Spontaneous symmetry breaking, Higgs mech, and particles getting masses?

[jeebs]

I'm trying to get a basic picture in my head of particles having mass. I always seem to come across the ridiculously vague statement that "the Higgs mechanism gives particles mass", and a passing mention of "spontaneous symmetry breaking". There is a lot of stuff confusing me at the minute so I'll try and lay out the main things.

On my course we're doing an introductory module on "beyond the standard model", and it starts out by mentioning how in the standard model, all the particles are massless, then something happens that bestows some particles with mass. This is my first little bit of confusion - how come they were thought of as having no mass? Particles clearly have mass so why was this ignored?

Then there was talk of a (currently) hypothetical "Higgs" field pervading all space, that has a "non-zero vacuum expectation value" - cue diagrams of the "wine bottle" potential, where some term is added to some Lagrangian and it shifts the minimum of the potential energy of the vacuum away from the zero value of the field.Apparently we give the non-zero shift to the neutral component, like
$$\left(\begin{array}{c}H^0\\H^-\end{array}\right) \rightarrow \left(\begin{array}{c}V\\0\end{array}\right) + \left(\begin{array}{c}H^0\\H^-\end{array}\right)$$
which keeps the vacuum charge neutral, which is obviously vital. That's fair enough.

Now apparently this is given the fancy name of "spontaneous symmetry breaking". Is it that the "symmetry" referred to is that all the particles are considered to be equally massless, but by some of them then being "given" mass that symmetry is gone?
Or, is it that the symmetry breaking refers to the neutral component of the Higgs field being modified, rather than having some other charged component?
(Here I'm picturing the example I keep reading of the ball positioned at the top of a 3D "wine bottle potential" type of hill, and it's got symmetry at the top of the hill, but if it rolls down in any direction, the symmetry is gone).

Now to something else that puzzles me. Apparently shifting the Higgs field to its non-zero vacuum expectation value, we have also done something called "electroweak symmetry breaking". This means nothing to me. This is no doubt a naive thing to say, but I do not see what the Higgs field has got to do with the electroweak interaction.

I mean, I get that something is symmetric with respect to a certain transformation, we say it is invariant under that transformation, but what has that got to do with what's going on here?
What has the Higgs field got to do with the electroweak interaction? Under what transformation is the electroweak interaction losing its invariance by invoking the Higgs field?

I'm struggling to tie these ideas together - What is the Higgs field actually doing beyond just stating that it's "giving particles mass" by "electroweak symmetry breaking"?
Sorry if this is a bit of a rambling question but I don't know how else to ask it.

 

[Bill_K]

Jeebs, the "particles that are initially massless" are the W and Z boson. Electroweak symmetry theorizes that the electromagnetic and weak interactions are aspects of the same force, mediated by the photon along with the W and Z. This symmetrical situation is broken by the Higgs because it gives mass to W and Z, making their influence short range. The photon remains massless and long range.

The Higgs also gives mass to the fermions, and of course that is necessary, but it is an added feature unconnected to the symmetry.

 

[jeebs]

ahhh right, this I was not aware of. thanks.

 

[Vanadium 50]

Bill_K is right. It's best to ignore the fermion case for the moment.

Pre-symmetry breaking, you have four massless fields: the w1, w2, w3 which form an SU(2) triplet and B, a U(1). You add a massless Higgs doublet, turn the crank, and you get a massive W+, W- and Z, a massive Higgs, and the photon stays massless. That's the trick - you don't want to give everything a mass.

One way of looking at the symmetry breaking is that it picks out the "direction" of the photon - exactly one linear combination of the w3 and b is massless. It turns out that this combination is "tilted" about 30 degrees from the B.

When you add a (phi+,phi0) Higgs doublet, it's actually a bit of a cheat. By putting electric charges on the fields, you are implicitly incorporating the answer. When I teach the Higgs mechanism, I do this twice: once with (phi+,phi0) and then again with a generic (phi_up, phi_down). The second time it's clear that there is always a direction with a massless field.

 

[jeebs]

so then, what was it that made people start out with these W1, W2, w3, B-boson, Left and right handed quark and lepton doublets and so on?
I'm suddenly confronted with a bunch of particles I've never heard of (not to mention SUSY partners) and it's making my head spin a bit.

Infact, as I'm writing this response I'm reading off those particles that I just mentioned from a table with a column entitled "interaction states", and close to that there is another column called "mass states" that contains all the particles I'm familiar with - leptons & quarks of the 3 generations, the photon, Z, W+, W- and g (plus 5 Higgs particles h0, H0, A0, H+ & H- that I'm not familiar with).

What is the difference between a mass state and an interaction state?

 

[Hans de Vries]

When you add a (phi+,phi0) Higgs doublet, it's actually a bit of a cheat. By putting electric charges on the fields, you are implicitly incorporating the answer.

Otherwise the theory would have to make some prediction about $\theta_w$ itself...

Regards, Hans

(The cancellation math is "effectively independent" of g,g' and just depending on the $\tau^i$)

 

[qsa]

Otherwise the theory would have to make some prediction about $\theta_w$ itself...

Regards, Hans

(The cancellation math is "effectively independent" of g,g' and just depending on the $\tau^i$)

Hans, could you please check your PM, I have send you a message. also your email does not work for some reason. Thanks

 

[arivero]

Otherwise the theory would have to make some prediction about $\theta_w$ itself...

Ah, your Weinberg angle. In a few hours, it could be even worse. Your "spin 1/2" and "spin 1" equations can be put as

s=1;c=s*(s+1);75.36*sqrt(-c+sqrt(c^2+4*c))
91.18560
s=1/2;c=s*(s+1);75.36*sqrt(-c+sqrt(c^2+4*c))
80.37219

So that you had one input and two outputs, and thus a prediction of $\theta_\omega$

But you didn't mention the negative sign solutions, perhaps because they implied a purely imaginary mass. But anyway, here they are:

s=1;c=s*(s+1);75.36*sqrt(c+sqrt(c^2+4*c))
176.15701
s=1/2;c=s*(s+1);75.36*sqrt(c+sqrt(c^2+4*c))
122.38614

To prove that we are not doing a postdiction, it is arxived as formula 9 in hep-ph/0606171

 

[Hans de Vries]

Ah, your Weinberg angle.

But anyway, here they are:

s=1;c=s*(s+1);75.36*sqrt(c+sqrt(c^2+4*c))
176.15701
s=1/2;c=s*(s+1);75.36*sqrt(c+sqrt(c^2+4*c))
122.38614

To prove that we are not doing a postdiction, it is arxived as formula 9 in hep-ph/0606171

That would be pretty close

Hans