# How many sigma did the discovery of the W boson have?

When the W boson was discovered in the 1980s, nobody spoke of sigmas. How many sigmas was it at that time?

-
Standard deviations of significance (i.e. "sigmas") were part of the process and the literature long before 1980. I wasn't around then, but @anna might be able to say a few words about when the consensus that five sigma were needed to claim discovery came to be. – dmckee Jul 2 '12 at 15:59
@dmckee i can't find any evidence for that. i find that that level of rigour began in the 90s, with LEP and Tevatron, prior to that there weren't many backgrounds to contend with. – innisfree Jan 31 '15 at 21:37
@innisfree Perhaps that is too strongly stated. People were using counting statistics to judge the reliability of their results all along. The rigor starting to come in as part of the fall out from the Oops, Leon particle and other embarrassments in the 70s. – dmckee Jan 31 '15 at 21:44
@dmckee in 1977/78 the group I was working with came up with a 4 sigma resonance of pi-mu. Everybody was enthusiastic, because up to then we thought 4sigma full proof. Then another group did not find it and the whole thing deflated and made everybody very cautious in the collaboration of new claims. At the time we did not know of the look elsewhere effect, and there were many cuts in isolating the events in the famous plot. The bump is still there in the tapes, as far as I know. – anna v Feb 1 '15 at 12:20
@innisfree the above is for you too – anna v Feb 1 '15 at 12:21

Look at figure 1.3 in this lecture.

The number of Z bosons, about 22, over an extrapolated background of 0, makes it a five sigma.

The W is more complicated, since it is detected by the Jacobean peak (search for Jacobean) of the seen electron, fig. 1.4, but still it is well over 5 sigma.

Actually when a phenomenon is way out of the possible background, even one event is significant beyond the statistics of one. Take the lambda baryon. Even if you see only one, there is not doubt of its existence. A pair production of a proton and a negative pion is not something that can be swept under the rug of statistics (except if it is a measurement error, which is a different story).

-
Thank you. But isn't a number of 22 over a zero background almost an infinite number of sigmas? – Clara Jul 3 '12 at 4:50
Well, the statistical error of 22 is the square root of 22 divided by 22, by construction, and that is the standard deviation. The reason it was not quoted is because it was irrelevant, models were predicting 0 at that part of phase space, and, as I said in the lamda example, statistical estimates are irrelevant and the observation is valid regardless (unless there is a conceptual or systematic or..error). – anna v Jul 3 '12 at 7:37
@clara i agree. if the null hypothesis was $n=0$ (and it's poisson i.e. the sd is $\delta n = 0$), the probability of seeing $22$ in the null hypothesis is $0$, so the significance is an infinite no. of sigmas. – innisfree Jan 31 '15 at 20:17
$0.5^{21} > 5 \sigma > 0.5^{22}$ - in other words if the true average is less than 0.5, the probability of detecting 22 is a less-than-five-sigma event (2.87E-7). I suspect that is where the number 22 came from? – Floris Feb 1 '15 at 1:32
@Floris There were 22 measured events. Any number of events has a square root which is taken as the sigma for the probability instead of 22 to have 0 events. I do not understand the rest of the discussion of innisfree either – anna v Feb 1 '15 at 4:30

The $W$-boson was discovered in 1983 at the UA1 collider

Experimental Observation of Isolated Large Transverse Energy Electrons with Associated Missing Energy at s**(1/2) = 540-GeV UA1 Collaboration (G. Arnison et al.), Phys.Lett. B122 (1983) 103-116, Experiment: CERN-UA-001, Feb 1983

There's no evidence in the paper of a precedent in high-energy physics that a discovery requires $5\sigma$ significance.

The experiment observed six candidate events ($o=6$). By contemporary standards, the discovery claim wasn't rigorous: the paper did not discuss the statistical significance of the observation. However, diligence was paid to possible backgrounds, with the conclusion that

none of the processes considered appear to be even near to becoming competitive.

I think it's reasonable to surmise that the mean number of expected events in the background only hypothesis was $0 < b \ll 1$. I suppose, then, this is a case of a colossal statistical significance, perhaps much more than $5\sigma$.

Let me sketch the meaning and calculation of a significance level. When searching for a particle, one claims a discovery if observations at least as "extreme" as those observed are unlikely to have been made in absence of that particle (the null hypothesis, backgrounds only). "As extreme" is formalized with a test-statistic, such as a chi-squared, though in this case as extreme means observing six or more events.

We can calculate this probability assuming that the background is Poisson distributed. For an example, let's take $b=10^{-2}$: $$p(o\ge6 | \text{background only hypothesis, expect } b \simeq 10^{-2}) \simeq 10^{-15}$$ This is essentially what is known as a $p$-value (the probability of making observations so extreme in the null hypothesis).

It is conventional in high-energy physics to convert $p$-values into significances (one-tailed $Z$-scores). The relationship is that is if $X$ follows a standard normal distribution, $$p(X > Z) = \text{p-value}$$ i.e. the the amount of probability in the right-hand side of a standard normal distribution for $X>Z$ is the $p$-value. With this rule, our $p$-value corresponds to a significance of about $8\sigma$. In fact, here is table of background levels $b$, $p$-values and $Z$-scores (thanks scipy.stats!):

b      p-value           z-score
1      0.000594184817582 3.24165698309
0.5    1.41649373223e-05 4.18649213413
0.1    1.27489869223e-09 5.95823304548
0.01   1.37703605634e-15 7.90157221605
0.001  1.38769893338e-21 9.4708634946
0.0001 1.38876984648e-27 10.8196771789
1e-05  1.38887698418e-33 12.0203550264
1e-06  1.38888769841e-39 13.1128980073
1e-07  1.38888876984e-45 14.1220534022
1e-08  1.38888887698e-51 15.0643755536
1e-09  1.3888888877e-57  15.9515803405
1e-10  1.38888888877e-63 16.7923185584


The $p$-value is $$p(\text{observing such an extreme outcome, } o \ge 6 | \text{background only hypothesis})$$ This is not at all equivalent to, $$p(\text{observing b events, as predicted by background only hypothesis}| \text{best-fitting signal hypothesis, } s = \hat s) = \frac{e^{-\hat s} \hat s^b}{b!}$$ which for $b=0$ and $s=6$ gives $\text{$p$-value} = 0.002$ corresponding to less than $3\sigma$. This is, however, the incorrect formula.

-
This talk gives 39 W candidate events in the sample hep.manchester.ac.uk/u/hanl/lecture/Lecture3_WZ.PDF . – anna v Feb 1 '15 at 12:03
Anyway, I am a contemporary of the discovery :) , and at that time we were computing standard deviations in a simple way. This business of emphasizing probability values came with the LHC and all the limits papers. We would get the background by Monte Carlo and do the statistics for normal distributions for the statistical error and then estimated the systematic error. – anna v Feb 1 '15 at 12:09
@annav fwiw that link also gives 6 events at the bottom of p16. the simple way you've described is wrong, i'm afraid. i hope it wasn't used in any analyses! – innisfree Feb 1 '15 at 19:21
that was the way data was analyzed afaik until papers with limits started appearing. – anna v Feb 1 '15 at 19:30
@annav do you have a ref with that method? it's a big blunder. i find it hard to believe it was used in a paper. – innisfree Feb 1 '15 at 19:34