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The top quark mass measured from its production rate June 26, 2007

Posted by dorigo in news, physics, science.

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).


1. Glen Behrend - June 28, 2007

It will be funny when they figure out they need millions of these collisions at the same time to allow they to really see whats on the other side of what they are doing.

Right now it’s kind of be like tring to look into my living room from behind a star through a screen one hundred times smaller then one pixel on your screen. But if you make the screen larger, you may see whats on the other side.

I think they have hit on gravitons, dangerous ground really.

2. dorigo - June 28, 2007

Dear Glen,

your comment makes little sense to me… Sorry! If you want to understand more about particle physics, however, hang around, or look at the collection of posts on the tab “physics made easy” of the main page.


3. Matti Pitkanen - June 29, 2007

Dear Tommaso,

your earlier posting about Rumsfeld hadrons and the new result about top quark mass served as a partial stimulus to look through the p-adic mass calculations for quarks and hadrons. TGD predicts the most recent top quark mass 169.1 GeV as the maximal value of this mass. I would expect the measured mass to reduce still by few GeV.

The p-adic mass formulas explain also the finding that Bc mass is average of Psi and Upsilon masses. The per mille accuracy is of course accidental. For details see my blog posting.


4. dorigo - June 29, 2007

Hi Matti,

169.1 is not far away from the current best estimate… In fact, it is 170.9+-1.8 as you probably know, so exactly just one standard deviation off the value you quote. Moreover, what mass are we talking about here ? Bare mass, the one you can get from an amplitude scan, or the one you reconstruct from kinematical fitting ? I think the difference is sizable, one cannot define the mass of top to an accuracy larger than some hundred MeV. But I’ll take your prediction, and let’s see which side the mass moves as we improve the precision. We expect a 1.2 GeV total error in a couple of years…

I will now have a look at your post.


5. Glen - August 6, 2007

I tend to think that gravitons actually seperate dimensional space, that’s all.

Question; When you do these experiments, are there un-accounted for energy or mass fluctuations? I think what we are missing is a tool to measure what is appearing as a result of these collisions, probably because we have no idea what is coming in to the space, or what is coming into the space cannot maintain integrity long enough to be measured.

6. Anthony Neil Barker - April 2, 2008

The problem with accelerating a particle is that the electron field is already in motion… I have come top see the electron as kinetic energy. I wrote a bit about it on my site neebert.net and I think its why we think time slows as we increase our speed. If the electron is measured at 50kms then I say we have accelerated a particle to the speed of light. It was energy in the electron in a particle we almost got to the speed of light. Anyway I think to get more proton – antiproton collisions is hard but maybe looking at things like where the moon and the sun are will help. If you are trying to be accurate something that pulls oceans around must be taken into the equation. Well I hope you don’t think I am lecturing, really I am not an expert, it’s just an intrest to me.

7. dorigo - April 2, 2008

Dear Anthony,

the pull of Moon and Sun is tiny, but it is taken into account in today’s particle accelerators. As for the rest of your comment, my honest advice is that if you are interested in these matters you should read more about it before trying your shot at fitting the bits of information together. Just my two cents…


8. Anthony Neil Barker - April 2, 2008

Thanks, good advice! Like I said I am no expert and I am glad to hear they take the moon etc into account. Your right I should read more but it’s a pain as I am dyslexic. Your two cents are much appreciated… I think I will leave particle collider’s to the experts…

9. Michael O'Halloran - July 3, 2008

Can I suggest that the mass of the top quark can be predicted if the proton is included in the analysis of quark configuration? Moreover, if this is done the relationships between the other quarks are predicted and clarified.

When the proton mass is included in the ranking of experimentally inferred quark masses, then the connections between the quark masses are revealed. Thus:

1. Proton mass is the square of the bottom quark; which is the square x 20 of the strange quark; which is the square x 20 of the up quark.

2. The top quark mass is a simple modular multiple of proton mass and the reciprocal of the Rydberg Constant.

MeV/c2 kg.10-27
up 1.5 2.673943431sq’d
= down 4 7.149973471 x 20
= strange 80 142.9994694sq’d x 10
= charm 1150 20448.84826 x 2

= bottom 40897.69651sq’d 7305
= proton 938.272 1672621580
x 1/109,737.31568639 x 2
= top 170900 ±1800 30484.0987

The Table leaves the MeV values unchanged from the usual listing and instead concentrates on the kg-mass values. There is an interesting transition in the Table from kg-mass to eV-mass and back to kg-mass with the bottom quark. I don’t fully understand the transition, though I have my suspicions. The top quark seems to be predicted to play a different role from that played by the other quarks.

I realize that the quark masses are usually portrayed as somewhat chaotic, but they do have structural form. The quark masses are usually referred to as ‘freely specified constants’, implying that we have no idea about why the numbers have the values they do, and that any other numbers would do, if experiment suggested some other set. The Table above would seem to suggest otherwise. In other words, I suggest that my theory above might remove the quarks at least from the list of ‘freely specified constants’ in the Standard Model.

What do you think?

10. Michael O'Halloran - July 3, 2008

Sorry; the Table layout has gone haywire in the sent version, it seems. So I will remove the MeV values (except for the bottom quark) and hope that you can read the Table more easily.

up 2.673943431 sq’d
= down 7.149973471 x 20
= strange 142.9994694 sq’d x 10
= charm 20448.84826 x 2
= bottom 40897.69651 sq’d (MeV) 7305 (kg)
= proton 1672621580
x 1/109,737.31568639 x 2
= top 30484.0987

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