Flavor constraints on scenarios with two or three heavy squark generations

Abstract

We re-assess constraints from flavor-changing neutral currents in the kaon system on supersymmetric scenarios with a light gluino, two heavy generations of squarks and a lighter third generation. We compute for the first time limits in scenarios with three heavy squark families, taking into account QCD corrections at the next-to-leading order. We compare our limits with those in the case of two heavy families. We use the mass insertion approximation and consider contributions from gluino exchange to constrain the mixing between the first and second squark generation. While it is not possible to perform a general analysis, we assess the relevance of each kind of flavor- and CP-violating parameters. We also provide ready to use magic numbers for the computation of the Wilson coefficients at 2 GeV for these scenarios.

Appendices

Appendix A: Experimental information

Values of experimental and lattice QCD parameters not given in the main part are listed in Table 1. We are aware that there have recently been some improvements in the determination of B _K [23–28], which have been averaged to a value of 0.7643±0.0097 [29]. Adopting this average would change the values of the SM predictions for Δm _K and ϵ obtained in [14], but we have checked that the impact on our results is negligible, since the limits are mainly determined by $\sigma_{\Delta m_{K}}$ and σ _ϵ .

Table 1 Experimental and lattice QCD values used for our analysis. The bag parameters B _i are given in the RI scheme at 2 GeV. ^∗We have computed with RunDec [32] the values of m _s and m _d in the RI scheme, using as input the values given by the PDG [31] in the $\overline{\mathrm{MS}}$ scheme

We use the values of the bag parameters B _i given by the ETM collaboration [15] because they provide the results in the RI scheme at 2 GeV. The RBC and UKQCD collaborations recently also reported new computations of the relevant matrix elements [30], which are in good agreement with the ETM calculations. Thus, we estimate that employing the results of [30] would change our limits by less than 10 %.

Appendix B: Beta functions

The two-loop β functions

$$ \beta_X = \mu\frac{dX}{d\mu} = \beta_X^{(1)} + \beta_X^{(2)} $$

(B.1)

for the strong gauge coupling and the gluino mass for the case of three heavy squark families are equal to those of the Split SUSY scenario. In the limit of pure QCD, they read [33]

(B.2)

For the case of one family of light sfermions and two heavy ones, we took the QCD limit of the two-loop β functions of the Effective SUSY scenario [33]. Due to the breaking of SUSY in the effective theory below $m_{\tilde{q}}$, the gluino–squark–quark couplings and the squark quartic couplings are no longer given by the gauge coupling. In principle, these couplings are also different for different squarks. However, as we consider only QCD, there is a single gluino–squark–quark coupling $\hat{g}_{s}$ and a single quartic coupling γ _s , which are related to the gauge coupling at $m_{\tilde{q}}$ by Eq. (B.4). The β functions relevant for our calculation are

(B.3)

The quartic coupling enters only via the two-loop part of $\beta_{\hat{g}_{s}}$. Hence, it is sufficient to consider its one-loop running in order to determine the two-loop running of g _s and M ₃.

As we have mentioned in Sect. 2.2, our inputs are the gluino pole mass $m_{\tilde{g}}$ and the scale $m_{\tilde{q}}$ at which the heavy squarks decouple. To begin with, we determine the value of $\alpha_{s}(m_{\tilde{g}})$ from the experimental value at M _Z and the SM β function. Then we convert $m_{\tilde{g}}$ to the $\overline{\mathrm{MS}}$ running mass $M_{3}(m_{\tilde{g}})$ via Eq. (14). Initially guessing values for $\hat{g}_{s}(m_{\tilde{g}})$ and $\gamma_{s}(m_{\tilde{g}})$, we run up to the scale $m_{\tilde{q}}$, where we apply the $\overline{\mathrm{MS}}$ matching conditions [34]

(B.4)

We then run down to $m_{\tilde{g}}$ and again set the QCD coupling to the correct value determined initially. This procedure is repeated until we obtain both the right value of $\alpha_{s}(m_{\tilde{g}})$ and couplings satisfying the matching conditions (B.4).

Appendix C: Magic numbers

3.1 C.1 Evolution below $m_{\tilde{g}}$

(C.1)

(C.2)

(C.3)

(C.4)

3.2 C.2 Evolution above $m_{\tilde{g}}$

(C.5)

These magic numbers correspond to LO effects. Consequently, they are independent of $\tilde{n}_{f}$, in particular of the number of heavy squarks.

3.2.1 C.2.1 Three heavy squark generations

(C.6)

(C.7)

(C.8)

3.2.2 C.2.2 Two heavy squark generations

(C.9)

(C.10)

(C.11)