What Triggers Electroweak Symmetry Breaking in the Standard Model?

[roger]

How is the electroweak symmetry actually broken in the standard model ?

Please could you explain the idea to me.




Thanks

Roger.

 

[selfAdjoint]

Roger this a question for another subforum. I am going to move this thread to particle physics, which I judge to be a closer match than quantum physics.

 

[samalkhaiat]

OK Roger, I got your message. I believe it is better to post my reply here so that other people can correct me if I make a mistake.But can you tell me why did you choose me for this job?
Roger, answering your question without using the appropriate math (group theory) is not an easy task. I will try my best to present the subject as simple as I possibly can. However, I cann't guarantee you a clear picture. This is because, one can not avoid the use of abstract notions here.
Ther are two concepts involved;
1. Hidden symmetry (spontaneous symmetry breakdown),
2. Higgs phenomenon.

In general, there is no reason why an invariance of the dynamics (Hamiltonian, equations of motion, etc) should also be an invariance of the ground (vacuum) state. So one (someone like Goldstone) can conjecture that the laws of nature may possesses hidden symmetries,i.e. symmetries which are not manifest to us because the ground state is not invariant under them. This is usualy called "spontaneous breakdown of symmetry",(SBDOS).
So let us look at a simple situation where this SBDOS occurs.
Imagine a big container of water.This water is homogeneous (all its points are equivalent) and isotropic (has no preferred directions).The state of the undisturbed water stay the same if you rotate the water by any angle. So we say; the undisturbed water is invariant under rotation (rotational symmetry).
Now, cool this water down until the 1st strand of ice starts to appear. Obviously, this defines a "direction" in the body of water(anisotropic), i.e.the rotational symmetry is nolonger manifest.The important point, you need to understand, here is;
the moment of breaking of the symmetric state of water, and the direction of the strand of ice are "accidental". They occur "spontaneously".Before the appearance of ice, we had an infinite number of equivalent directions (invariant vacuum states), to choose from, but the ice did choose "one" particular direction(non-symmetrical new vacuum state).
One obvious consequence of this is; these strands of ice are "hard" & "sharp".They can prick you! This "consequence" will give you "some sort" of "hint" as to why in the electroweak theory, the photons remain massless, while the vector mesons(w's & z) become massive. Understanding this point requires the so-called Higgs mechanism which I am going to explain by another, more abstract, example which ,I hope,will answer your question.
Now consider a solid and symmetrical Mexican Hat and small ball. If you carefully place the ball on the top of the central hill of the hat, it will stay in this symmetrical position but not for long, because this position is unstable and the ball will roll off to the non-symmetric lowest points in response to smallest perturbation. So an obviously non-symmetric state (the ball on one side of the central hill) arises from symmetric system (the ball at the top of the hill). As in the 1st example, the moment of breaking of the symmetric state and the point where the ball stops are "accidental" and occur "spontaneously".Now we make the following non-mathematical and not very accurate correspondence:
THE BALL<===>THE FIELD OF PARTICLES IN THE THEORY,
THE HAT<===>THE HIGGS FIELD POTENTIAL ENERGY.
which we will use in a moment.
As you might know, the electromagnetic interaction is the result of photons exchange, and the short range weak interaction is carried by the "massive" vector mesons(W's & Z).At very high temperatures, quarks & leptons possesses high energies. so they don't need to borrow the energy needed to exchange massive mesons. Indeed, creating these mesons would be as easy as that of photons.Thus, at sufficiently high energies, the interactions between quarks & leptons is the unified electroweak interaction.Its carriers, W's,Z and photon, have zero mass. Why? well, "local gauge invariance" requires massless vector fields!
At such energy scale even quarks and leptons have zero mass.why? why not!
Let us go back to our hat.The position of the ball on top of the hill corresponds to the degenerate (symmetric) vacuum.At higher energies the shape of the "hat" (potential) changes so that its slopes would rise immediatly from the centre, So this point would be a stable position of the ball (fields).As temperature decreases, the well shape changes to a mexican hat.i.e. with a hill at the central point. When the ball rolls to the lowest-energy position in the valley by the central hill (the new vacuum), its position is not symmetrical. So the new vacuum state does not have the old symmetry. Now the temperature has droped (the rolling off has occurred) the Higgs field split into components. The vector mesons "eat up" the massless component (the Goldstone boson) and grow "fat". This provides a short range for the weak interaction we see today.
Quarks and leptons also gain masses through their interaction with the non-symmetrical Higgs which formed the non-symmetrical vacuum. But the photon remains massless. TO KNOW HOW ALL THIS HAPPENS, YOU NEED MATHEMATICS!
Were it not for the Higgs, all particles would remain massless and the electroweak symmetry would survive.The symmetry that was so clear at high temperatures is now hidden.

It is nice te remember that, at the time of their formulations,both the theory of non-abelian gauge fiels and the theory of spontaneous symmetry breakdown were thought to be physically untenable, because both predicted "unobserved massless particles", the gauge vector mesons and the Goldstone bosons. Now we know that each of these diseases is the other's cure.


for references;
1) s.coleman, secret symmetry, in Laws of hadronic matter, academic press,1975.
2) I.Novikov, The river of time, camb.uni.press1998.



regards

sam

 

[roger]

What do you mean by ''degenerate (symmetric) vacuum '' exactly ?

And why do you refer to the w and z particles as vector mesosn ?

what causes the spontaneous breakdown of symmetry ?

Roger

 

[samalkhaiat]

What do you mean by ''degenerate (symmetric) vacuum '' exactly ?

WE SAY STATES ARE DEGENERATE IF THEY HAVE THE SAME QUANTUM NUMBERS.i.e. THEY BELONG TO THE SAME REPRESENTATION OF THE SYMMETRY GROUP IN QUESTION.JUST THINK OF THEM AS STAES HAVING THE SAME ENERGY.

And why do you refer to the w and z particles as vector mesosn ?

THESE ARE GAUGE PARTICLES,LIKE THE PHOTON, WITH SPIN=1. THEREFORE THEY ARE MESONS. IN GAUGE FIELD THEORIES, THE GAUGE FIELD FUNCTIONS, WHICH REPRESENT THE GAUGE PARTICLES, CARRY A SPACETIME INDEX.i.e.THEY ARE SPACETIME VECTORS.

what causes the spontaneous breakdown of symmetry ?

READ MY POST. YOU COULD SAY THE SAME THING WHICH CAUSES PHASE TRANSITION.

Roger, you didn't tell me why did you choose me to answer your question.

sam