Here is a list of noteworthy pieces I published on my new blog site in May. Those of you who have not yet updated their links to point there might benefit from it…

Four things about four generations -the three families of fermions in the Standard Model could be complemented by a fourth: a recent preprint discusses the possibility.

Fermi and Hess do not confirm a dark matter signal: a discussion of recent measurements of the electron and positron cosmic ray fluxes.

Nit-picking on the Omega_b Discovery: A discussion of the significance of the signal found by DZERO, attributed to a Omega_b particle.

Nit-picking on the Omega_b Baryon -part II: A pseudoexperiments approach to the assessment of the significance of the signal found by DZERO.

The real discovery of the Omega_b released by CDF today: Announcing the observation of the Omega_b by CDF.

CDF versus DZERO: and the winner is…: A comparison of the two “discoveries” of the Omega_b particle.

The Tevatron Higgs limits strenghtened by a new theoretical study: a discussion of a new calculation of Higgs cross sections, showing an increase in the predictions with respect to numbers used by Tevatron experiments.

Citizen Randall: a report of the giving of honorary citizenship in Padova to Lisa Randall.

Hadronic Dibosons seen -next stop: the Higgs: A report of the new observation of WW/WZ/ZZ decays where one of the bosons decays to jet pairs.

I read with interest -but it would probably be more honest to say I browsed, since I could understand less than 50%- a preprint released three days ago on “The hidden charm decay of Y(4140) by the rescattering mechanism“, by Xiang Liu, from Peking University (now at Coimbra, PT). The Y particle has been recently discovered by CDF.

The existence of the several new resonances of masses above 3 GeV recently unearthed by B factories and by the CDF experiment poses a challenge to our interpretation of these states as simple quark-antiquark bound states, because of their properties -in particular, their decay pattern and their natural widths.

Already with the first “exotic” meson discovered a few years ago (and recently measured with great precision by CDF), the X(3872), the puzzle was evident: at a mass almost coincident with twice the mass of conventional charmed mesons (states which are labeled “D”, which are composed of two quarks: a charm and a up or down quark, like or ), the X was immediately suggested to be a molecular state of two D particles. I wrote an account of the studies of the nature of the X particle a few years ago if you are interested -but mind you, the advancements in this research field are quick, and I believe the material I wrote back then is a bit aged by now.

The paper by Liu tries to determine whether the interpretation of the Y particle as a pure second radial excitation of P-wave charmonium (, with ) holds water once the observed branching ratio of the Y into the final state seen by CDF (), and the measured decay width, are compared to a theoretical calculation.

The nice thing about the decay of the Y into the observed final state is that it occurs only through a so-called “rescattering” mechanism, by means of the diagrams shown in the graph below (the ones shown refer to the J=0 hypothesis of the , but similar diagrams are discussed for the J=1 state in the paper).

As you can see, the Y produces the two final state particles by means of a triangle loop of D mesons. These diagrams usually describe rare processes, and in fact Liu’s calculations end up finding a small branching fraction. I am unable to delve into the details of the computation, so I will just state the result: the typical values of the branching ratio depend on a parameter, which, if taken in a “reasonable” range of values, provides estimates in the ballpark of a few . This appears inconsistent with the observation provided by the CDF experiment.

Clearly, work is in progress here, so I would abstain from concluding anything definite on the matter. So, for now, let us call this an indication that the simple interpretation of the Y as a excited charmonium state is problematic.

The CDF collaboration will present at a public venue (Fermilab’s Wilson Hall) its discovery of the new Y(4140) hadron, a mysterious particle created in B meson decays, and observed to decay strongly into a state, a pair of vector mesons. I have described that exciting discovery in a recent post.

From this site you can connect to streaming video (starting at 4.00PM CDT, or 9.00PM GMT – should last about 1.30 hours).

This morning CDF released the results of a search for narrow resonances produced in B meson decays, and in turn decaying into a pair of vector mesons: namely, . This Y state is a new particle whose exact composition is as of yet unknown, except that CDF has measured its mass (4144 MeV) and established that its decay appears to be mediated by strong interactions, given that the natural width of the state is in the range of a few MeV. I describe succintly the analysis below, but first let me make a few points on the relevance of area of investigation.

Heavy meson spectroscopy appears to be a really entertaining research field these days. While all eyes are pointed at the searches for the Higgs boson and supersymmetric particles, if not at even more exotic high-mass objects, and while careers are made and unmade on those uneventful searches, it is elsewhere that action develops. Just think about it: the baryon, the , those mysterious X and Y states which are still unknown in their quark composition. Such discoveries tell the tale of a very prolific research field: one where there is really a lot to understand.

Low-energy QCD  is still poorly known and not easily calculable. In frontier High-Energy Physics we bypassed the problem for the sake of studying high-energy phenomena by tuning our simulations such that their output well resembles the result of low-energy QCD processes in all cases where we need them -such as the details of parton fragmentation, or jet production, or transverse momentum effects in the production of massive bodies. However, we have not learnt much with our parametrizations:  those describe well what we already know, but they do not even come close to guessing whatever we do not know. Our understanding of low-energy QCD is starting to be a limiting factor in cosmological studies, such as in baryogenesis predictions. So by all means, let us pursue low-energy QCD in all the dirty corners of our produced datasets at hadron colliders!

CDF is actively pursuing this task. The outstanding spectroscopic capabilities of the detector, combined with the huge size of the dataset collected since 2002, allow searches for decays in the one-in-a-million range of branching ratios. The new discovery I am discussing today has indeed been made possible by pushing to the limit our search range.

The full decay chain which has been observed is the following: . That mesons decay to muon pairs is not a surprise, as is the decay to two charged kaons of the vector meson. Also the original decay of the B hadron into the final state is not new: it had been in fact observed previously. What had not been realized yet, because of the insufficient statistics and mass resolution, is that the and mesons produced in that reaction often “resonate” at a very definite mass value, indicating that in those instances the decay actually takes place in two steps as the chain of two two-body decays: and .

The new analysis by CDF is a pleasure to examine, because the already excellent momentum resolution of the charged particle tracking system gets boosted when constraints are placed on the combined mass of multi-body systems. Take the B meson, reconstructed with two muons and three charged tracks, each assumed to be a kaon: if you did not know that the muon pair comes from a nor that two of the kaons come from a , the mass resolution of the system would be in the few tens of MeV range. Instead, by forcing the momenta of the two muons to be consistent with the World average mass of the , , and by imposing that the two kaons make exactly the extremely well-known mass (), much of the uncertainty on the daughter particle momenta disappears, and the B meson becomes an extremely narrow signal: its mass resolution is just 5.9 MeV, a per-mille measurement event-by-event!

The selection of signal events requires several cleanup cuts, including mass window cuts around the J/Psi and phi masses, a decay length of the reconstructed B+ meson longer than 500 microns, and a cut on the log-likelihood ratio fed with dE/dx and time-of-flight information capable of discriminating kaon tracks from other hadrons. After those cuts, the B+ signal really stands above the flat background. There is a total of 78+-10 events in the sample after these cuts, and this is the largest sample of such decays ever isolated. It is shown above (left), together with the corresponding distribution in the candidate mass (right).

A Dalitz plot of the reconstructed decay candidates is shown in the figure on the right. A Dalitz plot is a scatterplot of the squared invariant mass of a subset of the particles emitted in the decay, versus the squared invariant mass of another subset. If the decay proceeds via the creation of an intermediate state, one may observe a horizontal or vertical cluster of events. Judge by yourself: do the points appear to spread evenly in the allowed phase space of the B+ decays ?

The answer is no: a significant structure is seen corresponding to a definite mass of the system. A histogram of the difference between the reconstructed mass of the system and the mass is shown in the plot below: a near-threshold structure appears at just above 1 GeV energy. An unbinned fit to a relativistic Breit-Wigner signal shape on top of the expected background shape shows a signal at a mass difference of , with a width of 11.7+-5.7 MeV.

The significance of the signal is, after taking account of trial factors, equal to 3.8 standard deviations. For the non-zero width hypothesis, the significance is of 3.4 standard deviations, implying that the newfound structure has strong decay. The mass of the new state is thus of 4143+-2.9 MeV.

The new state is above the threshold for decay to pair of charmed hadrons. The decay of the state appears to occur to a pair of vector mesons, , in close similarity to a previous state found at 3930 MeV, the Y(3930), which also decays to two vector mesons in . Therefore, the new state can be also called a Y(4140).

Although the significance of this new signal has not reached the coveted threshold of 5 standard deviations, there are few doubts about its nature. Being a die-hard sceptic, I did doubt about the reality of the signal shown above for a while when I first saw it, but I must admit that the analysis was really done with a lot of care. Besides, CDF now has tens of thousands of fully reconstructed B meson decays available, with which it is possible to study and understand even the most insignificant nuances to every effect, including reconstruction problems, fit method, track characteristics, kinematical biases, you name it. So I am bound to congratulate with the authors of this nice new analysis, which shows once more how the CDF experiment is producing star new results not just in the high-energy frontier, but as well as in low-energy spectroscopy. Well done, CDF!

Thanks to the many offers for help received a few days ago, when I asked for hints on possible functional forms to interpolate a histogram I was finding hard to fit, I have successfully solved the problem, and can now release the results of my study.

The issue is the following one: at the LHC, Z bosons are produced by electroweak interactions, through quark-antiquark annihilation. The colliding quarks have a variable energy, determined by probability density functions (PDF) which determine how much of the proton’s energy they carry; and the Z boson has a resonance shape which has a sizable width: 2.5 GeV, for a 91 GeV mass. The varying energy of the center of mass, determined by the random value of quark energies due to the PDF, “samples” the resonance curve, creating a distortion in the mass distribution of the produced Z bosons.

The above is not the end of the story, but just the beginning: in fact, there are electromagnetic corrections (QED) due to the radiation of photons, both “internally” and by the two muons into which the Z decays (I am focusing on that final state of Z production: a pair of high-momentum muons from ). Also, electromagnetic interactions cause a interference with Z production, because a virtual photon may produce the same final state (two muons) by means of the so-called “Drell-Yan” process. All these effects can only be accounted for by detailed Monte Carlo simulations.

Now, let us treat all of that as a black box: we only care to describe the mass distribution of muon pairs from Z production at the LHC, and we have a pretty good simulation program, Horace (developed by four physicists in Pavia University: C.M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini), which handles the effects discussed above. My problem is to describe with a simple function the produced Z boson lineshape (the mass distribution) in different bins of Z rapidity. Rapidity is a quantity connected to the momentum of the particle along the beam direction: since the colliding quarks have variable energies, the Z may have a high boost along that direction. And crucially, depending on Z rapidity, the lineshape varies.

In the post I published here a few days ago I presented the residual of lineshape fits which used the original resonance form, neglecting all PDF and QED effects. By fitting those residuals with a proper parametrized function, I was trying to arrive at a better parametrization of the full lineshape.

After many attempts, I can now release the results. The template for residuals is shown below, interpolated with the function I obtained from an advice by Lubos Motl:

After multiplying that function by the original Breit-Wigner resonance function, I could fit the 24 lineshapes extracted from a binning in rapidity. This produced additional residuals, which are of course much smaller than the first-order ones above, and have a sort of parabolic shape this time. A couple of them are shown on the right.

I then interpolated those residuals with parabolas, and extracted their fit parameters. Then, I could parametrize the parameters, as the graph below shows: the three degrees of freedom have roughly linear variations with Z rapidity. The graphs show the five parameter dependences on Z rapidity (left column) for lineshapes extracted with the CTEQ set of parton PDF; for MRST set (center column); and the ratio of the two parametrization (right column), which is not too different from 1.0.

Finally, the 24 fits which use the shape, with now all of the rapidity-dependent parameters fixed, are shown below (the graph shows only one fit, click to enlarge and see all of them together).

The function used is detailed in the slide below:

I am rather satisfied by the result, because the residuals of these final fits are really small, as shown on the right: they are certainly smaller than the uncertainties due to PDF and QED effects. The function above will now be used to derive a parametrization of the probability that we observe a dimuon pair with a given mass at a rapidity , as a function of the momentum scale in the tracker and the muon momentum resolution.

Here are the questions asked at an exam in Subnuclear Physics this morning:

• Draw the strong and electromagnetic coupling constants as a function of , explain their functional dependence using feynman graphs of the corrections to the photon and gluon propagators, write their formula, and compute the value of the constants at , given the values at (QED) and (QCD).
• The GIM mechanism: explain the need for a fourth quark using box diagrams of kaon decays to muon pairs. How does the charm contribution depend on its mass ? What conclusion could be drawn by that dependence in the case of B mixing measurements in the eighties ?
• Discuss a measurement of the top quark mass. For a dileptonic decay of top quark pairs, discuss the final state and its production rate.
• Discuss decay modes of W bosons and their branching fraction values. Discuss decay modes of Z bosons and their branching fraction values.

The student answered well all questions and he got 30/30 points.

Here is the second part of the list of useful physics posts I published on this site in 2008. As noted yesterday when I published the list for the first six months of 2008, this list does not include guest posts nor conference reports, which may be valuable but belong to a different place (and are linked from permanent pages above). In reverse chronological order:

December 29: a report on the first measurement of exclusive production of charmonium states in hadron-hadron collisions, by CDF.

December 19: a detailed description of the effects of parton distribution functions on the production of Z bosons at the LHC, and how these effects determine the observed mass of the produced Z bosons. On the same topic, there is a maybe simpler post from November 25th.

December 8: description of a new technique to measure the top quark mass in dileptonic decays by CDF.

November 28: a report on the measurement of extremely rare decays of B hadrons, and their implications.

November 19, November 20, November 20 again , November 21, and November 21 again: a five-post saga on the disagreement between Lubos Motl and yours truly on a detail on the multi-muon analysis by CDF, which becomes a endless diatriba since Lubos won’t listen to my attempts at making his brain work, and insists on his mistake. This leads to a back-and-forth between our blogs and a surprising happy ending when Motl finally apologizes for his mistake. Stuff for expert lubologists, but I could not help adding the above links to this summary. Beware, most of the fun is in the comments threads!

November 8, November 8 again, and November 12: a three-part discussion of the details in the surprising new measurement of anomalous multi-muon production published by CDF (whose summary is here). Warning: I intend to continue this series as I find the time, to complete the detailed description of this potentially groundbreaking study.

October 24: the analysis by which D0 extracts evidence for diboson production using the dilepton plus dijet final state, a difficult, background-ridden signature. The same search, performed by CDF, is reported in detail in a post published on October 13.

September 23: a description of an automated global search for new physics in CDF data, and its intriguing results.

September 19: the discovery of the baryon, an important find by the D0 experiment.

August 27: a report on the D0 measurement of the polarization of Upsilon mesons -states made up by a pair- and its relevance for our understanding of QCD.

August 21: a detailed discussion of the ingredients necessary to measure with the utmost precision the mass of the W boson at the Tevatron.

August 8: the new CDF measurement of the lifetime of the baryon, which had previously been in disagreement with theory.

August 7: a discussion of the new cross-section limits on Higgs boson production, and the first exclusion of the 170 GeV mass, by the two Tevatron experiments.

July 18: a search for narrow resonances decaying to muon pairs in CDF data excludes the tentative signal seen by CDF in Run I.

July 10: An important measurement by CDF on the correlated production of pairs of b-quark jets. This measurement is a cornerstone of the observation of anomalous multi-muon events that CDF published at the end of October 2008 (see above).

July 8: a report of a new technique to measure the top quark mass which is very important for the LHC, and the results obtained on CDF data. For a similar technique of relevance to LHC, also check this other CDF measurement.

This post is just a placeholder for a link and an invitation for you to join me and ask Marco Frasca to further his already enlightening discussion of glueballs, as I already did in the comments thread of his post.

The subject is indeed fascinating: gluonic matter. A condensate of bosons. Asymptotic states (particles) made of no fermions. Matter as we never experienced it.

Glueballs are (thought to be) bound states of gluons. The gluon, the carrier of strong interactions, is a massless boson, and it is charged with the attribute it mediates: colour. Because of the colour charge of gluons, these particles can interact with one another, giving rise to the fascinating properties of Quantum Chromodynamics, the asymptotic freedom properties of bound states of quarks, and the infrared slavery – the impossibility of obtaining free coloured objects (quarks or gluons). The non-abelian nature of the SU(3) group underlying quantum chromodynamics is a source of the difficulties of calculating the low-energy limit of the theory.

The self-interaction properties of gluons creates the possibility of a color-neutral state made of gluons only: glueballs. Glueball properties, however, cannot be computed with perturbation theory, and these remain very mysterious objects over thirty years after QCD was understood in its workings. Only with lattice calculations -you know, discretizing space and computing path integrals with finite sums- can the glueballs be understood. But Marco hints at methods that allow a deeper understanding: methods based on condensed matter physics. We are all ears!

It is my pleasure today to report on an extremely interesting measurement recently produced by the CDF experiment, one which is dear to me for several reasons.

First of all, the measurement involves the study of charmonium -a fascinating system made by a charm-anticharm quark pair, the two quarks orbiting around each other for a tiny fraction of a second before annihilating. Charmonium -in the form of the particle- is the state of matter discovered in November 1974 by Nobel Laureates Samuel Ting (at Brookhaven) and Burton Richter (at SLAC), and which immediately confirmed the reality of quarks -entities whose reality had kept many brilliant minds skeptical during the previous decade. The and the other charmonium states have been since then a gold mine for the study of Quantum Chromodynamics, of B hadron phenomenology (through the easy-to-spot decays of the type ), and an invaluable calibration tool for particle trackers. Charmonium may one day be even used with profit to measure with high precision the top-quark mass at the LHC!

Second, the study focuses on what is called “exclusive production”, a spectacular signature at hadron colliders: it involves the production of the particle without the disintegration of the projectiles, through a colourless exchange. Basically, the charmonium meson appears out of nothing, and in the company of exactly nothing else. It is as if by a stroke of a magic wand, a spark is exchanged between a proton and an antiproton coming close together, and a single particle is created. Those of you who know what kind of a mess is routinely yielded by proton-antiproton collisions at 2 TeV of energy in the core of CDF and D0 cannot but join my awe for the very concept of clean single-body production.

Third, exclusive production processes are quite rare in hadron collisions, and they are extremely interesting because they may, one day, provide the means by which we can access a direct measurement of the quantum numbers of the Higgs boson at the LHC. Also, their theoretical calculation is not on very solid ground, so a measurement is of great help. This one by CDF is the first ever on exclusive charmonium in hadron-hadron collisions.

Some notes on the production mechanisms

Before I come to describe the measurement details, let me present you with two Feynman diagrams of the simplest reactions responsible for the exclusive creation of a charmonium system. As I said above, the exchange between proton and antiproton is colourless: this, while still being a QCD interaction! But how is it possible for quantum chromodynamics to mediate a colourless reaction ? We know that the carrier of the strong force is the gluon, a bi-coloured object. On the other hand, if a proton emits or absorbs a gluon, it will never keep intact: it will become coloured, and explode like a firework. Unless…

Unless a second gluon immediately comes to the rescue, restoring the neutrality of the proton! This is what is needed for a strong interaction to produce a charmonium state: two gluons are emitted and absorbed by the proton and the antiproton, and the pair can thus leave the interaction point, slightly rattled but unharmed. The first pair of gluons meets halfways, creating a charm-quark loop. The loop then “closes” into a meson, a charmonium state with the right quantum numbers. This is the process shown on the right.

Notice that a two-gluon exchange cannot give rise to a meson. This is because of the invariance of strong interactions for the operation called “charge conjugation”. A discussion of the invariance properties of particle reactions under the operation of exchanging every particle with its antiparticle would take us too far, and is the proper subject of another post. I will just say here that the reaction is impossible: in quantum mechanics, one says that the “transition amplitude vanishes” for a two-gluon state converting into a vector particle.

Not so for the though, since that particle has zero angular momentum: it is not a vector, but a pseudoscalar meson. That is why we have examined production first. But exclusive production does occur anyway.

How, however, if we are to keep the two incoming baryons intact? Let us think at the problem from the point of view of the then. This particle is known to be quite slow to decay into hadrons. In fact, it has a lifetime much longer than similar hadronic states. So much so, in fact, that the very suppressed electromagnetic decay into electron or muon pairs gets a sizable share of the probability. An electromagnetic process competing with a strong process ? This is very weird: we know that the electromagnetic decay must have a rate proportional to the coupling constant , which is fifty times smaller than the strong coupling constant for such low-energy phenomena. However, how is quantum chromodynamics going to disintegrate a ?

Phase one is simple: the charm and the anticharm get together in the same point of space, annihilating. Phase two is the emission of gluons: not a single one though, because the is colourless! And not just two, because of the quantum mechanical selection rule concerning C-parity we mentioned above! The minimum number of gluons is three. That means that we have to take the third power of when computing the decay rate through QCD interactions, and so electromagnetic processes become competitive.

In reality, these three-gluon exchanges are even more suppressed than they ought to be. This is encoded in what is called “OZI rule”, by the name of Okubo, Zweig, and Iizuka: at any order of the strong coupling constant , “disconnected” strong interaction processes -ones that have no fermion lines propagating from the initial to the final state, but only gluons- occur with smaller frequency. This is a consequence of the variability of with the energy of the gluon emission: in disconnected diagrams gluons have to carry an energy at least equal to the mass of the particles they are going to materialize in the final state, and the corresponding value of the coupling constant is therefore evaluated at a higher energy, and is thus much smaller than the one involved in the emission of softer gluons in non-disconnected diagrams. Confused ? Not a problem, forget the argument and let us return to production.

The above discussion just meant to show that we need three gluons to produce one , since we now know that one such particle can decay to no less than three gluons. Now, let us take the protons -the projectiles producing the reaction- in the equation. Who is going to provide two gluons, and who is going to provide only one ? I can picture the proton and antiproton starting a bitter argument on this issue. The one who provides a single gluon is a dead baryon, since it will get its colourlessness unbalanced, and it will disgregate. This is bad, because the production process is not exclusive any longer if, together with a meson, we get additional hadrons.

How then can we produce a vector charmonium particle in an exclusive fashion ? Electrodynamics comes to the rescue. One of the two projectiles yields a pair of gluons, and the other emits a photon. Both can remain intact this way, and angular momentum does not object. The process is shown on the right: as you see, the charm-anticharm pair becomes a colourless one after the second gluon is exchanged, and a (or a meson, which has the same properties except angular momentum) arises.

The CDF search

Now that we know how proton-antiproton collisions are expected to yield exclusive charmonium, we may look for it more meaningfully.

The search uses an integrated luminosity of 1.5 inverse femtobarns of proton-antiproton collisions, recorded by the CDF experiment during Run II. The trigger used to collect exclusive production candidates selects data with two muon tracks of opposite charge, with no energy recorded in the BSC1 system, a set of scintillation counters covering the rapidity intervals . Such a requirement excludes data which may be similar kinematically to real exclusive production, but feature a few hadrons produced almost parallel to the proton-antiproton direction.

Offline, events are excluded if they contain any other detected particle but the two muons. However, the search for allows one electromagnetic energy cluster in the calorimeter, signalling the potential photon emitted along with two muons in the decay.

After those cuts, the only potential background to address is cosmic-ray muons that cross the detector, leaving a long track that may be fit as a pair of opposite-charge muons if it crosses the detector in proximity of the beam. These events are rejected by requiring that the two muons have an azimuthal opening angle smaller than 3.0 radians, and rejecting ones that have timing information incompatible with the instant when the proton and antiproton bunches cross in the detector.

The invariant mass of the 402 events surviving the selection is histogrammed in the graph below, if it is in the 3.0 – 4.0 GeV range (ignore the inset, which shows a fit to the background). A clear peak of and candidates is apparent on a flat background of non-resonant QED dimuon pairs; the set of events in the peak which contain one additional photon candidate make the signal (65 events).

The calculation of efficiencies is complicated by the fact that the exclusivity requirements remove bunch crossings which produced more than a single proton-antiproton interaction; this is a function of the number of particles in the bunches. After all is accounted properly, CDF can however determine the cross-section for charmonium production in exclusive processes with a good accuracy. The results are as follows:

• for

• for

• for

The above results are in agreement with theoretical calculations, which have very large uncertainties. This measurement was long overdue: it is a very important input for theoretical calculations of exclusive processes.

The kinematical characteristics of the resulting and decay signals are studied by comparing data with a Monte Carlo simulation of photoproduction, finding good agreement. Below are shown the transverse momentum distributions of the resonances in the data (red points), compared to simulation (black histograms).

In conclusion, I am happy to see these very intriguing signals appear in CDF data, and I congratulate with the main authors of the analysis, Long Zhang, Mike Albrow, and James Pinfold! You can find more information on the analysis in the public web page of the result.

I occasionally realize that this blog is a dead end of web surfers, not so much for its content -who cares about content these days- but for the lack of meaningful links. Apart from the blogroll column on the right, in fact, I am very, very lazy with html tags, and my posts have no “further reading” links at the bottom as those of some tidier bloggers.

So let me make an attempt at doing my share of sewing for the internet today. Here are a few blogs that have recently discussed things worth reading.

• David has recently discussed internet technologies at the italian Parliament
• Peter has issued six great posts on BRST theory.
• Jester has a very good post about the ATIC anomaly
• Alexey is opening a position for a postdoc.
• Kea is moving north!
• Marco has been excellently discussing the mysteries of low-mass hadronic states in a series of posts.
• The car has a full tank and new tires, the lady is there to ride it with you, and she can teach a thing or three about the universe in the meantime!
• What if you visited this advent series ?
• Lubos has just posted really cool pictures of weather stations data. With so many numbers to fiddle with, imagine how many things he can screw up! (sorry Lubos

The Z boson mass has been measured with exquisite precision in the nineties by the LEP experiments ALEPH, OPAL, DELPHI and L3, and by the SLD experiment at SLAC: we know its value to better than a few MeV precision. The PDG gives . Now, a precise Z mass is an important input to our theory, the Standard Model, and through its measurement, as well as that of other Z-related quantities that the four LEP experiments and SLD measured with great precision, a giant leap forward has been made in the understanding of the subtleties of electroweak interactions.

For an experimental physicist, however, the knowledge of the Z mass is more a tool for calibration purposes than a key to theoretical investigations. Indeed, as I have discussed elsewhere recently, I am working at the calibration of the CMS tracker using the decays of Z bosons, as well as of lower-mass resonances. We take decays, we measure muon tracks, determine the measured mass of the Z boson with them, and compare the latter to the world average. This provides us with precious information on the calibration of the momentum measurement of muon tracks.

In CMS we will quickly collect large numbers of Z bosons, so statistics is not an issue: we will be able to study the calibration of tracks very effectively with those events. However, when statistics is large, experimentalists start worrying about systematic uncertainties. Indeed, there are several effects that cause a difference between the mass value we reconstruct with muon tracks and the true value of the Z boson mass -the one so well determined which sits in the PDG.

I decided to study one of those effects today: the mass shift due to parton distribution functions (PDF). When you collide protons against other protons, what creates a Z boson is the hard interaction between a quark and an antiquark. These constituents of the projectiles carry a fraction of the total proton momentum, but this fraction -called parton distribution function- is unknown on an event-by event basis. By studying proton collisions in different conditions and environments for a long time, we have been able to extract functions which describe how likely it is that a quark q in the proton carries a fraction x of the proton’s momentum. As an example, if the proton travels at 5 TeV as in LHC, an x value of 0.1 means that the quark q will carry 500 GeV by itself.

Now, things are complicated, because each different quark q (u,d,s,c,b) has its own different parton distribution function. The proton contains two valence up-quarks and one valence down-quark: it has a (uud) composition. Those quarks carry a good part of the proton’s momentum, but a large share is due to the rest of partons the proton is made of: sea quark-antiquark pairs, and gluons. Protons do carry antiquarks of all kinds -five in total-, as well as gluons, and these, too, get their own distribution function. A plot of the parton distribution functions of the proton (with a logarithmic x-axis to enhance the low-x behavior) is shown on the right. Note the bumps of u- and d- quark distributions, in blue and green, respectively: those bumps are due to the valence quark contributions.

In reality, things are even more complicated than what I discussed above: you do not simply get away with one function per each of the 11 partons I mentioned thsi far, because these functions have a value which depends on the energy at which you probe the proton, : in a soft collision (which means a small ), is very different from what it is in a harder one, (with a larger ).

The reason for the weird behavior of parton distribution functions -their evolution with - is that quarks have the tendency of emitting gluons, becoming less energetic, and this tendency in turn depends on the energy Q at which they are studied. What is stated above is encoded in very famous functions called DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) equations. They are in a sense another consequence of the “asymptotic freedom” exhibited by strongly interacting particles: at high energy they behave as free particles, emitting little color radiation, while at low energy their interaction with the gluon field increases in strength. It is all due to the fact that the coupling constant of the theory, , is large at small Q. That constant is not a constant by any means!

You have every reason to be confused now: I was talking about calibrating the CMS tracker using muons, and now we are deep into Quantum ChromoDynamics. What gives ? Well: Z bosons are created by quark-antiquark annihilations, and those are found inside the colliding protons with probabilities which depend on their momentum fraction x, and on the total collision energy Q. Since the PDF of quarks and antiquarks peak at very small values of x, the probability of a collision yielding a Z boson -which has a respectable mass of 91 GeV- is small.  If the Z was lighter, more of them would be produced. Now, the Z boson is a resonance, and like every resonance, it has a finite width. What that means is that not all Z bosons have exactly the same mass: while the peak is at 91.186 GeV, the width is 2.5 GeV, which means that it is not infrequent for a Z boson to have a mass of  89, or 92 GeV, rather than the average value. This is described by the Z lineshape, a function called Breit-Wigner:

.

The function is shown below.

As you can see, there is a non-negligible probability that a Z boson has a mass quite different -even a few GeV off- from 91.19 GeV. Now, since Z bosons can be created at masses lower than , they will be privileged by parton distribution functions over masses higher than by the same amount, because .parton distribution functions are larger at lower x. This creates a bias: the perfectly symmetric Breit-Wigner lineshape gets distorted by the preference of partons to carry a lower fraction of the proton momentum.

The distorsion is very small, but it is very important to take it in account when one wants to use measured Z masses to precisely calibrate the track momentum measurement. To size up the effect of the PDF on the Z lineshape, one can compute an integral of the Breit-Wigner weighted with the PDF , by taking into account the different combinations of quarks which give rise to a Z boson in proton-proton collisions.

A Z can be produced by the following quark-antiquark interactions:

• : this can originate from a valence u-quark and a sea anti-u-quark, as well as from a sea u-quark and a sea anti-u-quark. The probability that this quark pair creates a Z depends on the coupling of u-quarks to the Z boson, and this probability is a function of some coefficient predicted by electroweak theory. It is proportional to 0.11784.
• : same as above, but the coupling is proportional to 0.15188.
• : these can only occur through sea-sea interactions. The coefficient is the same as for d-quarks.
• : these are due to the small charm component of the proton sea. They get the 0.11784 coefficient as u-quarks too.
• : these are tiny, but still exist. b-quarks couple to the Z with the 0.15188 factor.
• : these are basically zero.
• : gluon-gluon collisions cannot produce a Z boson, because they are vector particles as the Z (spin 1), and a vector-vector-vector vertex is zero by construction. Note that the same does not hold for the Higgs boson, which is a scalar (spin 0) particle: a vector-vector-scalar vertex is possible, and in fact it is the largest contribution to H production at the LHC.

Putting everything together, one may compute the shift in the lineshape of the Z, and plot it directly (right, on a logarithmic scale to show the effect on the tails), or as a function of the rapidity of the Z boson, a quantity labeled by the letter Y (the dependence is shown in the last graph of this post, below). Rapidity is a measure of how fast is the Z boson moving in the detector reference frame: when one of the partons has a much larger momentum fraction than the one it is colliding against, the produced Z boson has a large momentum in the direction of the more energetic parton.

The rapidity distribution of Z bosons is shown in the graph below, separately for Zbosons produced by valence-sea collisions (in red) and by sea-sea collisions (in blue).

A rapidity Y=0 means that the Z was produced at rest in the detector, +5 is a fast-forward-moving Z, and -5 is a Z moving in the opposite direction with as much speed. As you can see, the valence-sea interactions are the most asymmetric ones, predominantly producing a forward-moving Z boson.

On the right here I also plot with the same color-coding the x distribution of quarks taking part in the Z creation. The red distribution has both a very small-x and a very large-x component, highlighting the asymmetric production.

Despite being in black and white, the most interesting plot is however the following one. It shows the average mass of the Z bosons (on the vertical scale, in GeV) as a function of the Z rapidity. The downward shift from 91.186 GeV is relevant -about 0.25 GeV overall- but it increases at large values of rapidity, when one of the two partons has a very small value of x, so that the collision “samples” a rapidly varying PDF for that parton.

The plot on the left here is what is needed as an input for our calibration program: we will have to study how this dependence affects our determination of the momentum scale. A lot of work ahead, but a very enlightening one!

I was surprised by the recent measurement by the D0 collaboration of the Upsilon polarization, which finds a sizable effect which really disagrees with the CDF result, obtained six years ago and based on a 12 times smaller dataset from Run I.

D0 has a large acceptance to muons, and so can detect with good efficiency the decays. CDF has a slightly worse acceptance, but its momentum resolution is of a totally different class. Compare the mass distribution of Upsilon mesons published by D0 in their polarization paper, shown below (they refer to different bins of transverse momentum, left to right, and to different fit parametrizations, top to bottom; black points are the data, the red curve is the fit, and the black gaussians are the three Upsilon signals returned by the fit),

…with the Run I distribution by CDF on which their old result is based, in the plot below (where the data is the black histogram, and the curve is the fit):

Any questions ? Of course, the three Upsilon(1S), (2S), (3S) states merge together in a broad bump in the D0 signal plot, while they stand each on their own in the CDF plot. That’s resolution, baby. Muon momentum resolution is a thing on which Nobel prizes are done and undone.

Despite the lower resolution, D0 can statistically separate the three populations, and measure the Upsilon (1S) and (2S) polarization as a function of the meson transverse momentum. The polarization is defined by the number alpha:

,

where and are the cross-sections for producing a transversely and longitudinally-polarized meson, respectively. The polarization can be measured from the decay angle of the positively-charged muon in the Upsilon rest frame.

D0 had a total of 260,000 Upsilon decays to play with, and they produced a very detailed measurement of the behavior of and polarization as a function of the meson . This allows a comparison with NRQCD, a factorization approach to the calculation of quarkonium production processes which enshrines in universal non-perturbative color-octet matrix elements the non-computable part, and uses experimental data to fix them.

Confused? Don’t be. Let’s just say that NRQCD is a successful approach at determining several characteristics of charmonium production, and a test of its prediction of the dominance of at high transverse momentum -where gluon fragmentation is the main process for the production of quarkonium in the model- is quite useful.

Another thing to note is that understanding Upsilon production -particularly in the forward region- may be very important for the determination of parton-distribution functions of the proton at very small values of Bjorken x -the fraction of proton momentum carried by a parton. These measurements are very important for the LHC, where interesting physics phenomena will be dominated by very low x collisions.

So let me just jump to the results of the D0 analysis. The polarization plot for the 1S state is shown below. Black points are the D0 measurement, while the green ones show the old CDF result (by the way, it is a shame that CDF does not have a Run II measurement of the Upsilon polarization yet, and you can well say it is partly my fault, since a few years ago I wanted to do this measurement and then had to abandon it…).

As you can see, the D0 result shows a Pt dependence of polarization which is not well matched by the NRQCD predictions (the yellow band, which is only available above 8 GeV of Pt), nor by the two limiting cases of a kt-factorization model. What is worse, however, is that the result comes seriously at odds with the old CDF data points. Who is right and who is wrong ? Or are the two sets of points compatible ?

Well, this is one of those instances when counting standard deviations does not work. The two results have sizable correlated systematic uncertainties among the data points, so moving eight data points up by one sigma collectively may cost much less than standard deviations: in the limit that the systematics dominate and they are 100% correlated, moving all points up by just costs one standard deviation… In the case of the D0 points, however, one does not have this information from the paper. One learns that the signal modeling systematics amount to anywhere between 1 and 15%, with the bin with maximum uncertainty being the second one from left; and that background modeling systematics range from 4 and 21%, with the first bin being the worst one. As for the old CDF result, I could not find a detailed description of systematics either, but in that case the precision of the measurement is driven by statistics.

In any case, congratulations to D0 for producing this important new measurement. And I now hope CDF will follow suit with their large dataset of Upsilons too!

Quantum ChromoDynamics, the theory of strong interactions, is admittedly not considered the most exciting branch of particle physics at colliders these days. QCD processes make up 99.99% of what happens in hadronic collisions at the Tevatron, or what will happen at the LHC starting this fall: they are usually backgrounds to those much more interesting reactions involving electroweak bosons and leptons, or to the searches for the Higgs boson.

I would like to point out that QCD is in truth a wonderfully complex and beautiful theory, and that QCD measurements are very important. Only by understanding strong interactions in detail can we hope to find new phenomena lying underneath. And don’t even get me started on the need for more studies on low-energy strong interactions -they are really not well understood yet, and in fact precise measurements of strong interaction cross sections are direly needed in cosmology. But let me go back to high-energy high-energy physics.

Today I would like to discuss a precise measurement by CDF which will prove very useful when somebody -me and Mia, for instance- will start studying CMS data in search for the decay : the production of a higgs boson and its subsequent decay into a pair of Z bosons, with a final state including one leptonic Z very easy to identify, and a second one which can be separated from backgrounds by the identification of b-quark jets.

The signal is buried in a large background, namely production where the pair of b-quarks is not coming from a Z boson decay. How large ? Well, we have theoretical calculations, Monte Carlo simulations incorporating them, detector simulations… We have a pretty good idea, but unless we check that these calculations are precise, we are stuck with large systematic uncertainties. One good part of these is due to our limited knowledge of the probability to find a b-quark inside the proton, the b-quark PDF.

A recent result which improves matters has been obtained on the cross section for production by Andrew Mehta and Beate Heinemann, two very skilled colleagues from Liverpool and Berkeley, respectively. The comparison of the result with theoretical predictions provides a nice confirmation that the latter are in the right ball-park, and an estimate of the level of trust we can put in them. Let me try and describe very briefly how the measurement is produced.

Events with a leptonic Z boson decay are selected from 2.0 inverse femtobarns of proton-antiproton collisions produced by the Tevatron 1.96 TeV synchrotron in the core of the CDF detector. Both and decays are selected, for a total of about 200,000 events. Among these, the analysis selects those events containing one hadronic jet which has a secondary vertex reconstructed with its charged tracks: the vertex is the signal of the decay of a B-hadron, which contains the long-lived b-quark. By selecting jets with a secondary vertex, their b-purity is increased tremendously.

Below you can see the Z mass peak for events containing a b-quark jet accompanying the dilepton system. The black points are CDF data, the black line is the total of the various contributions, which include, together with the signal, few small backgrounds.

To compute a cross-section for production, there remains one step (ok, I am making things simpler than they really are, for the sake of clarity and space): understanding the fraction of these jets really due to b-quark hadronization. This can be accomplished by studying the invariant mass of all the measured charged tracks originating from the secondary vertex: the mass is larger for real b-quark jets and smaller for charm-quark jets or jets due to lighter quarks or gluons, for which the secondary vertex is due to a random mismeasurement of tracks rather than the true decay of a long-lived particle.

Above you can see a fit to the secondary vertex mass distrbution, with the three components. The cyan histogram represents the b-jet fraction, which has a larger vertex mass and amounts for 40% of the total. By measuring the fraction of b-jets one can proceed to measure the cross section, if one knows the efficiency of the selection of Z boson decays and the efficiency of the vertex-finding b-tag algorithm. What I am talking about is the following formula:

.

Don’t be scared: the ingredients have all been introduced to you already. is the cross-section of the process, i.e. the thing that is measured in the analysis. is the fraction of b-jets among those with a secondary vertex, and is extracted by the figure shown above. is the fraction of Z bosons which are detected and reconstructed from two observed muons or electrons; is the efficiency of finding the b-jet with the required energy and with a secondary vertex inside it. Finally, is the integrated luminosity of the data used, 2.0/fb.

What is the result ? CDF finds , a small number -eight times smaller than the cross-section for producing a pair of top quarks! Theory calculations at Next-to-Leading-Order (a good level of precision for this calculation) predict , a figure smaller but not utterly incompatible with the data.

Maybe the most interesting part of the measurement is the ratio between the measured cross section and the cross section for production of one Z boson alone. It is shown in the plot below as a function of the transverse energy of the b-jet, compared to three different Monte Carlo calculations. As you can see, the fraction of Z bosons which are produced together with a b-jet is tiny! The reason has to do with the smallness of the b-quark PDF.

Among the huge amount of beautiful new measurements produced at the Tevatron by the CDF and D0 experiments last month, just in time for showing at ICHEP 2008, the international conference in High-Energy Physics, there is one which does not make headlines, but it deserves one. It is the measurement of the top pair production cross section, a number which is by itself not terribly informative – it is basically only a check that perturbative calculations with Quantum Chromodynamics work well when they deal with an energy scale where the strong coupling constant is small enough. That is: the above is the only thing one gets from a precise measurement of the top cross section provided one is convinced that there is no other process, so far undiscovered, hiding in top production or top decay.

It is absolutely fair to ask oneself whether top pairs are produced at the Tevatron energy solely by quark-antiquark annihilation and gluon-gluon fusion, the two leading order QCD processes, or whether there is a heavy object X which decays to top quarks, thus enhancing the observed rate of top quarks over what QCD predicts. It is also perfectly legitimate to investigate whether the cross section is in line with predictions regardless of the final state in which one searches for top quarks: some non-standard decays of top could modify the mix. Further, one could hypothesize that the top quark dataset -the data enriched with top events which are used by the experiments to measure cross sections- contains some other process which messes up some of the measurements.

The above ideas are to me the most important reason for being interested, 14 years after I first got to know that the top quark existed, in the very precise new determinations of top quark pair cross section obtained by CDF. So let us look at the graph on the right, which details some of the recent determinations, which have been averaged into a result which carries a 8% total uncertainty, beating by 1% the most precise theoretical estimates (9% relative error).

One interesting thing to note is that the cross sections measured with SLT are higher than the average. SLT is the soft-lepton tagging algorithm, which tags b-quark jets coming from top decay through the identification of a muon or an electron embedded in the jet. In Run I, CDF measured a top cross section which was 9 picobarns when using SLT, while about 6 picobarns when using SVX tags -secondary vertices in the jets. Back then, the disagreement was the source of a huge controversy on the hypothesized presence of new physics in the sample of events containing SLT tags. The data did lend itself to some exotic interpretations, but things petered out after years of review and internal diatribas. Now, it does not look like there will be a reprise of that controversy, but the fact remains that SLT cross sections are still there: higher than they should be!

In any case, I salute this new, important result by the CDF top group, and by dozens of dedicated physicists who put their time and efforts into obtaining a very precise measurement. Now the ball is in the theorists’ court, to improve the precision on the theoretical estimate.

UPDATE – ok, a moment after posting the above piece, I looked back at the picture, and I realized that it is not true that the CDF determination is more accurate than theory. It is the theory band which has an 8% uncertainty if I am not mistaken, while CDF has the 9% measurement. That does not change much of the discussion, however, since once the result found by D0 is added to the above one, experiment does get the better hand.

UPDATE II: I also forgot to point interested readers to the public note describing the result!

Sometimes I come to think I need a secretary, or even better, a press office. It is such a tough job to keep up to date with the scientific publications popping out on a daily basis, that I sometimes have to completely leave my research aside to foster my own education.

When, however, a new important result escapes my attention for over one year, and the result comes directly from an experiment I am part of, I realize the task is beyond my forces.

Such was indeed the fate of the measurement of correlated b-quark pair production cross section, which CDF published in Phys. Rev. D77, 072004 (2008 ) a couple of months ago (get your preprint here), but which has been around for over a year now. It is especially annoying because it is a very careful measurement which probably settles the issue of b-quark pair cross section, a topic where collider experiments had in the past produced conflicting results. What’s more, the paper is the result of three years of work of a group of friends of mine. Shame on me!

Pairs of b-quarks are produced in hadron collisions by strong interactions (QCD, for Quantum Chromo-Dynamics), typically through the fusion of two gluons. While the production mechanism occurs at an energy high enough to warrant a perturbative calculation -because the strong coupling constant is small enough that it is meaningful to write down the cross section for the hard process as an infinite series of terms in powers of that quantity, the surrounding lower energy phenomena -what happens before the hard scattering, and what happens after it- are non-calculable, and they thus must be evaluated with a fair amount of assumptions.

Before the scattering, you need to “find” in the colliding bodies two suitable partons -mostly gluons, as I said above- of the right energy. The probability to find those partons in the proton and antiproton is a function of their energy, and is described by so-called “parton distribution functions“, which are determined by dedicated experiments.

Similarly non-calculable are the QCD processes responsible for the phase, called “fragmentation” that connects the outgoing b-quarks into a final state with a well-defined observed behavior. The energetic b-quark, leaving the interaction region, extends a color string until the latter “breaks” popping up quark-antiquark pairs which can then bind into color-neutral hadrons – one of them a B-hadron, containing the original b-quark. It is those hadrons which the experiment detects in the tracker and calorimeter system, collectively measuring their energy -or more customarily, their momentum transverse to the beam, .

All in all, theoretical calculations of b-quark pair production are a big headache. It is actually surprising that different calculations roughly agree, in fact, once the connection between quark energy and observed B hadron energy is treated with some amount of care. In fact, the comparison of calculations at different level of approximation (at “leading order” or “next-to-leading order” -LO, NLO) present a stable result, which can be therefore trusted to be correct to within 10-20%.

Experimentally, there have been five measurements of the production cross section obtained by CDF and D0 in the past. These consider different thresholds for the B hadrons, so to compare them it is meaningful to divide each by the corresponding theoretical prediction. Here are the past results on the experimental/theoretical ratios, based on Run I data:

• (CDF, jets with secondary vertex b-tags);

• (D0, jets with secondary vertex b-tags);
• (CDF, events with one semileptonic muon b-tag jet and the other containing a secondary vertex b-tag);
• (CDF, events with two semileptonic muon b-tag jets);
• (D0, events with two semileptonic muon b-tag jets).

In total, the average is with a 0.8 RMS and a poor overall agreement. Particularly nagging is the dimuon result by CDF, off from unity by more than three standard deviations. With the order-of-magnitude increase of Run II dataset statistics, the matter could and should be straightened!

The new measurement by CDF, based on 740 inverse picobarns of data -roughly eight times more than the former analysis- uses again dimuon events: events where two jets, back-to-back in azimuth, both contain a muon with an impact parameter consistent with the b-quark decay length. The impact parameter is the mismatch between the muon trajectory and the interaction vertex: a large value of is produced if the muon is not coming from the interaction vertex, but from the decay of a particle that travelled a sizable distance before disintegrating. By fitting the impact parameter distribution of muon tracks in dimuon events, CDF can determine the amount of events present in the data sample.

Above, the impact parameter, in centimeters, of muon tracks (black points with error bars) is compared to the sum of contributing processes. The $late x b \bar b$ component is shown in cyan. Ignore the blue bars below the main plot – they just show residuals from the fit results.

The measurement is not over once the sample composition of the data is assessed by the above fit, of course: backgrounds have to be shown to be modeled correctly, and the probability that a light hadron is misidentified as a muon must be taken into account. Checks of all kinds can and should be made to ensure a solid result.

As an example: a sizable amount of muons coming from cosmic rays that happened to cross the detector during collider operation need to be removed -they otherwise spoil the impact parameter distribution- by observing the correlation of the impact parameter of the two muons, as shown in the figure below. On the left you can see a population along the diagonal (cosmic muons) – the two muons have the same impact parameter because they are one single track reconstructed as two opposite ones. On the right, data after the requirement that the azimuthal angle between muons of opposite charge is smaller than 3.135 radians are totally devoid of the nasty background.

Other checks are made: promptly produced muons can be studied, and tails in their impact parameter distribution sized up, by selecting a sample of Upsilon meson decays. Every time I see a plot of the three resonances (see below) I am reminded of why I love particle physics!

So, the analysis is complex, as I said, but we need not delve into the details. So what is the result ? It is found that . The result is not terribly more accurate than the ones quoted above, but the error on R is dominated by the theoretical uncertainty (which is based on a next-to-leading order technique – ok, ignore this detail, and we can both be happy). Therefore, since R should be close to one, or actually exactly one if theory were perfect, can we deduce that the former CDF result () was plain wrong ? Well, probably yes. In principle, each of the five measurements could be wrong; or all of them, all for the same reason or each for a different one.

The theory predictions could also be the source of deviations from unity in the ratios. What matters, though, is that several of the details with which the present result has been obtained show that the margin for a mistake or a misinterpretation of backgrounds or other instrumental effects, in this case, is really narrow. I have finally -with one year of delay- carefully read the paper, and I am thoroughly convinced that there is no more mystery hiding in the correlated b-quark pair cross section at the Tevatron.

It only remains me to point those of you willing to know more about this measurement to the public page of the analysis, where tens of plots are available together with additional documentation.

Note: despite the technical nature of the matter, I have made an effort to keep this post to a level simple enough that non-scientists should be able to handle. Feedback is welcome!

Nowadays when you are presented with a statement about the inner consistency of the Standard Model of particle physics (SM), and on the range of mass values that a neutral Higgs boson may possess in order to
fit the observed value of several fundamental quantities -all related in a non-trivial way among each other- you are entitled to raise both eyebrows.

Indeed, theorists today speak of the SM as a still-dead entity, because they know it cannot be the ultimate theory. They say it only describes things so well because we have not tested it at energies and forces large
enough. They are quick to point out, if requested (or even without any prodding) that the SM is just “an effective theory”, meaning the opposite (with theorists this happens, at times -it is a lingo barrier rather than backward thinking). They may add that the SM fails to explain the smallness of the mass of the Higgs boson (which has in earnest no apparent reason to be as light as the model wants it), and it does not grant a high-energy unification of fundamental forces in a way which is pleasing to their eye, with the three coupling constants meeting at a single, very large energy scale.

The Standard Model does have those shortcomings. But it has survived more than thirty years of painstaking scrutiny. So your eyebrows have to come down once you realize that, despite all caveats, the predictive power of the combination of existing theory and excellent determination of its free parameters is astonishing. It ain’t no string theory!

There are dozens, but one might say hundreds, of experimental predictions that can be worked out, only to find the SM in exceedingly good health. It is thus not surprising that a handful of these predictions has shown some nagging disagreement with the data in the past. Among them, one might quote a few that are still unresolved today (each of them representing a deviation of measurement from theory by roughly two to three standard deviations): if you accept a list without explanation, I may quote the inconsistency of the measured value of the W boson mass with the observed ratio between neutrino charged-current and neutral-current interactions measured by the NuTeV experiment; the Z boson asymmetry measured by LEP, which shows a difference when measured in leptonic versus hadronic decays; the branching ratio of decays; and the anomalous magnetic moment of the muon.

The anomalous magnetic moment of leptons

A magnetic moment is a property of charged particles with a non-zero value of spin. Although quantum mechanics prevents us from drawing a perfect analogy, a spinning charged sphere develops a magnetic field, and so do charged elementary particles. For them the magnetic moment is easily computed as the product of charge by spin, divided by mass.

The so-called gyromagnetic ratio is then a pure number defined as the magnetic moment computed in units of its charge divided by twice its mass: for electrons (where is the electron charge). The magnitude of describes the magnetic properties of the electron. All charged leptons have gyromagnetic ratios very close to 2, but not quite equal to it. They are exactly 2 in Dirac’s theory of charged fermions, but quantum corrections cause a shift. The deviation of g from 2 -its residual from it- is called anomaly, and it is universally recognized as a very important number, . It is a crucial quantity in electrodynamics, and in particle physics in general, because it is a very small number which can be measured directly, and small non-zero numbers can usually be measured with high accuracy.

Indeed, the electron anomaly is measured with exquisite precision: it is found that , or to within 0.24 parts per billion! It is by its measurement that we know the value of the fine structure constant, -the fundamental quantity of quantum electrodynamics. Theoretical predictions for can be computed to fractions of a billionth too, so a direct comparison of the two provides a spectacular test of our understanding of the underlying physics.

For muons, has been measured with accuracy better than five parts in ten millions, and here theory and measurement differ by 3.4 standard deviations. A paper by Massimo Passera and collaborators, which I will describe in detail tomorrow, discusses the discrepancy and critically analyzes it in terms of the consistency of electroweak fits and low-energy measurements used as input. What I want to do today is to give some preliminary information about the problem, so that I have a chance of explaining to outsiders the details tomorrow.

Calculating the muon anomaly

So, what is it exactly that goes in the calculation of the muon anomaly? Well, it boils down to adding together the contributions of different processes which modify the Dirac picture of a muon (whose momentum is labeled “p” and then “p’”) emitting a photon, as in the graph on the right. At “leading order” -that is, when ignoring everything else but the bare-bone electromagnetic process of photon emission- the gyromagnetic ratio is 2 and the anomaly is zero. However, in subatomic physics every process that is not forbidden will happen, with a certain probability which is the square of the “amplitude” of the corresponding particle diagram. Looking beyond the bare-bone process, at “higher order” one needs to consider a huge number of other processes, such as the emission of a second photon by the muon line, with the former subsequently reabsorbed by the muon after the emission of the main outgoing photon, as in the top left graph of the figure below.

As the number of allowed vertices increases, there may be two photon emissions, and fancier things may start happening, as shown below:

Here we have to count at the same order (because they have the same number of vertices -points where three lines meet) diagrams where a single photon is emitted and reabsorbed, but the photon spends some time in the form of a virtual loop of charged leptons, as in the two graphs shown in the lower right above. At still higher orders the diagrams to consider are many more, but they respect the general structure with lines and blobs like those shown in the figures above.

Similarly, we can imagine that the incoming muon emits and reabsorbs a virtual Z boson; or the muon may emit a W boson and temporarily turn into a muon neutrino, as in the diagram in the center in the figure below. It is only by computing each and every possible virtual diagram, with all known particle interactions, that the total quantity we have to compute -the muon anomaly- comes out right.

Of course, the number of diagrams diverges as the number of “vertices” (points where particle lines intersect) increases. But physicists are good at performing approximations: by organizing the “higher order” corrections in series of the number of vertices, they can prove that each additional term in the series is a small correction to the former. So they just continue calculating more and more complex diagrams until they have to give up (since the number of diagrams to be computed grows factorially with the number of vertices), and their final result will be good to within the estimated contribution of the first neglected term in the series.

From the “nuts and bolts” description I gave above you have by now realized that in the “master” diagram we considered -photon emission from a charged muon line-, strong and electroweak physics enter by necessity at higher order in the perturbation series. Thanks to their electrical charge, even quarks may be produced by a photon fluctuation, and quarks are subject to strong interactions: what they may do, while they are alive in the red blob shown in the graph on the right, needs to concern us. The strength of those interactions will affect the final result for the muon anomaly despite the virtual nature of the quarks! The same goes with W and Z bosons which a muon line can emit (W bosons also connect to photon lines, thanks to their electric charge).

Thus,in the calculation of higher orders of the muon anomaly, there enter not only electrodynamics (which we claim to know inside-out), but also weak and strong interactions: the former are those mediated by the exchange of weak vector bosons (W and Z), the latter are instead those governing the dynamics of quarks -the constituents of nuclear matter- and gluons, the carrier of strong force.

The weakness of an interaction means that as we go to higher orders in the perturbation series diagrams with more vertices become very improbable, and the corrections they cause become small very quickly: the
series converges, and we can calculate it [Post-scriptum: The series does not actually converge - this is a mistake I prefer to not correct, see the comments thread below- but the calculation does work for quantum electrodynamics!]. But for quantum chromodynamics -QCD, the theory of strong interactions- this unfortunately does not happen! Alas, the basic QCD processes we need to compute are
“non perturbative”: higher-order contributions are large and cannot be neglected, no matter how you reorganize your perturbation series.

QCD is a wonderful theory, and high-energy processes can be computed with it with great precision, because at high energy the strong coupling constant (a number by which the probability of any QCD process has to be multiplied once for every particle vertex in the diagram describing the process) is small; but at low energy is large, and perturbation series diverge.

Because of that nasty property of QCD, a calculation of the muon anomaly needs to rely on approximations, modeling, and the knowledge of low-energy QCD. Some processes that help derive the quantities which are used in these approximations are those we measure in low-energy electron-positron scattering experiments: the cross section of these reactions determine how strong an impact some QCD virtual diagrams have in the muon g-2 calculation.

Ok, I think I have dumped above the preliminary information one needs to have in order to read the next, I hope enlightening, post, which will discuss the recent analysis by Massimo Passera and collaborators. In their paper they explain how the upper theoretical bound on the mass of the Higgs boson depends on the amount of uncertainty in low-energy hadronic cross sections one is willing to allow. Those who can’t wait for the post, and can read a hep-ph paper without assistance, are encouraged to get it here.

UPDATE: This link will bring you to the second part of this post.