Just before the PAC meeting of 15 June 2009, Peter Jacobs wrote to me the following mail asking more details about asHBT in the BE scan.  Uli had been asking similar questions.  Here is Peter's mail:

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Hi Nu, Jamie, and Mike,

In preparation for the PAC meeting next week, Uli and I have been discussing the azimuthally sensitive HBT measurement. We would like to understand the requirement of ~4M  events at lower energies in terms of BUR Fig 3-4.

Uli sent me a figure with hydro calculations with various EOS's, which around the CERES energy all have epsilon~0.2 or larger. This sets a rough minimum level of precision: what do you need to do to discriminate a hydro scenario from what CERES claims, i.e. discriminate epsilon~0.2 from epsilon~0?

So please comment on the following questions, and if possible have answers available for discussion on Monday:

(i) assuming the hydro value vs sqrt(sNN), what would be the error bar on epsilon for 1M, 2M, and 4M events at 17.3, 11.5 and 7.7 GeV?

(ii) how does the error scale with signal magnitude? (i.e. for the same number of events, if the signal is zero is the error different than if the signal is 0.2?)

(iii) assuming the hydro value for the signal, at what confidence level can you exclude the CERES measurement at 17 GeV for 1M, 2M and 4M (or 3.5M) events?

(iv) what are the ultimate systematic limitations on the measurement, beyond which additional statistics do not help?

Thanks,

Peter

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And here was my response:

Hi Peter et al (and Uli copied, too),

    My estimates (which is what they are) for the asHBT were based mostly on previous measurements by E895, STAR, and (somewhat) CERES.  I am attaching a .pdf file discussing some of this.  It was written maybe 1.5 years ago, updated March 08 and then a little bit last week.  You will note that it does not include CERES point.
    For a fuller discussion and plots of the recent results and models, check out my talk from last week at CPOD:
http://quark.phy.bnl.gov/~karsch/talks/Mon/Lisa.pdf

    Now then, the statistics in the table were designed essentially to give errorbars of about 0.02 in epsilon, when combining kT bins and some centrality bins.  This is important, because it means we cannot gain by combining bins in the energy scan-- I've already assumed we would.
   The values plotted on the famous plot using CERES and STAR points are kT-integrated and combining the following centrality bins:
STAR: 5-10,10-20,20-30%
CERES: 7.5-10,10-15,15-25%
these were chosen so as to compare to E895 7.5-30%
(Yes, I know these centrality cuts are not 100% identical, but it's the best I could do with the available results.)

with those combinations, here are the epsilons:
STAR = 0.094 +/- 0.02
CERES = 0.043 +/- 0.03
This means the errorbars just barely miss each other.  I attach a blow-up below to make the point.

    To answer Peter's questions:
* the uncertainties on the oscillations and magnitudes of the radii are dominated by statistics.  Now then, the *connection to epsilon* is through the "no-flow" formula
$\epsilon \approx \frac{R^2_{s,2}}{R^2_{s,0}}$,
which is estimated good to 30%.  But that is common to all datapoints (and calculations), so when one is looking at whether there is monotonic or not, it does not come into play.
Now then, the statistics are *pair* statistics, in particular close pairs.  For a fixed multiplicity, this means event statistics.  So, number of events, at a given energy, is what matters.  All scales with that

* the uncertainty does not depend on the magnitude of the effect.  Uncertainty on epsilon=0 should be same as uncertainty on epsilon=0.2.  Said another way, it is an absolute magnitude, not relative.
--------->>> Having said that, however, I need to say another thing: One can get an artificially very small or zero number from a *lot* of reasons (harder to get a "false positive").  Even though we will be very careful, asHBT is not exactly turn-key.  My new student Chris is finding 0.0 at 62.4 GeV (how exciting!), but initially saw 0.0 also at 200 GeV (clearly a f*ckup).  So, we are doing all sorts of tests at 62.4.  If we pare down the statistics to the "bare minimum" needed for a certain errorbar, I will not have the statistics to do systematic checks.  asHBT is not v2 or spectra (no offense to those...).  As I mention below, we maybe can reduce things a bit, but don't cut too far, especially if the signal might be small!

* other effects beyond statistics: well, if the reactionplane resolution is very bad, that builds in a systematic error.  But even at 200 GeV, we didn't find this significant.  I'll be honest and say I didn't think much about it, for *first* order RP.  But my feeling is that it will not be an issue.  I cannot prove that, though, sorry.  With that caveat, I'd say statistics is all.

* OK, so w,

--
errorbar on 7.7 GeV collisions
#events
1 M - 0.04
2 M - 0.03
4 M - 0.02

errorbar on 11.5 GeV collisions
#events
1 M - 0.04
2 M - 0.025
4 M - 0.018

errorbar on 17.3 GeV collisions - same as above (look, it doesn't change that much with energy...)
#events
1 M - 0.04
2 M - 0.025
4 M - 0.018

So, to answer Peter/Uli's question of "with how many sigmas can we distinguish hydro prediction of 0.2 and CERES value of 0.04? (Diff = 0.16)"
Well, for 17.3 GeV, with 1M, it is 4 sigma, with 2M it is 6 sigma, with 4 M it is 0.18 sigma.  Keep in mind we are asking for 3.5 M.
To make the same distinction at 7.7 GeV, with 1M it is very similar, as seen above.

    I would actually argue that the real thing we are looking for is whether epsilon at 17.3 GeV is *lower* than at 200 GeV, or *higher*. So, instead of 20%-4%=16%, we'd be looking at, say 14%-9%=5%=0.05.  Then, we are talking about:
with 1M events, it is 0.05/0.04=1.25 sigma with 2M events, it is 0.05/0.025=2 sigma, with 4M events it is 0.05=0.018=2.8 sigma.

    I hope this helps.  I'd say our 4M request makes sense.  If you cut us by factor of 2, we should be okay.  If you cut us by factor of 4, it is to the bone, and, given the point above about systematic studies and the fact that these are estimates, there is some danger that the measurement might not be definitive at all.  Don't cut us by more than that, please.

    I'll be available by email, but might not respond quickly.  If you just want a quick answer to something, my cellphone is 614.499.6005.

Thanks,
mike

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I had attached a .pdf file and a .png graphic, which are attached to this post (look at the bottom of the page)