Sum rules in the superpartner spectrum of the minimal supersymmetric standard model

Assuming that the string inspired, universal sum rules for soft supersymmetry-breaking terms, which have been recently found both in a wide class of four-dimensional superstrings and in supersymmertic gauge-Yukawa unified gauge models, are satisfied above and at the grand unification scale, we investigate their low energy consequences and derive sum rules in the superpartner spectrum of the minimal supersymmetric standard model. † Humboldt Fellow. ∗ On leave from: Department of Physics, Shinshu University, Matsumoto, 390 Japan. Partially supported by the Grants-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (No. 40211213). One of the most important issues in realistic supersymmetric theories is to understand how supersymmetry is broken and then to relate it with low energy physics. Though the recent exiting theoretical developments in supersymmetric gauge theories as well as superstrings [1], this problem has not been solved in a satisfactory fashion yet. It is however widely accepted that the supersymmetry breaking, whatever its origin is, appears as soft supersymmetry-breaking (SSB) terms in low energy effective theories, because the softness is a desirable property for not spoiling the supersymmetric solution of the naturalness problem of the standard model [2]. One might hope that the SSB terms have a minimal structure as it is suggested by N = 1 minimal supergravity [2], on one hand. It may be worthwhile, on the other hand, to find out relations among the SSB terms that have least dependence of the mechanism of supersymmetry breaking and are satisfied in a wide class of models. Phenomenological investigations and consequences based on these relations certainly would have a more general validity than those based on the assumption of the so-called universal SSB terms. At first sight one might think that once we deviate from the universality of the SSB terms, we would fall into the chaos of varieties [3]. Recent investigations on the SSB terms in 4D superstrings [4]-[8], however, have shown that it is possible to do systematic investigations of the SSB terms, and it has turned out to be also possible to parametrize the SSB terms in a simple way [6]–[8] so that one can easily find relations among them that are independent of the detailed nature of supersymmetry breaking. It has been then shown [9] that there exist symmetries in the effective N = 1 supergravity, which, along with a simple assumption on Yukawa couplings, lead to these relations. These symmetries happen to coincide with the Sand Tdualities, which are usually present in 4D superstrings [10]. Moreover it has turned out [9] that these relations are renormalization group (RG) invariant at the lowest nontrivial order in perturbation theory in all gauge-Yukawa unified (GYU) models [12], so that once they are satisfied at the string scale, they are satisfied at the grand unification scale, too. We call these relations sum rules for the SSB terms, which may be summarized as [8, 9, 11] h = −M Y , (1) M = mi +m 2 j +m 2 k. (2) Here M and mi stand for the unified gaugino mass and the soft scalar mass squared of the chiral superfield Φi, respectively. Y ijk is the dimensionless Yukawa coupling for the Φi Φj Φk term in the superpotential, while the h is the dimensional coupling for the trilinear term 2 of the corresponding scalar components. Higher order corrections to the above sum rules are model-dependent in general. We, furthermore, would like to recall that the assumption on the gauge-Yukawa unification in supersymmetric grand unified theories (GUTs), especially in the third generation sector, leads to a successful prediction of the top quark mass [13]. In this letter we are motivated by the desire to find out low energy consequences of the sum rules (1) and (2) which are assumed to be satisfied for the third generation sector of SU(5) type GUTs at the GUT scale MGUT. We will assume that between MGUT(∼ 10 16 GeV) and the supersymmetry braking scale MS(< 1 TeV) the minimal supersymmetric standard model (MSSM) describes particle physics, and we will derive sum rules in the superpartner spectrum (Eq. (13)) from the string inspired, universal relations (1) and (2). These sum rules are independent on the details of the SSB parameters as long as the sum rules (1) and (2) are satisfied at MGUT. Needless to say that these sum rules could be tested by future experiments, e.g., at LHC. Before we present the details of our investigations, we would like to briefly outline the basic nature that leads to the sum rules (1) and (2), both in 4D-superstring-inspired supergravity models and GYU models. To analyze how the sum rules (1) and (2) within the framework of effective N = 1 supergravity can be realized, one considers a non-canonical Kähler potential of the general form K = K̃(Φa,Φ ) + ∑ i K i(Φa,Φ )|Φ|, (3) where Φa’s and Φi’s are chiral superfields in the hidden and visible sectors, respectively. The basic assumptions are: (1) Supersymmetry is broken by the F -term condensations (〈Fa〉 6 = 0) of the hidden sector fields Φa. (2) The gaugino mass M stems from the gauge kinetic function f which depends only on the hidden sector fields, i.e. f = f(Φa). (3) We consider only those Yukawa couplings that have no field dependence. (4) The vacuum energy V0 vanishes. For the sum rules (1) and (2) to be satisfied under these assumptions, a certain relation among the Kähler potential K̃ in the hidden sector, the gauge kinetic function f and the Kähler metric has to exist, i.e., K(T )(Φa,Φ ) ≡ ln(K iK j jK k k ) = K̃ + lnRef + const. (4) for all {i, j, k} appearing in the sum rules (1) and (2), implying that the theory has two types of symmetries: The first one corresponds to the Kähler transformation together with the chiral 3 rotation of the matter multiplets, Φi → e Φi , Φ ∗i → eiΦ , K i → K i ie −(Mi+Mi) (5) K(T ) → K(T ) −M−M , f(Φa) → f(Φa) , W → e M W, (6) where Mi is a function of Φa and has to satisfy the constraint Mi + Mj + Mk = M for all possible set of {i, j, k} appearing in the sum rules (1) and (2). The second one is the invariance of the Kähler metric K (S) b under the SL(2, R) transformation of the gauge kinetic function f(Φa), where K(S) = − ln(f(Φa) + f(Φ )). For 4D string models, these symmetries appear as the target-space duality invariance and S-duality [10], respectively. In fact, Brignole et. al. [8] have already found these sum rules in their explicit computations in various orbifold models. In case that gauge symmetries break, we generally have D-term contributions to the soft scalar masses. Such D-term contributions, however, do not appear in the sum rules, because each D-term contribution is proportional to the charge of the matter field Φi [14]. The basic assumption in GYU models is that the Yukawa couplings Y ijk are expressed in terms of the gauge coupling g: Y ijk = ρg + . . . , (7) where ρ are constant independent of g and . . . stands for higher order terms. Eq. (7) is the power series solution to the reduction equation [15] β Y = βg dY /dg, where β Y and βg stand for the β functions of Y ijk and g, respectively. The next assumption is that the coefficients ρ satisfy the diagonality relation ρipqρ jpq ∝ δ i . This implies that the one-loop anomalous dimensions γ (1) j i /16π 2 for Φi’s become diagonal if the reduction solution (7) is inserted, i.e., γ (1) j i = γi δ j i g , where γi are constant independent of g. It can be then shown that the sum rule (1) as well as the relation (m)ji = m 2 i δ j i = κiM 2δ i with κi = γi/(T (R) − 3C(G)) are RG invariant in one-loop order, where T (R) and C(G) are the Dynkin index of the matter representation R and the quadratic Casimir of the adjoint representation of G, respectively. The sum rule (2) then follows from the consequence of the reduction of Y , i.e., γi+γj +γk = T (R)− 3C(G) for {i, j, k} appearing in the sum rule. To come to our main result of this letter, let us first describe the parameter space. Since we assume an SU(5) type GYU in the third generation, Eq. (7) takes the form gt = ρt g , gb = gτ = ρb g, (8) 4 at MGUT, where gi (i = t, b, τ) are the Yukawa couplings for the top, bottom quarks and the tau, and we ignore the Cabibbo-Kabayashi-Maskawa mixing of the quarks. For a given model, the ρ’s are fixed, but here we consider them as free parameters. It is more convenient to go from the parameter space (ρt , ρb) to another one (kt ≡ ρ 2 t , tan β), because we use the (physical) top quark mass Mt as input, i.e., Mt = (175.6± 5.5) GeV. Therefore, the unification condition of the gauge couplings of the MSSM, along with α EM(MZ) = 127.9 + (8/9π) ln(Mt/MZ) and also the tau mass Mτ = 1.777 GeV as low energy input parameters, fixes the allowed region in the kt − tanβ space, which is shown in Fig. 1, where we have used MS = 300 GeV. In the following analyses when varying tanβ, we use kt for Mt = 175GeV, while ignoring the bottom quark mass. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 kt tan β Mt = 180 [GeV]