Correction of observed decay rate

For C ppm diffusion is completely determined by scattering and formation, as scattering is too infrequent. Thus the shape of the distribution and of p(r,t) is independent of C. As seen from table 9, in the limit of small radial cuts the correction is simply , i.e. proportional to C$_{d}$. Thus measuring of the decay rate at two different deuterium concentrations C/C=C allows an extrapolation to C=0. The error of the extrapolated value has the following three sources of uncertainties:

$$\sigma _{1}=\frac{\Delta \lambda _{1}}{1-1/C}, \sigma _{2}=\... ...,\sigma _{3}= \frac{\Lambda _{d}(C_{d1})\Delta C}{1-1/C} s^{-1}$$

The first and second come directly from measurement errors at different deuterium concentrations and , respectively. The third one comes from the error of determination of the ratio C. The total error of extrapolated value of is . To decrease the error of extrapolation we need a measurement for large (up to ppm, where the correction is still linear to ). It is important that at the measurement at the second deuterium concentration could be done with only events without loss of precision of the extrapolated value. Then the resulting precision will depend only on the first and the third contributions. See table 10 for as the function of for two measurements, for ppm and 100 ppm.

Table 10: The total error of extrapolated $\lambda _{\mu p}(C_{d}=0)$ in case of $C_{d1}=1$ ppm, $C_{d2}=100$ ppm, $\sigma _{1}=4$ s$^{-1}$, $\sigma _{2}=100$ s$^{-1}$ in the limit of a tight radial vertex cut.

, s	4.04	4.28	5.86	8.14	10.7	14.7
C ,	0.1	1	3	5	7	10

In conclusion, for tight tracking we need an additional measurement of $\lambda$ at deuterium concentration $\sim 100$ ppm and a precison of better than $3\%$ for ratio of concentrations $C_{d2}/C_{d1}$. If we enlarge the tracking radius to 5 cm we only need about 6% precison for this ratio.