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Thou shalt have three generations March 25, 2008

Posted by dorigo in Blogroll, news, physics, science.

Ever since quark and lepton families were first discerned in the reality of subnuclear physics, the question of how many such generations of matter particles exist  has been in the mind of particle physicists.

Fermion “families”, or generations, are composed each by three pairs of quarks and a pair of leptons. It might be argued that the first generation alone – a up and a down quark for each of the three colors, an electron, and its neutrino – would be all that is needed to make stars burn, planets form, and coke taste better than pepsi.  But Nature (the bitch, not the magazine) appears to have decided otherwise, to the bewilderment of all of us – primus inter pares Isidor Isaac Rabi, who famously said of the freshly discovered muon: “Who ordered that ?”.

Three generations 

Three generations of quarks were recognized as the minimum in 1971 by Kobayashi and Maskawa to accommodate a complex phase in the quark mixing matrix responsible for weak interactions, and thus justify the 1964 observation of CP violation by Christensen, Cronin, Fitch and Turlay. Indeed, the discoveries of the charm quark in 1974, the tau lepton in 1975, the bottom quark in 1977, the top quark in 1994, and the tau neutrino in 2000 filled up a tidy three-generation scheme which is pleasing although mysterious. Why not a fourth generation ? Why on earth not a fifth ?

In the mind of particle physicists, three is a very round number: three is the number of color charges a quark may have; and electrons have an electric charge that is three times larger than that of down quarks. But while the two above “coincidences” are in fact deeply intertwined, the presence of additional fermion generations would make no apparent damage to the overall structure of the Standard Model.

However, there are experimental hints that point to three generations. The one which is most unequivocal  comes from the Large Electron-Positron (LEP) ring at CERN, the machine which had to be decommissioned in 2000 to leave room for the Large Hadron Collider, due to start slamming protons against protons at 14 TeV this fall. What did LEP find ? Aleph, Delphi, L3, and Opal – the four experiments housed along the LEP beam – detected millions of Z bosons produced in 91 GeV electron-positron collisions in the nineties. By an extremely precise measurement of the Z cross section as a function of beam energy – what is called the Z lineshape – they could determine a parameter of the Z boson called its natural width: the inverse of its lifetime. The Z boson decays to fermion-antifermion pairs, and its lifetime would be significantly shorter than it is if a fourth species of neutrino existed. The LEP experiments were thus able to determine that only three neutrinos exist with a mass smaller than half the Z mass. Below you can see the Z lineshape and the three different curves one would observe if the number of light neutrinos were two, three, or four.

3 nu lep

Caveats, assumptions, approximations…. Can we ever get rid of them ? Indeed, strictly speaking the LEP result says nothing on the number of generations! It just says that only the three lightest neutrinos have a mass smaller than 45 GeV. Given that in the nineties we had no experimental clue that neutrinos do have a non-zero mass, the LEP result seemed back then a quite compelling argument: if all neutrinos are massless, then there are three of them… But since we proved that they are massive, the LEP constraint on the possible theories of Nature has started to look quite soft. 

LEP, however, did much more than measure the Z lineshape. A large number of standard model parameters have been determined with great precision. Together with external information from other experiments -most notably those coming from the precise measurement of W and top quark masses- these parameters allow to place further constraints on the number and characteristics of supernumerary fermion generations. From the Particle Data Group (PDG) database one gathers that any additional quark or lepton doublet is ruled out if the difference in mass between up- and down-type fermions is larger than 85 GeV; and on the other hand, the analysis of so-called oblique parameters S,T,U allows to determine that no additional generations of fermions which are degenerate in mass exist.

Implications on Higgs phenomenology

It is important to stress that the above constraints are valid in the context of the Standard Model. More complicated frameworks might allow more generations! The Standard Model is a theory: a remarkably successful one indeed, but we have to keep in mind that it might one day be proven wrong… For that reason, direct experimental searches for new quarks and leptons are not a vacuous pastime. 

I will come back to direct searches at the end of this post, to describe the new result by CDF on the existence of a fourth-generation t’ quark. Suffices here to say that collider data are so far unable to exclude the existence of additional fermions. We can thus dream on for a moment, and discuss one further characteristics of a world with more quarks and leptons: their effect on the Higgs boson.

At a hadron collider, the majority of Higgs bosons are produced through a mechanism called gluon fusion: the two colliding hadrons emit an energetic gluon each, these “fuse” together, and a Higgs boson is generated. The fusion involves something called a fermion loop, as in the picture on the right: Higgs bosons do not “couple” to gluons, in fact, but rather to fermions, with a strength proportional to the square of the fermion mass. So gluons create a fermion loop, and the loop materializes a Higgs boson. 

To the fermion loop any existing fermion with mass smaller than M_H /2 contributes proportionally to its mass; but a fermion with a mass larger than half the Higgs boson mass will contribute a fixed, non-zero amount. Quite a remarkable fact: you could make this new hypothetical fermion as heavy as you want, and it would still influence the rate of production of Higgs bosons at hadron colliders!

In fact, the enhancement in production rate of Higgs bosons granted by a fourth heavy generation of fermions has already allowed the Tevatron experiments to exclude a Standard Model-like Higgs with mass close to 160 GeV in this four-generation scenario. As you might know, in fact, that mass region is the one where CDF and D0 are most sensitive to Higgs production: they are by now close to exclude a 160 GeV Higgs even in the regular, three-generation case.

An increase in Higgs  rate is always good news, but there is a catch – and a very important one! In fact, the fermion loop described above does not only occur at the production vertex: it influences also the Higgs decay modes. If a fourth generation of fermions makes the loop more probable, thus enhancing direct production through gluon fusion, it also makes the decay to two gluons more probable!

A decay to a gluon pair is horrible news! The two gluons would be utterly indistinguishable from QCD backgrounds at a hadron collider. Let us look at a typical scenario for the decay rates. In the plot below (taken from hep-ph/0706.3718) you can observe that a gluon decay enhancement would negatively affect the rate of observable decay signatures at low mass – to b-quark pairs, and also to pairs of photons. And the more high-mass generations you add, the worse things get.


The plot shows, as a function of Higgs mass, the fraction of decays to the possible final states. The purple curve shows the large fraction of virtually invisible H \to gg decays enhanced by the presence of a fourth-generation of quarks. Note that the gamma-gamma final state (in black) -the one on which LHC relies the most for a discovery of a light Higgs boson – is quite a bit less than one in a thousand in this scenario. 

What conclusions should we draw from the above picture ? I think just a simple lesson: when you buy a box of cereals you are allowed to read the ingredients – they are printed, albeit in small fonts, on the side of the box. Instead, when you examine a graph illustrating a particle physics result which claims to exclude the existence of a particle, you do not get the same treatment. A number of assumptions are implied and, if you are lucky, they will be listed in the accompanying paper, but they will not fit in the graph label nor in the caption. So, the next time you look at a Higgs mass exclusion plot, keep it in mind!

Direct searches for a fourth generation t’ quark

Finally, let me discuss the new CDF limit on the existence of heavy fourth-generation quarks. As I said above, direct search limits are not compelling on such animals. In the past CDF found a lower mass limit at 199 GeV on a heavy b’ quark decaying to a Z boson, b' \to b Z, using the fact that one would then observe a Z boson produced away from the interaction vertex. Other searches set less stringent limits. 199 GeV is heavy stuff if compared to the mass of a lightweight up quark (about 0.003 GeV), but not terribly so if compared with the next-of-kin top quark (172 GeV). So, there is room for improvement here!

And improvement it is. With 2.3 inverse femtobarns of Tevatron Run II proton-antiproton collisions, CDF -through the sapient hands of John Conway, Robin Erbacher, and their collaborators at UCDavis- has produced a new search for fourth-generation quarks. The search is in this case focused on top-like t’ quarks, which are assumed to decay 100% of the times in a W boson and a light quark: that is, one assumes that M_t' - M_b' < M_W, where b’ is the isodoublet partner of the sought t’. If the t’ is lighter than the sum of W and b’ mass, it will decay to a W and a down-type quark belonging to the first three generations: d,s, or b. Further assumptions (see, I stick to the cereal box rule) include a regular pair production of t’ and anti-t’ quarks via QCD processes. Also, the search focuses on t’ masses larger than 172 GeV.

The mass of the hypothetical t’ quark is reconstructed in events with a lepton plus jets topology in much the same way as is done for the mass measurement of the top quark. The analysis then uses the reconstructed mass along with the H_t variable, which is defined as the sum of transverse energies of all final state objects in the event: jets, missing Et, and the triggering charged lepton. A two-dimensional fit on these two variables is performed on the data, interpreting them as the sum of all standard model backgrounds (dominated by regular top-antitop production and W+jets production) and a possible t’ signal. The fit is performed for different masses of the t’ quark, using a Monte Carlo simulation of the expected shape of the mass and H_t distributions of the signal. 


The figure above shows the mass distribution of the data, overlaid with the backgrounds as interpreted by the fit: in blue top quark production, in green W+jets, in grey residual QCD backgrounds, and in yellow the t’ signal allowed by the fit assuming a t’ mass of 280 GeV.

The procedure discussed above extracts an upper limit on the abundance of signal allowed by the fit as a function of the unknown t’ mass. Such a curve can be translated, accounting for data luminosity and selection efficiency, into a limit on the production cross section of the new quark pair. Since the latter is well predicted by quantum chromodynamics as a function of t’ mass, the limit translates into a lower limit on the t’ mass. In the figure below you can indeed see, as a function of t’ mass, the cross section limit (red line) and the theoretical prediction (purple line) crossing at 284 GeV, which is the new lower limit on t’ mass.


Inquisitive minds will have by now started wondering about the “bump” in the mass distribution at about 400 GeV. Indeed, with some fantasy one could see the presence of a signal there. The significance of an excess in the tails of the H_t and mass distributions has been estimated by the authors, and is less than two standard deviations: no reason to get excited about it… Yet.

In any case, a look at the high-mass events is common practice. Here is the event display of one of them: in green the high-Pt muon, in red the missing transverse energy vector, and in grey tracks belonging to jets (seen in the calorimeter as blue/red energy depositions). The reconstructed t’ mass of this event is 45o GeV!


More event displays and other information can be found in the
public web page of the t’ search analysis


1. Kea - March 26, 2008

Hmm. Interesting. I see you mentioned the 400GeV bump on the Resonaances post about new physics.

2. Anonymous - March 26, 2008

If one restricts to b tags, does the bump go away?

Looking at the public web page, it is not possible to tell — and the “public” note requires a CDF password to access!

3. dorigo - March 26, 2008

Yep Kea, I did… Jester links to an enthusiastic CP-violation talk which is all about four generations, and which even mentions the “need” for a 400 GeVish new quark pair.

Anon, I will ask John about b-tags. But you must realize that we are talking about a handful of events. Whether they survive b-tagging or not says very little. For instance, if these were the decay of a real t’ quark, b-tagging would probably reduce them, although not just as much as if they were W+jet events (who knows what the fourth column of the CKM is ?). Since backgrounds at 400 GeV are half top and half W+jets, any result is possible.



4. Tripitaka - March 27, 2008

Fantastic post!
Goodness me, you do raise some worrying questions about the nature of discovery in this age of advanced research, worst case scenario being that the chosen assumptions may eliminate the possibility of detecting a real signal I guess!
One thing about nature is that it seems to have no problem with redundancy, why not 10 generations, or a hundred for that matter (though I must on the subject of redundancy that I recoil in horror at multiverse theories which really DO seem too wasteful!)
Anyway, we punters are cheering on the sidelines in the hope of some absurd exerimental result emerging that will set all the theorist on their asses… null Higs should do the trick?

5. dorigo - March 27, 2008

Hi Tripitaka, thank you…

Surely not finding the Higgs would make us pause. As would the discovery of a fourth generation of chiral matter… If there were infinite generations, I think a 120 GeV Higgs might well be unobservable even at the LHC (I should be able to do the computation but I am too lazy…), due to the asymptotic behavior of the H->gg branching fraction.


6. Alejandro Rivero - March 28, 2008

A 4th generation sounds too cheap even as a trick: 128 degrees of freedom in total, an exact power of two? C’mon.

7. island - March 28, 2008

Surely not finding the Higgs would make us pause

Probably because you’ll be so shocked that it is actually possible to hear me cheering from the other side of the Atlantic… 😉

8. dorigo - March 28, 2008

Hi Alejandro,

I actually do think there is nothing magical in three generations. I would think it much more natural if there were an infinite number… But then we’d have to part with asymptotic freedom 😦

Island, I do not think I’d be shocked – a non-find comes in very slowly, as opposed to a find. So we’d all have time to get accustomed to it, come to terms with it, and accept it.


9. Alejandro Rivero - March 29, 2008

Well, the practicioners of Maldacena’s conjecture would think more natural if there were an infinite number of colours.

But yes, I still should prefer an infinite number of generations (thus of flavour) than exactly 4 generations.

10. island - March 29, 2008

Island, I do not think I’d be shocked – a non-find comes in very slowly, as opposed to a find.

Yeah, “GITMO” has nothing on you guys… In fact, this whole last year is like one long water-boarding session, before the real torture starts… 😉

So we’d all have time to get accustomed to it, come to terms with it, and accept it.

Maybe you experimentalists would, but theorists will be in-line with the beam dump… 😉

11. Eric - March 29, 2008

Why not just prefer one generation rather than an infinite number. The structures in the universe as we know it don’t depend on the existence of more than one generation.

12. dorigo - March 29, 2008

Ok Alejandro, but three colors have been established, while evidence against N fermion families is still sketchy… Or am I wrong ?

Island, have you seen the cartoon posted by Marco here ?

Eric, one cannot prefer something which is ruled out by experiment. One can only choose between theories we have not yet killed!


13. Eric - March 29, 2008

Hi Tommaso,
Yes, I know but my point being that there are already two superfluous generations for which we have no explanation for, and so it doesn’t make any sense to prefer three, four, five, or an infinite number of generations. Of course, your other post on the Higgs mass suggesting that a signal may be emerging in the range 114-130 GeV also suggest only three generations of MSSM fermions, if true.

14. dorigo - March 29, 2008

Hi Eric,

I also have a point – an infinite number of generations to me looks quite simple, because they look like the energy levels of an electron in a bound system like an hydrogen atom: infinity takes away one degree of freedom. Not two, nor three, nor four. It can’t be one because of energy considerations, and thus there’s an infinity of replicas.

In any case, I suspect nature did choose, and it chose three. This is however a real mystery – much more mysterious to me than the number of colours, for reasons I mentioned in the post.


15. Eric - March 29, 2008

Actually, instead of energy levels of the electron in a bound system such as the hydrogen atom, I would prefer to say that the different generations are just different vibrational modes of the string.

16. island - March 30, 2008

Island, have you seen the cartoon posted by Marco here ?

Ha… that’s pretty funny and right on par with my thinking. Thanks, and thank you to Marco, as well.

17. Neil B. - April 3, 2008

Sorry to come late, but I remember hearing that more than three generations would have left cosmological consequences of a sort we don’t find. Well?

18. dorigo - April 7, 2008

Hi Neil,

sorry – I myself waited too long to answer. Your comment is vague, and I do not remember any real show-stopper against more than three generations from cosmological arguments – but I am probably not well informed. I will try to get back to you after I investigate further.


19. Matti Pitkänen - September 30, 2009

Just the same old boring stuff about p-adic length scale hypothesis predicting the possibility of mass scales coming as half octaves;-).

Primed tops could correspond to p-adic half-octaves of ordinary top rather than higher generations. m= sqrt(2)*m_t with m_t=170 GeV would give m=about 240 GeV and m= 2*m_t gives 340 GeV The lower limit for t’ mass is 280 GeV so that actual mass should be 340 GeV. m= 2*sqrt(2)m_t would give 480 GeV to be compared with 450 GeV mentioned above.

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