# Higgs Mechanism

1. Jan 10, 2010

### ryanwilk

Hi,

I'm doing an essay on the Higgs boson and I'm confused about what the "four degrees of freedom" of the Higgs field are. I know that the W and Z bosons become massive by absorbing 3 of the degrees of freedom and the remaining one becomes the Higgs boson but I don't really understand what this means. Are the degrees of freedom actual physical quantities?

Thanks.

2. Jan 10, 2010

### diazona

Not in the sense that you can measure them directly. They're really just scalar fields, variables in a Lagrangian.

How much do you know about quantum field theory? Or Lagrangian mechanics? I wouldn't want to get into a long detailed explanation that would go over your head

3. Jan 10, 2010

### ryanwilk

It's a L2 undergraduate essay so I don't really know much about either. I've attached what I've wrote so far about the Higgs mechanism. I just need a sentence or two where *** is, saying that the Higgs has 4 degrees of freedom and what this means...
I haven't mentioned the Lagrangian anywhere.

#### Attached Files:

• ###### higgs.JPG
File size:
64.3 KB
Views:
106
Last edited: Jan 10, 2010
4. Jan 10, 2010

### diazona

OK, let me see if I can fill in some details without getting too technical - not that you have to put this in your essay, it's just for your understanding.

In quantum field theory, basically everything is expressed in terms of fields, functions of spatial and temporal position. Kind of like the electric field and magnetic field (but those particular fields aren't the fundamental ones in QFT). The fields basically correspond to particles.

The Higgs mechanism requires a minimum of two of these fields, which could be labeled $\phi_1$ and $\phi_2$, or they could be combined into a single complex field $\phi = \phi_1 + i\phi_2$. Anyway, QFT has some sort of a potential energy that's associated with these fields. It's a function of the values of the fields. If you were to graph the potential $V$ as a function of the complex field $\phi$, it might look like this:

You'll notice that there is a whole ring of minimum-potential points, but the universe/particle/system/whatever you're dealing with can't be at every point on that ring simultaneously. It has to have some particular value of the field $\phi$ - it has to be at one particular point, just as if you dropped a ball into this graph it would come to rest at one particular point on the ring. Basically, the Higgs mechanism consists of making a particular choice of coordinates to put that one particular point at the origin of the graph. You define a new complex field $\eta$ which is a shifted version of $\phi$. That field $\eta$ is the Higgs field.

Now, when you use $\eta$ instead of $\phi$ in the mathematics of QFT, you find that your gauge fields (the ones corresponding to the W and Z bosons) acquire a mass. For a simple example, think of this (I'm switching to real, not complex, variables now):
$$[(1 + A)\phi]^2 = \phi^2 + 2A\phi^2 + A^2\phi^2$$
In QFT, any term that contains a field variable squared times a constant (like $m\phi^2$) tells us the mass of that field. In the above expression, $\phi$ has a mass of 1, but $A$ has no mass since there's no term of the form $m A^2$. But when you substitute $\phi \to \eta + D$ (this is how you shift the field), you get
$$[(1 + A)(\eta + D)]^2 = D^2+2 D^2 A+D^2 A^2+2 D \eta +4 D A \eta +2 D A^2 \eta +\eta ^2+2 A \eta ^2+A^2 \eta ^2$$
Now you see a term $D^2 A^2$, which means that now the particle corresponding to the field $A$ has a mass of $D^2$. That's basically the gist of the Higgs mechanism.

The Higgs mechanism that gives the weak bosons mass is just a more complicated version of that. It involves four scalar fields, that could be labeled $\phi_1$, $\phi_2$, $\phi_3$, and $\phi_4$. (If you were to read papers on the subject, you'd often see them combined into two complex fields.) That's where the four degrees of freedom come from. There are also the three gauge fields for the W and Z bosons, which correspond to $A$ in my example. I've only just started learning about this myself, so I couldn't really tell you much more than that, but hopefully that'll get you started

5. Jan 12, 2010

### ryanwilk

Thanks a lot. I think I get it now

6. Jan 15, 2010

### bapowell

Here's a short non-technical explanation if you're still interested:

The physical degrees of freedom associated with the Higgs are the symmetries of the model. Each symmetry of the model (really the Lagrangian of the model) has an associated Higgs field (eg the $$\phi_1$$ and $$\phi_2$$ that diazona is talking about).

Now, in the case of the Standard Model, the Higgs is associated with the electroweak symmetry -- this has 4 degrees of freedom (a higher dimensional analog of the figure provided by diazona, but conceptually equivalent). Early in the Universe's history, the Higgs field rolled down to one of the minima of the potential. This minimum no longer enjoys the full 4-dimensional symmetry of the theory, as you can see from looking at the lower dimensional analog. In fact, in the Standard Model, this minimum possesses only 1 symmetry, with 3 of the original 4 being "broken". These broken symmetries don't just disappear though -- they are what ultimately give rise to massive particles -- in this case, the three bosons of the electroweak force: Z and 2 W's.

7. Jan 24, 2010

### david2010

Is there a simple way of explaining to a senior high school student what (if anything) creates these scalar fields. At senior high school level: mass 'creates' the familiar newtonian gravitational field; charge 'creates' a classical electric field...therefore what 'creates' the Higgs field? Is an explanation at this level possible?

8. Jan 24, 2010

### ansgar

it has always been around, it is not a "force field"

9. Jan 24, 2010

### david2010

Thank you. It just 'is'. Unfortunately (some) students (possibly because of the way the force fields are taught at high school level) seem to look for 'something' to be the 'cause' of the field's existence. Is there any useful analogy I can employ?

10. Jan 25, 2010

### ansgar

Put it the other way around, masses are due to the field of gravity, and charges due to the field of electromagneism.. The concept of field was actually very debated in philosophical circumstances when it was introduced by guys like faraday etc in the 19th century.

So asking what is the cause of one another is like the chicken and the egg...

11. Jan 25, 2010

### bapowell

Quantum field theory teaches us that the fundamental entities of particle physics are fields. Of course, we're all familiar with gravitational fields and electromagnetic fields -- these are force fields, which, as has been mentioned in the post, are 'created' by something. But, we don't stop there. On equal footing with these force fields are electron fields, quark fields, and yes, the Higgs field. In quantum field theory (qft), we no longer think of, say, electrons as a fundamental entity, but as excitations of the electron field (just as photons are the excitations of the electromagnetic field). Nothing 'causes' the matter fields.

So, qft gives us a unified picture of both forces and matter -- they are all just fundamental fields, and the particles -- electrons, quarks, Higgs bosons, what have you -- are all just excitations of these fields. Bottom line: everything is a field -- some fields mediate forces, some do not. The confusion is probably a historical accident -- fields were invented to understand the influence of forces.

Last edited: Jan 25, 2010
12. Jan 25, 2010

### bapowell

Ansgar, this is not correct. Masses are not due to gravity. Within the standard model, masses are generated through interaction with the Higgs field. Gravity would be perfectly happy in a massless world, since it is energy that gravitates.

I also don't understand why you claim that charges are due to electromagnetism. Can you elaborate on this?

13. Jan 25, 2010

### ansgar

not all masses are created with interaction with the higgs field....

14. Jan 26, 2010