SUSY effects in $R_b$: revisited under current experimental constraints

In this note we revisit the SUSY effects in $R_b$ under current experimental constraints including the LHC Higgs data, the $B$-physics measurements, the dark matter relic density and direct detection limits, as well as the precision electroweak data. We first perform a scan to figure out the currently allowed parameter space and then display the SUSY effects in $R_b$. We find that although the SUSY parameter space has been severely restrained by current experimental data, both the general MSSM and the natural-SUSY scenario can still alter $R_b$ with a magnitude sizable enough to be observed at future $Z$-factories (ILC, CEPC, FCC-ee, Super $Z$-factory) which produce $10^9-10^{12}$ $Z$-bosons. To be specific, assuming a precise measurement $\delta R_b = 2.0 \times 10^{-5}$ at FCC-ee, we can probe a right-handed stop up to 530 GeV through chargino-stop loops, probe a sbottom to 850 GeV through neutralino-sbottom loops and a charged Higgs to 770 GeV through the Higgs-top quark loops for a large $\text{tan}\beta$. The full one-loop SUSY correction to $R_b$ can reach $1 \times 10^{-4}$ in natural SUSY and $2 \times 10^{-4}$ in the general MSSM.


I. INTRODUCTION
After the discovery of the 125 GeV Higgs boson [1,2], the primary task of the LHC is to hunt for new physics beyond the Standard Model (SM). Among various extensions of the SM, the low energy supersymmetry (SUSY) is the most appealing candidate 1 since it can solve the gauge hierarchy problem, naturally explain the cosmic cold dark matter and achieve the gauge coupling unification. The search for SUSY has long been performed both directly and indirectly. On the one hand, the colliders have directly searched for the sparticle productions. On the other hand, SUSY effects have been probed indirectly through precision measurements of some low energy observables.
R b ≡ Γ(Z →bb)/Γ(Z → hadrons) is a famous observable which is sensitive to new physics beyond the SM [4]. So far the most precise experimental value R exp b = 0.21629 ± 0.00066 comes from the LEP and SLC measurements [5], while the SM prediction is R SM b = 0.21579 [6]. The future Z-factories are expected to produce much more Z-bosons than the LEP experiment. For example, 10 9 , 10 10 and 10 12 Z-bosons are expected to be produced respectively at the International Linear Colldier (ILC) [7], the Circular Electron-Positron Collider (CEPC) [8], the Future Circular Collider (FCC-ee) [9] and the Super Z-factory [10]. This will allow for a more precise measurement of R b [11] and help pin down the involved new physics effects.
The SUSY effects in R b were calculated and discussed many years ago [12][13][14][15]. In this work we revisit these effects for two reasons: (i) The current experiments, especially the LHC experiments, have severely restrained the SUSY parameter space. It is intriguing to figure out the possible magnitude of the SUSY effects in the currently allowed parameter space; (ii) Given the possibility of some future Z-factories like ILC, CEPC or FCC-ee, a more precise measurement of R b will help reveal the SUSY effects although these effects may have already been restrained to be rather small by current experiments. In order to know if the SUSY effects are accessible in a future measurement of R b , we must figure out their currently allowed value.
This work is organized as follows. In Sec.II, we give a description of SUSY effects in R b . In Sec.III, we scan over the SUSY parameter space and display the SUSY effects in the allowed parameter space. Finally we give our conclusion in Sec. IV.

II. SUSY CORRECTIONS TO R b
Since the SUSY effects in R b have been calculated in the literature [12,13], here we only give a brief description. The dominant SUSY effects in R b are from the vertex corrections where v b = 1/2−2sinθ 2 w /3 and a b = 1/2 are respectively the vector and axial vector couplings of tree-level Zbb interaction,  [13,16,17] Here δg b λ (λ = L, R) is give by where Γ f λ (m 2 Z ) denotes the vertex loop contributions and Σ bλ (m 2 b ) is the counter term from the bottom quark self-energy. We perform straightforward loop calculations and confirm the expressions in [13]. The results can be expressed as where C g = 4/3 for the gluino loops and C g = 1 for other loops, and B 0 , B 1 and C 12 , C 23 , C 24 are Passarino-Veltman functions [18]. The notation (φ, ψ) represents (b,g) for gluino loops, (t,χ − ) for chargino loops, (b,χ 0 ) for neutralino loops, (H − , t) for charged Higgs loops and (h/a/G 0 , b) for neutral Higgs loops. In addition to R b , we also show the SUSY effects in the forward-backward asymmetry A b F B in the decay Z →bb: Its experimental value is 0.0992±0.0016 from the LEP experiment [5] while its SM prediction is 0.1032 ± 0.0004 [19]. In the future Z-factories, this forward-backward asymmetry will be measured together with R b , both of which will jointly allow for a revelation of SUSY effects.

A. SUSY parameter space
To clarify our numerical calculations we consider the general MSSM and the natural-SUSY scenario [20]. From the natural-SUSY results (the natural-SUSY parameter space is much smaller than the general MSSM), we can acquire the more detailed characters of each kind of loops, while from the general MSSM results we can obtain the more general size of SUSY loop effects.
For the natural-SUSY scenario, since in this scenario only the higgsino masses and the third-generation squark masses are assumed to be light, while other sparticles are assumed to be rather heavy and thus their effects in low energy observables are decoupled, in our scan we fix the soft-breaking mass parameters in the first two generation squark sector and the slepton sector at 5 TeV, and assume A t = A b . For the electroweak gaugino masses, inspired by the grand unification relation, we take M 1 : M 2 = 1 : 2 and fix M 2 at 2 TeV. The gluino mass is fixed at 2 TeV since it is supposed to be not too far above TeV scale in natural-SUSY. Other parameters vary as follows For the general MSSM, assuming A t = A b and M 1 : M 2 : M 3 = 1 : 2 : 6, we scan over the following parameter space 1 < tan β < 60, 100 GeV < µ < 1000 GeV, |A t | < 3 TeV, In our scan we consider the following experimental constraints: (1) The constraints on the Higgs sector from the LEP, Tevatron and LHC experiments. We use the package HiggsBounds-4.0.0 [21] to implement these constraints.
(3) The measurements of the precision electroweak observables. The SUSY predictions of ρ l , sin 2 θ l ef f and m W are required to be within the 2σ ranges of the experimental values [5].
(4) The dark matter constraints. We require the thermal relic density of the neutralino dark matter to be below the 2σ upper limit of the Planck value [24] and require the  dark matter-nucleon spin-independent scattering scross section σ SI r to satisfy the 95% C.L. limits of LUX [25]. We also consider the limits of spin-dependent dark matternucleon cross section σ SD r from the XENON100 experiment [26]. The relic density, σ SI r and σ SD r are calculated with the code MicrOmega v2.4 [27].
About the mass bounds from the LHC direct searches, in natural SUSY the higgsinos have very weak bounds because their pair productions only give missing energy and are rather difficult to detect (a mono-jet or mono-Z is needed in detection) [28], while for the stops the right-handed one is weakly bounded (its mass can be as light as 210 GeV for higgsinos heavier than 190 GeV) [29]. When we display the numerical results, we will not show a sharp LHC bound on stop or higgsino mass (we only consider the LEP bounds on stop and higgsinos). For each surviving sample we calculate the correction to R b and display the numerical results in the proceeding section. About the future precision of R b measurement, the CEPC would produce 10 10 Z-bosons and probably measure R b with an uncertainty of 1.7 × 10 −4 [8,11], while the FCC-ee could produce 10 12 Z-bosons and give a much better R b measurement at 10 −5 level [9]. In our figures, for illustration, we mark an uncertainty of 2 × 10 −5 [9,11]. The SUSY parameter space giving δR SUSY is suppressed (the lightest charginoχ ± 1 is dominated by higgsino component since the higgsino mass µ is much smaller than the gaugino masses M 1 and M 2 in natural SUSY). Only for a very large tan β can the coupling Y b be comparable to the corresponding right-handed stop coupling Y t = gm t /( √ 2m W sin β). Our numerical results shows that tan β is smaller than 35 (so that Y b /Y t < 1) fort 1 below 530 GeV (when tan β is larger,t 1 must be heavier to satisfy the experimental constraints). Note that, as commented in the preceding section, so far the right-handed stop mass in natural SUSY is weakly bounded by LHC experiments (its mass can be as light as 210 GeV for higgsinos heavier than 190 GeV) [29].
(b) As shown in Fig.8 (c) From Figs.11, 12 and 13 we see that the SUSY effects in R b and A b F B are correlated, as expected. Both observables can jointly probe the SUSY effects. While the chargino loop effects always enhance both quantities, the combined total effects of all loops can either enhance or reduce them. We also find that in the general MSSM without special naturalness requirement, both R b and A b F B are allowed to vary in a larger region than in natural SUSY, especially when tan β is small.
(d) From Figs.6-13 we see that in some currently allowed parameter space, the effects of natural SUSY may be accessible in the future R b measurement. If it can be measured with an uncertainty of 2 × 10 −5 , a large part of SUSY parameter space can be covered.
(e) We found that for natural SUSY the most stringent limits are from B-physics, while for the general MSSM the most stringent limits are from the dark matter-nucleon spinindependent scattering limits. The results are shown in Fig.14. Other constraints, such as the dark matter-nucleon spin-dependent scattering cross section, are also making impacts but not as stringent as these two.

IV. CONCLUSION
We revisited the SUSY effects in R b under current experimental constraints including the LHC Higgs data, the B-physics measurements, the dark matter relic density and direct detection limits, as well as the precision electroweak data. We scanned over the SUSY parameter space and in the allowed parameter space we displayed the SUSY effects in R b . We found that although the SUSY parameter space has been severely restrained by current experimental data, SUSY can still alter R b with a magnitude sizable enough to be observed at future Z-factories (ILC, CEPC, FCC-ee). Assuming a precise measurement δR b = 2.0×10 −5 at FCC-ee, we can probe the right-handed stop to 530 GeV through the chargino-stop loops, probe the sbottom to 850 GeV through the neutralino-sbottom loops and the charged Higgs to 770 GeV through the Higgs-top quark loops for a large tanβ. The full one-loop SUSY correction to R b can reach 1 × 10 −4 in natural SUSY and 2 × 10 −4 in the general MSSM.