Bottom-tau unification in a supersymmetric model with anomaly-mediation

We study the Yukawa unification, in particular, the unification of the Yukawa coupling constants of $b$ and $\tau$, in the framework of the supersymmetric (SUSY) model. We concentrate on the model in which the SUSY breaking scalar masses are of the order of the gravitino mass while the gaugino masses originate from the effect of anomaly mediation and hence are one-loop suppressed relative to the gravitino mass. We perform an accurate calculation of the Yukawa coupling constants of $b$ and $\tau$ at the grand unified theory (GUT) scale, including relevant renormalization group effects and threshold corrections. In particular, we study the renormalization group effects, taking into account the mass splittings among sfermions, gauginos, and the standard model particles. We found that the Yukawa coupling constant of $b$ at the GUT scale is about 70% of that of $\tau$ if there is no hierarchy between the sfermion masses and the gravitino mass. Our results suggest sizable threshold corrections to the Yukawa coupling constants at the GUT scale or significant suppressions of the sfermion masses relative to the gravitino mass.

Figure 1

The renormalization group runnings of the Yukawa coupling constants of $b$ and $\tau$, taking $\tan \beta =3.95$, $m_{3/2}=50\mathrm{TeV}$, $m_{\mathbf{10}}=m_{\overline{\mathbf{5}}}=m_{3/2}$, $m_{H\mathbf{5}}=m_{H\overline{\mathbf{5}}}=0.8m_{3/2}$, and $\mu >0$ (left), and $\tan \beta =40.0$, $m_{3/2}=250\mathrm{TeV}$, $m_{\mathbf{10}}=12\mathrm{TeV}$, $m_{\overline{\mathbf{5}}}=7\mathrm{TeV}$, $m_{H\mathbf{5}}=m_{H\overline{\mathbf{5}}}=2\mathrm{TeV}$, and $\mu >0$ (right). The solid lines are results with using the renormalization group analysis explained in Sec. 2, while the dotted lines are the case where the theory is directly matched to the MSSM at $Q=m_Z$ (see the main text). The vertical dotted lines are $Q=m_Z$, $M_{\tilde{G}}$, and $M_S$ to guide the eyes. The “jumps” of the coupling constants at $Q=m_Z$, $M_{\tilde{G}}$, and $M_S$ are due to the threshold corrections. The solid lines for $Q>M_S$ and the dotted lines for $Q>m_Z$ show the MSSM Yukawa coupling constants multiplied by $\cos \beta$, while the solid lines for $Q<M_S$ show the Yukawa coupling constants in the SM or $\tilde{G}\mathrm{SM}$.

Figure 2

The distribution of the sample points on $m_{\mathbf{10}}$ vs $\tan \beta$, $m_{\overline{\mathbf{5}}}$ vs $\tan \beta$, $m_{3/2}$ vs $\tan \beta$, and $m_{\tilde{B}}/m_{\tilde{W}}$ vs $m_{\tilde{g}}/m_{\tilde{W}}$ planes. The colors of the dots indicate the smallest value of $|R_{b\tau }-1|$: $|R_{b\tau }-1|<0.1$ (red dots), $0.1<|R_{b\tau }-1|<0.2$ (green dots), and $|R_{b\tau }-1|>0.2$ (blue dots).

Figure 3

The distribution of the sample points on $m_{\tilde{g}}$ vs $R_{b\tau }$, $\tan \beta$ vs $R_{b\tau }$, $m_{\mathbf{10}}$ vs $R_{b\tau }$, $m_{\overline{\mathbf{5}}}$ vs $R_{b\tau }$, $(m_{\mathbf{10}}/m_{3/2}{)}^2$ vs $R_{b\tau }$, and $(m_{\overline{\mathbf{5}}}/m_{3/2}{)}^2$ vs $R_{b\tau }$ planes. The colors of the dots show the largest value of $(m_{\mathbf{10}}/m_{3/2}{)}^2$: $(m_{\mathbf{10}}/m_{3/2}{)}^2>0.1$ (red dots), $0.01<(m_{\mathbf{10}}/m_{3/2}{)}^2<0.1$ (green dots), and $(m_{\mathbf{10}}/m_{3/2}{)}^2<0.01$ (blue dots).

Figure 4

$R_{b\tau }$ as a function of $m_{3/2}$, taking $m_{\overline{\mathbf{5}}}=m_{\mathbf{10}}$, and $m_{H\mathbf{5}}=m_{H\overline{\mathbf{5}}}=0.8m_{\mathbf{10}}$ for $\mu >0$ (left), $\mu <0$ (right). The value of $\tan \beta$ is taken to be 2.4, 3, 4, 6, 10, 20, and 40 as shown. The light gray (dotted) part of the lines corresponds to the regions excluded by the wino or gluino mass bounds. The marks on the lines show the points with $(m_{\mathbf{10}}/m_{3/2}{)}^2=1.0$ (star), 0.1 (up-pointing triangle), 0.03 (circle), 0.01 (square), and 0.003 (down-pointing triangle).

Figure 5

$R_{b\tau }$ as a function of $m_{3/2}$, taking $m_{\mathbf{10}}=m_{\overline{\mathbf{5}}}=m_{3/2}$, and $m_{H\mathbf{5}}=m_{H\overline{5}}=0.8m_{3/2}$. The upper and lower lines are for $\mu >0$ and $\mu <0$, respectively. The light gray (dotted) part of the lines corresponds to the regions excluded by the wino or gluino mass bounds. The cross mark on each line shows the point with $m_{\tilde{W}}=3.0\mathrm{TeV}$. The other marks on the lines show the points with $\tan \beta =4.0$ (star), 3.5 (down-pointing triangle), 3.0 (circle), and 2.4 (square).

Figure 6

Same as Fig. 4 for $\mu >0$, except for $m_h=123.09\mathrm{GeV}$ (left) and 127.09 GeV (right).

Figure 7

The distribution of the sample points on $m_{\tilde{g}}$ vs $|\delta y_{\overline{\mathbf{5}}}|$, $\tan \beta$ vs $|\delta y_{\overline{\mathbf{5}}}|$, $m_{\mathbf{10}}$ vs $|\delta y_{\overline{\mathbf{5}}}|$, $m_{\overline{\mathbf{5}}}$ vs $|\delta y_{\overline{\mathbf{5}}}|$, $(m_{\mathbf{10}}/m_{3/2}{)}^2$ vs $|\delta y_{\overline{\mathbf{5}}}|$, and $(m_{\overline{\mathbf{5}}}/m_{3/2}{)}^2$ vs $|\delta y_{\overline{\mathbf{5}}}|$ planes. The meanings of the colors are the same as those in Fig. 3.