1. K. Hagiwara et al., Phys. Rev. D 66 (2002) 010001.
2. The χ2 probability density distribution has ν, the number of degrees of freedom, as its mean value and has a variance equal to 2ν. To have an intuitive feeling for the goodness-of-fit, i.e., the probability that χ2>χmin2, we note that for the large number of degrees of freedom ν that we are considering in this note, the probability density distribution for χ2 is well approximated by a Gaussian, with a mean at ν and a width of 2ν, where 0<χ2<∞ (n.b., the usual lower limit of -∞ is truncated here to 0, since by definition χ2⩾0). In this approximation, we have the most probable situation if χmin2/ν=1, which corresponds to a goodness-of-fit probability of 0.5. The chance of having small χmin2∼0, corresponding to a goodness-of-fit probability ∼1, is exceedingly small. In our computer-generated example of a straight line fit with ν=103, the fit first can be considered to become poor—say by three standard deviations—when χmin2>146, yielding χmin2/ν>1.41. We found a renormalized χmin2/ν=1.01, indicating a very good fit.
3. In this context, a random distribution means a uniform distribution between a and b, generated by a random number generator that has a flat output between 0 and 1. A normally distributed (Gaussian) distribution means using a Gaussian random number generator that has as its output random numbers yi distributed normally about y¯, with a probability density 1/2πexp-12((y¯-yi)/σi)2, where σi represents the error (standard deviation) of the point yi.
4. The fact that rχ2 is greater than 1 is counter-intuitive. Consider the case of generating a Gaussian distribution with unit variance about the value y=0. If we were to define Δχi2≡(yi-0)2=yi2, with Δ being the cut Δχi2max, then the truncated differential probability distribution would be P(x)=1/2πexp(-x2/2) for -Δ⩽x⩽+Δ, whose rms value clearly is less than 1—after all, this distribution is truncated compared to its parent Gaussian distribution. However, that is not what we are doing. What we do is to first make a robust fit to each untruncated event that was Gaussianly generated with unit variance about the mean value zero. For every event we then find the value y0, its best fit parameter, which, although close to zero with a mean of zero, is non-zero. In order to obtain the truncated event whose width we sample with the next χ2 fit, we use Δχi2≡(yi-y0)2. It is the jitter in y0 about zero that is responsible for the rms width becoming greater than 1. This result is true even if the first fit to the untruncated data were a χ2 fit.
5. In deriving these equations, we have employed real analytic amplitudes derived using unitarity, analyticity, crossing symmetry, Regge theory and the Froissart bound.