The top quark mass measured from its production rate

CDF has recently used the production rate of top quarks to increase by 20% the precision of one of its top mass measurements. A nice new technique and a brand new result, of which I will say more at the end of this post: I wish to explain here a few basic facts about particle production in hadron collisions first.

When you collide a proton with an antiproton with enough energy, what is actually happening is that the two particles “see” each other as drops of liquid within which lie tiny hard spheres. The drops sometimes just bounce against each other without losing their integrity: it is what we call an elastic scattering; other times, they merge and recombine downstream -a diffractive event; or one of them can break up as a result of the encounter, while the other proceeds almost unaffected.  In these cases, the energy transfer between the bodies is usually very small. Instead, in the relatively infrequent cases when a pair of hard spheres (the constituent quarks or gluons) in the two projectiles hit each other head on -a hard scattering event-, the protons totally break up, and the hard scattering produces a large energy release which may result in the materialization of new particles. Those are the collisions we are most eager to study!

In fact, you can think of a hadron collider as a variable-energy quark-gluon collider. Since it is the hardest scatterings those which have a chance of producing the as of yet unknown -or least studied- highest mass particles, you would like to increase the rate of those events. But you can only do that by increasing the energy of the projectiles – protons and antiprotons, or increasing the collision rate.

Quarks and gluons inside the proton (or the antiproton) possess a variable fraction of the total energy carried by their container. Here, if one thinks in terms of “energy=speed” one gets confused: the bag has a certain energy by virtue of its own speed, and its contents must have the same speed, correct ? Well, yes. But they have a variable fraction of the total energy, nonetheless. Let’s not think in terms of speed, which is basically the speed of light anyway, and concentrate on the dynamics within the “bags”.

A proton is indeed a bag within which quarks and gluons are constantly materialized and destroyed in quantum fluctuations. When they tell you a proton is made by putting together two u-type quarks and one d-type quark, that is only part of the story in fact. You need those three, but once you put them together they will start exchanging virtual gluons. And the gluons themselves may split into pairs of quarks of any other kind. At any given instant, if you take a picture of a proton you portrait a soup of many bodies, but you cannot even say how many of them there are, since that depends on how close you looked. The proton has a different average structure if probed at different energy -and energy dictates the scale of your picture because the higher it is, the smaller scales you can discern.

Does that quantum orgy going on inside the bag mean we do not know how a proton is made up ? No, it only means we need to use probabilistic concepts to describe it. We do this by parametrizing the fraction of the total proton energy that a quark possesses as a Probability Density Function (PDF). We call x (or sometimes even “Bjorken x” in homage to James Bjorken, who introduced the concept with Feynman) the fraction, from zero to one, and U(x) is our up-quark PDF in the proton. If U(x) is large, that value of x is probable; if U(x) is tiny, we will seldom get such a u-quark to collide.

Of course, d-type quarks will have their own PDF, call it D(x). And gluons too: G(x). And wait – there’s more. In fact, if there are gluons, gluons may materialize into pairs of other quark types, albeit for very short instants. To understand the proton, we do need a S(x) for “sea quarks”, those produced by splitting of gluons. That includes quarks of all kinds.

By studying hadron collisions for thirty years, particle physicists have amassed such a huge amount of information on these quark and gluon PDF that we have now a quite detailed picture of how the proton is going to behave when we smash it against one of its peers. Having determined S(x) for x=0.001, for instance, we know what is the probability to find a c-quark with a thousandth of the energy of the proton containing it. Wow, finding charm quarks  inside the proton sounds even harder than squeezing orange juice from a stone! The charm quark has a mass larger than the proton mass itself… And yet yes, the former may occasionally be found within the latter. Prodigies of quantum dynamics! Below you can see some PDF functions for the proton constituents.

There is an additional complication, into which I will not delve today, due to the fact that the PDF of quarks and gluons depend on the energy of the proton, a variable we call Q^2 in this formalism. I just mention it to avoid hiding unwanted details and causing confusion in the well-read among you. 

I’m slowly diverging, so let us go back to what is important here: by knowing the PDF of the quarks and gluons inside the proton, we can compute the probability of a collision – the cross section for a given process. Now, it so happens that PDF functions are very large at low values of x, and are unnervingly small at high x, decreasing to zero very quickly as x approaches 1. What that means is that the higher you want x to be (so that you can produce more energy when you smash the quark against a colleague from the opposite projectile) the rarer it is to find it [and this only worsens as you increase the Q^2 of the proton – but let’s leave this aside].

This strong decrease of the probability to find large x constituents in the projectiles means that very high-energy collisions – those that have a chance of producing top quark pairs at the Tevatron – are very rare. Not just that: it means that it gets rapidly more infrequent as you require the energy to be higher, since a high energy collision requires both of the colliding quarks or gluons to have a very high x. To set the scale, x=0.5 is already quite uncommon, so that to get a collision whereby a x1=0.5 quark inside the proton happens to be the one which hits another x2=0.5 antiquark inside the antiproton is really really rare. And the total energy will be… E = x1*x2*Etot, when Etot is the Tevatron energy, 2 TeV: E is thus 500 GeV, just about right for producing two 170 GeV quarks with something to spare for the motion of the produced bodies.

Now, theoreticians have mastered the calculation of the cross section for producing heavy bodies in hadron collisions. By using the proton PDF functions, and the rules of quantum chromodynamics, they have a very precise function which says how frequent is the production of a heavy quark pair as a function of the quark mass. If we knew that function perfectly, then by measuring the frequency of top quark pair production we would get the mass straight away!

Unfortunately, we know PDF functions only with some approximation, and the top pair cross section is known with a precision of about 10%. That is enough to play the game anyway: CDF just demonstrated it is indeed possible. By measuring the rate of production of top quarks in the so-called “dilepton” final state – one which has very small backgrounds, and thus very small uncertainty on the rate – a very compex analysis allowed to fit that information together with the theoretical cross section predictions, along with direct determination of the mass of the top quarks in the sample. The result is a more precise top mass measurement: the statistical error decreases by 20%. The new measurement in this channel, using 1.2/fb of data, is

M(top) =170.0+4.2-3.9(stat)+-2.6(syst)+-2.4(theory) GeV,

where the last error accounts for the uncertainty on the theoretical estimate of the top pair production cross section. The total uncertainty is +5.5/-5.3 GeV, to be compared with what one would get with no cross section constraints (M(top)=169.1+6.1-5.8 GeV): a sizable decrease! 

The picture above shows the cross section as a function of top mass, with the separate determinations of both quantities hatched in blue. The theoretical band is also shown in solid blue. By adding the cross section constraint the mass measurement moves from the hatched vertical band to the point with black error bars. 

This measurement is thus a proof of principle that with the increased precision of theoretical estimates it is nowadays possible to use the production rate to determine the mass of a heavy particle (something that electron-positron colliders have been doing for a while, incidentally: there, no PDF play a role, and the relationship with production rate and mass is much easier to use).