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A mystery behind the Z width September 14, 2006

Posted by dorigo in internet, mathematics, news, physics, science.

A graph showing the natural width of non-strongly-decaying elementary particles as a function of their mass was linked in a comment to my post below about weak decays by Alejandro Rivero. Check it out here:


The plot is extremely informative. It represents years and years of elementary particle searches and measurements, obtained from our beloved source, the PDG. Particle widths are just a measure of how long it takes for them to disintegrate: the shorter their lifetime, the larger the width, which is the  uncertainty (actually the spread) in their mass, just as it happens when, asked to measure something, you come up with a worse measurement if you are given less time to complete it.

To discuss the graph in some detail, however, and to explain what is the mystery behind the Z width, I have to become a bit technical, and I think it will go beyond my ability to simplify for non-physicists, who will hopefully excuse me if they feel compelled to leave here.

In the plot, a line showing the scaling of a cubic law typical of electromagnetic decay relationship between widths and masses is drawn from the lightest particle in the pool, the neutral pion, all the way to the EW scale. The line ideally connects all particles with dominant electromagnetic decay, and qualitatively fits all of them, with the exception of the Upsilon (1S)-(2S)-(3S) triplet; amazingly, though, it hits the W and Z boson marks in the eye. 

A similar fifth-power law, which of course you would expect from dimensionality arguments alone (same kind of scaling as that of EM decays, with the additional power of two from the Fermi constant) is able to reproduce rather well all weak decaying resonances, by just extrapolating beyond the muon.

Of course, one would not expect elementary particles composed of quarks to line up too well – what matters is the mass of the decaying quark, at least in the spectator model, so the hadron mass is not the one which one would like to exhibit a scaling property.

The coincidence of the lining up of the Z with the line connecting electromagnetic decays is rather striking. Alejandro has in fact a paper out on this issue, on hep-ph/0603145:


Quoting from the paper:

“In total we have eight electromagnetically decaying strong particles (ten if we count W+ and psi(2S) following a scaling rule across four orders in mass and mysteriously fitting a purely electroweak quantity, the decay of Z° (that is controlled by sin(theta_W) basically).”

Quite remarkable indeed. In these times of scarce theory predictions, it is something to keep in mind…


1. dorigo - September 14, 2006

Courtesy A.Rivero, here are the instructions to get the above plot done, and customizable to your taste or new data:

–download mass_width_2006.csv from
–go to the download directory and invoke


–in gnuplot do the following

set datafile commentschars “*”
set datafile separator “,”
set logscale xy
set key bottom Right
plot “mass_width_2006.csv” using 1:4

… and feel free to browse and zoom using the right button in the
mouse. Bind keys for zoom management are so are
a `builtin-autoscale` (set autoscale keepfix; replot)
n `builtin-zoom-next` go to next zoom in the zoom stack
p `builtin-zoom-previous` go to previous zoom in the zoom stack
u `builtin-unzoom`
g `builtin-toggle-grid`

… it can be useful to plot the auxiliary lines for cubic and
quintic dependences

replot 2495.2*(x/91187.6)**3
replot 2.99591E-16*(x/105.658)**5

… and so on. The plot shows all the pdg data. The strongly decaying
particles, which cluster up in the shorter lifetimes, should be
removed by hand. In a very few cases the total electroweak decay rate for such particles is known, but again it must be introduced by hand.

respect to the 2004 data table, there is a new datapoint… for Y(3940).

3.943E+03 ,1.7E+01,1.7E+01,8.7E-05 ,3.4E-05,3.4E-05,? ,?,?
,?,?, , , 0, ,S,Y(3940) ,cC

What does it mean???? Well, if you go to the erratum webpage
you find
(September 6, 2006): Units of Y(3940) WIDTH should be MeV.
so it is 87 MeV, no 87 eV

2. Alejandro Rivero - September 15, 2006

Thanks for the separate post on this! I was afraid I was sort of hickjacking the thread on the top quark. A pair of points deserve comment:

1) It is funny that the argument for the scaling usually is run in reverse way: there is a lot of literature on the quintic scaling of weak decay, and then one argues that the electromagnetic must be cubic by removing the extra power two got from the Fermi Constant.
This reversion is because EM decay happens mostly with neutral particles composite of quark-antiquark and the calculation becomes very touchy, via the anomaly. Amazingly, the EM decay of Lambda (not quark-antiquark, but perhaps sort of effective quark-diquark) fits in the line too. And for particles with both EM and strong decays, in the rare cases where the EM decay can be separately measured, it fits too.
Simply to state this point about EM scaling deserved, IMO, a preprint.

2) Now, why is the coincidency with Z misterious? Well, the point is that while Z decay is calculated from first principles (SU(2)xU(1) coupling constants is all we need) the decay of EM particles gets a correction due to its SU(3) binding. This correction is unknown and we hide our ignorance in a parameter named “the decay constant”, f. Thus it seems not only that these constants can be related via a simple scaling (for sure this is a conjeture stated a hundred times) but, here is the surprise, that they adjust themselves to meet also with the Z0 width, which have no relevant SU(3) contribution. This is, a SU(3) quantity and a SU(2)xU(1) quantity seem to be counterbalanced one against other.

3. Alejandro Rivero - September 15, 2006

Of course, one would not expect elementary particles composed of quarks to line up too well – what matters is the mass of the decaying quark, at least in the spectator model, so the hadron mass is not the one which one would like to exhibit a scaling property.

Indeed, the jpg plot show horizontal lines at the onset of the masses of s and c quarks, as a kind of guide to the eye used to spectator decay 😉

4. Alejandro Rivero - September 16, 2006

Hmm now I am reading the Donoghue Golowich Holstein “Dynamics of the Standard Model” and I am afraid I have implied somewhere that also the top quark theoretical width follows the spectator model. This should happen if it were lighter than the W, but as it is heavier it decays to W and then its width depends on the cube of its mass. I will check Nucl Phys B 314, a about this.

5. dorigo - September 18, 2006

Hi Alejandro,

I have that book, and indeed it quotes a cubic relation between top mass and width. Also interesting is what is written in “QCD and collider physics” about the topic, on pages 340-342.

One thing to note: I gave you an upper limit to the lifetime earlier. Now, if one just looks at the width of the reconstructed top mass in the high-stat samples by CDF and D0, it becomes apparent that there also is a lower limit. That is, from all distributions one could indeed compute that the top has a width smaller than, say, 15 GeV or so, lest all distributions become impossible to understand.

Of course such a lower limit can only be interesting if one is to exclude some fancy model where the top is allowed to decay to as-of-yet-unknown additional bodies…


6. Alejandro Rivero - September 18, 2006

Hi Tomasso, I am going to check in the library.

Indeed 15 geV is not worthy a limit. It is interesting to add to the plot the theoretical value of the top and a short cubic line because being of the order of the GeV, it happens to live in the Z-W area.

Also you did a comment about plotting the theoretical formula for the width of the higgs as a funtion of its mass. Indeed it should be ilustrative… just that it open the box about which Higgs model to choose (minimal, type I, type II…), doesnt it?

7. Alejandro Rivero - September 18, 2006

Funny that “QCD and…” and “Dynamics of…” are one next the another here in the Newton Institute library. I have checked also the issue of higgs decay, well, asymtoticly mostly we have a cubic line paralell to the ones of Z and top, the order (1) part varying depending of the decays available at each level. A whole plot, including the already excluded range, could have the nice property of showing how the weak and the electromagnetic lines are interpolated, and the same could be told of the top quark line, jumping from quintic to cubic when W becomes available.

But this is mostly a didactical show about the electroweak model and the role of W, Z as symmetry breakers. Very didactical, although. The mistery part keeps being why the decay constants of pion, j/Psi etc are adjusted in patterns that really are su(2)xu(1) originated.

8. Alejandro Rivero - October 9, 2007

I think I have a clue, but I do not like it. On the other hands, hard stringers will not like it neither. The idea is that the Z,W and the mesons are respectively the closed and open sectors of a oriented string theory. Naively, a q-qbar meson could either disintegrate or to joint its extremes closing into a glueball, but this can not be the real history because if quark and antiquark are of different charge they should send a W particle too; it is easier to think that all the mesons can close into string loops but that these strings are directly the SU(2) x U(1) bosons.
Similarly, diquarks are unoriented string, and then can not close into themselves but only with the corresponding “antistring”; thus in the unoriented closed sector all the gauge bosons, including coloured gluons, can be generated.
This is an spin-off (pun intended) of my idea (today arXiv:0710.1526 )to predict the number of generations and light quarks from a QCD-scale superstring; so it is pro-string theory and anti-string theoreticians at the same time. Ugh.

9. dorigo - October 9, 2007

Hi Alejandro,

about your old comment #7: I am interested in the issue of the Higgs width now, and need to dust off my incomplete knowledge on it, because I will be teaching some parts of a course of particle physics in December, and it will touch the issue of Higgs decay. Do you have a reference for it, or were you basing your comment on the two books ?

Anyway. Your latest comment is intriguing, but I think it is above my head unless I put time I do not have into some reading… But I will take a look at your paper! If I can grasp something I’ll produce a post on it.


10. Guest post: Alejandro Rivero, “sBootstrap” « A Quantum Diaries Survivor - October 16, 2007

[…] Furthermore, it is worthwhile to think of the quark-antiquark composites as oriented open strings, while the quark-quark and antiquark-antiquark are different sectors of unoriented open strings. To close an unoriented string you need to zip it against another unoriented string from the opposite sector, and then the resulting closed string carries a charge of the kind colour+anticolour, similar to a gluon. On the other side, an oriented string can close upon itself, giving a closed string uncoloured but perhaps still with electric charge… pretty much as the electroweak bosons, and it could explain the strange similarity in decay rates between the Z, W particles and the most stable mesons, of which Dorigo was kind enough to speak time ago in this blog. […]

11. Alejandro Rivero - October 23, 2007

About that course on Particle Physics, an interesting exercise on decay rates is to calculate how the decay rate of an hypothetical massive lepton or quark changes in the area of mass around the W, say between 60 and 130 GeV. Most textbooks give either the large mass limit (for top quark decays) or the low mass limit (for muon and spectator decays). The intermediate range gives students a lot of fun with Mathematica, Maple or your preferred plotting package.

some references can be found starting in SPIRES from Phys. Rev. D 37, 2676 and Phys.Rev.D30:947-960,1984

12. dorigo - October 24, 2007


I think you overestimate the level of my course 🙂 I think I will be happy if I can make the branching ratio curve understood qualitatively! And the cut of the course is rather experimental, so I won’t make many calculations, unfortunately… But I will still give a look at the two PRDs, for which I thank you!


13. Alejandro Rivero - October 24, 2007

What is funny of these papers is that they are written with some experimental perspective, in the sense that they do not buy the GSW model but instead they keep Fermi Constant separate from the mass of the W, not introducing the Weinberg angle nor the SU(2)xU(1) couplings.

14. Alejandro Rivero - October 26, 2007

The plot can be also used to give an illustration of electroweak symmetry breaking or restoring: the cubic line is (unexplained, but it coincides to be) the decay rate for a massive SM fermion into massless ones, when the mass of W is near zero but you keep Fermi constant fixed. The quintic line is the decay rate when the mass of W has its actual value.

15. dorigo - October 26, 2007

That is food for thought… I will think about it, I would like to show this plot but before I do I need to understand some of its nuisances… My course is rather standard, but I do like the idea of adding some bold ideas.


16. Alejandro Rivero - October 27, 2007

For standard ideas, I am inspecting with some detail the calculation of beta decay of the top (or any other fermion of the SM) in this thread: http://www.physicsforums.com/showthread.php?p=1482998

17. dorigo - October 29, 2007

I have read the thread, very interesting stuff! Please continue… I might end up using it for my course!


18. carlbrannen - October 31, 2007

Alejandro, thanks for the exposition on PF pointing here.

What I find interesting is that so many of the baryons (and mesons) fit so nicely on the same cubic line with the W,Z. Makes it seem easy to speculate that they’re made from the same preons, combined with the same forces.

I’ve managed to convince myself that the quarks, leptons, and gauge bosons are all made from the same number of preons, 12 each, counting left and right handed contributions.

19. dorigo - October 31, 2007

Hi Carl,

I find your hypothesis intriguing but my limited knowledge is forced to deem it far-fetched. I will look a bit deeper into your posts when I have time… This fall is being horrible as far as free time goes.

20. Alejandro Rivero - October 31, 2007

Carl, yep, in part the bold claim of the sBootstrap post is supported because this one invites to think that if some preon is in play in W,Z, then it is the same preon game to be played in the hadrons, and we know that the hadrons are composed of quarks and QCD strings. So my conjecture at the end of the other post, that perhaps the W and Z were just a string without the quarks.
A problem with stringers is that they keep unfocused (or focused on publication) and thus they never finish [almost] anything, so no real calculation is done of a quantum string terminated in fermions, nor to speak of it susy version. Also, a baryon should be a quantum string terminated between a fermion and a boson, and this is not ever mentioned in string theory.
In any case, there is a very good reason for preonic theories to work in the same way that theories of quarks and leptons, and it is that they need to cancel their own anomalies.

21. carlbrannen - October 31, 2007

Alejandro, things are moving very quickly now. There are too many directions to push ideas.

Some time ago, Michael Rios pointed out to me that the Koide formula makes the most sense if mass is treated as a vector. I guess other people have mentioned this too, perhaps.

Let me try and put it in E&M terms. Suppose the preons each contribute a certain amount of field strength. We are in an s-wave state (for point particles and also for the s-wave baryon resonances) so we can ignore spatial dependency. The total field is the sum of all the fields, contributed by valence and sea.

Since the E & B fields have three real components each, each preon’s field is defined as a 6-vector. The total energy in the field is given by the sum of squared amplitudes, as is usual in E&M. Then Koide’s square root mass formula becomes a statement about the relationship between the fields of the sea preons and that of the valence preons.

22. Alejandro Rivero - November 2, 2008

Let me note that the only particle outside from the mistery line, Ypsilon, is got theoretically as explained in http://pdg.lbl.gov/2008/reviews/upsi_m049.pdf

Thus I am left wondering if the new physics in https://dorigo.wordpress.com/2008/10/31/cdf-publishes-multi-muons/ alters this calculation, or even if it can be used to provide an alteration making the whole line consistent.

23. Alejandro Rivero - November 21, 2008

Revisiting comment 1: it is not needed for the argument here, but generically it is possible to plot errorbars from the csv file. It is convenient to separate the plot according particle status (col 103): R is “Established” and common, D is “Established but ommited from booklet”, S is “not well Stablished”, and F is “Further mesons”, poorly observed or coming from a single group, unconfirmed. Besides, some particles do not have error bars. Considering all of this, a valid plotting routine in unix with gnuplot is:

set datafile commentschars “*”
set datafile separator “,”
set logscale xy
set key bottom Right
plot ‘< grep ,R, mass_width_2008.csv | grep -v ,-1 ‘ using 1:4:($1-$3):($1+$2):($4-$6):($4+$5) with xyerrorbars
replot ‘< grep ,D, mass_width_2008.csv | grep -v ,-1 ‘ using 1:4:($1-$3):($1+$2):($4-$6):($4+$5) with xyerrorbars
replot ‘< grep ,S, mass_width_2008.csv | grep -v ,-1 ‘ using 1:4:($1-$3):($1+$2):($4-$6):($4+$5) with xyerrorbars
replot ‘< grep -v ,[RDS], mass_width_2008.csv | grep -v ,-1 ‘ using 1:4:($1-$3):($1+$2):($4-$6):($4+$5) with xyerrorbars
replot ‘< grep ,-1 mass_width_2008.csv ‘ using 1:4
replot 2495.2*(x/91187.6)**3

If you dislike grep, or you are in a windows machine, you must use instead some conditional operators of gnuplot itself.

Note that if you follow this procedure with the current data file, http://pdg.lbl.gov/2008/mcdata/mass_width_2008.csv , you will find two “not well Stablished” particles with a huge errorbar: K(1630) and Xi(c)(3123). The former is a 1998 measurement still sitting in the PDG book; the later is part of a recent study of BaBar. You can safely grep away both of them, or set a -1 for the error bars.

24. Alejandro Rivero - November 21, 2008

More plot issues. 1) This one puts the dots first, then overprints the errorbars in “box” format. For most of the RD data the errors are barely visible.

plot [100:100000] ‘< grep ,[RD], mass_width_2008.csv | grep -v ,-1 ‘ using 1:4 with points
replot ‘< grep ,[RD], mass_width_2008.csv | grep -v ,-1 ‘ using 1:4:($1-$3):($1+$2):($4-$6):($4+$5) with boxxyerrorbars

2) For the study in the present blogpost, the strong decays are unimportant. Thus we could discard the particles whose main decay is via QCD. In linux, I do this selection with
grep -v -E [\(][0-9]\{3,4\}[\)] mass_width_2008.csv
but mileage can vary. Besides, some electromagnetic or electroweak partial decays of those particles are huge enough to appear in the plot by themselves, so a more careful crafting is needed.

3) Names. In some configurations of gnuplot, you can plot “with labels”. In our case, with a log scale around, it can be:
replot ‘…’ using ($1*1.05):($4*1.02):17 with labels left rotate by -30

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