Advances in High Energy Physics

Volume 2015 (2015), Article ID 785217, 6 pages

http://dx.doi.org/10.1155/2015/785217

## Singlet Extensions of the MSSM with Symmetry

^{1}Physik-Department T30, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany^{2}Excellence Cluster Universe, Boltzmannstraße 2, 85748 Garching, Germany^{3}Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany

Received 3 March 2015; Accepted 9 June 2015

Academic Editor: Kai Schmidt-Hoberg

Copyright © 2015 Michael Ratz and Patrick K. S. Vaudrevange. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We discuss singlet extensions of the MSSM with symmetry. We show that holomorphic zeros can avoid a potentially large coefficient of the term linear in the singlet. The emerging model has both an effective term and a supersymmetric mass term for the singlet which are controlled by the gravitino mass. The term turns out to be suppressed against by about one or two orders of magnitude. We argue that this class of models might provide us with a solution to the little hierarchy problem of the MSSM.

#### 1. Purpose of This Paper

The symmetry [1, 2] provides us with compelling solutions of the and proton decay problems of the minimal supersymmetric extension of the standard model (MSSM). This symmetry appears anomalous, but the anomaly is cancelled by the (discrete) Green-Schwarz (GS) mechanism [3] in such a way that it does not spoil gauge coupling unification (see, e.g., [4] for a discussion). More precisely, if one extends the MSSM by a symmetry (continuous or discrete) that solves the problem and (i) demands anomaly freedom (while allowing GS anomaly cancellation), (ii) demands that the usual Yukawa couplings and the Weinberg operator be allowed, (iii) demands consistency with (10) grand unification, and (iv) demands precision gauge coupling unification, then this is the unique solution [2] (see also [5] for an alternative proof). By relaxing (iii) to consistency with (5), one obtains four additional symmetries [6]. Further, can be thought of as a discrete remnant of the Lorentz symmetry of compact extra dimensions; that is, it has a simple geometric interpretation and can arise in explicit string-derived models with the precise MSSM matter content [7]. The charge assignment is very simple: MSSM matter superfields have charge 1 while the Higgs superfields have 0, and the superpotential carries charge 2.

However, if one attempts to construct singlet extensions of the MSSM, one faces the problem that the presence of superpotential coupling of the singlet to the Higgs bilinear implies that also a linear term in the singlet is allowed by all symmetries. In more detail, since the Higgs bilinear has charge 0, the singlet needs to carry charge 2 in order to match the charge 2 of the superpotential. Then the desired term is allowed. However, in this case one might expect to have a problematic, unsuppressed linear term in in the (effective) superpotential, with of the order of the fundamental scale. In order to forbid this linear term, one may try to add a new symmetry. It is quite straightforward to see that an ordinary symmetry cannot forbid this linear term and be, at the same time, consistent with criteria (i)–(iv) above: in order to forbid the linear term, the singlet needs to carry a nontrivial charge under the new symmetry. But, as we want the term , this implies that also carries a nontrivial charge. Consequently, the new symmetry would yield a solution to the problem. However, this is not possible: as stated above, one can prove that (under our assumptions) the unique solution to the problem is , and this symmetry does not forbid the linear term.

In this paper, we take an alternative route and describe how one can get rid of the linear term (1) by employing holomorphic zeros [8] associated with an additional pseudoanomalous gauge symmetry.

#### 2. Forbidding the Linear Term in the (G)NMSSM

##### 2.1. Setup

Consider a singlet extension of the MSSM with a singlet and an additional symmetry. is pseudoanomalous symmetry, whose anomaly is cancelled by the GS mechanism. Such factors often arise in string compactifications and are accompanied by nontrivial Fayet-Iliopoulos (FI) term [9] , which arises at 1-loop [10]. The FI term of the is assumed to be cancelled by a nontrivial vacuum expectation value (VEV) of a “flavon” , which carries negative charge and charge 0. Without loss of generality, we can normalize such that has charge and . (Of course, in true string-derived models the situation is usually more complicated: in approximately 500 out of a total of 11940 MSSM-like models from [11] the FI term can be cancelled with one field only. In all other models, one would have to identify with an appropriate monomial of MSSM singlet fields (see Appendix A for details).) For the sake of definiteness, we assume thatwhere the Planck scale is identified with the “fundamental scale.” In this case, can be used as Froggatt-Nielsen symmetry [12] to explain the flavor structure of quarks and leptons. However, this assumption is not crucial for the subsequent discussion, yet this is what one gets in explicit orbifold compactifications of the heterotic string which exhibit the exact MSSM spectrum at energies below the compactification scale.

Further, also the anomaly of is assumed to be cancelled by the GS mechanism with the GS axion being contained in the dilaton or another superfield, which we will denote by . Since the mixed and anomalies are universal, the GS mechanism does not interfere with the beautiful picture of MSSM gauge coupling unification (see, e.g., [4]). The “nonperturbative” term carries the same charge as the superpotential, namely, 2. It might be thought of as some nonperturbative hidden sector (see, e.g., [13]). Further, will also carry positive charge such that holomorphic zeros get lifted by “nonperturbative” terms. More details on the charge of can be found in Appendix B (see, e.g., [6, 14]). In more detail, we demand that be allowed, which is equivalent to the statement that carries charge (Note that may also be fractional even if the charges of all “fundamental” fields are integer, for instance, if one assumes that is given by the Affleck-Dine-Seiberg superpotential [15]. Examples for such terms can be found, e.g., in [13].) may be thought of as gaugino condensate [16] or some other nonperturbative physics, such as the one discussed in [17], which is involved in spontaneous supersymmetry breaking. We discuss this in more detail in Appendix B. Inserting the VEV we obtain in Planck units. (Note that (3) is not the “full” hidden sector superpotential. One must, of course, make sure that does not attain an -term VEV, and one needs to cancel the vacuum energy. A detailed discussion of these issues is, however, beyond the scope of the present paper.) This implies, in particular, that That is, symmetry breaking is controlled by the gravitino mass, as it should be, and due to the presence of we obtain a Froggatt-Nielsen-like [12] modification of the terms. However, in contrast to the usual Froggatt-Nielsen mechanism, it yields in our setup an enhancement rather than a suppression factor for the lifting of the holomorphic zeros by nonperturbative effects.

##### 2.2. Charges and Allowed Terms in the Superpotential

We summarize the and charges in Table 1.