A Quantum Diaries Survivor

I cannot refrain from commenting this news, which I read today on www.repubblica.it .

Maurizio Pusceddu, a 29-year-old italian man from Sardinia (a large island in the Mar Tirreno), who raped and brutalized for weeks his ex-girlfriend in Hannover, was found guilty last year of rape by a jury in Buckleburg. The sentence for the barbarous acts was however softened by the extenuating circumstance of his “ethnic and cultural background“. The story -and the motivation of the sentence- was made public only now, one year after the verdict, because the lawyer of the man has requested his transfer to an italian prison.

I do not know whether to laugh or to cry. Maybe doing both would be appropriate. I think I understand the logic behind the sentence, but the distorted view the judge has of Sardinia, which has never seen a Taliban domination, is amazing. Here is an approximate translation of the relevant part of the sentence:

“In this context, one has also to keep in mind the particular ethnic and cultural background of the accused. He is from Sardinia. The situation of the role of man and woman existing in his country cannot be ascribed as an excuse, but it has to be taken into account as an extenuating circumstance.”

About one year ago D0, the Tevatron experiment competitor of CDF, announced they had obtained for the first time evidence for the long-sought standard model process whereby a single top quark is created in proton-antiproton collisions. Top quarks are less frequently produced alone than in pairs, and the event then possesses fewer characteristics useful to distinguish it from the large backgrounds. Indeed, the analysis methods used by D0 involved neural networks, multivariate approaches, the heaviest machinery.

CDF of course did the same, but was less lucky last year. Experimental searches nowadays have reached a comfortable level of belief in their Monte Carlo simulations, and they usually present along with the significance of the observed effect also an estimate of the “expected” significance, obtained through the application of the same analysis methodology to large pools of simulated experiments. Last year CDF should have found a better result than D0 in terms of significance of the signal but, although both experiments were expecting to be on the verge of finding the coveted “three-sigma” effect, D0 fell on the right side of the net and CDF on the wrong one.

WHATSA THREE-SIGMA ?

“Three-sigma” means that you observe an effect which, if attributed to known processes other than the one you were looking for, happens by random fluctuations only about three times in a thousand. To make a simple example, suppose you count events with certain characteristics in a given dataset, expecting to see 100 from known background sources. You see 130: that is a surplus of +30 events, which is unlikely to be due to a fluctuation in the sample size. Usually, event counts follow Poisson statistics, which basically says that the variance of the 100 events is nothing else than , i.e. 10. A Poisson distribution centered at 100 with a width of 10 is basically a gaussian function, which dies out quickly as you move away from 100 on either side. How quickly ? Well, you expect 68% of the distribution to be contained in the [90,110] interval - “1-sigma”; 97% to be within [80-120] - “2-sigma”; and 99.7% to be within [70-130] - a “three-sigma” inteval.

Now, we could go on, but a “three-sigma” or larger effect is usually called “evidence” by particle physicists looking for a particle decay signal. It means the data really fights with the interpretation of containing only background processes you have already accounted for when you estimated your central value (100 in the case above). A separate word is reserved for “five-sigma” effects, which are a really tiny probability of being due to accidental background fluctuations: in that case the effect is called “observation” of the sought particle.

Back to CDF, D0, and their single top searches: you now see what it means to be “lucky” in a search for a particle in a counting experiment: if your data contains background plus signal, and you expect that the size of the signal is sufficient to reach the “three-sigma” excess with respect to background alone, you are not necessarily going to find exactly that: your data might have fluctuated high or low, and the “evidence” might turn in a more robust or weaker signal.

CDF in fact expected to see a 2.6 sigma effect last year, but they got much less than that. Now, with 50% more data as much data analyzed, they expected to find a round 3.0-sigma effect in one search for single top events using a very refined matrix-element technique. And they found a 3.1-sigma excess, finally. I will describe the analysis in short below, but first I want to discuss the production of single top events.

TWO WORDS ABOUT TOP PRODUCTION

1- pair production

At the Tevatron proton-antiproton collider, top quarks are produced in pairs by strong interactions through the diagrams shown below. The most probable process -responsible for the creation of 85% of the top pairs- occurs when a quark in the proton hits an antiquark of the same flavor in the antiproton. The two annihilate, producing a highly off-shell, “time-like” gluon which materializes in a top-antitop pair. “Time-like” just means that it is drawn as a particle propagating in the direction of time.

The other pair-producing process is called gluon fusion, which accounts for the remaining 15% of the top pair production cross section. Two energetic gluons from the proton and antiproton fuse together. The rest is similar to the former diagram.

In the two diagrams time flows from left to right, and space is represented by the orthogonal direction. That is the reason for calling them “time-like”, but it is a detail on which I will not elaborate, although the way diagrams are drawn has a lot to do with the resulting computation of the probability of the processes. Instead, I note in passing that the relative importance of the two diagrams strictly depends on the fact that we are asking what is the initial state of the collision given that a top pair has been produced: in fact, most proton-antiproton collisions involve gluons rather than quarks. However, at the Tevatron the energy necessary to produce two top quarks is a sizable fraction of the total available: and if you fish out of the proton a constituent carrying a large momentum fraction, it is likely to be a quark. At the LHC, curiously, the relative importance of the two diagrams is reversed: 15% quark-antiquark annihilation, 85% gluon fusion. A coincidence, due to the much smaller energy fraction required to make a top quark pair there.

Now, how can we produce a single top quark ? These things do get produced in pairs in strong interactions. Strong interactions are flavor-blind: since they are mediated by gluon exchange, and gluons are only sensitive to the color charge of bodies they interact with, they do not distinguish top quarks from other ones of the same color. You cannot create a top-anticharm quark pair (say) from a gluon, because gluons are not able to change the flavor of quarks.

“Wait a moment”, you could now say. “Ok, you convinced me that with QCD you cannot create single top quarks. But is it not possible to find a top quark inside one of the two projectiles -say the proton, and propagate it to the final state through a space-like graph ? The thing would exchange color quantum numbers with a parton in the antiproton, retaining its flavor. In the final state we would have one top quark and another parton.” (see plot on the right).

That is a good idea, but unfortunately while it is indeed possible to find in a proton a quark (or whatever other particle) which has normally a mass larger than the proton altogether, for a top quark this probability is extremely small. In order for the energy of the system to be conserved, your virtual top quark must be far off its mass shell. And the farther it is, the shorter the time it may exist inside the proton. Now, when you hit the proton with another hadronic particle, what you are effectively doing is “illuminating” it with a stream of partons. You are taking a snapshot of the instantaneous composition of quarks and gluons contained in the pictured body. A virtual top quark will almost never show up in your snapshot, because of the almost vanishing probability of popping out of the vacuum – or equivalently, the small time on average it spent on duty.

Since I felt inspired today, I cooked up a picture showing a proton and what you may find inside at a given instant. You have three “valence” quarks - the red, green, and blue points - providing the proton with its invariant characteristics: zero net color, unit electric charge (1=2/3+2/3-1/3), and half unit of spin (1/2=1/2+1/2-1/2, say, in the direction chosen as your measurement axis). You have gluons flying around, and being emitted and absorbed by themselves or by other quarks (the bicolored wiggly lines). And you have quark-antiquark pairs popping out of the vacuum for brief instants (the points close together).  

Then, in the second picture you see the momentary creation of a virtual top-antitop pair. One of them is drawn larger, for no particular reason other than drawing attention. Also, another criticism is that quark-antiquark pairs popping out of the vacuum ought to have vacuum quantum numbers, so no net color; however, they can be the result of a virtual splitting of gluons, so that is not a real error. In any case, now that I look at the picture in detail, it is not that interesting. Let us move on.

By now, you should have accepted that QCD allows you to get only an even number of top quarks (when I say top quarks, I count antiparticles too) out of a proton-antiproton collision. Four is actually possible, but very, very unlikely because of the large energy required. So let us instead find out what it takes to produce a single top.

2- single top production

The only way you can end up with a single top quark is by electroweak interactions. The W boson carries “charged-current” weak interactions, and indeed, it has the ability to create pairs of quarks of different flavor: weak interactions do not conserve flavor quantum numbers as QCD. You can then envision the processes pictured below, both yielding a single top quark (plus something else) in the final state. The first is “space-like” interaction between a gluon from one projectile and a W boson from the other, ; the second is the production of a W boson which subsequently decays into a top-antibottom pair, . In both cases, you end up with a top quark line in the final state (in red). Also note that the presence of two W-t-b vertices in both diagrams. We will come back to their significance later. Oh, also note that due to a unfortunate labeling (by Mandelstam) of s-channel and t-channel, they are the opposite of what you would think: s-channel is time-like, and t-channel is space-like… It sucks, I know!

I can hear somebody scream: wait, how can in the s-channel diagram (right) a W (whose mass is about 80.4 GeV) decay into a top quark (whose mass is more than double, 171 GeV), with a b-quark (mass 4.5 GeV) thrown in to boot ? Well, the W in the process is not on mass shell: it is produced with a high virtuality –that is to say, with a mass quite different from the nominal one. It does not matter, as long as it is an internal leg in the diagram. It happens, we can compute it, it does not violate any rule.

If you compute the cross section - i.e., the probability - for producing a single top quark by the diagrams shown above at the Tevatron, you come up with a number which is half the one for pair production: roughly 3 picobarns. That means that in a dataset of 1.5 inverse femtobarns (a femtobarn is a thousandth of a picobarn) you expect to have produced no less than 4500 single top events! How come, then, that we are only seeing a three-sigma evidence ? 4500 events are a lot of dough. The question is answered below, where I describe the troubles with the analysis.

THE SEARCH FOR SINGLE TOP IN CDF

Well, not “the search”, but “this search”. Indeed, we have searched for single top production for many years now, and with many different techniques. The one I am reporting about today is the most successful so far, the one that allowed us to find the long-sought evidence.

If you look at the production diagrams, you see that a single top quark is usually accompanied by a b-quark. You thus get either four hadronic jets in the case the top decays hadronically: with each of the three quarks producing a hadronic jet, plus a fourth from the other b-quark; otherwise, as shown in the diagram on the right, you get two b-quark jets and a lepton-neutrino pair from the W decay: .

The 4-jet final state is impossible to detect, since it is mimicked by strong interaction processes yielding four partons in the final state, and these have collectively a cross section that is four orders of magnitude larger. So you are left with the leptonic final state: a so-called “W+2 jets” signature. The W signal is not hard to extract from the data, since high-energy electrons and muons are seen with ease in CDF, and the energetic neutrino also leaves a striking imbalance in the transverse energy budget. However, discriminating from $latex p \bar p \to W b \bar b$ - a process that can happen without the creation of top - is hard. And not much easier is to discriminate the signal from top pair production: because the additional W boson can escape undetected.

So we have large backgrounds, and their kinematics is not very different from that of our signal. Usually, one would try to find the best kinematical variables and use their value to select a signal-enriched sample. But that is by and large the past! New technologies and more confidence in the Monte Carlo simulations of signal and background processes allow much more refined techniques.

In the new and very successful CDF analysis, authored by my ex-co-convener (or co-ex-convener ?)  of the jet energy and resolution working group Florencia Canelli (now FNAL), together with Peter Dong, Rainer Wallny, and Bernd Stelzer (all from UCLA), the knowledge of the signal production mechanism is exploited to the utmost, by taking for each event the kinematics of all measured objects (jets, lepton, missing energy) and computing the probability that it arises from any of the possible configurations and energies with which single top events are produced. That is to say, use is made of the matrix element of the sought process  as a probabilistic weight for the event, once experimental transfer functions that modify energies and angles of the detected final state bodies have been taken into account.

The same is done for all the main backgrounds: , which produces b-jets with a similar rate, and other processes yielding W bosons and jets.

A Neural network classifier is used to determine the likelihood that there is b-quark content in the jets. The output is a number , which is used together with the matrix element information in a global discriminant:

This event-probability discriminant is shown for the various processes in the plot below.

 

You can see that EPD is close to 1 for most of the single top signal, while backgrounds are more likely to peak at zero.

Once computed for the 1078 CDF data events passing a selection requiring a W+2 jet topology, the EPD is fit as a sum of signal and the concurring backgrounds. Backgrounds are constrained in normalization to the Monte Carlo prediction for their rates, and systematic uncertainties are taken into account with nuisance parameters in the fit. The output is shown below: the red area corresponds to single top events, and it amounts to a cross section $\sigma(t) = 3.0^{+1.2}_{-1.1} pb$.

The inset shows the region at high EPD, where in red you see the excess due to single top events. The experimental data is shown by the black points with error bars. 

From this measurement, it is straightforward to derive a measurement of the Cabibbo-Kobayashi-Maskawa matrix element , a number that specifies how likely it is that a W boson couple to a t and a b quark line. The cross section for single top production is in fact proportional to the square of that element. CDF finds , where the second uncertainty is theoretical and it arises from the uncertainty in top cross section dependence on top quark mass, and other modeling details (fragmentation and renormalization scales, alpha_s value).

One more nice result in the bag for CDF, and a sigh of relief… This signal had to wait for a long time to emerge!

UPDATE: Tony Smith, in a comment below, asks for the distribution of reconstructed top quark mass of candidate events with a high value of EDT, a plot which last year caused some discussion (echoed  for the D0 analysis), given that it showed some excess at 140 GeV which could fit Tony’s hypothesis of a top quark at that mass value. Here is the updated plot:

For a ghost signal, I must say this 140 GeV top quark issue is hard to die… One bin up, one bin down, and there still is something to talk about! And it all started about 15 years ago… For more information you can read some details of the analysis in teh conference note of the analysis.