# The electroweak force?

1. Jun 20, 2004

### Ductaper

in particle accelerators, we've seen the electromagnetic force and the weak force unified. How? Do W+, W-, Z, and photons dissappear, to be replaced by another fermion, or what?

2. Jun 21, 2004

### zefram_c

Re: EW force

No, those particles do not disappear; they remain the mediators of the EW force at all energies as far as we know. The unification is more subtle. In a typical quantum field theory, you have the matter fields (fermions) interacting with the mediators (bosons). These interactions are described a constant known as the coupling strength, which sets the intrinsic strength of the interaction. The constant may be different for all pairs of fermion/boson, and a good theory should predict all of them from a smaller number of parameters via some symmetry arguments. There is also another part to the theory that describes the behavior of the bosons (eg. whether they interact amongs themselves or not. In pure EM and weak theory, photons do not directly interact with themselves but the Ws and Zs do).
What the unification does is it describes *all* couplings of the W,Z and photons to matter by only two parameters: one coupling strength and one mixing angle. For example, the coupling of the Z is different for all basic fermions (which amounts to 4 different couplings), but it turns out all of those couplings can be described by the charge of the fermion and the mixing angle in the theory. Furthermore, you can see the 'unification' at work in the sense that the coupling to the Z (supposed to be a weak force phenomenon) depends on the charge (an electromagnetic quantity). Another way in which the unification takes place can be seen in one predicted type of interaction that involves a W+, a W-, a photon and a Z. This can be interpreted to say a photon and a Z can directly interact to produce a W+ and W-. But the Z is a neutral particle, so a photon would not interact with it outside the unification framework.

Sorry if this is too long or unclear; it is my first post so feedback would be appreciated.

3. Jun 27, 2004

### turin

I would like to hear a little more.

What is the weak force? What interaction does it describe exactly? Can you give a typical/common example of two particles (fermions?) interacting weakly?

4. Jun 27, 2004

### zefram_c

The weak force is the mechanism responsible for beta decays (you can look these up on your own). It was first discovered on the atomic level, where a nucleus can change one of its constituent protons into a neutron, or a neutron into a proton (nevermind that this was first observed before neutrons were known; the change in number of protons is enough to make the transition visible since the new atom has a different chemistry). Today we regard this interaction at happening at the quark level, where one quark changes flavor and becomes another. This is known as the charged current interaction, where particles change identity. In non-unified contexts, the weak force mediates the reactions like the following:

$$n \rightarrow p + e + \bar{\nu}$$
$${\mu} \rightarrow e + \bar{\nu}_e + {\nu}_{\mu}$$

There are of course any number of related reactions by moving terms around. You might say that all these are decay reactions, and not fermions interacting weakly. So here are some that do:

$$p + e \rightarrow n + \nu$$ (Electron capture)
$$\nu_\mu + d \rightarrow p + n + \nu_\mu$$ (Dissociation of deuteron nucleus by high-energy muon neutrino)

The second reaction is an example of the neutral current interaction; it is a type of weak interaction that does not change particle types. Like EM, it permits particle-antiparticle annihilation/creation, but does not otherwise change the fermions.

The mediators of the weak interaction are the W's and Z. The W's have charge +1 or -1 and mediate the charged currents, the Z is neutral and mediates the neutral current.

This is not meant to be a complete description of the weak interaction; it has some quirks that are not found in any of the others (strong, EM) and I don't know if you have the background to understand them.

5. Jun 27, 2004

### turin

I probably don't have the background, because I have seen these types of (equations?), but I still am not seeing where the W's and Z's come into the picture (not like I can see where the photons and gluons come into the picture).

Is everything supposed to be made up of, or at least randomly emit, these W and Z particles?

6. Jun 28, 2004

### heardie

the force is mediated by (virtual?) W and Z particles I believe

7. Jun 28, 2004

### zefram_c

Hm. It is quite difficult to explain the concept of mediators without resorting to quantum field theory, but I will try. In our theories for the fundamental forces, the fermions do not interact directly amongst themselves; they interact via mediator boson fields. You can write some "half-reactions", in which fermions emit/absorb mediators, as follows:
$$e^- \rightarrow W^- + {\nu}$$
$$e^- \rightarrow e^- + Z$$
$$e^- \rightarrow e^- + {\gamma}$$
In the first bit, the electron converted into a neutrino through emission of a W-. In the other two, the electron emitted a Z boson and a photon. Now these processes cannot be observed in the lab because they violate conservation of energy/momentum. In the electron's rest frame, its total energy is $$m_e c^2$$. It cannot emit a photon or a Z and recoil, as the energy of the recoil electron is already greater than $$m_e c^2$$. Nor can it emit a W, which has a rest mass of 90GeV compared to the electron's 0.5MeV. However, we can connect two of these processes, like so:
$$e^- \rightarrow {\nu} + W^-$$
$$W^- + p \rightarrow n$$
This now represents electron capture, a real and observable event.
In the above process, the electron and neutrino are "real" particles - they obey the proper relativistic energy-momentum relation E2 = (pc)2+(mc2)2. The W does not. Since 4-momentum is conserved at the emission and absorption of the W, it can only pick up the difference and for certain it violates the above expression. This W is called a virtual particle, and we cannot observe it directly.
You can have a naive picture by thinking of an electron as surrounded by a cloud of such 'virtual' particles; they are constantly being emitted and reabsorbed. Now the more the energy-momentum relation is violated, the shorter the time between emission and reabsorption. This is because the uncertainty principle naively allows one to 'borrow' the energy required to create the W/Z, as long as it is returned in a time that is inversely proportional to the borrowed energy. If some other particle happens to wander in the vicinity and sees the field, it may pick up the emitted quantum. This is now a real process: either particles have changed type, or at least exchanged energy and momentum (aka they exerted a force on each other)
To answer your other question, the electron is not 'made up' of such particles. As far as we know, electrons, quarks and these mediators have no substructure (although there are theories that suggest they might). It is, however, correct to naively suppose that the particles that make up matter are surrounded by clouds of virtual particles.
Please note however that this is only a 'naive' picture. However, this understanding is ok for a qualitative description and only if you want to work in the field do you need to tackle the field theoretic ideas. (Of course, curiosity is encouraged, and if requested I can try to explain the QFT way of looking at things).

Last edited: Jun 28, 2004
8. Jun 28, 2004

### turin

Firstly, I want to let you know that I appreciate the way you are explaining this. I think that you filled in all of the blanks that I asked about, and now you have led me well into some new questions.

This is one of those things that I hope to get cleared up. If I take for granted that there are these bosonic field quanta that account for beta decay and such, I still don't understand why they are emitted in the first place. Why is the electron "unhappy" about being an electron so much that it violates a natural principle? Is this one of those questions that is beyond the scope of physics? Is there any kind of mechanism that excites the electron so that it will emit a weak boson, or is the process completely random?

I am familiar with the dispersion relation from relativity, but I don't understand quite what you were saying here about the W. What difference does it pick up (negative energy)?

Let's do it.

9. Jun 28, 2004

### zefram_c

Aha! Now we are getting even deeper into the field formalism. I'm not even sure if I can explain everything in a single post .
You will need a qualitative (non-computational, since there's no reason for me to post equations that you can easily find) understanding of Lagrangian mechanics, and quite a bit of quantum mechanics.
There is a simple, and almost magical, way to derive all of the three interactions. You start with the Lagrangian for a *free* fermion. You then require that the Lagrangian have a symmetry known as local gauge invariance. This roughly means that you must be able to choose a different phase for the quantum field at each point in space. It immediately becomes obvious that the free fermion Lagrangians do not have this symmetry. The way to rescue it is to introduce additional terms in the Lagrangian to absorb the variation. These terms are suggestive of boson fields, and to complete the Lagrangian one includes the free particle component for the boson fields. The result is a system of coupled fermion and boson fields, and the couplings, when the correct gauge symmetry has been imposed, describe exactly the three forces that we observe. (I should note in passing that one can then convert from/to a Lagrangian to/from a Hamiltonian formulation; it is a mathematical procedure that doesn't change the physics.) You can try to find this procedure applied to create the electromagnetic interaction (the other ones are just as simple conceptually, but the mathematics is prohibitive) I cannot give you a definite reason why we should demand that the local gauge symmetries hold; the fact is that this creates an extremely accurate and predictive theory. Also, local gauge theories are one of the few ways to write a self-consistent theory. If you try to write a 'naive' theory of the weak interaction with massive mediators W and Z without resorting to the gauge principle, the theory will contain nasty mathematical divergences that cannot be consistently removed.
Now why would an electron be unhappy being an electron? In a simple Dirac field (ie obeying the free Dirac equation), it is perfectly happy being an electron: the wavefunction of an electron is an eigenfunction of the Dirac Hamiltonian. In other words, what we would call a free particle is a quantum of a Dirac field. But we just found that the Dirac Hamiltonian is incomplete: there are the new terms to consider. When you consider the *full* Hamiltonian which includes the gauge fields (the technical term for the fields we introduced to gain our symmetry, they correspond to the physical bosons we observe), you find that the free electron is NO LONGER an eigenfunction. Its wavefunction then evolves according to the full Hamiltonian, which means that there is a non-zero possibility that gauge particles are present. In this picture, you don't need the cloud of virtual particles: it is simply the case that the true 'stationary' states of the system are not those of free particles, but mixtures of the above fields.

10. Jun 28, 2004

### zefram_c

Now onto the second step. We have our full Lagrangian that describes the fermions and their interactions... so let's solve it! There's a slight problem: we cannot. The mathematics is such that exact solutions are not available, and we must resort to approximations. The most commonly used technique is perturbation theory via the Feynman diagrams.
You can think of the perturbation series in a manner similar to a power series expansion. Close to x=0, you can approximate sin x = x and cos x = 1 - x2 / 2. If you want more accuracy, you can include more terms. Now the nice thing about the cos and sin functions is that they are analytic. You can write the power series expansion at x=0 and, if you include enough terms, you can find cos (anything) to arbitrary accuracy. Of course you might need to include thousands of terms to calculate cos (10000) by this method, but it can be done. Our gauge theories have the same property, which is to say they work at high energies (large x) as well.
[Here it *really* helps if you've seen Feynman diagrams before]
The perturbation series works the same way. You start by considering 'free' particles as corresponding to the constant (zeroth order) term of the expansion. In the next order, you picture that a single field quantum is exchanged between the particles. In higher orders, more field quanta are exchanged. Now every time a quantum is emitted or absorbed, a dimensionless constant is introduced. This is the coupling constant and it describes the intrinsic strength of the theory. In a higher order calculation, the constant appears with increasingly large power.
(To exemplify: In E&M, the single photon exchange is the dominant channel at low energies, and it results in the Coulomb potential and classical E&M. But higher order corrections are needed to explain, eg, the Lamb shift in hydrogen or the electron's magnetic moment)
Now each of these orders gives you a magnitude. The *net* magnitude is obtained by adding all the magnitudes to each order. In practice, you hope that magnitudes become negligible at higher orders; this is indeed so most of the time but it breaks down for QCD at low energies and this plagues QCD to this day. The net magnitude describes the interaction.
Now of course Nature doesn't know anything about perturbation theory and Feynman diagrams. The interaction proceeds in a single form, but we cannot know it. Our diagrams with exchanges of virtual quanta are the results of the approximation procedure. I'm not sure if those virtual quanta have any detectable effects, but one thing is certain: we cannot observe them directly without altering the process.
To answer your other question: the rules for the diagrams require conservation of 4-momentum whenever interactions occur. Let's consider electron-muon scattering by Z exchange. The electron emits a Z and recoils, the muon absorbs the Z and recoils. Both the electron and the muon are considered to be 'free' in their initial and final states, so they must surely obey the momentum-energy equation I posted. It follows that the 4-momentum of the Z is completely determined and it *cannot* obey the relation for a proper Z boson. But this is not a problem, as our theory dictates that there is no way to observe the virtual Z.
I know this is pretty intense stuff, so bear with me

11. Jun 28, 2004

### turin

Check!

Check! (I think)

I think I've seen this one before (someone tried to explain it to me on another forum).

I don't think I am quite appreciating this issue.

Independently? That sounds absurd. Wouldn't that give you some crazy (or at least ill-defined) momentum behavior? (i.e. discontinuities)

So the Lagrangian of which you previously spoke is the "Dirac Lagrangian" (which could be acquired from the Dirac Hamiltonian)? I don't know much about the Dirac equation. That is where my formal QM instruction came to an end. I tried to read the chapter in Shankar about it (Ch 2#), but I couldn't follow his justifications (rather hand-wavy).

I don't follow this at all.

This brings me to a new level of clarity. Excellent.

12. Jun 28, 2004

### jtolliver

The gauge choice is required to be continuous.
However our wavefunctions are not scalars; they are spinors(or actually pairs of spinors that are swapped under certain discrete symmetries; one reason this is required is so that we have a representation of the symmetries).

since the particles have multiple components it seems reasonable to assume that its equtions of motion mix the components. the Klein-Gordon equation says
$$p_\mu p^\mu = -m^2$$
to get an equation that mixes the components of the Dirac spinor from this we take the 'square root' of both sides. we get
$$\gamma^\mu p_\mu = i m$$
where the gamma matrices are chosen such that $\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 g_{\mu \nu}$ ($g_{\mu \nu}$ is 0 unless $mu$ and $\nu$ are equal, -1 if both indices are 0 and 1 otherwise), to see that this satisfies the Klein-Gordon equation subtract im from both sides, square it and subtract m^2 from both sides. It turns out there is no way of chosing 2 by 2 matrices satisfying these constraints, so instead we require that it consist of 2 spinors, which are mixed by at least one of the gamma matrices.

13. Jun 29, 2004

### zefram_c

After a crazy election night, time to get back to some physics...

Thanks jtolliver for the catch: of course the gauge function has to be continuous. You can't expect to allow finite changes in the gauge at an infinitesimal scale, the math wouldn't allow it. But you can intuitively justify that since points in space at a fixed time are not causally connected according to SR so it should be possible to impose different gauge choices at separated points under some restrictions.

$$\gamma^\mu p_\mu = i m$$ is the Dirac equation for the fermions. All we need is to create a Lagrangian that reproduces it when you apply the Lagrange equations. That Lagrangian is given in this lecture on page 11:

http://www-hep.uta.edu/~yu/teaching/spring03-5326/lectures/phys5326-spring03-030503-post.pdf

You should keep in mind that the lagrangian was contrived to generate the Dirac equation. Basically you have to postulate something, and the Dirac equation is what works for free particles. Then you introduce the other fields as I described.
First, take a cursory look here:
http://www.physics.orst.edu/~stetza/COURSES/ph656/Chapter6.pdf [Broken]

This will give you the Lagrangian (and Hamiltonian) for the free system. I referred to the Hamiltonian since I figured it would be easier to explain why a single electron isn't a stationary state; all QM courses will tell you that this happens iff the wave function is an eigenfunction of H. Otherwise we don't really need it.

Also look (don't bother trying to get too deep into the math) at his perturbation series expansion (6.15). This will generate our Feynman diagrams.

Let's look at a later lecture:

http://www.physics.orst.edu/~stetza/COURSES/ph656/Chapter9.pdf [Broken]

Start at the very first rule (p5): "draw all possible diagrams". There's infinitely many of them! Each corresponds to a term in the perturbation you saw.

Then consider rule 6. The factor e is the dimensionless parameter that sets the strength of the interaction; here it's the charge of the electron in whatever system of units he chooses. It appears "at each vertex", and all factors multiply. So for higher order diagrams, which have more vertices, it appears to higher powers. For electron-photon interactions, this comes to about 1/137; you can see that the higher order terms will quickly drop to zero (unless the results diverge, in which case we need to do some math tricks).

Finally, Feynman diagrams correspond to probability amplitudes; to get the net amplitude you add the results for each diagram, and keep doing so until the desired numerical accuracy.

You would be much better off to learn this theory from a book. The text I used in fourth year was David Griffith's "Introduction to Elementary Particles"; the math is involved enough to beat the qualitative descriptions we've been giving but not too hard to follow if you have experience with QM. There's also plenty of qualitative descriptions and history; only problem is that the book hasn't been updated since the late 80's I think so it doesn't cover things like the top quark (found in 1993?).

Last edited by a moderator: May 1, 2017
14. Jun 29, 2004

### styler

The top was formally "discovered" by D0 in '95 based on data collected from a run in 93/94.
Griffiths latest edition was updated in 1987 I think.
It is still good book from which to learn the basics but no top quark...course no neutrino oscillations either.

Last edited: Jun 29, 2004
15. Jun 29, 2004

### zefram_c

I don't think you can find any introductory book that includes neutrino oscillations yet... they were only confirmed a couple of years ago. But they are covered in courses. A more theoretical book that includes the top quark and a section on neutrino oscillations (though not their confirmation) is "Particle Physics: 2nd ed." by Martin & Shaw. But if you have little experience with the field, it's better to start with Griffiths since this one jumps straight into relativistic wave equations.

16. Jun 29, 2004

### turin

Thanks to all three of y'all. I didn't know there was a Griffiths book on particle physics; I've always heard good things about Griffiths texts (though I've never actually used one). I haven't read those links, yet, but I am definitely going to do so when I find some time.

I think I'm starting to understand the guage issue now. It's not so much that the phase does oscillate wildly against space, it is that we observe the relative phase as a degree of freedom or dynamical variable? Therefore, we must include the influence in the Hamiltonian/Lagrangian? I am still confused why it was pounded into my head when I took QM that the phase is not physical (&psi; can be multiplied by an arbitrary factor of ei&phi;). I thought that I understood that on an intuitive level. Is the guage issue a different issue?

jtolliver,
You have reminded me of what I believe to be my biggest stumbling block in QM:
Why is the wavefunction a spinor? Is this emperical? I remember reading about some experiment that had a beam of electrons passing through a magnetic field. The beam split into discrete spots on the detecting screen. This is supposed to demonstrates the spinor nature of the electron's wave function. I don't quite follow how this demonstrates the spinor nature of the electron's wave function.

17. Jun 29, 2004

### styler

Yeah, so i was being sort of semi-facetious....trying to make the point that despite the pop scinece notion that we are so near to a "final theory" or whatever, particle physics is, as any science should be, very much an evolving discipline.
As such, it won't be possible to write a book that gives all the up to date stuff whose relevance is lasting (or whose obsolescence isn't always looming)
You might argue that nobody would have been super surprised if the top had been found in 1987 in time for Griffith's publishing but massive neutrinos and oscillations would have been contentious then to say the least, i think. (true? (i was just in jr high then, so maybe its unfair of me to speculate upon a part of history i never personally observed))
So i was actually sort of defending Griffith's book by saying it was reasonably comprehensive for its time. And this is a lot to say for an introductory book if you can couple the statement with a one about its readbality.
As far as that goes i say some of its is excellent. Ironically, the topic of this thread, the weak interactions, are covered in a GWS chapter which is pretty hard to read if you are a beginner (plus not much mention of IVB and why its was insufficient in a book that's loaded with history). The typos in the text are also numerous and a few of them are really pretty non-trivial.
But i think the progression of topics is a good one and thsi book is waaaay better than his silly quantum mechanics book. So its not a bad place to start looking at particles for someone new to the subject.

Last edited: Jul 2, 2004
18. Jun 29, 2004

### styler

This is an interesting point of view. I always find it funny that physicists/authors recommend a shying from differential equations for newbies to advanced topics in the field when undergraduate physics curricula put so much emphasis on differential equations (and not analysis or algebra that can genralize them).
I actually found myself inclined to agree with your recomendation but then i wondered if positing a diff eq and then dealing with consequnces might be more palatable to some. For me, the phenomoenolgy was always harder to remember the how the equations worked (course i was always wondering where they came from....but that's another story)

19. Jun 30, 2004

### zefram_c

Well styler, I can certainly agree that it might be more palatable to some to get straight into a big ugly mess of formulas. But my suggestion was well meaning. It's hard to keep motivated to work through ugly math when you don't understand the purpose behind it all, or at least I find it so. If someone were to ask me to work through a Feynman diagram without telling me why we do that, I don't know if I'd carry it through. In a course situation, things are different: students can get their hands dirty knowing there's the instructor to fall back on, to explain the how and the why. But our posts cannot pretend to be a substitute for a formal course. If people become interested in the field as a result of our conversations, then a course that includes all the gory details is what they would benefit most from. But if they're just curious about the way our theories describe Nature (and the originating question seems to fit in that category), then a conceptual read a la Griffiths would be best. I think that even freshmen can understand a good part of that, and a senior undergraduate can probably follow almost everything. (I agree that the very section on GWS theory is pretty hard to understand, but the last chapter on lagrangians and gauge fields including a simple form of the Higgs mechanism is quite illuminating)

And speaking of gauge fields, there's a quite lovely book on that topic called "Gauge Theories in Particle Physics"; don't recall the authors offhand but it's in the "Graduate Student Series in Physics". Hard to miss, the cover has a picture of a bottle with an embedded Feynman diagram for a weak force process.

20. Jun 30, 2004

### turin

I hope that this is not absolute. My university offers no such course. You have all done a great job at explaining this crap so far. When I get the Griffiths text I will no doubt post a barrage of personal confusion that I trust y'all to mitigate.