03-17-2002. Jianglai An update of the lumi analysis. We discovered a non-negligible non-linear reponse of lumi with the change of the beam current in the 2nd pass "production" lumi analysis. As a matter of fact, this should have been our first item to look into but we have gone through the hard way ... Below are some technical details. I was looking at the stability of the correlation slopes between lumi yield and other beam parameters for the Jan. runs --- and they did not look very good. (The correlation vector (Y,P) and the covariance matrix (Pi,Pj) were calculated in the analysis code and in the end the matrix was inverted to obtained the slopes run by run.) "lumi_slope_vs_run.ps" --- shows the 6 slopes vs run number. I converted the slopes to a dimensionless forms (multiplying by Sigma_P/Sigma_Y where Sigma_P and Sigma_Y are the variances of the beam parameter and lumi yield, respectively) to show the relative significance of different beam parameters. I haven't done the error propagation of the slopes so the mean quoted by the fit could well be biased. At any rate, one may discover that the yield vs charge slope dominates over the others. At this stage I realized the might be a non-linear component in the lumi reponse. (Naively I thought this was an integrating system ==> perfectly linearity.) "lumi_nolin2.ps" --- a closer look of the LUMI1 and BCM1 during run 16100. The left panel shows the raw LUMI vs raw BCM1. In order to extract the linear component right, we actually need to perform the linear fit right at the beam current zero point. This was a little tricky, especially that we had no great statistics to make a good fit at low beam current points. For some runs I just could not make any sensible fit. This plot was made with run 16100 (it was a 36.5 uA run with some beam trips), where the linear fit was obtained in the range ~ [0, 10 uA]. Then the residual of LUMI against the fit was plotted on the right pad. One can see clearly that it is off zero by ~ 6000 channels. Comparing with the average S-P (~ 50,000 channels), the non-linearity component is about 12%! Also the slope obtained from [0,10uA] is an overestimate of the tangential slope at I=0, so the actual non-linear component could be even larger. BTW a saturation of the V/F channel, or target density fluctuation can't explain this, since the slope is in the wrong way ... I was still a little suspicious, so I went on more checks. Two more approaches were used. One was to fit the normalized yield as a function of beam charge and extrapolate the quadratic component to 40 uA (method 2). The other is by fitting raw LUMI vs raw BCM1 with a parabola and extract the 2nd order component to 40 uA(method 3). In fact all these methods are in principle equivalent. Just want to make sure that I wouldn't be fooled. Various runs were examined and below is a fact sheet: Run f_non_lin(method 2) f_non_lin(method 3) 15835 15.3% 16.3% 15836 15.9% 15.6% 15848 16.8% 17.2% 15941 14.8% 12.7% 15948 18.1% 17.6% 15949 15.8% 16.5% 16100 15.5% 15.7% So looked like the results were consistent using three different approaches. Looks absurb at the 1st glance but I guess it is real. "lumi_asym_correlation.ps" --- plots the correlation between the lumi asymmetry and the helicity correlated beam parameters difference/asymmetry. The In/Out state of IHWP have been obsorbed in the asymmetries/differences. Not surprisingly and sadly, a 9% correlation between A_lumi and A_q has been observed (almost the same level as Wells slope!). Actually I guess part of the reason that we got away with 9% of asymmetry slope was that most of the 2nd half runs were 35 uA runs so (35/40)^2*15% ~ 11.5% ... I am not sure what else could be argued from other plots since most fits look non-sense, except for the one with Diff_y. The slope of that is 0.05ppm/nm. Comparing with the golden number of the FPD --- 0.7%/mm maximum, we can see that the lumi yield is much more sensitive to the beam motion. To conclude, the observed non-linear effect could be due to the non-linear nature of PMT we used during the experiment. If that's the case, a vacuum photo-diode is the right way to go ...