Soft electroweak breaking from hard supersymmetry breaking

We present a class of four-dimensional models, with a non-supersymmetric spectrum, in which the radiative corrections to the Higgs mass are not sensitive, at least at one-loop, to the UV completion of the theory. At one loop, Yukawa interactions of the top quark contribute to a finite and negative Higgs squared mass which triggers the electroweak symmetry breaking, as in softly broken supersymmetric theories, while gauge interactions lead to a logarithmic cutoff dependent correction that can remain subdominant. Our construction relies on a hard supersymmetry breaking localized in the theory space of deconstruction models and predicts, within a renormalizable setup, analogous physics as five-dimensional scenarios of Scherk–Schwarz supersymmetry breaking. The electroweak symmetry breaking can be calculated in terms of the deconstruction scale, replication number, top-quark mass and electroweak gauge couplings. For mtop ∼ 170 Gev, the Higgs mass varies from 158 GeV for N = 2 to 178 GeV for N = 10. The weak scale is unlikely to be a fundamental scale of physics. Its calculation in terms of more fundamental scales is one of the central problems in particle physics. The problem is aggravated by the fact that in the Standard Model (SM) the Higgs field mass parameter gets radiative corrections that grow quadratically with the scale Λ of new physics: mh ∼ Λ. Thus, any attempt to calculate mh in terms of a more fundamental scale Λ and to make it stable against radiative corrections needs a mechanism of suppression of the quadratic dependence on Λ; the higher the scale Λ is, the stronger suppression is required. An attractive solution to the stability aspect of the hierarchy problem is provided by softly broken supersymmetry. Quantum corrections to the weak scale depend quadratically onMSUSY and only logarithmically on the cutoff scale Λ: δm2 ∼ M SUSY ln Λ (therefore Λ can be taken as high as the Planck scale). However, the generation of the weak scale vh ≪ Λ is overshadowed by the μ-problem, i.e., by the question why the supersymmetric parameter μ is of the order of the weak scale. Also, some other aspects of softly broken supersymmetric theories are sufficiently troublesome to justify the quest for alternative routes of solving the hierarchy problem. An important new element in attempts to solve the hierarchy problem is the idea of large (TeV−1 size) extra dimensions [1], realized by low scale string theories or at the level of effective theories [2]. One possibility, which does not require supersymmetry, is to consider the Higgs boson to be a component of the gauge field propagating in extra dimensions. The Higgs potential, by higher-dimensional gauge invariance, does not depend on the cutoff scale and is calculable in terms of the compactification scaleMc ∼ 1/R [3]. Another possibility [4, 5, 6] is to break supersymmetry via the Scherk–Schwarz mechanism [7]. The non-local character of this mechanism ensures that at least one-loop corrections to the Higgs mass are finite [5, 8]. In the effective theory supersymmetry is broken in a hard-way and it is conceivable that divergences re-appear at higher-loop level. However, large extra dimensions and related low value of the cutoff scale Λ change qualitatively the hierarchy problem in the sense that calculating mh in terms of R and Λ does not require as strong suppression of quadratic divergences as for the canonical case with Λ = MPL. From a phenomenological point of view, with the cutoff scale close to the compactification scale, one-loop finiteness of the leading corrections to the Higgs potential is sufficient for the cutoff dependence to be very weak. This point of view is taken in the model of Ref. [5]. The electroweak symmetry breaking (EWSB) is triggered by the top/stop loops. Although the gauge interactions contribute to a quadratically divergent result [9], the finiteness of the leading corrections still allows to make predictions about the Higgs boson mass [10]. The obvious advantage of this scenario is that the full Higgs potential and the superpartner masses are calculable to a good precision in terms of one dimensionful parameter — the compactification radius, and the soft breaking masses are not necessary. Recently, a new idea called deconstruction appeared [11, 12] which allows to realize the physics of extra dimensions in a strictly four-dimensional set-up. Soon after, the 4D analogue of the mechanism [3] was constructed [13] where the Higgs boson mass is protected from receiving divergent radiative corrections by the pseudo-Goldstone mechanism (see also [14] for another deconstruction model of electroweak symmetry breaking and [15] for a deconstruction model where the radiative corrections are highly suppressed as a result of the topological nature of the supersymmetry breaking). Although the deconstruction models yield no unambiguous predictions about the fundamental scale, the low-scale unification [16] suggests that the fundamental scale could be much lower than the Planck scale. Thus, similarly as in the large extra 1 dimensions models, less suppression of the quadratic divergence is required to alleviate the hierarchy problem. In this paper we investigate the four-dimensional analogue of the Scherk– Schwarz mechanism and take the model proposed by Barbieri, Hall and Nomura (BHN) [5] as our reference point. We do not aim at constructing a complete and phenomenologically viable model, which would give the Standard Model as its low-energy approximation. We rather aim at analyzing the general situation, when divergences in non-supersymmetric theories are considerably softened. More precisely, we start withN = 1 supersymmetric models consisting of a chain of N gauge groups which communicate to each other through N−1 bifundamental link-Higgs fields Φi [17]. The matter and Higgs fields are also replicated and represented by a set of N chiral superfields transforming in fundamental representation of the corresponding gauge group. When the linkHiggs fields acquire vacuum expectation values (vev) the mass pattern of the gauge, Higgs and matter fields is similar as in the theories with extra dimensions. We are mainly interested in the models in which the low-energy spectrum (zero-modes of the mass matrix) shows no sign of supersymmetry but still the radiative corrections to the mass parameters are weakly dependent on the cutoff scale. "! # "! # "! # "! # SU(2) SU(2) SU(2) SU(2) Φ1 ΦN−1 H1 H̃1 Q1 Q̃1 H2 H̃2 Q2 Q̃2 HN−1 H̃N−1 QN−1 Q̃N−1 HN H̃N QN Q̃N Figure 1: The quiver diagram of the model. Each circle (site) represents an SU(2) N = 1 Yang-Mills multiplet. Each line pointing outwards a circle represents a chiral multiplet in the fundamental representation of the group while a line pointing towards a circle stands for a chiral multiplet in the anti-fundamental representation of the group. First, recall how the 5D supersymmetric Yang-Mills theories on S1/Z2 are realized in the 4D set-up [17]. We write a supersymmetric Lagrangian for a chain of N gauge multiplets (Ai , χ a i ) (with common gauge coupling g0) and N chiral, link-Higgs multiplets (Φi,Ψi). In this paper all the gauge groups are SU(2) and the diagonal subgroup is identified with the SM weak hypercharge group. The link fields are 2×2 complex matrices transforming in the fundamental representation of the i-th and antifundamental of the (i+1)-th gauge group . The orbifolding procedure is accounted for by the fact that the first and the last gauge groups are not linked, thus the quiver diagram has ’topology’ of the line segment (see Fig. 1). The product group is spontaneously broken by the link-Higgs fields which acquire a common expectation value 〈Φi〉 = v1. Diagonalizing the mass matrix, one finds that the spectrum in the large N limit is the same as in the 5D super-YM theory compactified on S1/Z2. In particular, the link-Higgs degrees of freedom account for completing N = 1 gauge multiplets up to N = 2 at every massive level. 2 To realize the SM matter and Higgs fields in the bulk we need to deconstruct 5D hypermultiplets. To this end, to every gauge group we attach a set of chiral multiplets: ’Higgs doublets’ Hi = (hi, ψHi) and ’quark doublets’ Qi = (qi, ψQi), in the fundamental of the i-th group and their mirror partners with opposite quantum numbers H̃i = (h̃i, ψ̃Hi), Qi = (q̃i, ψ̃Qi) which complete the spectrum to N = 2 hypermultiplets. The superpotential is chosen as: W = ( N−1 ∑ i=1 y i H̃iΦiHi+1 − N ∑ i=1 mi H̃iHi + N−1 ∑ i=1 y i Q̃iΦiQi+1 − N ∑