SI-HEP-2015-26
QFET-2015-33
Phenomenological Implications of an
$\bm{SU(5)\times S_{4}\times U(1)}$ SUSY GUT of Flavour

The fermion structure was already scrutinised in [22], and we have completed this analysis by including the effects of canonical normalisation. In the basis with canonical kinetic terms, that is after redefining the superfields such that the Kähler metrics are identified with the unit matrix, the Yukawa matrix for the up-type quarks reads

$\displaystyle Y^{u}_{\text{GUT}}$

$\displaystyle\approx$

$\displaystyle\left(\begin{array}[]{ccc}y_{u}\,\lambda^{8}&-\frac{1}{2}k_{2}\,y_{c}\,\lambda^{8}&-\frac{1}{2}k_{4}\,y_{t}e^{i(\theta^{d}_{3}-\theta^{d}_{2})}\,\lambda^{6}\\ -\frac{1}{2}k_{2}\,y_{c}\lambda^{8}&y_{c}\,\lambda^{4}&-\frac{1}{2}k_{3}\,y_{t}e^{-i5\theta^{d}_{2}}\lambda^{5}\\ \!\!-\frac{1}{2}k_{4}\,y_{t}e^{-i(3\theta^{d}_{2}+2\theta^{d}_{3})}\,\lambda^{6}&~~-\frac{1}{2}k_{3}\,y_{t}e^{-i(7\theta^{d}_{2}+3\theta^{d}_{3})}\lambda^{5}&y_{t}\end{array}\right)\ ,~~~~$

(2.4)

where $y_{f}$ and $k_{i}$ are real order one coefficients, with the former stemming from the Yukawa part of the superpotential of the theory and the latter from the Kähler potential. In particular, $k_{2},k_{3}$ and $k_{4}$ appear in the non-canonical Kähler metric of the $SU(5)$ ${\bf{10}}$-plets, in the (12), (23) and (13) elements, respectively.

The Yukawa matrices for the down-type quarks and charged leptons take the form

	$\displaystyle Y^{d}_{{\text{GUT}}}$	$\displaystyle\!\approx\!$	$\displaystyle\left(\begin{array}[]{ccc}z^{d}_{1}e^{-i\theta^{d}_{2}}\lambda^{8}&\tilde{x}_{2}\lambda^{5}&\!-\tilde{x}_{2}e^{i(3\theta^{d}_{2}+2\theta^{d}_{3})}\lambda^{5}\\ -\tilde{x}_{2}\lambda^{5}&~~y_{s}e^{-i\theta^{d}_{2}}\lambda^{4}&\!-y_{s}e^{2i(\theta^{d}_{2}+\theta^{d}_{3})}\lambda^{4}\\ \!\left(z^{d}_{3}\!-\!\frac{K_{3}}{2}y_{b}\right)e^{-i(3\theta^{d}_{2}+2\theta^{d}_{3})}\lambda^{6}\!&~\left(z^{d}_{2}\!-\!\frac{K_{3}}{2}y_{b}\right)e^{-i(3\theta^{d}_{2}+2\theta^{d}_{3})}\lambda^{6}\!\!&y_{b}\lambda^{2}\end{array}\right)\!,~~~~~~~~$		(2.8)
	$\displaystyle Y^{e}_{{\text{GUT}}}$	$\displaystyle\!\approx\!$	$\displaystyle\left(\begin{array}[]{ccc}-3z^{d}_{1}e^{-i\theta^{d}_{2}}\lambda^{8}&-\tilde{x}_{2}\lambda^{5}&~~\left(z^{d}_{3}-\frac{K_{3}}{2}y_{b}\right)\lambda^{6}\\ \tilde{x}_{2}\lambda^{5}&-3\,e^{-i\theta^{d}_{2}}y_{s}\lambda^{4}&~~\left(z^{d}_{2}-\frac{K_{3}}{2}y_{b}\right)\lambda^{6}\\ -\tilde{x}_{2}\lambda^{5}&~~3\,e^{-i\theta^{d}_{2}}y_{s}\lambda^{4}&~~y_{b}\lambda^{2}\end{array}\right)\!.$		(2.12)

Again, these expressions are given in the canonical basis and all coefficients are real and of order one. $\tilde{x}_{2}$, $y_{f}$ and $z^{d}_{i}$ arise from the superpotential operators and $K_{3}$ from the Kähler potential, where it enters symmetrically in all off-diagonal elements of the non-canonical Kähler metric of the $SU(5)$ $\bf{\bar{5}}$-plets.

Finally, the Dirac neutrino Yukawa matrix in the canonical basis is given by

$\displaystyle Y^{\nu}$

$\displaystyle\!\approx\!$

$\displaystyle\left(\begin{array}[]{ccc}y_{D}&-\frac{y_{D}(K_{3}+K^{N}_{3})}{2}\lambda^{4}&\left(z^{D}_{1}-\frac{y_{D}(K_{3}+K^{N}_{3})}{2}\right)\lambda^{4}\\ -\frac{y_{D}(K_{3}+K^{N}_{3})}{2}\lambda^{4}&\left(z^{D}_{1}-\frac{y_{D}(K_{3}+K^{N}_{3})}{2}\right)\lambda^{4}\!\!&y_{D}\\ \!\left(z^{D}_{1}-\frac{y_{D}(K_{3}+K^{N}_{3})}{2}\right)\lambda^{4}&y_{D}&-\frac{y_{D}(K_{3}+K^{N}_{3})}{2}\lambda^{4}\end{array}\right),~~~~~~~~~$

(2.16)

which is real up to LO in $\lambda$. The parameters $y_{D}$ and $z^{D}_{1}$ originate from the superpotential, while $K^{N}_{3}$ is associated to the Kähler metric of the right-handed neutrinos. Note that this metric is identical to that of the $SU(5)$ $\bf{\bar{5}}$-plets, up to renaming the order one coefficients, see [1] for details.

Transforming the left- and right-handed superfields $f_{L,R}$ by unitary matrices $U^{f}_{L,R}$, we obtain the canonically normalised diagonal and positive Yukawas in the SCKM basis

$$\tilde{Y}^{u}_{\text{GUT}}\approx\left(\begin{array}[]{ccc}y_{u}\lambda^{8}&0&0\\ 0&y_{c}\lambda^{4}&0\\ 0&0&y_{t}\end{array}\right),\qquad\tilde{Y}^{d}_{\text{GUT}}\approx\left(\begin{array}[]{ccc}\frac{\tilde{x}_{2}^{2}}{y_{s}}\lambda^{6}&0&0\\ 0&y_{s}\lambda^{4}&0\\ 0&0&y_{b}\lambda^{2}\end{array}\right),$$

(2.17)

$$\tilde{Y}^{e}_{\text{GUT}}\approx\left(\begin{array}[]{ccc}\frac{\tilde{x}_{2}^{2}}{3y_{s}}\lambda^{6}&0&0\\ 0&3y_{s}\lambda^{4}&0\\ 0&0&y_{b}\lambda^{2}\end{array}\right).$$

(2.18)

Up to phase convention, the CKM matrix is given by $V_{\text{CKM}_{\text{GUT}}}=(U^{u}_{L})^{T}U^{d*}_{L}$, leading to the mixing angles

$$\sin(\theta^{q}_{13})_{\text{GUT}}\approx\frac{\tilde{x}_{2}}{y_{b}}\lambda^{3}\ ,\qquad\tan(\theta^{q}_{23})_{\text{GUT}}\approx\frac{y_{s}}{y_{b}}\lambda^{2}\ ,\qquad\tan(\theta^{q}_{12})_{\text{GUT}}\approx\frac{\tilde{x}_{2}}{y_{s}}\lambda\ .$$

(2.19)

The mixing arises purely from the down-type quark sector and incorporates the Gatto-Sartori-Tonin relation [23] $\theta^{q}_{12}\approx\sqrt{m_{d}/m_{s}}$. The amount of CP violation is given by the Jarlskog invariant [24]

$\displaystyle J^{q}_{\text{CP}_{\text{GUT}}}$

$\displaystyle\approx$

$\displaystyle\lambda^{7}\frac{\tilde{x}_{2}^{3}}{y_{b}^{2}y_{s}}\sin\theta^{d}_{2}\ .$

(2.20)

These results are in agreement with the LO expressions derived in [22], where canonical normalisation effects were ignored. As discussed in [1], the LO results for the quark and charged lepton masses and mixing angles remain unaffected by the process of canonicalising the kinetic terms. We point out that these 13 observables of the charged fermion sector are given in terms of only 8 input parameters ($\lambda$, $y_{u,c,t}$, $y_{s,b}$, $\tilde{x}_{2}$ and $\theta_{2}^{d}$) at LO.

The soft trilinear $A$-terms and the Yukawa couplings originate in the same superpotential terms. Hence, they have a similar flavour structure and, in the basis of canonical kinetic terms, the soft flavour matrices $A^{f}_{\text{GUT}}/A_{0}$, where $A_{0}$ denotes the scale of the trilinear terms, can be deduced from Eqs. (2.4-2.16) by simply replacing $y_{u}\to a_{u}\,e^{i(\theta^{a}_{u}-\theta^{y}_{u})}$, $y_{c}\to a_{c}\,e^{i(\theta^{a}_{c}-\theta^{y}_{c})}$, $y_{t}\to a_{t}$, $y_{s}\to a_{s}\,e^{i(\theta^{a}_{s}-\theta^{y}_{s})}$, $y_{b}\to a_{b}\,e^{i(\theta^{a}_{b}-\theta^{y}_{b})}$, $\tilde{x}_{2}\to\tilde{x}^{a}_{2}\,e^{i(\theta^{\tilde{x}_{a}}_{2}-\theta^{\tilde{x}}_{2})}$, $z^{f}_{i}\to z^{f_{a}}_{i}\,e^{i(\theta^{z_{f_{a}}}_{i}-\theta^{z_{f}}_{i})}$ and $y_{D}\to\alpha_{D}$. Here, the Yukawa phases are all given in terms of $\theta^{d}_{2},~\theta^{d}_{3}$ as follows: $\theta^{y}_{u}=\theta^{y}_{c}=\theta^{y}_{s}=\theta^{z_{d}}_{1}=2\theta^{d}_{2}+3\theta^{d}_{3}$, $\theta^{y}_{b}=\theta^{z_{d}}_{2}=\theta^{z_{d}}_{3}=\theta^{d}_{3}$ and $\theta^{\tilde{x}}_{2}=3(\theta^{d}_{2}+\theta^{d}_{3})$. On the other hand, the trilinear phases $\theta^{a}_{f},~\theta^{\tilde{x}_{a}}_{2},~\theta^{z_{f_{a}}}_{i}$ are kept free.

Turning to the soft scalar mass squared matrices in the canonical basis, we find

$\displaystyle\frac{M^{2}_{T_{\text{GUT}}}}{m_{0}^{2}}$

$\displaystyle\approx$

$\displaystyle\left(\begin{array}[]{ccc}b_{01}&~~(b_{2}-b_{01}k_{2})\lambda^{4}&~~~e^{i(\theta^{d}_{2}-\theta^{d}_{3})}(b_{4}-\frac{k_{4}(b_{01}+b_{02})}{2})\lambda^{6}\\ \cdot&~~b_{01}&~~~e^{5i\theta^{d}_{2}}(b_{3}-\frac{k_{3}(b_{01}+b_{02})}{2})\lambda^{5}\\ \cdot&\cdot&~~b_{02}\end{array}\right)\ ,$

(2.24)

for the $SU(5)$ ${\bf{10}}$-plets as well as

$\displaystyle\frac{M^{2}_{F(N)_{\text{GUT}}}}{m_{0}^{2}}$

$\displaystyle\approx$

$\displaystyle\left(\begin{array}[]{ccc}B^{(N)}_{0}&~~~(B^{(N)}_{3}-K^{(N)}_{3})\lambda^{4}&~~~(B^{(N)}_{3}-K^{(N)}_{3})\lambda^{4}\\ \cdot&~~~B^{(N)}_{0}&~~~(B^{(N)}_{3}-K^{(N)}_{3})\lambda^{4}\\ \cdot&\cdot&B^{(N)}_{0}\end{array}\right),$

(2.28)

for the $SU(5)$ ${\bf{5}}$-plets and the right-handed neutrinos, with the latter being associated to the coefficients with index $N$. For convenience, we absorb the universal order one parameter $B_{0}$ on the diagonal into the soft SUSY breaking mass $m_{0}$, so that the leading contribution to the diagonal entries of $M^{2}_{F_{\text{GUT}}}/m_{0}^{2}$ is one.

In order to study the phenomenological implications of the soft SUSY breaking sector, it is useful to rotate all quantities into the physical basis where the Yukawa matrices are diagonal and positive, i.e. the SCKM basis. Any misalignment between the fermion and sfermion flavour matrices constitutes a source of flavour violation, with the off-diagonal entries of the sfermionic mass matrices contributing to FCNCs. The sfermion mass matrices are given as

$$m^{2}_{\tilde{f}_{LL}}\!\!=(\tilde{m}_{f}^{2})_{LL}+\tilde{Y}_{f}\tilde{Y}_{f}^{\dagger}\upsilon^{2}_{u,d}\ ,~\quad m^{2}_{\tilde{f}_{RR}}\!\!=(\tilde{m}_{f}^{2})_{RR}+\tilde{Y}_{f}^{\dagger}\tilde{Y}_{f}\upsilon^{2}_{u,d}\ ,~\quad m^{2}_{\tilde{f}_{LR}}\!\!=\tilde{A}_{f}\upsilon_{u,d}-\mu\tilde{Y}_{f}\upsilon_{d,u}\ ,$$

(2.29)

where $\tilde{m}^{2}_{f}$ and $\tilde{A}_{f}$ denote the soft flavour matrices in the SCKM basis, and $\tilde{Y}_{f}$ are the diagonal Yukawa matrices. $\mu$ is the (real) higgsino mass parameter, and the VEVs of the two neutral Higgses are defined as

$\displaystyle\upsilon_{u}$

$\displaystyle=$

$\displaystyle\frac{\upsilon}{\sqrt{1+t_{\beta}^{2}}}\,t_{\beta}\ ,~~~~~~\upsilon_{d}=\frac{\upsilon}{\sqrt{1+t_{\beta}^{2}}}\ ,$

(2.30)

where $t_{\beta}\equiv\tan\beta=\frac{\upsilon_{u}}{\upsilon_{d}}$ and $\upsilon=\sqrt{\upsilon_{u}^{2}+\upsilon_{d}^{2}}=174$ GeV. The indices $L$ and $R$ refer to the chirality of the corresponding SM fermions and $m^{2}_{\tilde{f}_{RL}}\equiv(m^{2}_{\tilde{f}_{LR}})^{\dagger}$. With these definitions, the amount of flavour violation can be measured in terms of the dimensionless mass insertion parameters [11]

$\displaystyle(\delta^{f}_{LL})_{ij}=\frac{(m^{2}_{\tilde{f}_{LL}})_{ij}}{\langle m_{\tilde{f}}\rangle^{2}_{LL}}\ ,\qquad(\delta^{f}_{RR})_{ij}=\frac{(m^{2}_{\tilde{f}_{RR}})_{ij}}{\langle m_{\tilde{f}}\rangle^{2}_{RR}}\ ,\qquad(\delta^{f}_{LR})_{ij}=\frac{(m^{2}_{\tilde{f}_{LR}})_{ij}}{\langle m_{\tilde{f}}\rangle^{2}_{LR}}\ ,$

(2.31)

where the average masses in the denominators are defined by

$$\langle m_{\tilde{f}}\rangle^{2}_{AB}=\sqrt{(m^{2}_{\tilde{f}_{AA}})_{ii}(m^{2}_{\tilde{f}_{BB}})_{jj}}\ .$$

(2.32)

We mention in passing that the phase structure of the mass insertion parameters depends on the choice of the phase conventions of the CKM and PMNS matrices. In [1], we have worked out the expressions in Eq. (2.31) explicitly for our model at the GUT scale, choosing a phase convention in which $V_{\text{CKM}}$ and $U_{\text{PMNS}}$ take their standard form.

The effects of RG running down to the low energy scales where experiments are performed were also estimated, using the leading logarithmic approximation. Introducing the parameters

$\displaystyle\eta$

$\displaystyle=$

$\displaystyle\frac{1}{16\pi^{2}}\ln\left(\frac{M_{\text{GUT}}}{M_{\text{low}}}\right),~~~\eta_{N}=\frac{1}{16\pi^{2}}\ln\left(\frac{M_{\text{GUT}}}{M_{\text{R}}}\right),$

(2.33)

we performed a two-stage running ($i$) from $M_{\text{GUT}}$ to $M_{R}$, where the right-handed neutrinos are integrated out, and ($ii$) from $M_{R}$ to $M_{\text{SUSY}}\sim M_{\text{W}}\equiv M_{\text{low}}$. For $M_{\text{GUT}}\approx 2\times 10^{16}$ GeV, $M_{\text{R}}\approx 10^{14}$ GeV and $M_{\text{low}}\approx 10^{3}$ GeV, $\eta\approx 0.19$ is of the order of our expansion parameter $\lambda\approx 0.22$ and $\eta_{N}\approx 0.03$. In terms of their $\lambda$-suppression, the resulting flavour structures of the low energy mass insertion parameters $\delta$ read

	$\displaystyle\delta^{u}_{LL}\sim\left(\begin{array}[]{ccc}1&\lambda^{4}&\lambda^{6}\\ \cdot&1&\lambda^{5}\\ \cdot&\cdot&1\end{array}\right),~~~\delta^{u}_{RR}\sim\left(\begin{array}[]{ccc}1&\lambda^{4}&\lambda^{6}\\ \cdot&1&\lambda^{5}\\ \cdot&\cdot&1\end{array}\right),~~~\delta^{u}_{LR}\sim\left(\begin{array}[]{ccc}\lambda^{8}&0&\lambda^{7}\\ 0&\lambda^{4}&\lambda^{6}\\ 0&\lambda^{7}&1\end{array}\right),~~~~$		(2.43)
	$\displaystyle\delta^{d}_{LL}\sim\left(\begin{array}[]{ccc}1&\lambda^{3}&\lambda^{4}\\ \cdot&1&\lambda^{2}\\ \cdot&\cdot&1\end{array}\right),~~~\delta^{d}_{RR}\sim\left(\begin{array}[]{ccc}1&\lambda^{4}&\lambda^{4}\\ \cdot&1&\lambda^{4}\\ \cdot&\cdot&1\end{array}\right),~~~\delta^{d}_{LR}\sim\left(\begin{array}[]{ccc}\lambda^{6}&\lambda^{5}&\lambda^{5}\\ \lambda^{5}&\lambda^{4}&\lambda^{4}\\ \lambda^{6}&\lambda^{6}&\lambda^{2}\end{array}\right),~~~~$		(2.53)
	$\displaystyle\delta^{e}_{LL}\sim\left(\begin{array}[]{ccc}1&\lambda^{4}&\lambda^{4}\\ \cdot&1&\lambda^{4}\\ \cdot&\cdot&1\end{array}\right),~~~\delta^{e}_{RR}\sim\left(\begin{array}[]{ccc}1&\lambda^{3}&\lambda^{4}\\ \cdot&1&\lambda^{2}\\ \cdot&\cdot&1\end{array}\right),~~~\delta^{e}_{LR}\sim\left(\begin{array}[]{ccc}\lambda^{6}&\lambda^{5}&\lambda^{6}\\ \lambda^{5}&\lambda^{4}&\lambda^{6}\\ \lambda^{5}&\lambda^{4}&\lambda^{2}\end{array}\right).~~~~$		(2.63)

Appendix A provides the explicit expressions for each entry in terms of the parameters of the model.

Numerical results for the running quark and charged lepton masses as well as for the quark mixing angles at the GUT scale can be found in [25]. The matching conditions from the SM to the Minimal Supersymmetric Standard Model (MSSM), imposed at the SUSY scale, take the form

	$\displaystyle y_{u,c,t}^{\text{SM}}$	$\displaystyle\approx$	$\displaystyle\,y_{u,c,t}^{\text{MSSM}}\sin\bar{\beta},$
	$\displaystyle y_{d,s}^{\text{SM}}$	$\displaystyle\approx$	$\displaystyle(1+\bar{\eta}_{q})\,y_{d,s}^{\text{MSSM}}\cos\bar{\beta},$
	$\displaystyle y_{b}^{\text{SM}}$	$\displaystyle\approx$	$\displaystyle(1+\bar{\eta}_{b})\,y_{b}^{\text{MSSM}}\cos\bar{\beta},$
	$\displaystyle y_{e,\mu}^{\text{SM}}$	$\displaystyle\approx$	$\displaystyle(1+\bar{\eta}_{l})\,y_{e,\mu}^{\text{MSSM}}\cos\bar{\beta},$
	$\displaystyle y_{\tau}^{\text{SM}}$	$\displaystyle\approx$	$\displaystyle y_{\tau}^{\text{MSSM}}\cos\bar{\beta},$		(3.1)

for the singular values of the Yukawa matrices. Similarly, we have for the CKM mixing

$\displaystyle\theta^{q,\text{SM}}_{i3}\approx\frac{1+\bar{\eta}_{q}}{1+\bar{\eta}_{b}}\theta^{q,\text{MSSM}}_{i3},\qquad\theta^{q,\text{SM}}_{12}\approx\theta^{q,\text{MSSM}}_{12},\qquad\delta^{q,\text{SM}}\approx\delta^{q,\text{MSSM}}.$

(3.2)

Here

$\displaystyle\bar{\eta}_{q}=\eta_{q}-\eta_{l}^{\prime},\qquad\bar{\eta}_{b}=\eta_{q}^{\prime}+\eta_{A}-\eta_{l}^{\prime},\qquad\bar{\eta}_{l}=\eta_{l}-\eta_{l}^{\prime}\,,$

(3.3)

represent SUSY radiative threshold corrections that are parametrised by $\eta_{i}=\epsilon_{i}\,\tan\beta$, with explicit expressions for $\epsilon_{i}$ available in [26]. The unprimed $\eta$ parameters correspond to corrections to the first two generations, the primed ones to the third generation, and the one with index “$A$” to a correction due to the soft SUSY breaking trilinear terms. The parameter $\bar{\beta}$ follows from the absorption of $\eta_{l}^{\prime}$ into $\beta$,

$\displaystyle\cos\bar{\beta}\equiv(1+\eta_{l}^{\prime})\cos\beta,~~~~~\sin\bar{\beta}\approx\sin\beta,$

(3.4)

with the approximation being valid for $\tan\beta\gtrsim 5$. In the limit where threshold effects for the charged leptons are neglected, $\tan\bar{\beta}$ simply reduces to $\tan\beta$.

Our model predicts $\hat{y}_{{b,\tau}}=y_{b}\,\lambda^{2}$, where the hat indicates the diagonalised Yukawa sector at the GUT scale. As a consequence, very large values of $\tan{\beta}$ are excluded, and we only study the parameter space in which $\tan{\beta}\in[5,25]$, keeping the value of $y_{b}$ below four. In order to obtain viable ranges for our Yukawa input parameters, we plot $y_{u,c,t,b}$, $(\tilde{x}_{2}/y_{s})^{2}$ and $(1+\bar{\eta}_{l})y_{s}$ against $\tan\bar{\beta}$ using the results for the diagonalised Yukawa sector at the GUT scale provided in [25]. We remark that $y_{b}$, $y_{s}$ and $\tilde{x}_{2}$ are extracted from the lepton sector. We fit the resulting curves using the relative uncertainties $\sigma(y_{u})/y_{u}=31\%$, $\sigma(y_{c})/y_{c}=3.5\%$, $\sigma(y_{t})/y_{t}=10\%$, $\sigma(y_{b})/y_{b}=0.6\%$, see [25]. Concerning $y_{s}$ and $\tilde{x}_{2}$, we take $\sigma(y_{s})/y_{s}=10\%$ and $\sigma(\tilde{x}_{2})/\tilde{x}_{2}=10\%$, allowing for higher order corrections to the mass ratios that would reduce the discrepancy between the values of $\tilde{x}_{2}/y_{s}$ predicted from the lepton and the quark sectors and maximise the GUT scale value of $(\hat{y}_{\mu}\,\hat{y}_{d})/(\hat{y}_{s}\,\hat{y}_{e})$. Due to the implementation of the Georgi-Jarlskog relation [27], it is equal to 9 in our model at LO, while its preferred range is $10.7^{+1.8}_{-0.8}$ [25], which is independent of threshold corrections and also not sensitive to a change of the SUSY scale.

We estimate the low energy Yukawa couplings using the leading logarithmic approximation as described in [1]. Clearly, the resulting low energy Yukawa matrices are only valid up to that approximation. Mindful of such limitations, we obtain

	$\displaystyle\tilde{Y}^{u}_{{\text{low}}}$	$\displaystyle\approx$	$\displaystyle\text{Diag}\,\Big{[}(1+R^{y}_{u})\,y_{u}\,\lambda^{8},(1+R^{y}_{u})\,y_{c}\,\lambda^{4},(1+R^{y}_{t})\,y_{t}\Big{]},$		(3.5)
	$\displaystyle\tilde{Y}^{d}_{{\text{low}}}$	$\displaystyle\approx$	$\displaystyle\text{Diag}\,\Big{[}(1+R^{y}_{d})\frac{\tilde{x}_{2}^{2}}{y_{s}}\lambda^{6},(1+R^{y}_{d})\,y_{s}\,\lambda^{4},(1+R^{y}_{b})\,y_{b}\,\lambda^{2}\Big{]},$		(3.6)
	$\displaystyle\tilde{Y}^{e}_{{\text{low}}}$	$\displaystyle\approx$	$\displaystyle\text{Diag}\,\Big{[}(1+R^{y}_{e})\frac{\tilde{x}_{2}^{2}}{3\,y_{s}}\lambda^{6},(1+R^{y}_{e})3y_{s}\,\lambda^{4},(1+R^{y}_{e})\,y_{b}\,\lambda^{2}\Big{]},$		(3.7)

where the corrections from the RG running are encoded in the parameters $R^{y}_{f}$

	$\displaystyle R^{y}_{u}$	$\displaystyle=$	$\displaystyle\eta\left(\frac{46}{5}g_{U}^{2}-3y_{t}^{2}\right)-3\eta_{N}\,y_{D}^{2},\qquad R^{y}_{t}=R^{y}_{u}-3\,\eta\,y_{t}^{2},$		(3.8)
	$\displaystyle R^{y}_{d}$	$\displaystyle=$	$\displaystyle\eta\frac{44}{5}g_{U}^{2},\qquad R^{y}_{b}=R^{y}_{d}-\eta\,y_{t}^{2},\qquad R^{y}_{e}=\eta\frac{24}{5}g_{U}^{2}-\eta_{N}\,y_{D}^{2}.$		(3.9)

Here, $g_{U}\approx\sqrt{0.52}$ denotes the universal gauge coupling constant at the GUT scale. Our scan produces the following values for the right-hand sides of Eq. (3.1)

	$\displaystyle\tilde{Y}^{u}_{\text{low}_{11}}\sin\bar{\beta}\in[3.4,6.9]\times 10^{-6}\!,~\tilde{Y}^{u}_{\text{low}_{22}}\sin\bar{\beta}\in[2.34,2.65]\times 10^{-3}\!,~\tilde{Y}^{u}_{\text{low}_{33}}\sin\bar{\beta}\in[0.77,0.89],$
	$\displaystyle\tilde{Y}^{d}_{\text{low}_{11}}\cos\bar{\beta}(1+\bar{\eta}_{q})\in[0.9,1.6]\times 10^{-5},~~~~~\tilde{Y}^{d}_{\text{low}_{22}}\cos\bar{\beta}(1+\bar{\eta}_{q})\in[2.2,3.5]\times 10^{-4},$
	$\displaystyle\tilde{Y}^{d}_{\text{low}_{33}}\cos\bar{\beta}(1+\bar{\eta}_{b})\in[1.17,1.6]\times 10^{-2},$
	$\displaystyle\tilde{Y}^{e}_{\text{low}_{11}}\cos\bar{\beta}(1+\bar{\eta}_{l})\in[2.4,3.8]\times 10^{-6},~~~~~\tilde{Y}^{e}_{\text{low}_{22}}\cos\bar{\beta}(1+\bar{\eta}_{l})\in[5.6,7.7]\times 10^{-4},$
	$\displaystyle\tilde{Y}^{e}_{\text{low}_{33}}\cos\bar{\beta}\in[1.06,1.14]\times 10^{-2},$		(3.10)

which have to be compared to the SM values, taken from Table 2 of [25],

	$\displaystyle y_{u}^{\text{SM}}$	$\displaystyle\in$	$\displaystyle[3.40,7.60]\times 10^{-6},~~~y_{c}^{\text{SM}}\in[2.69,3.20]\times 10^{-3},~~~y_{t}^{\text{SM}}\in[0.78,0.88],$		(3.11)
	$\displaystyle y_{d}^{\text{SM}}$	$\displaystyle\in$	$\displaystyle[1.15,1.56]\times 10^{-5},~~~y_{s}^{\text{SM}}\in[2.29,2.84]\times 10^{-4},~~~y_{b}^{\text{SM}}\in[1.21,1.42]\times 10^{-2},$
	$\displaystyle y_{e}^{\text{SM}}$	$\displaystyle\in$	$\displaystyle[2.85,2.88]\times 10^{-6},~~~y_{\mu}^{\text{SM}}\in[6.01,6.08]\times 10^{-4},~~~y_{\tau}^{\text{SM}}\in[1.02,1.03]\times 10^{-2}.~~~~~~$

The corresponding ranges of the order one input parameters of the Yukawa sector are listed in the first five rows of the first column of Table 1. All other coefficients that are not fixed by this fit, are scanned over the interval $\pm[0.5,2]$, with the following exceptions: we allow the absolute value of the Dirac neutrino Yukawa coupling $y_{D}$ to be as small as $0.2$ but not larger than $0.6$, such that it does not exceed the maximum allowed value of $y_{t}$. We also relax the lower bounds on $|\tilde{x}^{a}_{2}|$, $|a_{s}|$ and $|a_{u}|$ and extend the upper bound on $|a_{b}|$, such that they are allowed to get the same values as the corresponding Yukawa coefficients. The coefficients $c_{H_{u}}$ and $c_{H_{d}}$ of the soft Higgs mass squares,

$\displaystyle m^{2}_{{H_{u}}_{\text{GUT}}}$

$\displaystyle=$

$\displaystyle c_{H_{u}}\,m_{0}^{2},\qquad m^{2}_{{H_{d}}_{\text{GUT}}}=c_{H_{d}}\,m_{0}^{2}\ ,~~$

(3.12)

are taken to be positive, just like the coefficients $b_{01}$, $b_{02}$ and $B_{0}^{(N)}$ of the leading order diagonal elements of the soft scalar mass squared matrices. Phases are generally allowed to take arbitrary values within $[0,2\pi]$. As mentioned earlier, $\tan\beta$ is varied between 5 and 25. Concerning the CMSSM parameters, we define

$\displaystyle\alpha_{0}$

$\displaystyle\equiv$

$\displaystyle A_{0}/m_{0},\qquad x\equiv(M_{1/2}/m_{0})^{2},$

(3.13)

and scan over $M_{1/2}\in[0.3,5]$ TeV, $m_{0}\in[0.05,5]$ TeV as well as $\alpha_{0}\in[-3,3]$ in order to avoid charge and colour breaking minima.⁶⁶6In our numerical scan, we have checked that the potentials are always bounded from below and that the corresponding minima do not break charge or colour [28].

The $\mu$ parameter, which we take as real, is given at the electroweak scale by the relation⁷⁷7The lack of any evidence for low energy supersymmetry requires a certain amount of cancellation between the terms of Eq. (3.14), see e.g. [29].

$\displaystyle\frac{M_{Z}^{2}}{2}$

$\displaystyle=$

$\displaystyle\frac{m_{H_{d}}^{2}+\Sigma^{d}_{d}-(m_{H_{u}}^{2}+\Sigma^{u}_{u})t_{\beta}^{2}}{t_{\beta}^{2}-1}-\mu^{2},$

(3.14)

where $M_{Z}$ denotes the $Z$ boson mass [30]. $\Sigma^{u}_{u}$ and $\Sigma^{d}_{d}$ are radiative corrections, with the most important contributions coming from the stops

	$\displaystyle\Sigma^{u}_{u}\left(\tilde{t}_{1,2}\right)$	$\displaystyle=$	$\displaystyle\frac{3}{16\pi^{2}}F(m^{2}_{\tilde{t}_{1,2}})\left(Y_{t}^{2}-g_{Z}^{2}\mp\frac{A_{t}^{2}-8g_{Z}^{2}\left(\frac{1}{4}-\frac{2}{3}x_{W}\right)\Delta_{t}}{m^{2}_{\tilde{t}_{2}}-m^{2}_{\tilde{t}_{1}}}\right),$		(3.15)
	$\displaystyle\Sigma^{d}_{d}\left(\tilde{t}_{1,2}\right)$	$\displaystyle=$	$\displaystyle\frac{3}{16\pi^{2}}F(m^{2}_{\tilde{t}_{1,2}})\left(g_{Z}^{2}\mp\frac{Y_{t}^{2}\mu^{2}+8g_{Z}^{2}\left(\frac{1}{4}-\frac{2}{3}x_{W}\right)\Delta_{t}}{m^{2}_{\tilde{t}_{2}}-m^{2}_{\tilde{t}_{1}}}\right).$		(3.16)

In these expressions, $Y_{t}$, $A_{t}$ and $\mu$ denote the low energy Yukawa and trilinear couplings and the low energy $\mu$ parameter, respectively. Moreover

	$\displaystyle m^{2}_{\tilde{t}_{1,2}}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(m^{2}_{\tilde{t}_{LL}}+m^{2}_{\tilde{t}_{RR}}\mp\sqrt{4\,m^{2}_{\tilde{t}_{LR}}+(m^{2}_{\tilde{t}_{LL}}-m^{2}_{\tilde{t}_{RR}})^{2}}\right),$
	$\displaystyle F(m^{2})$	$\displaystyle=$	$\displaystyle m^{2}\left(\log\left(\frac{m^{2}}{M_{S}^{2}}\right)-1\right)\!,~~~~~~\Delta_{t}=\frac{1}{2}\left(m^{2}_{\tilde{t}_{LL}}-m^{2}_{\tilde{t}_{RR}}\right)+M_{Z}^{2}\cos(2\beta)\left(\frac{1}{4}-\frac{2}{3}x_{W}\right)\!,$
	$\displaystyle x_{W}$	$\displaystyle=$	$\displaystyle\sin^{2}\theta_{W},\qquad g_{Z}^{2}=\frac{M_{Z}^{2}}{4\upsilon^{2}},\qquad M_{S}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}},$		(3.17)

with $\theta_{W}$ denoting the Weinberg angle. $m^{2}_{\tilde{t}_{LL}}$, $m^{2}_{\tilde{t}_{RR}}$ and $m^{2}_{\tilde{t}_{LR}}$ are the low energy (33) elements of the squark mass matrices defined in Eq. (2.29). The so-determined $\mu$ parameter can then be used to calculate the physical Higgs mass. Adopting the approximate formulas of Section 2.4 of [31], we demand that the resulting Higgs mass lies within the interval $[110,135]$ GeV. Additionally, we impose cuts on the SUSY parameters from direct searches by requiring that the first and the second generation squark masses are larger than $1.4$ TeV.

In this section, we analyse the predictions for the low energy mass insertion parameters $\delta$ whose explicit expressions are given in Appendix A. Tables 2-6 provide naive expectations for the individual $\delta$s, where we take into account the $\lambda$-suppression and the main effects of the RG running, while setting any order one coefficients to one. Clearly, we still expect to see a spread within a few orders of magnitude due to the variation of the SUSY scale and the order one coefficients. The third columns of Tables 2-6 list existing experimental bounds. The full ranges of our $\delta$s arising from scanning over the input parameters, given in Table 1, are depicted in Figures 1-3.