A Quantum Diaries Survivor

Today we had our meeting of the CMS analysis group in Padova, a monthly recurrence where we get adjourned of the various efforts going on. It was my turn to chair the meeting (I am co-convener of the meeting with Ezio Torassa and we alternate), and I had put together a tightly packed agenda, which included updates on the global cosmic runs (weeks of data taking when muons from cosmic rays are collected and used to understand the detector response), the tracker checkout (issues with the final commissioning of the silicon tracker), the trigger studies for SLHC (or how to measure muon momenta accurately enough to prevent being overwhelmed by the huge rate of fake muons of low transverse momentum, when we will take data with CMS at a luminosity of ), plus analyses of the decay, ttH production, and dimuon mass spectra.

Ignazio Lazzizzera, from the associated group of Trento, presented some kinematical distributions of muon tracks extracted from minimum bias Monte Carlo that will be used for SLHC studies. Minimum bias is a jargon that particle physicists use to describe events that withstood no selection whatsoever: events which suffered the minimum possible bias by the fact of having been collected by the detector. Such a collection of events is useful to understand what our “priors” are: at the full LHC luminosity (just a factor 10 below SLHC ones), every 25 nanoseconds we will have 20 proton-proton collisions to deal with, and only very rarely these interactions originate a high-momentum muon, which tags a potentially very interesting event. We have to rely on these minimum bias simulations to understand how easy it is for a light hadron -a pion or a kaon- to fool our detection system and be identified as a muon by our trigger, if we want to understand our chances of tuning trigger cuts and select good muons with high efficiency without being drowned in impossibly high rates from fake muons.

As Ignazio showed the plot below, which is the distribution of rapidity of simulated muon tracks in minimum bias data, I jumped on my chair. What was going on ? The two-humped distribution resembled a camel’s back!

To let you understand why such a distribution is unphysical, I need to take a step back. When you collide protons with other protons at high energy, what you are actually doing is creating hard interactions about proton constituents: quarks and gluons. Each of these constituents of a high-energy proton carries a fraction of the proton momentum: the two streams of “partons” (i.e. quarks or gluons) travel together in the positive and negative direction along the z axis - the beam direction- inside each proton; but some carry a larger, and many a smaller fraction of the total protons momentum.

Because of the variable amount of momentum carried by each parton, the collision center-of-momentum reference frame is not at rest in the detector reference frame: if a 90mph truck hits a 50mph compact car head on the debris will fly away following the truck direction!

What governs the probability that quarks and gluons carry a certain momentum fraction of the proton containing them are some functions called “Parton Distribution Functions“. They are shown below for the different constituents of protons.

As you see, it is increasingly probable (in a measured described by the PDF xf(x)) to find a parton carrying a smaller and smaller momentum fraction x (forget the u-distribution, which has a local maximum due to valence quarks: we are discussing the low-x tail of these shapes, since we are discussing not-so-high-energy interactions which constitute the bulk of collisions). Is this enough to figure out what will be the distribution of the debris, and in particular, the motion of the most energetic particles produced in the collision in the detector frame ?

Well, basically yes. If we label the momentum fractions of the colliding partons (which can be assumed massless for all practical purposes at LHC), the center-of-mass energy will be their geometric average times the 14 TeV globally possessed by the colliding protons. The motion of the center-of-momentum frame in the detector frame will instead be described by rapidity - the quantity , which reduces to .

Rapidity is, for the muons, the quantity plotted in the two-humped histogram above. Can there be a hole at zero in this distribution ? Not really! It does not take complicated math to realize that if you pick at random two values from a monotonous function, their values are most likely to be close to each other, and so their ratio will be close to one more often than not. The logarithm of one is zero, and at zero there cannot be a minimum! The distribution has to have a single maximum at zero rapidity instead!

You might find the above reasoning rather complicated. It is. However, had you worked at a hadron collider for 16 years, you would not need the math at all: the rapidity distribution of any physics process is (with very few exceptions) a broad distribution with a maximum at zero, unless the data have been biased by selection cuts.

I could thus explain what was going on in the distribution Ignazio was showing: the data he was plotting had been stripped of events which could fire the CMS trigger -that is, events with high-Pt, central muons in our case. Take a dromedar, substract stuff in the middle (the muons which are central), and you are left with a camel!

It remains to be seen why the minimum bias Monte Carlo had been selected this way. I suppose one such sample is rather useless for trigger studies!

On Saturday, May 15th, a conference called “sci.bzaar.net” will take place in Milano. It will bring together a restricted group of researchers, psychologists, bloggers, designers, physicists, writers, philosophers, computer scientists and web experts, who will discuss scientific divulgation, production of knowledge, and open culture in the academic world.

I will not be there in person, but a video I produced for the event will be shown - and I will connect with skype or some other means to take questions. You can see the agenda of the workshop here.

In addition, I produced for the web site of the event another short video where I discuss the importance of horizontality in a blog aimed at scientific divulgation. Unfortunately, I only have a version in Italian so far (the event is aimed at an italian public). I will paste below a writeup as I have the time, but if you are interested you can see me in the 7-minutes video here (beware though, it is kind of heavy - 500 Mbytes!).

Here is an excerpt of the latest LHC schedule for the following few months, as agreed in a meeting at CERN chaired by the Director-General, with the experiments and LHC machine heads.

Based on the good progress for the cool down of the LHC sectors, and on the powering tests from two sectors, the following planning was arrived at:

1. End of June: The LHC is expected to be cooled down. [...]
2. Mid of July: The experimental caverns will be closed [...]
3. End of July: First particles may be injected, and the commissioning with beams and collisions will start.
4. It is expected that it will take about 2 months to have first collisions at 10 TeV.
5. Energy of the 2008 run: Agreed to be 10 TeV. The machine considers this to be a safe setting to optimize up-time of the machine util the winter shut-down (starting likely around end of November).[...]
6. The winter shut-down will then be used to commissioning and train the magnets up to full current, such that the 2009 run will start at the full 14 TeV design energy.

The above means that the machine will deliver collisions from the end of September on, for at most nine weeks in 2008. More safely, one can assume 6 full weeks of data-taking. What luminosity do we expect to collect ?

A state-of-the-art estimate was made by a colleague, who used his past experience with LEP as well as the information on the current limitations of the RF system -which will make the proton bunches shorter than planned (RMS of 5.4 cm), and with a transverse size of 46 microns. At the lower energy the low-beta squeeze will also be loosened from 2 to 3 meters. These figures reduce the instantaneous luminosity, and the estimate for 6 weeks of collisions are of about 40 inverse picobarns of data in 2008.

If ATLAS and CMS will be fully on during the weeks of collisions, these 40 inverse picobarns will fruit, in my opinion:

• A top pair production cross section with 10-15% accuracy
• A sizable sample of vector boson decays to leptons, very useful for calibrations and checks of lepton efficiency studies
• The first estimates of b-tagging and tau-tagging capabilities of current algorithms
• no information on the Higgs
• no SUSY discovery (of course!)

All the above will have a chance of being ready for the 2009 winter conferences, if all goes well…

Here is a selected list of interesting links from blogs I read:

• Bee at Backreaction has the most complete list of reasons why you should not be bothered by the LHC destroying the Earth. Instructive, entertaining, to the point. With useful furthering of the matter in the comments thread.
• Peter at Not Even Wrong has two interesting posts out. In one he reports about Witten’s take on dark energy. In the other the question on what string theorists would do if their pet theory was proven wrong is discussed. Don’t miss the comments thread.
• Carl at Mass explains in detail why the current cosmology does not explain the angular correlations in the fluctuations of cosmic microwave background for large angles, while a changing speed of light would fit the data better. Controversial!
• Lubos at the Reference Frame discusses whether a theory that makes no predictions is to be preferred or disfavored, in relation to one that is more predictive. He also has a poll. Let’s all ask him to add a bullet, “A and B are equally unlikely because they are both favored by Lubos”,
• Jester at Resonaances has a short but poignant post on how to be a good crackpot. Recommended.
• Kea at Arcadian Functor has reached lesson 182 in category theory. Her explanations make you believe you know those things, and there are a bunch of graphs you cannot miss. Esthetically pleasing.
• Chad at Uncertain Principles has one of his imperdible dog dialogues out. Highly recommended.

Long overdue, here is the final part of a long post on the searches for new particles that may be the solution of a long-standing problem in astrophysics today: the missing mass in our Universe.

The large majority of cosmologists have become convinced, through the analysis of masses of data collected in the last two decades, that four-fifths of the matter in the Universe is non-baryonic. If we neglect particles which can only be created in high-energy collisions and decay in ridiculously small amounts of time, Baryons exists in just two forms: protons and neutrons. These make up the nuclei of atoms, and provide the fuel for stars to shine as they fuse into helium nuclei.

Non-baryonic matter does exist, and we know it well: we have electrons and neutrinos; but these are irrelevant. Electrons weigh less than a thousandth of a proton -and there are just as many electrons as protons around, to a very good approximation. As for neutrinos, despite our ignorance on their mass, they cannot make up the deficit of mass observed in the rotation speed of galaxies (exhibit one in support to Dark Matter: the speed of rotation does not decrease as much as it should if their mass was concentrated in stars) or in clusters of galaxies (exhibit two: gravitational effects we may detect visually do not match the observed distribution of galaxies in these agglomerates).

One intriguing solution to the problem lies in hypothesizing that a massive particle called neutralino wanders around in huge amounts, slow and unbothered by its close encounters with ordinary matter. Neutralinos would be electrically neutral, they would not interact strongly with matter, and they would be perfectly stable, lest they violate a very convenient quantum-mechanical conservation law. For more details on these hypotheses, see part II of this post.

So how can collider experiments detect this evanescent particle ? By producing pairs of higher-mass supersymmetric particles, which would chain-decay into non-supersymmetric ones plus a pair of those lightest supersymmetric particles, LSP. On the right you can see a decay chain whereby a gluino - a SUSY particle produced in large amounts in hadron collisions, due to its strongly interacting nature - emits a squark, the squark in turn emits another quark and decays into an excited neutralino, this emits a slepton, and the slepton ends up producing the lightest neutralino. All in all, from each of these chains (one per decay of each of the produced gluinos) one should observe two jets of hadrons from the quark hadronization, two leptons, and some missing energy. The missing transverse energy stolen by each neutralino would add as two vectors add in a plane: only rarely they would cancel each other out. In the graph below, for instance, two neutralinos leaving in different directions (the two dashed lines pointing towards the upper and lower left, in the transverse cut-away view of the ATLAS detector) would create a missing transverse energy vector pointing roughly mid-way between their exit directions.

The Tevatron experiments have searched for these events in their Run II data. The search in CDF considered the signature of two, three, or four hadronic jets plus a significant amount of missing energy from the neutralinos. This signature can be mimicked very effectively by the frequent, generic production of many jets by quantum chromodynamics interactions between quarks and gluons; the missing energy is thus required to be large and significant to suppress these processes.

The CDF experiment applied three different sets of selection cuts on their data to seek sensitivity to different regions of the parameter space of Supersymmetry. Indeed, as the mass of gluinos, squarks, and sleptons varies, so does the visible final state. For instance, if squarks and gluinos have a similar mass one is unlikely to detect a hadronic jet from the quark that is emitted in the transformation of the former into the latter. The signature pf pair-produced gluinos then more closely resembles one with only two jets and missing energy.

The figure on the right shows the final selection of the data in one of the three search regions. It is clear that known Standard Model processes provide a good modeling of the observed distribution of missing transverse energy in the data (black points with error bars), whereas a supersymmetric signal (the empty histogram in green, overlaid to SM contributions) would have instead stood out and created a disagreement.

From the distributions an upper limit can be extracted on the amount of signal contained in the data, and from the latter a limit is obtained in the cross section of gluino pair production: this translates into a mass exclusion range for squarks and gluinos. The final summarizing plot is shown below.

The plane is spanned by the mass of the two hypothetical particles. Colored areas have been excluded by different experiments; the CDF search extends the excluded region by the size of the red-painted area. We thus learn that gluinos cannot be lighter than 300 GeV, whatever the squark mass, otherwise CDF would have seen a bunch of anomalous events with large missing energy and jets.

The Tevatron protons and antiprotons do not have enough energy to investigate supersymmetric particles of mass much larger than the limit discussed above: so if Supersymmetry is the right theory of Nature, it may turn out to be the job of the Large Hadron Collider to discover it. With its 7-fold increase in energy and hundred-fold increase in interaction rates, the LHC is expected to provide a clear-cut answer: discover supersymmetry, or rule it out for good. As you can see in the plot below (where the plane is spanned by two convenient parameters among the multitude of choices: and ), the discovery reach of the CMS experiment extends to mass values in excess of a TeV - where supersymmetric particles would be close to useless, because they would not have a chance to solve the problems of electroweak symmetry breaking for which they were once invented.

The graph is complicated and it requires some more explanation: the blue areas are excluded by theoretical constraints and experimental searches, and the green area is also excluded. The colored wavy lines show instead the limits that CMS will be able to set in the plane -intending it will exclude anything to the left of the curves - with different searches, labeled by their respective “smoking guns”. The red curve is labeled for missing transverse energy, and it is one of the most performant in excluding the parameter space.

So, indeed, CMS and ATLAS will have an easy way to find signals of supersymmetry across the table -the wide space of parameters: they just need to study their distribution of missing transverse energy, just as we saw CDF do in the analysis mentioned above. The fanthom signal of a neutralino, which cannot interact with the detector and leaves unseen, turns out to be more striking at the end of the day than the multitude of jets and charged leptons the pyroclastic Supersymmetric production events would give rise to. Seeing events with a large amount of missing transverse energy would not allow us to determine which form of supersymmetry we are dealing with - whether a minimal supersymmetric extension of the Standard Model with two higgs boson doublets, or more complicated schemes. However, it would still allow us to claim that we have evidence for THE candidate particle which constitutes 80% of the stuff the Universe is made of.

I need to warn the reader here: of course, ATLAS and CMS have already studied dozens of methods, some of which are quite complicated, to extract very detailed information on Supersymmetry and very clean signatures of its presence from LHC data. These analyses focus on kinematical properties of the supersymmetric decays which are very model-dependent, and very complicated to explain. Although I reported about these methods in my seminar, I take the liberty here of jumping ahead a little…

So what instead if SUSY is not, after all, the right idea ?

Despite the general enthusiasm of theorists, phenomenologists, and other assorted believers, in fact, we have to keep a cool mind. Let’s review the cost of the purchase we have to make if we are to marry Supersymmetry:

• twenty brand-new particles, never before seen
• at least 104 new parameters, whose value is unknown and to be determined by improbable experiments
• a strict conservation of R-parity, the number you get by adding together spin, baryon, and lepton number in a suitable combination - the combination allows the proton and the lightest neutralino to remain stable
• We also have to agree that despite the fact that in principle the Tevatron and LEP colliders could have well stumbled into Supersymmetry, they haven’t - new physics chose to hide in the far away corner, just like the small coin that you dropped from your pocket.

Some of us think the above is too much to buy, for a theory which “solves” the mystery of a unnaturally small mass of the Higgs boson (provided the Higgs exists and is light as every evidence still suggests) and which collapses two crossings between running coupling constants into one single point. Ockham’s razor comes a-slashing: “entia non sunt multiplicanda praeter necessitatem“, one must not multiply entities. The most economical explanation is the best one… The razor cuts unnecessary entities.

One should mention, at the end of this long post which focused on the searches for just one candidate for dark matter - the one which hadron colliders may have a chance to find, the neutralino - that there is a long list of alternatives, of many flavors: kaluza-klein gravitons, sneutrinos, gravitinos, little higgses, axions, primordial black holes, charged massive particles, heavy neutrinos, sterile neutrinos, you name them.

It is for this very reason that in the end, LHC searches will require to follow the very important two-step procedure outlined by M.Mangano in a recent paper: first establish that an anomaly exists in the data, and only after it has been demonstrated to be utterly unexplainable by known phenomena, proceed with an exotic explanation.

To conclude, dark matter candidates have been searched at past and present collider experiments with no success. LHC appears to have the right energy and the potential to finally discover the source of this astounding enigma. In any case, we will know in a few years whether Supersymmetry is real or just a crazy concoction. If SUSY exists, new accelerators will be needed to investigate it in detail, but if it doesn’t, particle physics may be at a dead end. Despite this threatening possibility, we have extremely exciting years ahead of us!

Upon coming back from a shor vacation on the Alps, I rushed to connect my laptop to the internet. And one of the first things I did was to check for recent results by CDF. The experiment has been producing new beautiful results at an impressive pace during the last few months: it is as if the work of years of preparations, refining algorithms, tools, thinking hard at new methods, and a parallel strong push for the collection and processing of data had converged to a singularity, and now results are popping up like flowers in a garden.

My interest in the new analyses is boosted by the fact that in less than a month I will be describing them at PPC2008, a conference in Albuquerque where I am going to give a talk on new CDF results. So it is about time for me to start thinking about the organization of my talk.

As I browsed the recent talks in the Higgs Discovery Group, I found a new blessed (i.e., internally approved for public consumption) result that warmed my heart. It is the first Run II limit on associated Higgs boson production based on the 4-jet signature of WH or ZH decay. This signature arises when the Higgs boson is produced by the process called “higgs-strahlung” off a virtual W or Z boson, and both bosons then decay in a pair of hadronic jets (see picture). The Higgs, if it is lighter than 135 GeV, most of the times decays to a pair of b-quarks (in red), while W and Z bosons decay to all available quarks (in blue) more democratically.

Hadronic decays of vector bosons are the most common ones: W bosons decay to two quark jets 66% of the time, and Z bosons 70% of the time. So, with a large fraction of Higgs bosons also materializing into two jets, looking for four-jet final states to see a WH or ZH signal might look like a no-brainer. Quite the contrary!

Indeed, the 4-jet final state has always been considered absolutely hopeless. 4-jet events are among the most common final states of a proton-antiproton collision, and the kinematic handles one can use to try and discriminate associated WH or ZH production from generic QCD 4-jet production are absolutely insufficient. One can consider the invariant mass of pairs of jets, in the knowledge that W, Z, H all have a well-defined mass, while QCD produces jet pairs without any constraint on their common mass.

Hopeless, in particle physics, is a very attractive word for some of us. Out-smarting our colleagues is one of the highest forms of satisfaction in a scientific workplace… So, after my group demonstrated against all odds the possibility to see top pair decays in their 6-jet final state (one that arises when both W bosons emitted in the chain decay to jet pairs), in 1996, we started thinking at what would be the best way to exploit the experience we had formed in reconstructing high-mass states with jets.

One branch had already born fruit: my PhD was already in full swing, and I would show a first signal of decays soon thereafter. But that is another long story. Instead, in 1998 we started working at the idea of reconstructing the WH or ZH signal in events with four hadronic jets. In Run I the analysis had already been undertaken by Juan Valls and Jorge Troconiz, and they had indeed produced a fine piece of physics, with a limit on Higgs production which challenged those in the “golden” leptonic channels.

We aimed at Run II, and started working at the most critical issue: the one of triggering on 4-jet events with b-quarks. The multijet trigger which had been the basis of both the and the analyses was very inefficient on the latter signal, because of inefficiencies in the online jet reconstruction.

Enter the SVT (silicon vertex tracker), a fantastic device which measures online the impact parameter of tracks, allowing the collection of B-decays with high efficiency. SVT had been designed for B-physics purposes and was thus aimed at low-energy events, so we needed to verify it would work fine for 4-jet events too. This implied determining that those complicated, high-track-multiplicity events were reconstructable in the 20 microseconds available for a trigger decision at Level 2; and then designing a set of selection cuts that would allow the maximum efficiency on signal events while keeping the data acquisition rate at an acceptable level. In parallel, we also studied alternative strategies involving the semileptonic decay of B-hadrons, by combining jet signatures with soft lepton detection.

This job kept us busy for three years, and fruited a graduation to Giorgio Cortiana, a PhD to Luca Scodellaro and Mario Paolo Giordani (and I am certainly forgetting some other students). But as Run II started for real, and multijet events started being collected with high efficiency, we gradually lost interest: Luca Scodellaro’s analysis had shown that the signal was really, really hopeless. Too hopeless even for us - or maybe we were already growing old and disillusioned ?

The recent analysis by Song-Ming Wang, Rong-Shyang Lu, and Ankush Mitra (Academia Sinica), Daniel Whiteson (UC Irvine), and Aart Heijboer and Joe Kroll (University of Pennsylvania) shows otherwise. Sure, they do not reach a sensitivity sufficient to exclude Standard Model production of WH and ZH events in any region of Higgs masses, but they nevertheless extract an excellent result which will be successfully combined with the other searches, improving the global Tevatron limits on Higgs production. Since this post has become much longer than I wanted, I will only describe it shortly, and jump to the results.

The analysis selects events with four jets, two of which have to contain a signal of B-hadron decay, and then uses a Matrix-Element approach to determine the probability that the observed final state is the result of the decay of a WH or ZH pair, and the probability that it is instead due to background processes. The information is merged in a discriminant which separates the processes on a statistical basis. One thus ends up fitting the distribution of the discriminant as a sum of background and signal, as in the plot below.

To put in evidence the small contribution from top pair production (in blue), diboson and single top (in green), and WH/ZH processes (in red), a logarithmic plot is appropriate:

As you see, the signal would contribute mainly in the right part of the distribution, but with a tiny fraction of the events: Standard Model predicts a contribution of less than 10 events in a sample of more than 20,000.

The maximum amount of signal allowed by the fit determines a limit on the production cross-section of Higgs and vector bosons. The limit on the cross-section depends on the Higgs boson mass for two reasons: one is the increase in collection efficiency as the Higgs mass grows, and the other is the decrease in Higgs branching fraction to b-jet pairs. In the end, one obtains a limit on the ratio between cross section and SM expected cross section, as a function of Higgs mass. The limit is always larger than 1 -it actually is higher than 30- so no Higgs mass is excluded by this search. It is shown below with a red line; the limit the analysis would predict to set, based on pseudo-experiments, is shown by the hatched black line and 1-sigma and 2-sigma yellow and green bands.

This result really makes me feel that the work we did eight years ago was not wasted!

If you just visited this blog (that is as I post this message, between 11.40 and 11.50PM on May 1st), you have a 10% chance of having generated its 500,000th view. Sorry, no red carpet, band with trumpets, or prize.

I believe about a third of the visitors are colleagues with some degree of parenthood -meaning they work in the same field I do, or similar ones. The rest are a 50-50 mix of non-physicists who are just interested in science, and occasional visitors who are not likely to hang around.

While I do enjoy the increased interaction I obtained in these years with fellow physicists, particularly theorists and people from whom I have a chance of learning something new, the class of readers that are dearest to me are the non-physicists who try to understand physics. It is to them that this blog is mostly aimed at.
Of course, I not always manage to write something that is both at the right level and interesting enough for them, but I do try to.

In any case, I thank all of you who visit this blog occasionally or regularly for giving me the encouragement and the stimulus to make this site worth the time I spend making it better and keeping it -hopefully- interesting and informative. I also use this occasion to encourage any of you who has something potentially worth a post, to submit it to me. You can get a feeling of what guest posts here may be by looking at the “guest post” page up here.

During the last ten years I have graduated 11 undergraduate students in Physics, plus tutored four PhD students through to their title. Despite this variety of personalities that have crossed my path in forming their credentials as physicists, there is one single example of “my student” which stands above all, for continuity and results, and that example is Giorgio Cortiana.

Giorgio joined our group in 2000 as a summer student at Fermilab, and he worked during the months of August and September with me at the design of a trigger we were putting together to collect Higgs bosons in the forthcoming Run II. Following the positive experience, he asked our group for a thesis in CDF, and worked with me at the same topic, a multijet trigger for Higgs events.

He graduated with the highest score, and entered Padova’s PhD program at the end of 2002. CDF data was just starting to pour in in reasonable amounts, and Giorgio’s PhD time span was well-placed to allow us to invent something new. We started working at a search for top pair decays including tau leptons and jets, a channel nobody had ever considered due to its apparent trouble -a huge background from QCD events. We, however, were soon convinced our search could yield a pleasant surprise.

And indeed we struck gold when, in early 2004, we found out that by extending the search to an inclusive signature of missing transverse energy and jets -which allowed to include events where one of the top quarks decayed to an electron or a muon which failed the tight lepton identification criteria- we soon obtained a large signal of events that other searches had totally ignored.

With the data we had selected, Giorgio and I obtained CDF’s third-best measurement of the top pair production cross-section, and we soon published a paper on Physics Review Letters. In the meantime, Giorgio also obtained his PhD, which was soon followed by a research grant to continue working with our group in Padova. The plan of the grant was to measure the top quark mass with the decays he had collected in the inclusive missing Et plus jets search: he did it very effectively, and he published another nice paper in record time. While he was doing that, he also had his hands full in a new re-design of the CDF calorimeter trigger, again focused on a more efficient collection of Higgs events. He took an important role in the project as responsible for the monitoring of the trigger, and his group completed the task in due time: CDF now has a much more effective identification of jets at trigger level 2, and this means a sizable increase in Higgs sensitivity.

Despite these successes, we had to witness once again how Italy is not generous with young researchers. Bright, young and able, with the highest academic title in his pocket Giorgio -as hundreds like him- is deprived of job security, and has to accept a salary which in other countries would be refused by a graduate student. So he recently started looking for a better position outside Italy, and he of course found one very soon. He gave a farewell seminar in Padova last week (if I have a chance I will describe his interesting talk here), and he is now off to Munich, where he is joining the ATLAS group. ALAS, I would say, since I at least hoped he would end up in a CMS group instead: that would have allowed me to continue collaborating with him…

The best of luck to Giorgio then. I am sure he will be appreciated in his new group. In the meantime, I have to reckon with a thinning group of collaborators: Julien left three months ago… To ATLAS too!

Jeff is a physics professor at the University of Cassino, and a long-time colleague and friend of mine. He worked in the SLD and CDF collaborations as a particle physicist, but later moved on to study radiation damage on silicon detectors for particle and astroparticle applications.

Besides admiring him for his wicked sense of humor, which he uses to make the workplace around him always a pleasant place, I have the highest esteem of Jeff as a professor, because he is quite skilled in explaining physics concepts in simple terms. He always looks for the most intuitive way to understand things, as you might appreciate in the contribution he offers below.

The following describes a very elegant and simple derivation of the relativistic formula for the addition of velocities, .

It is due to David Mermin. I fell in love with it and have been telling it for the past four years now to the students of my general physics course. The students are first year telecommunications and electrical engineering students. Before sitting in on my course all of them have heard about Einstein and most of them heard the expression “the velocity of light is constant”. I do not have the time to discuss special relativity in detail. My course is quite traditional. I discuss reference frames, inertial frames, Galilean transformations and covariance of Newton’s laws. I then point out that when describing mechanical waves the frame that is stationary respect to the medium is a special reference! In particular the wave motion can be made to disappear by moving respect to the medium with a velocity equal to that of the wave. It is clear at this point that the constancy of the velocity of light cannot be understood by assuming Newton’s laws and then modeling light as a mechanical wave in a medium (the ether). I then restate the constancy of the velocity of light and begin Mermin’s derivation.

The derivation uses:

• only one reference frame (no use of Lorentz transformations),
• simple kinematics (always good to brush up on),
• the constancy of the velocity of light (something that every telecommunications and electrical engineering student should know),
• the idea that some things are invariant; i.e. while many quantities are relative, observers will agree on some absolutes.

Consider a train of length L moving along the x-axis at a constant velocity v respect to an inertial frame of reference (the observer watching the events unfold). At the trailing end of the train a loaded gun is aimed in the forward direction and fired at time : the bullet and flash of light emerge and travel in the forward direction with different speeds: w the velocity of the bullet, c the velocity of light. A mirror at the front end of the train reflects the light back towards the advancing bullet. Let f be the fraction of the length of train that the reflected light travels before meeting up with the bullet. The constancy of light (Einstein’s dictum) tells us that the velocity of light in the forward direction is equal to the velocity of light in the backward direction; i.e. .

The space-time plot looks like this:

Let be the time for the light flash to reach the forward-going mirror and be the time the reflected light needs to return from the mirror and meet up with the forward-moving bullet. Simple kinematics allows us to label the space-time plot:

Simple algebra:

It is important to note that the expression for f we just obtained is valid if the velocity of light in the forward and backward direction are equal. Note:

• A classical pre-Einstein physicist would say this expression is valid only if the observer is stationary respect to the ether frame.
• On the other hand Einstein says that any inertial observer would use the same velocity of light; i.e. Einstein tells us that this expression is valid for any observer (generic inertial frame).

Following Einstein we consider a particular observer (frame), one that is moving along with the train. For this observer the velocity of the train is . For clarity let us use the symbol u to indicate the velocity of the bullet with respect to this observer; i.e. with respect to the train.

Suppose the train has 10 windows and the reflected light and the bullet meet up at the third window from the front (). It is important to realize that all observers will agree on the value of f. The fraction f is an invariant!

The constancy of the velocity of light allows us to impose the invariance of f the following way:

Q.E.D. !

I have wanted to write some version of the present post for a while, and so it is a relief to publish it at last. In fact, it is rather strange to have completely avoided discussing in my blog the problem I have invested the best part of my research time in the last three months -plus a fair share of last year’s thinking-, and it was due time that I filled that void somehow.

Unfortunately, strange as it may seem, there are topics in my research activities that are hard to explain in simple terms. The problem I have been working on is not difficult to state, nor too difficult to solve, but it is extremely complicated and varied, so that a comprehensive description is challenging. However, I want to make an attempt…

The problem I have been dealing with, together with a small and focused group of bright colleagues (Sara Bolognesi, Marco De Mattia, and Chiara Mariotti: lads from Padova and ladies from Torino University) is the one of calibrating the momentum of charged tracks detected by the CMS experiment at CERN.

After being produced in a proton-proton collision in the core of CMS, charged particles have their position measured in a dozen layers of silicon detector before they hit the calorimeter system; the few penetrating ones surviving the encounter with trillions of heavy nuclei are also detected by the large set of muon chambers situated outside them. With the information provided by the silicon detectors -and, for muon candidates, by the muon system- a very performant and refined software algorithm reconstructs and fits the trajectory of the track, providing a measurement of the five parameters describing the helical trajectory; most notably, the curvature inside the solenoid, which yields a precise determination of transverse momentum through the formula (where B is the magnetic field intensity -about 4 Tesla- and is transverse momentum).

There are a number of reasons why a precise determination of the momentum of charged tracks is crucial. Let me just flash a few:

1. Charged particles are measured with a better precision than neutral ones, and a careful determination of their momentum allows to calibrate in turn other parts of the detector.
2. Some physics measurements such as the mass of the W boson rely heavily on track momentum.
3. The identification of a high-mass resonance -say a new Z’ boson- may require the reconstruction of its decay, and a scale error on the momentum of those high-energy tracks translates in a worse resolution in the Z’ mass, and a diminished discovery reach.
4. B-physics crucially needs charged tracks to be precisely reconstructed in order for exclusive B decays to be extracted from backgrounds.

So how do we do it ?

We use resonances. A few neutral particles -vector mesons and the Z boson- decay to pairs of muons, and they can thus be extracted with small backgrounds from the data (events with two muons are easy to collect with CMS, and muons have the benefit that they are “perfect” tracks in several ways). We know the mass of these particles with great accuracy, thanks to previous experiments:

• The Z boson mass is known to be , a 0.023% precision.
• The Y(1S), the ground state of the vector meson family, has its mass known as , a 0.0028% measurement.
• The Y(2S) mass is , a 0.0031% measurement.
• The Y(3S) mass is , a 0.005% measurement.
• The J/Psi, the ground state of the vector meson family, has its mass known as , a 0.0004% measurement.
• The Psi(2S) has mass , a 0.001% measurement.

All the above particles are easy to trigger on, collect, reconstruct, and measure. With CMS we expect to collect thousands of these decays every day of running. Their mass can be measured on a event-by-event basis by reconstructing the momentum of the two muons they decayed into, using the relativistic equation

where M is the resonance mass, E is the muon energy, and P is the muon momentum vector.

By comparing the average mass of each reconstructed resonance to the reference values above, we get to know the scale of our momentum measurement, ; every time we measure a momentum we then do , forget P, use P’, and we are done. Easy enough, wouldnt’ you agree ?

Sure. Easy enough. But actually kind of lame. With the millions of dimuon resonances we collect, can’t we do something better ? Our detector is, in fact, a quite complicated set of devices. The momentum scale -or, to be precise, the bias on the momentum measurement- depends on very subtle effects, such as tiny distorsions in the magnetic field generated by the 4-Tesla solenoid, occasional mis-alignment (by a few microns, that is) of one of the thousands silicon sensors, erratic behavior of the reconstruction algorithm in very particular regions of the detector. We can, and we must, check the bias on our measured momentum more closely, because it in turn gives us a chance to verify the B field map, check the alignments, validate the reconstruction code.

In the simplified formulas described above to determine a corrected momentum P’, you might have noticed that we used the invariant mass of the two muons making the resonance, rather than each muon separately. Indeed, the decayed particle is not produced at rest in the laboratory frame of reference, so we cannot expect that the two muons share evenly their parent’s energy, M/2 each. Only by combining their momenta can we get a number to compare to the reference value. Or is there a smarter way ?

There is a smarter way. Strangely enough, to my knowledge it has not been used in the past for this application. Let me explain in short what it is. I will try to make this as simple as possible, but not simpler - in Einstein’s style.

In the formula for the relativistic mass above enters the energy and momentum -or better, if you allow a slip into special relativity jargon, quadrimomentum. We can, in purely symbolic terms, write:

where we have made explicit the fact that the computed invariant mass is a function f of the quadrimomenta of the two muons, and that each of the two quadrimomenta is in turn a function of many (i, in the formula) other variables, collected in two i-dimensional vectors x . These variables are the measured characteristics of the track: its angles, the region of the detector it crosses, its electric charge, you name them.

Still here ? Ok. The next step is to realize that what we really would love to have is a measurement of the momentum as a function of the particular characteristics x of the track, and not just , which only depends on the curvature . Through a knowledge of we could get sensitive to the effects mentioned above -B field distorsions, alignment errors, reconstruction biases.

There is a simple way: we can compute the probability that we observe a mass M, if the reference value is , as a function of the measured quantities of each muon, by assuming a functional form for the way the momentum P depends on the parameters. So let us write:

where the new function g( ) describes how the momenta vary with the vector of measured track parameters , and is a vector of unknown variables describing the function g( ).

(To let you understand what the heck I am talking about, assume that your detector measures a track momentum with a bias depending on momentum itself:

,

with , and , . This function describes momenta which are underestimated by 0.2% for small P, correctly estimated for P=10 GeV, and overestimated by 1% for every additional increase of P by 50 GeV. )

Using the parametrization, we compute for each event the measured mass as a function of the variables . WIth these numbers we finally form a likelihood function:

which of course implicitly depends on the functional form we have chosen for g. By maximizing L as a function of the parameters, we obtain their most likely values, and we are done: we get to know how our track momentum depends on its characteristics .

In the discussioon above I have not given much emphasis on the fact that the true form of the “bias function” g( ) is not known. One can in fact test different hypotheses with the data, and the value of the likelihood will be a measure of how well they describe the experimental situation. There’s more: the likelihood can be studied as a function of each of the components of the vector x, allowing to spot biases which require a more subtle parametrization.

The above discussion is a simplified view of the problem: In reality, things are much more complicated. Here is a short list of details I hid under the carpet above:

• We model the probability to observe a given mass in the likelihood function by convoluting a Lorentzian function (the Breit-Wigner, which is the true form of the mass distribution of the resonance) with a gaussian resolution function; the gaussian has parameters which also get fit simultaneously with the bias parameters . The figure below shows the probability distribution function of a measurement of mass M and resolution for a Z boson: for each point in the plane, defined by the two values , the probability is the height of the surface. Notice how the probability grows as the resolution increases, for values of mass very far from the true resonance mass (for instance, for a mass of 71 GeV-the left boundary of the surface), while the opposite happens for values of mass close to it.

• the fitter also assumes a functional form for the background (which is unavoidably included in the dataset containing the resonances), and fits it together with the bias and resolution parameters;
• Each of the six considered resonances can be fit individually, or all together. The window around the peaks defining events used or not used in the computation requires an optimization;
• The fitter iterates several times the whole procedure: after bias parameters are extracted, momenta get corrected, and a new parameter extraction must return values which are compatible with no bias.
• And so on…

The algorithm is indeed quite complicated. I spent the last three months implementing the fitting of resolution and background, and the algorithm is not yet complete but it now works well. It is particularly satisfactory to be able to launch the program on a set of resonances, and extract all at once not just the parameters that allow momenta to be corrected, but also a precise estimate of momentum resolution as a function of track kinematics - something that would once require detailed studies with simulations. All is now squeezed out of the data!

The work is far from over. With the help of my colleagues, we will test the code on a very large sample of simulated events in the next few months, to be ready for the data which will hopefully start pouring in this fall… But the work will only be started then: we plan to fit chunks of data on a monthly basis, checking the stability of the detector and the track reconstruction, and producing a correction function to be used by all analyses in need of a precise momentum measurement… It really is a long-term plan!