## History of Electroweak Symmetry Breaking

Physics in the 50s had great success with quantum electrodynamics and its perturbative methods because the coupling constant was smaller than 1: 1/137. However, for other interactions, perturbation theory was not working due to interaction strength and people looked at alternative theories, like S-matrix and Regge poles which ultimately lead to dead ends in physics.

If you look at strong interactions, the proton and the neutron are very similar and people naturally looked at the SU(2) symmetry. However this symmetry is broken by electromagnetism and people started thinking about how to break symmetries. Also from strong interactions the SU(3) symmetry was developed by Gell-Mann's eightfold way which made a successful prediction for a new particle. Today we know this is an approximate symmetry which comes from up, down, and strange quarks.

In 1954 Yang and Mills had their seminal paper in gauge theory. The same result was obtained by Ronald Show, a grad student under Abdus Salam, but he only wrote it in his PhD thesis and was not taken seriously. The problem of Yang-Mills theory is that it predicts a new infinite range interaction which does not exist in nature. Adding mass to the interaction restricts the range due to uncertainty principle, but adding a mass term makes the theory non-renormalizable.

Around the same time, Fermi developed his weak interaction V-A 4-point interaction theory and Schwinger suggested in 1957 what is now called the W+, W- weak bosons.

It was known that the weak interaction violates parity and was short range and the search was on for how to introduce this into the theory.

In 1961 Glashow proposed a solution to the parity problem by mixing Z0 with W0 and proposing the SU(2)xU(1) symmetry. Salam and Windberg independently proposed the same thing in 1964, and the W mass was put in by hand.

For the mass problem responsible for the short range of the interaction, Nambu proposed spontaneous symmetry breaking in 1960. Condensed matter physics were very familiar with spontaneous symmetry breaking as the explanation for plasmons in superconductivity.

The basic idea of spontaneous symmetry breaking is that the ground state does not share the system symmetry. A typical example is a ball of water which freezes: during crystallization the rotational symmetry is lost. In quantum field theory there was the Goldstone theory with its Mexican hat potential:

The radial motion generate an effective mass term (because locally one approximate the radial motion with a parabola), but the motion around the center corresponds to a zero mass particle: the Goldstone boson. Since the Goldstone boson was not observed in nature, this was a major roadblock to adding mass to non-abelian gauge theories.

In 1964 Gerald Guralnik at Imperial College, collaborated with Walk Gilbert - a student of Abdus Salam, and a US visitor: Richard Hagen came with the idea of the Higgs mechanism to combine the massless gauge theory with the massive Goldstone boson. The same mechanism was proposed also independently by Peter Higgs/ Guralnik, Hagen, and Kibble/, and by Englert and Brout.

The problem was how to avoid the unobserved Goldstone boson. If you impose a continuity equation you get a charge by integrating the current density. However you need to consider the surface at infinity and due to relativity and microcausality in Coulomb gauge charge does not exists as a self-adjoint operator and this avoids the presence of the Goldstone boson. The key is the presence or absence of long range forces which interfere with the Goldstone theorem.

Then the electroweak unification and successes followed: Weinberg in 1967 and Salam in 1967 and 1968 proposed the electroweak theory, in  1971 't Hooft proved its renormalizability. In 1973 Z0 neutral currents were observed in CERN, and in 1983 W and Z bosons were observed in CERN as well.

70's and 80's saw the development of quantum chromodynamic based on SU(3) and the Standard model based on SU(3)xSU(2)xU(1) emerged.

After 1983 the only missing piece of the puzzle was the Higgs boson. Originally this played a minor role, the big deal was the Higgs mechanics. In 2012 the Higgs boson was confirmed experimentally and Englert and Higgs were awarded the Nobel Prize.

So what next? Grand unification of electroweak and strong force and supersymmetry (SUSY)? With SUSY the three coupling constants for electromagnetism, weak and strong force converge exactly and this is very powerful evidence. Unfortunately there is no current experimental evidence for SUSY.

Then there is a big gap between the Standard Model and M-theory/quantum gravity. To put it in perspective, strings to Standard Model is like atoms to our Solar System. Or if an atom is blown to the size of the observable universe, a string in string theory is the size of a tree on Earth.

## The Sleeping Beauty Problem

I just came back from the DICE2014 conference and as I recover from the jet lag and prepare posts about the conference I'll present the last topic in the statistics mini-series, the sleeping beauty problem. Unlike the Monty Hall problem, there is no consensus on the right solution even among the experts which makes this problem that much more interesting.

So here is the setting: Sleeping Beauty participates in an experiment. Every day she is explained the process and she is asked about the the credence (degree of belief) that a certain fair coin landed heads or tails. So what is the big deal you may ask? The coin is fair and this means that it lands heads 50% of the time, and tails 50% of the time. However, there is a clever catch.

Whenever the sleeping beauty is put to sleep she takes an amnesia drug which erase all her prior memory. If the coins lands tails, she will be woken up Monday and Tuesday but if the coins lands heads, she will be woken up only Monday. On Wednesday the experiment ends.

So now for the majority opinion: the thirders:

To make this very clear, let's change the experiment and keep waking up the Sleeping Beauty a million time if the coin lands tails, and only once if the coin lands heads. If she is woken up on a day at random, the chances are really small that she hit the jackpot and was woken up in the one and only Monday. So being woken up more times when the coin lands tails, means that the in the original problem the credence the coin landed heads should be one third. If you play this game many times and attach a payout for the correct guess, you maximize the payout overall if your credence is one third.

Now for the opposing minority point of view: the halfers

On Sunday, the credence is obviously 50-50 because the coin is fair. Even the thirders agree with this. However, From Sunday to Monday, no new information is gained, therefore the credence should be unchanged and the overall credence should remain 50% throughout the experiment. If you adopt the thirder position you should explain how does the credence change if no new information is injected into the problem.

So which position would you take? There were all sorts of approaches to convince the other side, but no-one had succeeded so far.

## The Monty Hall Problem

Continuing the discussion about probabilities and their intuition, here is a classical problem: the Monty Hall problem.

The setting is as follows: you are presented with three doors. Behind each door there is either a goat or a car. There are two goats and only one car. You get to pick a door, and someone who knows where the car is located, opens one of the two remaining doors and reveals a goat. Now there are two doors left, the one which you pick, and another one. Behind those two doors there is either a goat or a car.

Then you are given a choice: switch the door, or stay with the original one. What should you do?

Now there are two schools of thought:
• stay because it makes no difference, your new odds are 50/50.
• switch because it increases your odds
Before answering the question, to build up the intuition on the correct answer, let's consider a similar problem:

Instead of 3 doors, consider 1 million doors, 999,999 goats and one car. You pick one door at random, and the chances to get the car are obviously 1 in a million. Then the host of the game, knowing the car location, opens 999,998 doors revealing 999,998 goats. Sticking with your original choice, you still have 1/1,000,000 chances to get the car, switching increases your chances to 999,999/1,000,000. There is no such thing as 50/50 in this problem (or for the original problem). For the original problem switching increases your odds from 1/2 to 2/3. Still not convinced? Use 100 billion doors instead. You are more likely to be killed by lightening than finding the car on the first try. Switching the doors is a sure way of getting the car.

The incorrect solution of 50/50 comes from a naive and faulty application of Bayes' theorem of information update. Granted the 1/3-2/3 odds are not intuitive and there are a variety of ways to convince yourself this is the correct answer, including playing this game with a friend many times.

One thing to keep in mind is that the game show host (Monty Hall) is biased because he does know where the car is and he is always avoiding it. If the game show host would be unbiased and by luck would not reveal the car, then the odds would be 50/50 in that case. An unbiased host would sometimes reveal the car accidentally. It is the bias of the game show host which tricks our intuition to make us believe in a fair 50/50 solution. The answer is not a fair 50/50 because the game show host bias spoils the fairness overall.

The amazing thing is that despite all explanations, about half of the population strongly defends one position, and half strongly defends the other position. If you think the correct answer is 50/50, please argue your point of view here and I'll attempt to convince you otherwise.

Next time we'll continue discussing probabilities with another problem: the sleeping beauty problem. Unlike the Monty Hall problem, the sleeping beauty problem lacks consensus even among experts.

## Casualties of modern society mathematical illiteracy:

### Lucia de Berk, Colin Norris, Ben Geen, Susan Nelles

A good friend of mine, Michael Slott, once made the following observation: claiming illiteracy carries a social stigma, but stating that "I don't know math" is perfectly acceptable. Granted, math deals with abstractions you don't really learn until you assimilate the content and pure memorization is not enough.  But given the proper motivation, it is not that hard to understand it.

In his last interview, Carl Sagan noted: "people read the stock market quotations, and financial papers." Also: "people are able to look at sports statistics. look how many people can do that."

The trouble comes when emotions enter the picture and under the right circumstances it can lead to really bad consequences.

Statistics can be counter-intuitive. For example, rolling the dice and getting six times in a row 1-1, for the seventh time you may think you are in a lucky streak and the chance to get the same thing is really high. Casinos know very well how to take advantage of our emotions and lack of statistical intuition, and so are the state lottery systems.

Now in statistics there is a common mistake: the prosecutor's fallacy. To explain this we first need to explain Bayes' theorem:

$$P(A|B) = \frac{P(B|A) P(A)}{P(B)}$$

The simplest way to understand this is to derive it as follows:

P(A|B) is the probability to have A given B
P(B|A) is the probability to have B given A

Then:

P(A|B) * P(B) = the probability to get both A and B = $$P(A \cap B)$$

also

P(B|A) * P(A) = the probability to get both B and A = $$P(B \cap A)$$

Because $$P(A \cap B) = P(B \cap A)$$ we have:  P(A|B) * P(B) = P(B|A) * P(A) q.e.d.

Now let A be the evidence (E) of guilt, and B the innocence (I). Prosecutor's fallacy is ignoring Bayes' theorem and concluding that:

P(I|E) is small because P(E|I) is small

This is best illustrated with a lottery example. Winning the lottery has a very small probability, but does winning it make you a cheater? Prosecutor's fallacy implies that:

P(honest person given that you won the lottery) = P(honest | won the lottery) = P(winning the lottery | honest person) = P(winning the lottery by a honest person) = tiny percentage

Therefore you are a cheater.

Now onto Lucia de B case. She was a nurse in a children's hospital and some of the kids there died while under her care. Some doctors called the police to investigate and she was ultimately convicted based on the Prosecutor's fallacy which yielded a 1 in 342 million chances that a nurse is present by chance at the scene for the unexplained deaths . In fact the correct calculation was 1 in 25.  Fortunately the conviction was ultimately overturned, but other similar cases were subsequently discovered.

In Netherlands there was Lucia de Berk, in UK Colin Noris and Ben Geen, in Canada Susan Nelles

Are there any unknown US cases?

So how do this happen? In a stressful hospital situation something out of the ordinary occurs. Then someone who stands out from the crowd is associated with the incidents.

The situation snowballs and all past unexplained events are associated in an "aha moment" with the "serial killer".

Side note: I recall a personal experience with this pattern matching psychological effect. In 2002 in Washington DC area there were two snipers who randomly shot people. Police were clueless and at some point they announced on the radio that the snipers were firing from a white van. The next day I was amazed to realize how many white vans were on the road, and probably all the white vans were pulled over by the police that weak. In the end it turned up that the white van information was incorrect. Now if you did not experience this kind of police drama, recall the last time you bought a new car. You tend to notice so many cars on the road identical with yours.

The case goes to trial and faulty prosecutor reasoning is not properly countered by the defense due to sheer math ignorance and/or incompetence. The "serial killer" is unfairly convicted and only the dedication of honest experts and the public pressure can sometimes overturn the conviction.

When emotions run high, logic takes a back seat. In US there was a racially motivated nurse cases, in France there was the famous Dreyfus case. We should "J'accuse" elementary mathematical ignorance.

## DICE 2014

This is a very short post. Soon I will attend the DICE2104 conference in Castiglioncello (Tuscany) Italy. One presentation I am looking forward is "History of electroweak symmetry breaking" by Tom Kibble, one of the co-discoverers of the Higgs boson.

I applied for this conference well past the deadline and there were no more presentation slots available and I am only presenting a poster. Here it is below (please zoom in the browser to see it better: CTRL + does the trick.)

After the conference I'll have plenty of material to cover, including interesting tidbits on gauge theory. In the meantime I am preparing an interesting statistical post which includes dramatic real-life implications. Please stay tuned.