In 1989 the CDF experiment was sitting on its first precious bounty of proton-antiproton collisions, delivered by the Tevatron collider at the unprecedented energy of 1.8 TeV. One of the first measurements that was produced was the measurement of the mass of the Z boson, which was at the time known with scarce precision by the analysis of a handful of candidates produced by the CERN SppS collider, at a third of the Tevatron energy.

Additional motivation to perform a precise measurement of the Z boson mass was the competition with the Mark II experiment at SLAC. In those years there was open rivalry between the two laboratories, SLAC and Fermilab. Mark II was about to produce its first Z mass measurement with electron-positron collisions, and it was clear that either CDF quickly produced a determination of the Z mass, or there would be no business left for them once the SLC collider would start doing the same at full power (and Mark II, in turn, would soon lose the battle with the four CERN experiments ALEPH, DELPHI, L3, and OPAL - but that's another story).

The story of the very important and precise measurement that CDF did in the summer of 1989 is full of interesting anecdotes. Here I want to tell you briefly one of those, loosely taking it from a more complete text I am writing for my book on the experiment.

One of the crucial tasks to be carried out was to calibrate the momentum scale of the tracking chamber to as high a precision as possible: only once that is done can the Z mass be precisely measured from the momenta of the two leptons it decays into (either an electron-positron pair, or a muon pair). The calibration is a necessary step: if you step on a scale to measure your body weight, you cannot be sure of the reading unless you have previously checked that the scale is calibrated, e.g. shown to return a reading of 100 kg or so if you put a 100 kg reference weight on it.

In principle the magnetic field inside the solenoid where the tracking chamber resided was known to a few parts in a hundred thousand or so. Thanks to that, from the curvature of charged tracks one could determine the particles’ momenta with high precision. To calibrate the measurement one could use the tracks of the decay of J/ψ mesons: those particles yield electron and muon pairs when they decay, and their mass was already known with sub-MeV precision from the measurements performed in the seventies, so it was a perfect “reference” for the calibration procedure.

CDF had a large sample of identified J/ψ decays, but something apparently was wrong: the mass of the J/ψ came out 3 MeV lower than expected. This was a 0.1% downward bias, since the particle, which is a bound state of a charm-anticharm quark pair, weighs 3097 MeV. Arguably a smallish offset. But what was its source ? Steve Errede and Bob Wagner, two of the physicists who participated in the collective effort to deliver a quick and precise Z mass measurement, could not believe that the magnetic field map could be wrong by a part in a thousand; and on the other hand track curvatures were very precisely measured by the many ionization deposits in the tracking chamber. They decided to ask the in-house theorist, Michelangelo Mangano, whether electromagnetic radiation effects could affect the J/ψ mass measurement at that level.

Michelangelo was a young and brilliant theorist who had joined the previous year the CDF experiment with the Pisa group, originally to work on measurements of quantum chromodynamics. He had previously worked as a post-doctoral scientist at Fermilab in the theory division, where he had developed techniques for the calculation of strong interaction processes. This made him an invaluable resource for the experimental studies that the QCD group in CDF was starting to carry out. At that time QCD was by no means a very well-understood theory, and those measurements were extremely interesting: the Tevatron was stepping for the first time in a totally new energy regime, where the understanding of strong interactions needed to be tested with experimental data.

The basic processes which allowed QCD studies were ones yielding many hadronic jets, whose energy was back then also still in the need of a precise calibration; but for jets, the typical errors were of tens of GeV: thousands of times larger than those that Errede and Wagner were puzzling over ! When they explained their problem to Michelangelo, he could not help bursting off with a sound laugh: “3 MeV ? M-e-V ? Are you guys kidding me ?”.

But indeed, that was a quite significant offset in the case of track calibration; a failure to understand it would force Errede and Wagner to ascribe the bias to a systematic uncertainty associated with the momentum scale. And due to the fact that tracks with higher momentum are increasingly harder to bend in a magnetic field, a scale error of 0.1% on the momenta of the decay products of the J/ψ, which were in the 5 GeV ballpark, would automatically become a 1% error on the measured momentum of the 50 GeV tracks emitted in the decay of the Z, which weighs 91 GeV: a total disaster, making a competitive determination of the Z mass totally impossible. CDF was aiming for a measurement with a precision of a few hundred MeV, whereas a 1% uncertainty from the momentum scale alone would make the mass error as large as one GeV.

Michelangelo spent some time working at the problem, but he found no clue of what could be going on: there seemed to be nothing unaccounted for in the J/ψ mass measurement from a theoretical standpoint. Of course final state QED radiation did affect the measurable mass of the J/ψ meson, but the effect was well understood and under control. The problem had to be elsewhere.

In fact, it was fortunately discovered soon thereafter to be of a quite different nature: it was to be ascribed to the habit of experimentalists to round off numbers. If you draw a circle on a sheet of paper and give it to an experimentalist and a theorist, asking them to tell you what is its circumference, the theorist will tell you it is pi times its diameter and will stop there: a perfectly correct but useless answer. The experimentalist will instead measure the diameter as well as she can with a ruler, and she will then multiply that by 3.14, reporting the result with three digits of accuracy; she will not bother to use a dozen digits in the expression of pi, since she knows that the experimental error on the diameter estimate (a sixteenth of an inch or so) dominates the uncertainty in the answer: three significant digits for pi are more than sufficient. But if later another experimentalist measured the diameter with a digital microscope, getting a value with eight significant digits, and proceeded to multiply that by 3.14, forgetting that pi is in fact 3.1415926..., that would be a big mistake !

A similar thing was happening with the reconstruction code that spewed out particle momenta from fits to the track trajectories measured in the tracking chamber. The conversion from curvature to momentum required an implicit multiplication by the speed of light, which is c=299,792,458 meters per second. Whoever had written the routine had probably had better things to worry about at the time than looking up the exact value in a data book, and had inserted a good approximate value for c: 300,000,000 m/s ! The difference was less than one per mille, but it was a huge one considering the high precision of the tracking measurement. That error had percolated up to become a nagging problem with the momentum scale !

Additional motivation to perform a precise measurement of the Z boson mass was the competition with the Mark II experiment at SLAC. In those years there was open rivalry between the two laboratories, SLAC and Fermilab. Mark II was about to produce its first Z mass measurement with electron-positron collisions, and it was clear that either CDF quickly produced a determination of the Z mass, or there would be no business left for them once the SLC collider would start doing the same at full power (and Mark II, in turn, would soon lose the battle with the four CERN experiments ALEPH, DELPHI, L3, and OPAL - but that's another story).

The story of the very important and precise measurement that CDF did in the summer of 1989 is full of interesting anecdotes. Here I want to tell you briefly one of those, loosely taking it from a more complete text I am writing for my book on the experiment.

One of the crucial tasks to be carried out was to calibrate the momentum scale of the tracking chamber to as high a precision as possible: only once that is done can the Z mass be precisely measured from the momenta of the two leptons it decays into (either an electron-positron pair, or a muon pair). The calibration is a necessary step: if you step on a scale to measure your body weight, you cannot be sure of the reading unless you have previously checked that the scale is calibrated, e.g. shown to return a reading of 100 kg or so if you put a 100 kg reference weight on it.

In principle the magnetic field inside the solenoid where the tracking chamber resided was known to a few parts in a hundred thousand or so. Thanks to that, from the curvature of charged tracks one could determine the particles’ momenta with high precision. To calibrate the measurement one could use the tracks of the decay of J/ψ mesons: those particles yield electron and muon pairs when they decay, and their mass was already known with sub-MeV precision from the measurements performed in the seventies, so it was a perfect “reference” for the calibration procedure.

CDF had a large sample of identified J/ψ decays, but something apparently was wrong: the mass of the J/ψ came out 3 MeV lower than expected. This was a 0.1% downward bias, since the particle, which is a bound state of a charm-anticharm quark pair, weighs 3097 MeV. Arguably a smallish offset. But what was its source ? Steve Errede and Bob Wagner, two of the physicists who participated in the collective effort to deliver a quick and precise Z mass measurement, could not believe that the magnetic field map could be wrong by a part in a thousand; and on the other hand track curvatures were very precisely measured by the many ionization deposits in the tracking chamber. They decided to ask the in-house theorist, Michelangelo Mangano, whether electromagnetic radiation effects could affect the J/ψ mass measurement at that level.

Michelangelo was a young and brilliant theorist who had joined the previous year the CDF experiment with the Pisa group, originally to work on measurements of quantum chromodynamics. He had previously worked as a post-doctoral scientist at Fermilab in the theory division, where he had developed techniques for the calculation of strong interaction processes. This made him an invaluable resource for the experimental studies that the QCD group in CDF was starting to carry out. At that time QCD was by no means a very well-understood theory, and those measurements were extremely interesting: the Tevatron was stepping for the first time in a totally new energy regime, where the understanding of strong interactions needed to be tested with experimental data.

The basic processes which allowed QCD studies were ones yielding many hadronic jets, whose energy was back then also still in the need of a precise calibration; but for jets, the typical errors were of tens of GeV: thousands of times larger than those that Errede and Wagner were puzzling over ! When they explained their problem to Michelangelo, he could not help bursting off with a sound laugh: “3 MeV ? M-e-V ? Are you guys kidding me ?”.

But indeed, that was a quite significant offset in the case of track calibration; a failure to understand it would force Errede and Wagner to ascribe the bias to a systematic uncertainty associated with the momentum scale. And due to the fact that tracks with higher momentum are increasingly harder to bend in a magnetic field, a scale error of 0.1% on the momenta of the decay products of the J/ψ, which were in the 5 GeV ballpark, would automatically become a 1% error on the measured momentum of the 50 GeV tracks emitted in the decay of the Z, which weighs 91 GeV: a total disaster, making a competitive determination of the Z mass totally impossible. CDF was aiming for a measurement with a precision of a few hundred MeV, whereas a 1% uncertainty from the momentum scale alone would make the mass error as large as one GeV.

Michelangelo spent some time working at the problem, but he found no clue of what could be going on: there seemed to be nothing unaccounted for in the J/ψ mass measurement from a theoretical standpoint. Of course final state QED radiation did affect the measurable mass of the J/ψ meson, but the effect was well understood and under control. The problem had to be elsewhere.

In fact, it was fortunately discovered soon thereafter to be of a quite different nature: it was to be ascribed to the habit of experimentalists to round off numbers. If you draw a circle on a sheet of paper and give it to an experimentalist and a theorist, asking them to tell you what is its circumference, the theorist will tell you it is pi times its diameter and will stop there: a perfectly correct but useless answer. The experimentalist will instead measure the diameter as well as she can with a ruler, and she will then multiply that by 3.14, reporting the result with three digits of accuracy; she will not bother to use a dozen digits in the expression of pi, since she knows that the experimental error on the diameter estimate (a sixteenth of an inch or so) dominates the uncertainty in the answer: three significant digits for pi are more than sufficient. But if later another experimentalist measured the diameter with a digital microscope, getting a value with eight significant digits, and proceeded to multiply that by 3.14, forgetting that pi is in fact 3.1415926..., that would be a big mistake !

A similar thing was happening with the reconstruction code that spewed out particle momenta from fits to the track trajectories measured in the tracking chamber. The conversion from curvature to momentum required an implicit multiplication by the speed of light, which is c=299,792,458 meters per second. Whoever had written the routine had probably had better things to worry about at the time than looking up the exact value in a data book, and had inserted a good approximate value for c: 300,000,000 m/s ! The difference was less than one per mille, but it was a huge one considering the high precision of the tracking measurement. That error had percolated up to become a nagging problem with the momentum scale !