Closing the light sbottom mass window from a compilation of e+e- ->hadron data

The e+e- ->hadron cross section data from PEP, PETRA, TRISTAN, SLC and LEP, at centre-of-mass energies between 20 to 209 GeV, are analysed to search for the production of a pair of light sbottoms decaying hadronically via R-parity-violating couplings. This analysis allows the 95%C.L. exclusion of such a particle if its mass is below 7.5 GeV/c2. The light sbottom mass window is closed.


Introduction
At the end of the last millenium, the Tevatron Collaborations [1,2] came out with a bottom quark production cross section at √ s = 1.8 TeV in excess of the theoretical prediction by about a factor of two. Refined parton density functions and other theoretical improvements, e.g., in the b-quark fragmentation function, have recently been shown to account for the difference in the data recorded at √ s = 1.96 TeV [3].
A more exotic model [4], in which a pair of gluinos with mass 12 to 16 GeV/c 2 is produced in pp collisions, with subsequent decays into a bottom quark and a light sbottom, with mass below 6 GeV/c 2 , has been shown to also fit the excess well. In this model, the sbottom must either be long-lived or decay via R-parity-violating coupling to light quarks, e.g.,b →ūs, to comply with various experimental constraints. Long-lived sbottoms have recently been excluded up to masses of 92 GeV/c 2 by ALEPH [5] in direct searches for e + e − → qqqq and e + e − →qq, but R-parity-violating prompt hadronic decays have not been addressed by the ALEPH analysis.
A light, hadronically decaying sbottom would increase the e + e − → hadron cross section above thebb production threshold by up to a quarter of the e + e − → bb cross section, i.e., about 2% far from the Z peak and 5% at the Z peak. For this reason, the measurements of the hadronic cross section at centre-of-mass energies from 20 to 209 GeV (i.e., well above the known bb resonances) from PEP [6,7], PETRA [8]- [13], TRISTAN [14]- [20] and LEP/SLC [21], are re-analysed in this letter to search for a possible consistent excess.
This letter is organized as follows. A compilation of the data is presented in a synthetic manner in Section 2 to allow easy re-interpretation in the future. The global fit of the data is described in Section 3. The results of the analysis are given in Section 4 and the conclusions are listed in Section 5.
where s is the e + e − centre-of-mass energy squared and α QED is the fine structure constant. The latest TOPAZ [16,17] and VENUS [20] publications report directly the value of σ 0 had instead. In both cases, the latter includes a correction that unfolds the effects of initial state radiation (ISR), while still reflecting the running of the fine structure constant with the centre-of-mass energy [22].
The R and σ 0 had data are listed in Table 1 (PEP, PETRA) and in Table 2 (TRISTAN), as obtained from a comparison of two recent compilations [23,24] and the original pub- Table 1: The ratio R and the effective Born hadronic cross section, σ 0 had , from the PEP and PETRA experiments, with increasing centre-of-mass energy ( √ s). The expected statistical (σ stat ), point-to-point systematic (σ ptp ) and normalization systematic (∆ norm ) uncertainties are also given (in %). The latter is correlated between all energy points in a given publication. An additional normalization error ∆ QED = 0.1%, fully correlated between all measurements, is to be added to account for missing QED higher orders. The last column points to the original publication.  had , from the TRISTAN experiments, with increasing centre-of-mass energy ( √ s). The expected statistical (σ stat ), point-to-point systematic (σ ptp ) and normalization systematic (∆ norm ) uncertainties are also given (in %). The latter is correlated between all energy points in a given publication. An additional normalization error ∆ QED = 0.1%, fully correlated between all measurements, is to be added to account for missing QED higher orders. The last column points to the original publication. lications [6]- [20]. In these tables, only the final -and most accurate -result for each experiment and each centre-of-mass energy is reported. (Superseded data are reported in both Refs. [23] and [24], but are not always clearly flagged as such therein.) Other refinements were considered in this letter for a rigorous statistical treatment of the data, and are described in the following. First, in each experiment, the systematic uncertainty was divided into a point-to-point contribution, σ ptp , and an overall normalization error, ∆ norm , as is done in most of the orginal publications. The point-to-point systematic uncertainties are uncorrelated (related to, e.g., the limited simulated statistics, or the statistical uncertainty on the measured luminosity), are assumed to have a Gaussian probability density function and are taken directly from the original publications.
In contrast, the overall normalization error definition varies among the publications, being either the largest possible variation interval (e.g., between several sets of selection criteria, different ways of determining the luminosity, or various quark fragmentation models) or half this interval. Here, the definition was unified in such a way that the overall normalization can vary by −∆ norm and +∆ norm , with a uniform probability over the whole interval. This overall normalization error is 100% correlated between the different centre-of-mass energy points reported in each given publication.
Third, the published values of ∆ norm often contain an estimate of the effect of missing higher-order QED corrections in the ISR unfolding procedure, at the level of a couple of percent. Indeed, at the time of PEP, PETRA and TRISTAN, the Monte Carlo programs used to simulate the e + e − → qq and e + e − → e + e − processes were limited to O(α QED ). The missing orders have a potential effect on the measured value of σ 0 had via the prediction of both the hadronic cross section and the Bhabha scattering cross section: the former is used to correct the measured σ had for QED effects, and the latter to determine the integrated luminosity. Altogether, the published cross section values would have to be corrected as follows, where the indices (1) and (all) refer to the cross section prediction up to the QED first order (used in the original publications) and with all orders, respectively.
With the programs that have been developed for LEP, it is now possible to evaluate this correction with a better accuracy than that assumed twenty years ago. The e + e − → qq and e + e − → e + e − cross sections were determined here with and without QED higher orders by ZFITTER [25] with an emulation of the kinematical cuts described in the original publications. It was found that the corrections to Bhabha scattering and hadron production essentially cancel in the ratio of Eq. 2. The remaining contribution of QED higher orders is at the 0.1% level, almost independently of the event selection and the centre-of-mass energy.
The large uncertainties related to the missing QED higher orders were therefore taken out from the original values of ∆ norm . While the aforementioned 0.1% contribution could be simply corrected for in σ 0 had , a new normalization error ∆ QED = 0.1% was added instead (assumed to be 100% correlated between all PEP, PETRA and TRISTAN measurements) to conservatively account for the yet missing orders in ZFITTER.
Finally, early TRISTAN data [14,15,18,19] are also corrected in the original publications for other electroweak effects, dominated by the top quark contribution (with a (m top /m Z ) 2 dependence at first order). These small corrections (between +0.1% and +0.7% at √ s = 60 GeV, depending on the top quark mass chosen to determine the correction) were unfolded here (i) to have a consistent data set to work with; and (ii) for a sound comparison with the ZFITTER prediction, which includes first-and higher-order electroweak contributions as well. The latest TRISTAN data [16,17,20] were, more adequately, corrected for QED effects only. The electroweak effect correction needs therefore not be unfolded in that case.
For practical reasons, the measurements of Tables 1 and 2 were clustered in few centreof-mass bins as indicated by the horizontal separation lines in these two tables. The ratio R values were averaged in each bin according to the total uncertainties, i.e., with a weight proportional to the inverse of σ 2 tot = R 2 × σ 2 stat + σ 2 ptp + ∆ 2 norm /3 + ∆ 2 QED /3 . The corresponding averaged Born effective cross sections (σ 0 had ) and centre-of-mass energy values are displayed in Table 3. The R values found for PEP/PETRA were found to agree with those of an earlier combination [8]. The effective Born hadronic cross section (σ 0 th ) predicted by ZFITTER [25] is also shown in Table 3. Computer-readable files for these data will be transmitted to the Review of Particle Physics and are available at http://janot.web.cern.ch/janot/HadronicData/.
The ratio and the difference of these measured cross sections and those predicted by ZFITTER are displayed in Fig. 1 as a function of the centre-of-mass energy. When no systematic uncertainties are assigned to the theoretical prediction, the average ratio appears to exceed the prediction by (0.79 ± 0.52)%, i.e., by 1.5 standard deviations. This excess is, however, about 2.4 standard deviations below the prediction of an additional light sbottom pair production (here with a mass of 6 GeV/c 2 ), which would amount to about 2% of the total cross section.  The experimental correlations between the different bins, essential for a rigourous statistical treatment of the data, were determined following the lines of Ref. [8]. In practice, a Monte Carlo technique relying on the generation of many gedanken experiments was used to determine the probability density functions of the measured R ratio values listed in Tables 1 and 2. In each gedanken experiment, 108 R values were generated around the measured central value, smeared by (i) a Gaussian distribution with a width equal to the quadratic sum of σ stat and σ ptp ; (ii) a uniform distribution in the [−∆ norm , +∆ norm ] interval, identical for all energy points of a given publication; and (iii) a uniform distribution in the [−∆ QED , +∆ QED ] interval, identical for all 108 measurements.
As above, an average value R i was determined in each centre-of-mass-energy bin i for each gedanken experiment. This allowed the R i values of Table 3 and their uncertainties to be confirmed, when averaging over a large number of gedanken experiments. Similarly, the uncertainties of the cross-products R i × R j led to the correlation matrices shown in Tables 4 and 5, for PEP and PETRA on the one hand, and for TRISTAN on the other. The cross-correlations between PEP, PETRA and TRISTAN (induced solely by ∆ QED ) were found to be smaller than 5 10 −4 and were therefore neglected in the following.   The precise measurements of LEP and SLC and their correlations [21] are summarized in Table 6. Most of these Z observables would be modified in case of an additional New Physics contribution to hadronic Z decays. Let ε had NP be the ratio of this new partial width Γ NP to the total decay width of the Z without this new contribution. As was shown in Ref. [27], the Z total width Γ Z , the ratio R ℓ of the hadronic to the leptonic branching fractions, and the peak cross section σ 0 had are modified as follows,

The LEP 1 and SLC data
In Ref. [27], the new hadronic decay channel considered was flavour-democratic. The individual branching fractions into the different quark flavours were therefore not modified by this new contribution. In the case of a sbottom pair production with hadronic R-parityviolating decays into light quarks exclusively, the ratio of the bb branching ratio to the hadronic branching ratio, R b , is also modified according to while (g V /g A ) remains untouched.
These observables would also be modified by the virtual corrections arising from the New Physics responsible for the additional hadronic contribution. As in Ref. [27], the value of ε had NP was fitted to the measurement of the five observables together with the generic contribution of these virtual effects. The result is which corresponds to an additional hadronic contribution of It allows a 95% C.L. upper limit of 56 pb to be set on the cross section, at the Z peak, of any additional hadronic contribution to the Z decays into light quarks only. The resonant contribution of the sbottom pair production cross section [28] with mb = 6 GeV/c 2 is shown in Fig. 2 as a function of the mixing angle cos θ mix between the two sbottom states b L andb R , superpartners of the left-handed and right-handed bottom quarks, respectively. For cos θ mix ≃ 0.39, the coupling between the Z and the lighter sbottom vanishes.
For mb = 6 GeV/c 2 , the Z data allow all values of cos θ mix below 0.22 and above 0.52 to be excluded at the 95% confidence level. These data are therefore incompatible with a light sbottom pair production, unless the coupling to the Z is negligibly small.

The LEP 2 data
The preliminary LEP 2 hadronic cross section data were taken from Ref. [21]. The measured cross sections σ had and the standard model predictions σ th are summarized in Table 7. These data are displayed in Fig. 3 and the correlation matrix is given in Table 8. When no systematic uncertainties are assigned to the theoretical prediction, the average ratio appears to exceed the prediction by (1.5 ± 0.9)%, i.e., by 1.7 standard deviations. This excess, although not significant, is compatible with, and actually slightly larger than an additional light sbottom pair production with cos θ mix = 0.39. The latter would amount to about 1% of the total cross section in this centre-of-mass energy range.

Global fit
When no systematic uncertainties are assigned to the standard model prediction, the data can be combined in a global negative log-likelihood L(cos θ mix , α) as follows, where S is the covariance matrix of the N (= 28) measurements of PEP, PETRA, TRIS-TAN, LEP 1, SLC and LEP 2 as compiled in Section 2, θ mix is the mixing angle in the sbottom sector and α is an arbitrary normalization constant of the sbottom pair production cross section, σ NP,i . The likelihood is then minimized with respect to cos θ mix and to α to find the best fit to the data. A fitted value of α compatible with unity and incompatible with 0. would be the sign of New Physics, while a value compatible with 0., but incompatible with 1., would allow this New Physics to be excluded with a certain level of confidence. (This same technique can be applied for any kind of New Physics leading to hadronic final states in e + e − collisions.) For α = 1 and mb = 6 GeV/c 2 , the negative log-likelihood is displayed in Fig. 4a as a function of cos θ mix . Not surprisingly, the Z peak data (Section 2.2) constrain the coupling of the sbottom to the Z to be vanishingly small, cos θ mix = 0.39 ± 0.07. The value of the mixing angle was therefore fixed to cos θ mix = 0.39. The combined negative log-likelihood and those for PEP/PETRA, TRISTAN and LEP 2 data are shown in Fig. 4b as a function of α. (For LEP 1 and SLC, the likelihood does not depend on α, because of the vanishing sbottom cross section for cos θ mix = 0.39.) The values of α for which the different negative log-likelihood functions are minimized are indicated in Table 9, together with the corresponding 68% confidence intervals and the 95% C.L. upper limits on α. (This one-sided upper limit is the α value for which the negative log-likelihood increases by 1.64 2 /2 with respect to the minimum.) As was already alluded to in Section 2.1, the lower energy data do not favour the sbottom hypothesis (α = 1). They are, instead, compatible with the standard model (α = 0) within one standard deviation or thereabout. A slight excess in the LEP 2 data, at the 1.7σ level (Section 2.3), translates as such to the combined result. The latter, however, excludes the sbottom hypothesis with mb = 6 GeV/c 2 at more than 95% C.L., when no systematic uncertainty is assigned to the standard model prediction.
The main sources of uncertainty for the theoretical prediction of the e + e − → qq cross section are (i) the knowledge and the running of the strong coupling constant α S ; (ii) the running of the electromagnetic coupling constant α QED ; and (iii) the theoretical accuracy of the prediction from the ZFITTER program. As in Ref. [27], the values and the uncertainties of the strong and electromagnetic coupling constants were taken to be α S (m Z ) = 0.1183 ± 0.0020 [29] and α(m Z ) −1 = 128.95 ± 0.05 [21], (10) leading to uncertainties in the hadronic cross section prediction of 0.15% and 0.08%, respectively. The missing higher orders in ZFITTER are estimated to contribute another 0.1%. These numbers add quadratically to a total systematic uncertainty η th of the order of 0.2%, in agreement with the estimate of Ref. [21] (η th = 0.26%) for LEP 2 data.
If this common systematic uncertainty is assumed to have a Gaussian probability density function, the negative log-likelihood can be modified as follows, to account for the full correlation between all centre-of-mass energies: where ρ th is the actual theoritical bias of the standard model prediction, to be fitted from the data.
It is reasonable, however, to take into account the non-Gaussian nature of uncertainties of theoretical origin. For example, the missing higher orders in ZFITTER may turn into a bias of −0.1%, 0% or 0.1% with an equal probability. (In fact, the least likely value is certainly 0%, as missing orders are expected to contribute a finite amount to the cross section.) Similarly, the uncertainty on the absolute value of α S (m Z ) is dominated by theory, and cannot be considered as Gaussian. It is therefore probably more adequate to assume a probability density function as displayed in Fig. 5, i.e., flat between −η th and +η th , and with a Gaussian shape outside this interval. The likelihood was therefore further modified by changing the ρ 2 th /2η 2 th term to This negative log-likelihood was then minimized with respect to the theoretical bias ρ th , for each value of α, with Gaussian and non-Gaussian uncertainties. The result is displayed in Fig. 6 in the two configurations as a function of α, for cos θ mix = 0.39 and mb = 6 GeV/c 2 . The values of α for which the negative log-likelihood is minimized are indicated in Table 10, together with the corresponding 68% confidence intervals and the 95% C.L. upper limits on α. It can be seen that the upper limit on α depends very little on the way the common systematic uncertainties are dealt with. The most conservative approach is chosen here to derive the final results.

Results
The same procedure was repeated by varying the sbottom mass from 0 to 12 GeV/c 2 . For each mass, the 95% C.L. upper limit on α was determined as explained above. A sbottom with a given mass is excluded if this upper limit is smaller than unity. Figure 7 shows the 95% C.L. upper limit on α for cos θ mix = 0.39 as a function of the sbottom mass, with Gaussian and non-Gaussian uncertainties. (In the latter configuration, the non-Gaussian nature of the likelihood was taken into account in the determination of the limit.) Sbottom masses below 7.5 GeV/c 2 are excluded at the 95% confidence level.  Because cos θ mix is very much constrained by the Z peak data, the upper limit on α is expected to be smaller than that shown in Fig. 7 for any other value of the mixing angle. As a check, the procedure was repeated again by varying cos θ mix from 0 to 1, with non-Gaussian uncertainties. The resulting sbottom mass lower limit is shown in Fig. 8 as a function of cos θ mix , and is indeed at least 7.5 GeV/c 2 over the whole range. (The region excluded by LEP 2 data at large values of cos θ mix is probably over-optimistic, as four-jet events -expected from such heavy sbottom pair as well as W pair productionare rejected from the qq event samples selected above the WW threshold.) It is worth mentioning that the presence of a light sbottom would slow down the running of α S with the centre-of-mass energy. (It would be even more so with an additional  light gluino.) Starting from the value accurately measured in τ decays [30], (the only measurement not affected by a sbottom heavier than 2 GeV/c 2 and lighter than 5.5 GeV/c 2 , and corresponding to α S (m Z ) = 0.121 ± 0.003 in the standard model), this slower running would lead to values of α S larger than assumed in this letter, at all centre-of-mass energies. The total New Physics contribution (from the direct sbottom production and the increase of α S ) would further increase the effect on the total hadronic cross section expected at PEP, PETRA, TRISTAN, SLC and LEP. The 7.5 GeV/c 2 lower limit on the sbottom mass is therefore probably very conservative.

Conclusion
The e + e − → hadron cross section data collected well above the bb resonances have been compiled and analysed to search for an anomalous production of hadronic events. Altogether, the PEP, PETRA, TRISTAN, LEP 1, SLC and LEP 2 data allow a light sbottom decaying hadronically to be excluded at 95% C.L. for any mixing angle, if its mass is below 7.5 GeV/c 2 . When combined with the result of Ref. [5] in which a stable sbottom with mass below 92 GeV/c 2 is excluded, this analysis definitely invalidates the model of Ref. [4] with a 12-16 GeV/c 2 gluino and a 2-5.5 GeV/c 2 sbottom.