A Heterotic Standard Model

Within the context of the E8 ×E8 heterotic superstring compactified on a smooth CalabiYau threefold with an SU(4) gauge instanton, we show the existence of simple, realistic N = 1 supersymmetric vacua that are compatible with low energy particle physics. The observable sector of these vacua has gauge group SU(3)C ×SU(2)L ×U(1)Y ×U(1)B−L, three families of quarks and leptons, each with an additional right-handed neutrino, two Higgs-Higgs conjugate pairs, a small number of uncharged moduli and no exotic matter. The hidden sector contains non-Abelian gauge fields and moduli. In the strong coupling case there is no exotic matter, whereas for weak coupling there are a small number of additional matter multiplets in the hidden sector. The construction exploits a mechanism for “splitting” multiplets. The minimal nature and rarity of these vacua suggest the possible theoretical and experimental relevance of spontaneously broken U(1)B−L gauge symmetry and two Higgs-Higgs conjugate pairs. The U(1)B−L symmetry helps to naturally suppress the rate of nucleon decay. ∗vbraun, yanghe, [email protected]; [email protected] The discovery of non-vanishing neutrino masses indicates that, in supersymmetric theories without exotic multiplets, a right-handed neutrino must be added to each family of quarks and leptons [1]. It is well known that this augmented family fits exactly into the 16 spin representation of Spin(10), making this group very compelling from the point of view of grand unification and string theory. Within the context of N = 1 supersymmetric E8 × E8 heterotic string vacua, a Spin(10) group can arise from the spontaneous breaking of the observable sector E8 group by an SU(4) gauge instanton on an internal Calabi-Yau threefold [2]. The Spin(10) group is then broken by a Wilson line to a gauge group containing SU(3)C × SU(2)L × U(1)Y as a factor [3]. To achieve this, the Calabi-Yau manifold must have, minimally, a fundamental group Z3 × Z3. Until now, such vacua could not be constructed since a) Calabi-Yau threefolds with fundamental group Z3 ×Z3 were not known and b) it was unknown how to find SU(4) gauge instantons on such manifolds. Recently, the first problem was rectified in [4]. We have now solved the second problem, exhibiting a large class of SU(4) gauge instantons on the Calabi-Yau manifolds presented in [4]. Generalizing the results in [5, 6], these instantons are obtained as connections on stable, holomorphic vector bundles with structure group SU(4). The technical details will be given elsewhere [7]. In addition to these considerations, we also use a natural method for “splitting” multiplets that was introduced for general bundles in [6]. In this paper, we present the results of our search for realistic vacua in this context. The results are very encouraging. We find N = 1 supersymmetric vacua whose minimal observable sector, for both the weakly and strongly coupled heterotic string, has the following properties. • Observable Sector: Weak and Strong Coupling 1. Gauge group SU(3)C × SU(2)L × U(1)Y × U(1)B−L. 2. Three families of quarks and leptons, each with a right-handed neutrino. 3. Two Higgs-Higgs conjugate pairs. 4. Six geometric moduli and a small number of vector bundle moduli. 5. No exotic matter fields. These are, to our knowledge, the first vacua in any string theory context whose observable sector contains no exotic matter. We emphasize that, although very similar to the supersymmetric standard model, our observable sector differs in three significant ways. First, there is an extra right-handed neutrino in each family. Closely related to this is the appearance of an additional gauged B − L symmetry. Finally, we find, not one, but two Higgs-Higgs conjugate pairs. 1 The structure of the hidden sector depends on whether one is in the weakly or strongly coupled regime of the heterotic string. In the strongly coupled context, we find the following minimal hidden sector. • Hidden Sector: Strong Coupling 1. Gauge group E7 × U(6). 2. A small number of vector bundle moduli. 3. No matter fields. Again, note that this hidden sector has no exotic matter. Combining this with the above, we have demonstrated, within the context of the strongly coupled heterotic string, the existence of realistic vacua containing no exotic matter fields. We emphasize that the hidden sector gauge group E7 × U(6) is sufficiently large to allow acceptable supersymmetry breaking via condensation of its gauginos. In the weakly coupled context, we find the following minimal hidden sector. (This is also a valid vacuum in the strongly coupled case). • Hidden Sector: Weak Coupling 1. Gauge group Spin(12). 2. A small number of vector bundle moduli. 3. Two matter field multiplets in the 12 of Spin(12). Note that, in this case, there are a small number of exotic matter multiplets in the hidden sector. Again, the hidden sector gauge group Spin(12) is sufficiently large to allow acceptable supersymmetry breaking via gaugino condensation. The vacua presented above are the result of an extensive search within the wide context made precise in [7]. They appear to be the minimal vacua, all others containing exotic matter fields, either in the observable sector, the hidden sector, or both, usually with a large number of Higgs-Higgs conjugate pairs. We have been unable to find any vacuum in this context with only a single pair of Higgs-Higgs conjugate fields. Furthermore, to our knowledge, phenomenological vacua in all other string contexts [6, 8, 10, 11, 12] have substantial amounts of exotic matter, both in the observable and hidden sectors. For all these reasons, we refer to the class of vacua presented in this paper as a heterotic standard model and speculate that it may be of phenomenological significance. In 2 particular, it would seem to motivate renewed interest, both theoretical and experimental, in its characteristic properties; namely, 1) the physics of a U(1)B−L gauge symmetry spontaneously broken at, or above, the electroweak scale and 2) the physics of two pairs of Higgs-Higgs conjugate fields, particularly their experimental implications for flavor changing neutral currents. It is immediately clear that the B − L symmetry will help to naturally suppress the rate of nucleon decay. This potentially resolves a long-standing problem in phenomenological string vacua. At the least, our results go a long way toward demonstrating that realistic particle physics can be the low energy manifestation of the E8 × E8 heterotic superstring, as originally envisaged in [8, 9]. We now specify, in more detail, the properties of the these minimal vacua and indicate how they are determined. Following [8], the requisite Calabi-Yau threefold, X, is constructed as follows. We begin by considering a simply connected Calabi-Yau threefold, X̃, which is an elliptic fibration over a rational elliptic surface, dP9. In a six-dimensional region of moduli space, such manifolds can be shown to admit a Z3 × Z3 group action which is fixed point free. It follows that