Supersymmetric models with minimal flavour violation and their running

TL;DR

The paper refines the MFV framework in the MSSM by introducing a Cabibbo-angle counting rule and a corresponding MFV basis that makes the flavour structure transparent under renormalization group running. It derives one-loop RGEs projected onto the MFV basis, and provides analytic solutions in the moderate $\tan\beta$ regime, revealing a quasi-fixed-point behavior where flavour-violating parameters converge to scale-dependent but stable values, largely driven by the gluino-dominated running of flavour-blind terms. The study extends to large $\tan\beta$, illustrating via SPS-4 that more MFV coefficients can become order one and that CP-violating phases can propagate more, though the overall MFV structure remains predictive. Overall, MFV appears RG-invariant under the proposed counting and basis, with high-scale MFV imposing strong low-energy constraints on flavour observables, while deviations at high scale can signal MFV emergence or breakdown at different scales.

Abstract

We revisit the formulation of the principle of minimal flavor violation (MFV) in the minimal supersymmetric extension of the standard model, both at moderate and large tan(beta), and with or without new CP-violating phases. We introduce a counting rule which keeps track of the highly hierarchical structure of the Yukawa matrices. In this manner, we are able to control systematically which terms can be discarded in the soft SUSY breaking part of the Lagrangian. We argue that for the implementation of this counting rule, it is convenient to introduce a new basis of matrices in which both the squark (and slepton) mass terms as well as the trilinear couplings can be expanded. We derive the RGE for the MFV parameters and show that the beta functions also respect the counting rule. For moderate tan(beta), we provide explicit analytic solutions of these RGE and illustrate their behaviour by analyzing the neighbourhood (also switching on new phases) of the SPS-1a benchmark point. We then show that even in the case of large tan(beta), the RGE remain valid and that the analytic solutions obtained for moderate tan(beta) still allow us to understand the most important features of the running of the parameters, as illustrated with the help of the SPS-4 benchmark point.

Figures (5)

• Figure 1: Upper four plots: RGE evolution of the MFV parameters for the SPS-1a benchmark point. The solid curves always show the evolution with mSUGRA type of initial conditions. In the upper-left panel, the dashed (dotted) lines show the evolution when $x_{1(2)}(M_{\text{GUT}})=\pm M_{0}^{2}$. In all other cases, for each parameter, only three curves are shown -- the upper and lower ones (always shown as dashed) correspond to different initial conditions for that single parameter. Lower two plots: the real and imaginary parts of the mass insertions $\delta_{1(2)}\equiv(\delta_{RL}^{U(D)})^{32}/V_{ts}=(\delta _{RL}^{U(D)})^{31}/V_{td}$. For these plots, the initial conditions are varied independently for $\mathbf{A}^{U}$ and $\mathbf{A}^{D}$, allowing for a CP-phase as $A_{0}=re^{i\phi}$ with $r=0\rightarrow200$ GeV and $\phi$ between $\pm180{{}^{\circ}}$. Dashed lines in the $\delta_{2}$ plot show the impact of the initial conditions for $\mathbf{A}^{U}$, while the sensitivity of $\delta_{1}$ to those for $\mathbf{A}^{D}$ is negligible. The behaviours of the other mass-insertions, Eqs. (\ref{['MIA1']}, \ref{['MIA2']}, \ref{['MIA3']}), can easily be obtained from those of the parameters shown in the upper four plots. In particular, note that $y_{2}\left( M_{Z}\right) \ll x_{1}\left( M_{Z}\right)$, both because of the $\tan\beta$ suppression, and because of RGE effects.
• Figure 2: RGE evolution of the MFV parameters for the SPS-1a benchmark point, if the boundary conditions are fixed at the low scale. The dashed (dot-dashed) lines correspond to the change of boundary conditions $x_{1(2)}(M_{Z}) =-x_{1(2)}^{\text{SPS-1a}}(M_{Z})$. The curves quitting the plot reach $x_{1}/\tilde{a}_{1}\approx18$ and $x_{2}/\tilde{a}_{2}\approx40$ at the GUT scale.
• Figure 3: Upper four plots: RGE evolution of the MFV parameters for the SPS-4 benchmark point. The solid curves always show the evolution of the mSUGRA type of initial conditions. In the upper-left panel the dashed (dotted) lines show the evolution when $x_{1(2)}(M_{\text{GUT}})=\pm M_{0}^{2}$. In all other cases, for each parameter only three curves are shown -- the upper and lower ones (always shown as dashed) correspond to different initial conditions for that single parameter. Lower two plots: the mass-insertions $\delta_{1(2)}\equiv (\delta_{RL}^{U(D)})^{32}/V_{ts}=(\delta_{RL}^{U(D)})^{31}/V_{td}$, with the initial conditions at the GUT scales varied as explained in the text, but allowing in addition for a large CP-phase (between $\pm180{{}^{\circ}}$). In this case, $\delta_{1}\sim\tilde{a}_{4}$ is entirely radiatively generated, since $\tilde{a}_{4}=0$ is set to zero at the GUT scale. The behaviours of the other mass-insertions, Eqs. (\ref{['MIA1']}, \ref{['MIA2']}, \ref{['MIA3']}), can easily be obtained from those of the parameters shown in the upper four plots. In particular, note that though $y_{2}\left( M_{Z}\right)$ is still smaller than $x_{1}\left( M_{Z}\right)$, it is much less suppressed by RGE effects than in the SPS-1a case, and so is its CP-violating phase.
• Figure 4: RGE evolution of the MFV parameters for the SPS-1a benchmark point, but with $\tan\beta=50$. As in Fig. \ref{['fig:sps1']}, the initial conditions at the GUT scale are varied, but now allowing larger range for the $y_{i}$, which can be of $\mathcal{O}(1)$. Still, the quasi fixed-point behaviour is obviously largely independent of $\tan\beta$. The corresponding plot for $\delta_{2}$ is similarly close to the $\tan\beta=10$ one. Those for $w_{1,2}$ are not shown because these parameters stay very small, even with $\tan\beta=50$.
• Figure 5: Up: RGE evolution of the Yukawa couplings, in terms of their skeleton structures given in Eq.(\ref{['eq:Yusc']}), for $\tan\beta=10$ and $50$. The running obtained by solving the RGE of Eq.(\ref{['eq:ysRGE']}) or the full running projected back on Eq.(\ref{['eq:Yusc']}) are indistinguishable. Down: the running of the parameters with large $\tan\beta$ initial conditions, as specified in Eq. (\ref{['EffLTanB']}). In the right plot, starting with $c_{b}\left( M_{Z}\right) <0$, the RGE drives it positive, hence the zero at about $10^{11}$ GeV.