ELEM PARTICLE PHYS 1
ELEM PARTICLE PHYS 1 PHZ 6355
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This 23 page Class Notes was uploaded by Maureen Roob IV on Friday September 18, 2015. The Class Notes belongs to PHZ 6355 at University of Florida taught by Andrey Korytov in Fall. Since its upload, it has received 5 views. For similar materials see /class/206872/phz-6355-university-of-florida in Physics 2 at University of Florida.
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Date Created: 09/18/15
Introduction to W m quot1 my Particle Physics Note 13 Page 1 0f23 Data Analysis Basics Probability Distributions Poisson distribution Gaussian distribution Central Limit Theorem Propagation of errors Averaging with proper weights Statistics Estimates of average sigma and errors on the estimates Confronting Data and Theory Best Estimates of Theory Parameters Max Likelihood Method lVIin x2 Method and dealing with x2 values Signal in Presence of background Statistical Signi cance of an observed signal Enhancing signal over background Confidence Levels when a signal is not seen Systematic Errors Crosschecks Traps of Wishful Thinking Examples of low statistic discoveries Introduction to W m quot1 try Particle Physics Note 13 Page 2 0f23 Probability Distributions Poisson distribution random independent of each other events occurring at rate V Therefore during time At one should be expecting to detect on average rt VAt events However the actually detected number of events k in a concrete experiment may be different k rt Probab111ty of detect1ng k events Pkri Pk Fe quot Average k ri Variance Dispersion 0392 ltk rt2gt ri RMS root of mean squared or rootmeansquared 1 ltk r02 J Gaussian distribution is a good approximation for many typical measurement errors Its importance is largely derived from the central limit theorem see below Probability of measuring x within the range from x1 and x2 is P J pxdx Hu Where px is probability density px e 2 72 l27r039 Probability to be within 16 is 68 Probability to be within i2039 is 95 Probability to be within i3039 is 997 Central Limit Theorem if one has ri independent variables x1 xquot having probability distribution functions of any shape but with finite means y and variances 02 the sum X 2x at ri gtoo will have the Gaussian distribution with the mean equal sum of y and the 2 varlance equal to sum ofoi Poisson distribution of large n ngtgt1 is very close to Gaussian with xon 62n Average 033 Average 50 Pk and Ggtlt kandx Introduction to W m quot1 my Particle Physics Note 13 Page 3 of 23 Propagation of errors mfx if x has a small uncertainty ox one can estimate omfxox quotF1796 y if x and y have small uncertainties ox and cry and no correlations 2 2 2 am 020 jg0 Averaging Assume that there are two measurements of x x1 and x2 that have estimated or known errors 01 and 02 One can easily calculate that the best estimate of the value of x and the error on this estimate are 02 1 xm wlxl w2x2 where w1 2 2 and w2 2 2 0391 0392 0391 02 2 Z Z 0102 m 03912 02 Trivial consequences a lousy measurement can be ignored it hardly adds any weight for the estimate and does not improve the error on the estimate two equally goodbad measurements should be counted with equal weighs and the error from two measurements is l 2 better than from a single measurements Introduction to W m quot1 my Particle Physics Note 13 Page 4 of23 Statistics Given the nite number of measurements a estimate probability distribution function parameters eg mean width and b evaluate errors on the estimations number of events measurements giving particular value 13 5 7 91113151719212325272931 measurement Value Assume that the true probability distribution has mean X0 and dispersion DO3902 l N Best est1mate of mean xm x1 11 1 N Best estimate of dispersion 039 N 12 q xm 2 11 Estimate on error in xm 69cm VN 0quot Estimate on error in am 60m for Gaussian distribution and large N W Introduction to W m quot1 my Particle Physics Note 13 Page 5 of 23 Confronting Data and Theory Best Estimates of Theory Parameters The primary questions one must answer are is the theory consistent with data b what are the best estimates on theoretical parameters what are the errors on the estimates d are there any indications that experimental data are not selfconsistent O N V V Max Likelihood Method Generic Example 0 Data a set of y measurements at x points with 0 known y l y error distribution functions probability of measuring y when the true value is y o and no correlations between points 0 Theory with parameters a yFx a Probability to get a particular set of measurements y for a given choice of parameters a dP Hdp 139fy Fxa dy 139fy Fxa 139dy Ly a139dy Lyl l a iLikelihood function We will choose the best possible theoretical parameter by maximizing the probability dP or equivalently the Likelihood function Note it is often more convenient to maximize the log of L lnLyI 61 instead of Lithe answer would be the same as the logfunction is monotonous Case of Gaussian errors Maximum Likelihood method is equivalent to the Minimum X2 method mm W 1n139fy Fxa21nfy Fxa yrw aw 1 Zln e 4270 y Fx 612 Const A Z 202 x l Const 2 2 Introduction to Elementary Particle Physics Note I 3 nge 6 0f 23 0 Statistical expectations for x2 and what if you get something very different 012 llI Vx VI I 139llH1 I 2i5llll llll HH HH HH HH HH HH HH HH 03910 n 10 20 2 008 g 157 61quot I 0a 2 7 550 10 ofarea Xn g CL1039 10 o I 004 Fri 90 002L MW y quot 00 lHH llllillllillllillllillllillllillllilll 0 5 3910 15 20 25 30 0 10 40 50 X2 Degrees of freedom 17 2 o I nmemwmem kpmmem nd f number of degrees of eedom 0 Large x2 I Theory does not describe Data I Errors are underestimated I There are large negative correlations systematic errors 0 Small x2 I Errors are overestimated I There are large positive correlations systematic errors 0 Other crosschecks for hidden systematic errors 0 Estimation of errors on parameter estimates from x2 o a aaiaa 2f ajH 0 When using the x2 minimization method is wrong 0 Errors are not Gaussian eg I Gaussian with long tails I Small statistics must use Poisson errors I Flat error distribution for digitized signal bin width gtgt noise 0 Errors have correlations I Both Max Likelihood and Min x2 Methods can be appropriately modi ed Introduction to W m m my Particle Physics Note 13 Page 7 of 23 Signal in presence of background Statistical Signi cance You expect b events background and observe n0 events and n0 is greater than b What is the signi cance of this observation Have you discovered a new process that would account for the observed access of events Or maybe this excess is a plain statistical uke Signi cance S is introduced to quantify the probability of a statistical uctuation to observe n0 events or more when you expect only b events It maps a probability of a statistical uctuation into a number of Gaussian sigmas m m 1 7L2 pn2nb pkb 62 dx 0 EU 27 39 39 1 2 3 4 5 lprobabilityp value 16 23 014 3x1039 3x1039 Signi cance estimators poor man solutions signal nmmm b s kagd J3 J3 39 This is a very popular estimator but it has a very lousy performance small statistics For blt100 this S1 estimator breaks down and gives too large values overestimates signi cance 0 For large N S1 The best simple estimator is SM 2n01nlsb 2s lt arises from comparing probabilities of the fact that the number of observed n0 is due to backgroundsignal or due to background only a socalled likelihood ratio SM 1 21nQ where Q M 1W0 b This estimator is very close to the true signi cance even for a very small statistics does not deviate by more than 02 or so Introduction to W m quot1 my Particle Physics Note 13 Page 8 of 23 Two plots below show histograms of reconstructed invariant masses for positivenegative charged particles in reactions p pa e39 a thing Number of events Number of events What is signi cance of the excess in the bin at Mass70 in the left and righthand histograms The answer will depend strongly on whether you know a priori the mass of this resonance Assuming you knew that the resonance mass was predicted to be exactly M7l and it would be very narrow much narrower than the bins used in these histograms AM4 Then using bins other than the one centered at M7l one can estimate background rate to be B100 counts Assuming that the background in bin at M7l is the same as in the other bins it is expected to uctuate with o100B10 The excess of events in the resonancecontaining bin in the rst case is 817210072 or 726 which can be written as SxE 72039 The second histogram gives 25 excess events or S JE 25039 Probabilities p of such upward uctuations are lt103912 and 06 Both numbers are very small and one can feel con dent enough to claim the discovery of the predicted resonance If one did not know at what mass the resonance might show up the signi cance of the peaks would be very different Now we need to take into account that there are 20 bins and chances that at least one of them would uctuate upward as measured would be larger that the probability of a particular a priori predetermined bin Probabilities of none of the bin with at background uctuating upward as shown is lp20 Therefore probability of at least one bin uctuating upward is llp20 which gives 103911 and 12 One can see that the statistical signi cance of the discovery in the second case is not as striking and one would have to collect more data Introduction to W m quot1 my Particle Physics Note 13 Page 9 of 23 Enhancing Signal over Background Collecting more data Collecting more data implies a reduction in relative statistical errors resulting in a cleaner signal identi cation same histogram assuming that signal was real in the second histogram collect 10 times more data the background would be B100gtlt101000 events the excess would also grow 10fold 825X10250 events Then signal signi cance per bin would be SxB2501000796 Data cuts of ine selectioncuts One can enhance signal signi cance by using some special criteria that allow one to suppress background by a large factor while leaving the signal events relatively intact For example if background charged tracks are mostly pions one can use electronpion separation criteria e g electromagnetic calorimeter Let s assume that such criteria allow to cut pions by a factor of 10 while remain F90 ef cient to electronspositrons So statistics will be reduced but with very different factors for background and signal same histogram and assuming that signal was real the background would be Bnew BUM 10 events the excess would also decrease SneWSoldX822 events Then signal signi cance per bin would be SneWWBnew SneWVBHCWX8f76 Note once statistics becomes very small one must not use PWN Trigger online selectioncuts 1 Often one is limited not by a number events that can be produced but by the number of events one can record Then online selectioncuts trigger conditions can be applied to enhance the statistical signi cance of the signal being looked for For instance identi cation of electrons discussed above can and is often done online Introduction to W m quot1 my Particle Physics Note 13 Page 10 0f23 Signal in presence of background Exclusion Limits Let s assume you look for black holes at LHC You wait for a year and do not nd any How would you quantify the outcome of your search Failed to nd black holes sounds too lame Maybe one can say that based on the experimental data you are 99 con dent that the following statement is correct Black hole production at LHC if possible at all has a cross section is smaller than somany femptobarns The 99 con dence level CL means that you are allow yourself a 1 chance to be wrong in what you are stating In the following I give examples on how to set such exclusion limits using two different approaches By a direct analogy with the signi cance de nitions one may try to construct a probability of observing fewer than no events in assumption that the signal was s pnSn0bspkbs 0 If for a given signal s this probability is smaller than 0c the signal at that strength would be excluded This sounds good but has one unfortunate pitfall If you happened to be unlucky and you see quite fewer than b events than you would formally exclude even s0 which is a logical nonsense Introduction to W m m my Particle Physics Note 13 Page 11 of23 Method A Another way of asking a similar question Given we observed n0 events what are the odds to observe n5 n0 due to the bkgdsignal hypothesis or the bkgd only hypothesis This is sort of making bets on two possible hypotheses nu quotn b n Zpn bs S e w r n0 quot0 no nu nu bnn 7 2pm b 2 e b n0 n0 no There are conventional names for the two sums and their ratio CL Zpn bs CLb Zpn b CLS CL r quot0 quot0 CLb Assuming that that signal s20 the odds de ned this way range from 0 to 1 Introduction to W m quot1 my Particle Physics Note 13 Page 12 of23 Method B Another way of estimating an exclusion limit from not observing a signal is based on a socalled Bayes theorem Lyl 61 39 Ma where ILyl coma da paly palyiprobability that theory s parameter is a eg black hole production cross section given we have a set of measurements yi LOlailikelihood function of getting a set of measurements y if the theory s parameter is a Meyia priory probability distribution function for the theoretical parameter a which might be based on theoretical reasoning practical considerations or plain common sense At the end it always boils down to some a priori believes For example an a priori probability distribution function for signal rate can be naturally assumed to be the stepfunction zero for negative values and uniformly distributed for positive values However what is at in one parameterization may not be at in an another e g one can assume that it is the matrix element that must have at distribution in this case the rate will be zero for negative values and NOT at for positive values Bayes theorem shows this arbitrariness explicitly Using the probability distribution function paly obtained this way one can exclude regions of the parameter space with some predefined probability of making an error 0L Malinda 0 x Introduction to W m quot1 my Particle Physics Note 13 Page 13 of23 0 Given we observed 710 events what is the pdf s for the value of s in the signal hypothesis Assume at prior for 320 b Syn e l7s fsps bn0 W Pquotolb339 3 mpn0lbs nnggbn Jpn0bsvrsrds Ipn0lbsgtds 2787b 0 0 n0n 0 Exclude all sgtsx in the tail of s nu bsxn erwmx w I J39 fsds a nu bn 17 e n0 7 5x 1 Bkgd b 3 n00 n03 n06 3 quot5 a 0 5 10 15 20 signal s We can say that the a probability that signal is larger than sx is very small 0L popular choices are 1 5 and therefore we exclude this possibility with 10L con dence level 99 or 95 A scienti c paper may read in this case as follows we excluded signal sgtsx with a 95 CL NOTE Both presented approaches give identical results when we use a at prior on the signal event count in the Bayesian approach Introduction to W m quot1 my Particle Physics Note 13 Page 14 of23 Example mu m 392quot m 7 12345578ain iziamisiewiaiazn xi 7 Number of events cross Section pb Mass The plot on the left shows a histogram of reconstructed invariant masses for positivenegative charged particles in reactions p pa 8 e39 anything Assume that experimental setup was such that if resonances were to be produced at all one would record on average 1 electron positron pair per each 1 pb of the resonance production cross section The plot on the right shows the CL contour line in this case of signal cross section being higher than the line For calculating these limits I used 7K6const for all values including negative ones Note that the line is the function of mass and the wiggling results from the actual numbers of observed counts Introduction to W m quot1 my Particle Physics Note 13 Page 15 of23 Systematic errors estimation of biases biases due to theory background level andor shape signal shape biases due to event selectioncuts either at trigger or of ine levels biases due to reconstruction and corrections apparatus effects why error function tails are so dangerous in new physics searches biases due to the analysis methodology eg ignoring correlations between errors Crosschecks A good data analysis presents a large number of crosschecks and auxiliary measurements to show that an experimenter understands what heshe is doing Introduction to W m m Irv Particle Physics Note 13 Page 16 0f23 Traps ofwishful thinking posteriori adjustments Histogram Binning The choice of in width is usually based on the expected statistics of events and detector resolution however there are no strict rules And there is always a freedom of shifting bins left and right Although a priori many of possible choices are equally valid one nds that by tweaking them posteriori one can quotenhancequot the apparent statistical signi cance of a signal especially in the cases of small number of events and marginal significance Below are four histograms with the same bin width but with different offsets The data used are exactly the same set of points generated to be randomly distributed with the density of 25 events per unit of Mass One can see that by shifting bins leftright the accidental quotpea quot around Mass70 can be tuned to vary from 25 to 12 over the average background of 100 SWB is 12 to 256 Another quotoptimizationquot can be done by choosing how many bins are to be used for estimating the background By using i4 bins around the quotpeakquot at M70 in the second histogram one can take advantage of statistical downward uctuation around M55 This choice would give the average Background975 and consequently quotpeakquot significance SVB125975975286 One can play the game further and pick the quotoptima quot bin width Selection Cuts Similarly quotoptimizationquot of event selection cuts will quotenhancequot the desired signal if the optimization is based on promoting the significance of the signal posteriori rather than on a priori physics considerations Dismissing quotbadquot data Another trap one can notice that removing a particular subset of data say taken on Mondays or with crystal sample 1 or at the beginning of each data collection run or anything else makes quotsignalquot more prominent Typically this prompts one to think what may have gone wrong on Mondays that lead to quotbadquot data rather than to think what may have gone wrong on TuesdayFridays that lead to quottoo goodquot data The errors of both types do happen but such biases in thinking lead to finding real errors of the first kind more often Sometimes explanations may end up being merely plausible Obviously this may lead to biases toward quotdiscoveryquot That search is just one of many There are many ongoing searches 100s all proceeding at the same time and coming up for publications every year If one chooses 99 CL of observing signal as a sufficiently convincing criterion then heshe should not be surprised to see a few breakthroughs every year Introduction to W m quot1 my Particle Physics Note 13 Page 1 7 0f23 Solutions if you do analysis Binning When at risk typically when you expect to have or actually have small statistics use methods devised for unbinned data analysis Signi cance evaluation Never use SWB for small statisticsiuse Poisson probabilities In general make the best effort to nd the correct error distribution functions Presence of systematic errors may drastically effect calculations Selection Cuts To optimize the cuts use a priori considerations Monte Carlo generated events and if absolutely needed only a small fraction of data eg 20 apply the optimized cuts to the rest of the data no further tuning of cuts is allowed after opening the quotboxquot with the remaining data the results should include the fraction of data used for cut optimization Dismissing quotbadquot data No recipe Be aware Rules of thumb 3Gmight be a real thing or might be a statistical uke worth publishing do NOT claim a discovery more data andor independent experiments are needed 56time to get serious independent experiments are needed Introduction to Elementarv Particle Physics Note 13 Page 18 0f23 bins 0481 Q g I E I E n E E in 2U an Am an EU 7n an an Mass F cm m e R bins 15911 2 5 gt E a A E E U 20 3U 4U an EU in an an Mass 9 bins 261014 Number nl mm bins 371115 Number n wens Int t0 pigmentWV Particle thsics Note I 3 Page I9 on3 Examples of 10W statistic discoveries DoubleB neutrinoless decay by HeidelbergMoscow Experiment 76Ge gt 76Se 2e39 This implies that neutrino is its own antiparticle a la photon Energy of two electrons is known Q M76Ge M76Se 203900i005 keV Paper of January 2001 claimed the discovery of neutrinoless doubleB decay A good fraction of the collaboration did not sign the paper counts energy keV Figure 2 Sum spectrum of the 75G5 detectors Nr 1235 over the period August 1990 to May 2000 46i502kgyt The curve results from Bayesian inference in the way explained in the text It corresponds to a halflife Tg72075 1833X 1025 y 95 cl E 5 g E loo E a E so 3 E 2 n m 40 20 2050 2060 2770 0 energy keV energy keV Large Window scan gives CL70 for observing a nonzero signal at Q2039 keV Smaller optimized Window scan gives CL97 for observing a nonzero signal Introduction to Elementary Particle Physics Note 13 Page 20 0f 23 Examples of 10W statistic discoveries Signs of leptoquark at HERA ZEUS 199497 Preliminary No 3 gt 10 d lt5 3 9 E 102 39 l m 10 1 101 mm 11 3quot 249 m m r I 11Hulnnlunulinl 75 100 125 150 175 200 225 250 275 300 MejGeVc2 Introduction to Elementa Particle Physics Note 13 Page 2 of 23 Examples of low statistic discoveries Higgs at LEP One does not know Where it is but if it is there to be observed it must be at the very tail otherwise it would have been seen before 2000 ALEPH 46 for Higgs signal present All four collaborations combined 296 2002 More thorough reanalysis of the same data ALEPH 36 All four collaborations lt26 or by inverting logic there is no Higgs with MHlt114 GeV at 95 CL Online Higgs Analyses Two independent streams NN19 variables and Cuts 6039 BEHOLD 60 BEHOLD L 15415 phquot L 1545 p0 gt Nmmoo 5 5 Nmzl9900 8 5039 desms 5 5 5039 wa219611 NN Analyses CUT Analyses 189209 GeV 189209 GeV 39 39 g 40 z 40 395 L5 30 3039 El n l lag airsLN M 50 6390 7390 8390 9390 1001i01i010 quot50 6390 7390 8390 9390 1001i01i010 mmnm12m34912 GeVc2 MHGeVlcz Introduction to W m m Irv Particle Physics Note 13 Page 22 of 23 Examples 0f10W statistic discoveries Top quark at Tevatron real thing that started from 286 997 CL Debate at CDF over the title key word Discovery Study Evidence Observation Search for The jokes were Evidence for Study Observation of Search for The nal compromise was Evidencefor top quarkproduction in p barp collisions at sqrts 18 TeV Abstract The probability that the measured yield is consistent with the background is 026 Though the statistics are too limited to establish rmly the existence of the top quark a natural interpretation of the excess is that it is due to tt bar production Subsequent papers based on much larger statistics con rmed the signal and were titled Discovery of Introduction to W m quot1 my Particle Physics Note 13 Page 23 of 23 Appendix Deriving Poisson distribution and its parameters 0 Assume average rate of M events per second 0 Large time interval T expect average nMT 0 Calculate probabilities to get none one two three etc events I The time interval can be broken in MTdt small intervals probability to get one event during dt p1pdtLLTM probability of getting more than 1 is vanishing in comparison to p1 at M gt 00 probability to get no events during very short time dt p0lMdt probability to get no events during T P0p0MlMdtMlMT MM gt e39HT e39n probability to get 1 event P1CMlrp1gtp0M391 MrLLTMrp0Mp0 gt ne39n probability to get k event PkCMkrp1krpoaI39k gt k n in Pk He I Crosscheck Average 0gtP0 er1 krPk 2 kgtnkkre39n nre39quot2 nk39lkl nre39r en 11 o RMS root ofmean squared 1lltk n2gt J k2 kZP kin e w k quot ne quot k Eac w w nkil w nkil w nkil ne quot k ll ne quot k l ne quot i k l i k l k1 w kil w kil ne quotZnn ne quot quot k2 16 2 m k l w nk w nk nze quotZ ne quotZ n2 n 160 k 160 k ltk n2gtltk2gt n2 nzn n2 n
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