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Systematic errors

Here I give a brief summary of the systematic errors in the measurements of the DWT texture correlation observable $P_{true} - P_{mix}$. Static texture and dynamic background texture present the largest problem in the search for the phase transition-related dynamic texture via power spectra of local fluctuations. The method of solving the problem is comparison with the reference sample created by event mixing. Thus the $P_{true} - P_{mix}$ observable was created.

By static texture we mean texture which reproduces its pattern event after event. This can be either because it is coupled with detector channels (dead pads, geometry distortion, channel cross-talk, etc) or because of static physics features such as $\,dN/\,d\eta$ shape. We eliminate the static texture from the texture correlation observable by empty target subtraction (Subsection 6.7.2) and by subtraction of mixed events power spectra (Subsection 6.6.2). For comparison with models, a Monte Carlo simulation of the Si detector is used (Section 6.8). It includes the known static texture effects and undergoes the same procedure to remove the effects. The ``irreducible remainder'' is the residual effect which may

1. survive the elimination procedure
2. emerge as a difference between the data, subjected to the elimination procedure, and the MC analyzed in the same manner.

Table 6.2 lists the sources of static texture and summarizes the methods of their treatment.

Table: Sources of background texture (dynamic and static) and their treatment. The irreducible remainder estimate is quoted for diagonal texture correlation in the $326<\,dN/\,d\eta<398$ bin, and is expressed in the units of $\sigma^2/\langle dE_{MIP} \rangle^2$; see text for information on how it was obtained.

	Correction				Irredu-
		event mixing			cible
Source	subtract	subtract	preserve	do	remainder
	empty target	mixed events	sectors	MC	estimate
stat. fluctuations	N/A	yes	N/A	yes	0.
$\,dN/\,d\eta$ shape,	N/A	yes	OK	yes	0.
offset, dead pads
finite beam	N/A	N/A	N/A	yes	$0.14$
Xsection $1\times2$ mm
background hits	yes	yes	yes	can't	$>0.070$,
channel Xtalk	N/A	yes	yes	can't	$<0.37$
8.5% for neighbours

We group the background texture sources according to similarity of manifestation and treatment, into

• statistical fluctuations
• static texture
• background dynamic texture

The statistical fluctuation is the most trivial item in this list. Both event mixing (provided that mixing is done within the proper multiplicity class) and MC comparison solve this problem. The statistical fluctuations do not result in irreducible systematic errors.

The static texture group includes:

• geometrical offset of the detector with respect to the beam's ``center of gravity'' in the vertical plane (determined in Section 6.4)
• dead pads (Section 6.1)
• $\,dN/\,d\eta$ shape - a genuine large scale multiparticle correlation sensitive to the physics of the early stage of the collision

Cleanliness of the static texture elimination via event mixing has been checked by simulating the contributing effects separately. First, by running the detector response simulation on MC-generated events without the beam/detector offset and with a beam of 0 thickness it was ascertained that the remaining dynamic texture is very small compared with the systematic errors due to the background Si hits and the beam geometrical cross-section, for all scales and all directional modes $\lambda$. Due to the finite size of the multiplicity bin, the mixed events consist of subevents coming from events of different total multiplicity. With the sector-wise mixing, this causes an additional sector-to-sector variation of amplitude in the mixed events, thus resulting in an enhancement of $P^{\zeta}_{mix}$ primarily on the finest scale, with respect to $P^{\zeta}_{true}$. (If the mixing is done ring-wise, rather than sector-wise, the same effect is seen in $P^{\eta}_{mix}$, rather than $P^{\zeta}_{mix}$.) On Fig. 7.1, this effect can be seen as the $P^{\zeta}_{true}-P^{\zeta}_{mix}$ values progressively grow negative with multiplicity in the finest scale plot. However, as can be seen on the same figure, the effect is small compared with the total systematic errorbars shown as boxes.

The background dynamic texture group includes:

• elliptic and directed flow
• finiteness of the beam cross-section
• background hits in the Si (Subsection 6.7.1)
• channel-to-channel cross-talk (Section 6.5)

Elliptic and directed flow, observed at SPS [10], are large scale dynamic texture phenomena of primarily azimuthal (elliptic) and diagonal (directed flow) modes. Because both reaction plane and direction angle vary event by event, the respective dynamic textures can not be subtracted by event mixing, unless the events are classified according to their reaction plane orientation and the direction angle, with mixing and $P_{true} - P_{mix}$ subtraction done within those classes. Neither reaction plane nor direction angle was reconstructed in the present analysis, and the $P_{true} - P_{mix}$ (especially that of the azimuthal and diagonal modes on the coarse scale) retain the elliptic/directed flow contribution. The effects of flow on dynamic texture observables are smaller than other texture effects, so they can not be singled out and quantified in this analysis.

The finite beam cross-section effect belongs to this group, despite the fact that a very similar effect of geometrical detecor/beam offset has been classified as static texture. An effect must survive mixing with its strength unaltered in order to be fully subtracted via event mixing. Preserving the effect of the random variations in the $Pb+Pb$ vertex on the power spectra in the mixed events requires classification of events according to the vertex position and mixing only within such classes. This requires knowledge of the vertex for each event, which is not available in this experiment. Therefore, MC simulation of the beam profile remains the only way to quantify false texture arising from vertex variations. MC studies with event generators show that the beam spatial extent and the resulting vertex variation is the source of the growth of the coarse scale azimuthal texture correlation with multiplicity (see Fig. 7.1). Uncertainty in our knowledge of the beam's geometrical cross-section must be propagated into a systematic error on $P_{true} - P_{mix}$. Here is how it was done:

1. run MC for beam thickness $1\times2$ mm, with the calibrated detector offset
2. run MC for beam thickness $1.5\times3$ mm, with the calibrated detector offset
3. take 50% of the difference between steps 1 and 2 as the estimate of the error

The other two effects in this group are difficult to separate and simulate and the error estimate reflects the combined effect. The systematic errors were evaluated by removing the $Pb$ target and switching magnetic field polarity to expose the given side of the detector to $\delta$-electrons (from the air and T0), while minimizing nuclear interactions. This gives an ``analog'' generator of uncorrelated noise. The runs used for this purpose are the positive field polarity runs listed in Table 6.1. All correlations (i.e. deviations of $P^\lambda(m)_{true}$ from $P^\lambda(m)_{mix}$) in this noise generator are treated as systematic uncertainties. Thus this component of the systematic error gets a sign, and the systematic errors are asymmetric. The effect of increasing texture correlation (for diagonal and azimuthal modes) with multiplicity on the coarse scale, attributed to the geometrical offset of the detector with respect to the beam (the leading one in the static group), is present in the switched polarity empty target runs as well. For this reason, it was impossible to disentangle the background dynamic contribution on the coarsest scale. In Table 6.2, the ``irreducible remainder estimate'' for the diagonal, coarse scale is bracketed with two numbers, which form the lower and upper estimates. The lower estimate is obtained by taking the scale one unit finer and quoting its number. This, indeed, sets the lower limit because the deviations of $P^\lambda(m)_{true}$ from $P^\lambda(m)_{mix}$ generally grow with scale coarseness. The upper limit is set by ascribing the entire texture correlation, observed in the $\delta$-electron data, to the background hits and channel cross-talk, and ignoring the fact that significant portion of it must be due to the vertex fluctuation (finite beam profile). This upper limit is likely to be a gross overestimation, and in Fig. 7.1 we show systematic errors, obtained by adding in quadrature the finite beam error with the background hit error.