Oblique Parameters

The data are precise enough to constrain additional parameters describing physics beyond the SM. Of particular interest is the $\rho$-parameter, defined by $$\rho_0 = {M_W^2 \over M_Z^2 \hat{c}^2_Z \hat{\rho} (m_t,M_H)},$$which is a measure of the neutral to charged current interaction strength. The SM contributions are absorbed in $\hat{\rho}$, so that in the SM $\rho_0 = 1$, by definition. Examples for sources of $\hat{\rho} \neq 1$ include non-degenerate extra fermion or boson doublets, and non-standard Higgs representations.

In a fit to all data with $\rho_0$ as an extra fit parameter, the correlation of M_H with m_t is lifted, and replaced by a strong (73%) correlation with $\rho_0$. As a result upper limits on M_H are weak when $\rho_0$ is allowed. Indeed, $\chi^2 (M_H)$ is very shallow with $$\Delta \chi^2 = \chi^2 (1 \mbox{ TeV}) - \chi^2 (M_Z) = 4.5,$$and its minimum is at M_H = 46  GeV, which is already excluded. We obtain,  $$\begin{array} {lcr} \rho_0 &=& 0.9996^{+0.0009}_{-0.0006}, \vs... ...pm 4.8 \mbox{ GeV}, \\ \alpha_s &=& 0.1212 \pm 0.0031,\end{array}$$in excellent agreement with the SM. The central values are for M_H = M_Z , and the uncertainties are $1 \sigma$ errors and include the range, $M_Z \leq M_H \leq 167$ GeV, in which the minimum $\chi^2$ varies within one unit. Note, that the uncertainties for $\ln M_H$ and $\rho_0$ are non-Gaussian: at the $2 \sigma$ level ($\Delta \chi^2 \leq 4$), Higgs masses up to 800 GeV are allowed, and we find $$\rho_0 = 0.9996^{+0.0031}_{-0.0013} \mbox{ ($2 \sigma$)}.$$This implies strong constraints on the mass splittings of extra fermion and boson doublets [76], $$\Delta m^2 = m_1^2 + m_2^2 - \frac{4 m_1^2 m_2^2}{m_1^2 - m_2^2} \ln {m_1\over m_2} \geq (m_1 - m_2)^2,$$namely, at the $1 \sigma$ and $2 \sigma$ levels, respectively,  $$\sum\limits_i {C_i\over 3} \Delta m^2_i < \mbox{ (38 GeV)}^2 \mbox{ and (93 GeV)}^2,$$where C_i is the color factor. Due to the general condition (42) in the MSSM, stronger $2 \sigma$ constraints result here, $$\rho_0 \mbox{ (MSSM) } = 0.9996^{+0.0017}_{-0.0013} \mbox{ ($2 \sigma$)}.$$The constraints (48) would therefore change to $$\sum\limits_i {C_i\over 3} \Delta m^2_i < \mbox{ (38 GeV)}^2 \mbox{ and (64 GeV)}^2 \mbox{ (MSSM)}.$$

Similarly, constraints on heavy degenerate chiral fermions can be obtained through the S parameter [77], defined through a difference of Z boson self-energies, $$\frac{\hat\alpha (M_Z) }{4 \hat{s}_Z^2 \hat{c}_Z^2} S \equiv \frac{\Pi^{\rm new}_{ZZ} (M_Z^2) - \Pi^{\rm new}_{ZZ} (0) }{M_Z^2}.$$The superscripts indicate that S includes new physics contributions only. Likewise, $T = (1 - \rho_0^{-1})/\hat\alpha$ and the U parameter to be discussed below, also vanish in the SM. A fit to all data with S allowed yields,  $$\begin{array} {lcr} S &=& -0.20^{+0.24}_{-0.17}, \vspace{4pt} ... ...pm 4.8 \mbox{ GeV}, \\ \alpha_s &=& 0.1221 \pm 0.0035.\end{array}$$In the presence of S , constraints on M_H virtually disappear. In fact, S and M_H are almost perfectly anticorrelated ($-92\%$). A heavy degenerate ordinary or mirror family contributes $2/3\pi$ to S . By requiring $M_Z \leq M_H \leq 1$ TeV, we find with $3 \sigma$ confidence, $$S = -0.20^{+0.40}_{-0.33} \mbox{ ($3 \sigma$)}.$$A fourth sequential fermion family is excluded at the 99.8% CL.

New physics contributions to the third oblique parameter, U , which is defined through $$\frac{\hat\alpha(M_Z)}{4 \hat{s}_Z^2} \left( S + U \right) \equ... ...frac{\Pi^{\rm new}_{WW} (M_W^2) - \Pi^{\rm new}_{WW} (0) }{M_W^2},$$are usually expected to be small. A fit to all data with U allowed,  $$\begin{array} {lcr} U &=& 0.09 \pm 0.19, \vspace{2pt} \\ M_H ... ...pm 4.9 \mbox{ GeV}, \\ \alpha_s &=& 0.1207 \pm 0.0030,\end{array}$$reveals perfect agreement with the SM prediction U = 0 . Notice, that allowing U has little effect on the extracted M_H , as it has only small correlations with the SM parameters.

A simultaneous fit to S , T , and U can be performed only relative to a specified M_H . If one fixes M_H = 600  GeV, as is appropriate in QCD-like technicolor models, one finds $$\begin{array} {rcr} S &=& -0.27 \pm 0.12, \\ T &=& 0.00 \pm 0.15, \\ U &=& 0.19 \pm 0.21.\end{array}$$Notice, that in such a fit the S parameter is significantly smaller than zero. From this an isodoublet of technifermions, assuming N_TC = 4 technicolors, is excluded by almost 6 standard deviations, and a full technigeneration by more than $15\sigma$. However, the QCD-like models are excluded on other grounds, such as FCNC. In particular, in models of walking technicolor S can be smaller or even negative [78].

The allowed range of the oblique parameters in the context of SUSY is obtained by demanding $M_Z \leq M_H \leq 150$ GeV, which yields, $$\begin{array} {rcr} S &=& -0.17^{+0.17}_{-0.12}, \vspace{4pt} ... ...^{+0.15}_{-0.18}, \vspace{4pt} \\ U &=& 0.19 \pm 0.21.\end{array}$$Note the $2 \sigma$ upper limit $T \leq 0.14$. Allowing supersymmetric contributions to R_b , which can be mediated by light top squark and chargino loops, this limit would tighten further to  $$T \leq 0.12 \mbox{ ($2 \sigma$)}.$$These results are to be compared with the predictions of various scenarios for the mediation of SUSY breaking from the hidden to the observable sector. For example, in the minimal supergravity model with universal soft SUSY breaking terms, there are regions of parameter space in which T can be as large as 0.20, so they have to be excluded. Of course, there are in general also (smaller) contributions to S and U , as well as non-oblique corrections, so much more parameter space can be excluded than what is suggested by the constraint (58). A systematic analysis of precision data in the MSSM, and a discussion of the excluded parameter space can be found in Ref. [79].