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 \section{Relative Luminosity Determination} Bunch-by-bunch beam luminosities are measured using the BBCs and ZDCs and recorded via the Scaler Boards. The detectors are described in Sections~\ref{sec:bbc}, \ref{sec:zdc}, and \ref{sec:scalers}. The BBC measurements are more precise and are employed in the calculation of $$A_{LL}$$; the ZDCs provide a cross-check essential for evaluating systematic uncertainties. Various bits of information related to BBC coincidences were encoded on several Scaler Boards during the 2005 and 2006 RHIC runs. A detailed QA and comparison of the different boards led to the conclusion that the fine-grained timing information on boards 5 and 6 should be used to calculate the relative luminosities. These boards allocated enough bits to track the BBC coincidences in 16 buckets in $$\Delta T$$, the time difference between a hit in the East BBC and the West BBC. Board 5 was configured to integrate throughout each $$\sim$$ 40 minute long STAR run, while board 6 took samples every 250 seconds to allow a determination of the relative luminosity stability throughout a run. When possible, the relative luminosities for a run are calculated using the measurements from board 5. A small number of otherwise-acceptable 2006 runs do not have reliable scaler information from board 5; in these cases the analysis relies on the data from board 6. The bunch crossing spin assignments are obtained using the procedure described in Section \ref{sec:spindb}. The coincidence count for each bunch crossing is restricted to a set of time buckets chosen to approximate a 60~cm cut on the $z$ position of the vertex; in the 2005 analysis buckets 7, 8, and 9 are selected, while the 2006 analysis adds bucket 6 into the sum. The relative luminosity $$R$$ for a run is then simply the sum of BBC coincidences in the selected set of time buckets for the bunch crossings with $$++$$ or $$--$$ spin assignments divided by the same quantity for bunch crossings determined to be in a $$+-$$ or $$-+$$ spin configuration. % remove runs less than 60 seconds long % remove runs where the time-integrated BBC coincidence in bit16 < 10k % remove runs with large number of counts in abort gaps, maybe % require data from 5/6/11/12, as well as multiple scaler files for the sampling boards % require that the relative luminosity is fairly stable throughout a fill -- |max(R3) - min(R3)| < 0.0001 % use bit16 on boards 11 and 12, but per-timebin info on boards 5 and 6 (guessing that 11 and 12 don't have timebin info) % some discrepancies between STAR run stop time and scaler timestamp. STAR started "getting ahead" of scaler system. Was anything done about this? Not clear. I don't think so % what was the difference between release 1 and release 2? \begin{figure} \includegraphics[width=1.0\textwidth]{figures/relative_luminosities} \caption{Distributions of per-run values for $R = \frac{\mathcal{L}_{++}}{\mathcal{L}_{+-}}$} \end{figure} The statistical uncertainties on $A_{LL}$ assume perfect knowledge of the relative luminosity of the different spin states. That simplification is addressed via the systematic uncertainty evaluation described below. \subsection{Uncertainty Evaluation Using the ZDCs} % \textit{Note: \href{http://mare.tamu.edu/star/2005n06Jets/2005relLumSys_mar29_2008/}{analysis by Murad Sarsour}} We can quantify the precision with which we understand the relative luminosities obtained from the BBCs by using an independent luminosity monitor, the ZDCs. In the absence of non-statistical fluctuations, the uncertainty on R will be dominated by the statistics in the ZDCs, which count at a much lower rate than the BBCs during proton-proton running. A couple of problems in the ZDC data need to be corrected before a comparison to the BBCs can be trusted. The first problem is due to the killer bit'' algorithm, which suppressed signals in the ZDCs for 10 bunch crossings after an initial signal. The algorithm is used in heavy ion running to prevent ringing in the calorimeters from generating false signals, but in pp running it biases the ZDC counts. Bunch crossings immediately following abort gaps (where the killer bit is more likely to be off) end up with more ZDC counts than crossings in the middle of a filled set of bunches. As a result, the ratio of relative luminosities obtained from the ZDC and BBC will not be flat, see Figure \ref{fig:zdctobbc6170012zoom}. \begin{figure} \includegraphics[width=1.0\textwidth]{figures/ZDCtoBBC_r7138003} \caption{Ratio of uncorrected ZDC and BBC coincidences versus bunch crossing. The ratio is larger in bunch crossings immediately following abort gaps.} \label{fig:zdctobbc6170012zoom} \end{figure} The procedure developed to correct for this effect requires scaling the counts for a given bunch crossing by a factor that takes into account the frequency with which the previous ten bunch crossings had a signal. For the ZDC singles rates, the formula for the corrected counts $n_{j}$ in a given bunch crossing $j$ is % \begin{equation} n_{j}^{corrected} = n_{j} * \frac{N_{cycles}}{N_{cycles} - \sum_{i=1}^{10}n_{j-i}} \end{equation} % where $N_{cycles}$ is the number of times the beam cycled through STAR in the run. Figure \ref{fig:zdc-singles-ratio} shows the effect of applying the correction for a sample run. \begin{figure} \subfloat{ \includegraphics[width=0.5\textwidth]{figures/ZDCtoBBC_r7133049ER} } \subfloat{ \includegraphics[width=0.5\textwidth]{figures/ZDCtoBBC_r7133049WR} } \caption{Change in the ZDC singles rates after applying the killer bit correction.} \label{fig:zdc-singles-ratio} \end{figure} The formula to correct the ZDC coincidence counts is complicated by the need to track the killer bits for the two detectors simultaneously. The formula for the corrected coincidence counts $$c_{j}^{corrected}$$ in a bunch crossing given raw singles counts $$e_{j}$$ (ZCDE) and $$w_{j}$$ (ZDCW) and coincidence counts $$c_{j}$$ is % \begin{align} &\alpha_{j} = N_{cycles} - \sum_{i=1}^{10}c_{j-i} \notag\\ &\beta_{j} = E_{j-10} + W_{j-10} + E_{j-9}*\left(1 - \frac{W_{j-10}}{\alpha_{j} - E_{j-10}}\right) + W_{j-9}*\left(1 - \frac{E_{j-10}}{\alpha_{j} - W_{j-10}}\right) + ...\notag\\ &c_{j}^{corrected} = c_{j} * \frac{N_{cycles}}{\alpha_{j} - \beta_{j}} \end{align} % where $$E(W)_{j-i} \equiv e(w)_{j-i} - c_{j-i}$$ is the ZDCE(W) singles count minus the coincidence count for the j-ith bunch crossing. The effect of the killer bit correction on the coincidence distributions is shown in Figure \ref{fig:coinRat6143016}. \begin{figure} \begin{center} \includegraphics[width=0.6\textwidth]{figures/ZDCtoBBC_r7133049coin} \end{center} \caption{Change in the ZDC coincidence rates after applying the killer bit correction.} \label{fig:coinRat6143016} \end{figure} \begin{figure} \begin{center} \includegraphics[width=0.8\textwidth]{figures/c7308} \end{center} \caption{Example of a coherent spin pattern and even-odd ZDC rate oscillation. In this case, the ZDC rate is always higher when the spin of the blue beam is down.} \label{fig:c7308} \end{figure} % More Details on even-odd effect % http://cyclotron.tamu.edu/star/2006Jets/jul31_2007/ % http://cyclotron.tamu.edu/star/2006Jets/aug2_2007/ - figure 4 on this page plots the asymmetry between the ZDC/BBC ratio for even bxings and the ZDC/BBC ratio for odd bxings. The asymmetry can be positive or negative even within a spin pattern, basically no correlation. But at the bottom of the page, Murad clearly shows that the even/odd asymmetry arises from the ZDCs, not the BBCs, when he plots a particular jet asymmetry using R_ZDC and then R_BBC, and compares the sign of that asymmetry to the sign of the asymmetry in Figure 4. % the cut on |F*(S-1)| < 0.002 is strictly a 2005 thing, in 2006 the ZDC coincidences are actually renormalized! Uber-sketchy if you ask me. The second problem in need of correction has come to be known as the even-odd'' effect. The ZDC coincidence rates are often different for even-numbered and odd-numbered bunch crossings, due to oscillations in the electronics pedestals in specific ADC channels of the CDB boards used to read out the ZDCs. This oscillation can introduce a false asymmetry if it aligns coherently with a particular spin pattern. For instance, in Figure \ref{fig:c7308} the ZDC coincidence rates are always higher when the spin of the blue beam is down. To quantify the bias this introduces on $A_{LL}$, we can define the fractional overlap between the even-odd ZDC oscillation and relevant portion of the spin pattern for $A_{LL}$ using a 120 element vector $|EO\rangle = |+1,-1,+1,-1,...\rangle$ and another 120 element vector $|LL\rangle$ whose elements are 1 if the bunch crossing is UU or DD, -1 if UD or DU, and 0 otherwise. The inner product of these vectors measures the susceptibility of $A_{LL}$ for that spin pattern to any even-odd oscillation. % \begin{figure} % \includegraphics[width=1.0\textwidth]{figures/fevfod} % \caption{Magnitude of even-odd rate asymmetry versus time in the 2005 RHIC run.} % \label{fig:fevfod} % \end{figure} It turns out that $A_{LL}$ is less biased by the even-odd rate oscillation in the ZDC than, say, the blue beam single-spin asymmetry. Figure \ref{fig:cll} plots the fill-by-fill change in $A_{LL}$ if the ZDC is used for relative luminosities instead of the BBC against the the product of the fractional overlap $F \equiv \langle EO | LL \rangle$ and the magnitude of the even-odd oscillation $S-1$. Placing a cut on $|F*(S-1)| < 0.002$ is well-motivated. For fills without reliable ZDC information, we assume a conservative $|S-1| = 0.03$. \begin{figure} \begin{center} \includegraphics[]{figures/cll} \end{center} \caption{Change in $A_{LL}$ versus the product of the even-odd rate oscillation amplitude and the fractional overlap $\langle EO | LL \rangle$. Deviations from 0 on the x-axis indicate fills where $A_{LL}$ is biased by the even-odd effect.} \label{fig:cll} \end{figure} After correcting for the killer bits and rejecting the fills that fail the even-odd oscillation cut the ZDC coincidences counts for even and odd bunch crossings are separately normalized using the following normalization factors: % \begin{equation} f_{even} = \frac{\langle ZDC/BBC \rangle}{\langle ZDC_{even}/BBC_{even} \rangle}, ~~~~ f_{odd} = \frac{\langle ZDC/BBC \rangle}{\langle ZDC_{odd}/BBC_{odd} \rangle}; \end{equation} % that is, the ZDC coincidence counts for even(odd)-numbered bunch crossings in a run are rescaled by the mean ZDC/BBC ratio for the run divided by the mean ZDC/BBC ratio for even(odd)-numbered bunch crossings in the run. Finally, the uncertainty on $A_{LL}$ due to the uncertainty in the relative luminosities is calculated as the change in $$A_{LL}$$ when the relative luminosities are supplied by the normalized ZDCs instead of the BBCs, and is equal to $9.4\times10^{-4}$. % \subsection{Beam Background Bias} % % % \textit{Note: \href{http://www.star.bnl.gov/protected/spin/kowalik/2005/r-lumi/bkg_sys.html}{analysis by Kasia Kowalik}} % % The relative luminosities obtained from the BBCs might also be biased by false % signals generated by beam-gas background. We can try to quantify this by % studying the coincidence rate in crossings where one of the two beams has an % unfilled bunch (abort gaps''). The beam-gas background is assumed to be % crossing- and spin-independent, but it can be different in each beam. It follows % that the per-crossing coincidence rate due to beam-gas in each beam is just the % average number of BBC coincidences found in the abort gaps for that beam. In % Figure \ref{fig:bkg-yellow-blue}, the x-axis is the background rate divided by % the total rate, defined as the average number of coincidences per bunch crossing % with a spin state of $$++$$, $$+-$$, $$-+$$, or $$--$$. The two histograms are % incremented for each STAR run. We see that the background rate in the BBCs due % to beam-gas is typically less than 0.1\% of the total rate. % % \begin{figure} % \includegraphics[width=1.0\textwidth]{figures/bkg-yellow-blue} % \caption{Fraction of the total coincidence rate attributed to beam gas in % each beam. The histograms are incremented once for each STAR run.} % \label{fig:bkg-yellow-blue} % \end{figure} % % Given run-dependent background fractions for both beams, it's possible to % calculate background-subtracted relative luminosities. Figure % \ref{fig:r-lumi-sys-bkg} shows the difference between the raw relative % luminosity and the background-subtracted version. The background-corrected % relative luminosities yield an $A_{LL}$ that differs from the original by % $3.0\times10^{-4}$, so we use that as the uncertainty for this source of % systematic error. % % \begin{figure} % \begin{center} % \includegraphics[width=0.6\textwidth]{figures/r-lumi-sys-bkg} % \end{center} % \caption{Change in the relative luminosities after correcting for beam-gas % background.} % \label{fig:r-lumi-sys-bkg} % \end{figure}