersymmet- a fermion number operator, counting each mode of f once ric left-moving sector includes the two transverse spacetime (and if f is complex, f ∗ minus once). For Ramond fermions fermionic coordinates ψμ and 18 internal worldsheet real with α( f ) = 1 the vacuum is a doubly degenerate spinor |±, χ I ∗ fermions , whereas the right-moving bosonic sector con- annihilated by the zero modes f0 and f0 , and with fermion 123 Eur.
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