Weinberg-Salam Model

[Markus Hanke]

So I have recently ( 2-3 weeks ago ) begun teaching myself QFT in earnest. Truth be told, it is hard going - while the basic concepts are straightforward and easy enough to understand, the finer details are most definitely not. The trouble I am having is that a lot of texts seem to focus much on the mathematical details ( for obvious reasons ), at the expense of the bigger picture. I am currently struggling through the Weinberg-Salam part of the SM, and I pretty much had to piece together the bigger picture from all the maths involved in it. I wonder if someone here can confirm whether or not my understanding is correct :

 

So the basic idea is that we have a Lagrangian that has a leptonic part, a gauge part, and an interaction part. The leptonic part is just a sum of Dirac fields for the electron, muon, and tau, together with their associated neutrinos; since the neutrino is taken as massless, there is a bit of an asymmetry here in that the right-handed fields have no neutrino part. The spinor fields are hence of the form

 

[latex]\displaystyle{\Psi =\binom{v_e}{e_L}+\binom{0}{e_R}}[/latex]

 

and likewise for muon and tau. We then introduce the electroweak charges ( electric, isospin, hyper charge ), and assign them to the various particles in such a way that charged currents couple to left-handed particles, and to right-handed anti-particles. The neutrino interacts weakly, but not with the photon; the left-handed electron/tau/muon interact weakly and electromagnetically, and their right-handed versions couple to hyper charge and electric charge, but not isospin. So far so good.

 

Here's where it gets a little tricky now. We have an overall symmetry group for the Lagrangian which is

 

[latex]\displaystyle{SU(2)\otimes U(1)}[/latex]

 

This means we need four gauge fields, three to cover SU(2), and one to cover U(1), corresponding to the three components of isospin, plus hypercharge. Those fields are

 

[latex]\displaystyle{U(1)\Rightarrow B_{\mu}}[/latex]

 

and

 

[latex]\displaystyle{SU(2)\Rightarrow W_{\mu}^{1},W_{\mu}^{2},W_{\mu}^{3}}[/latex]

 

We introduce field strength tensors for these, compute the corresponding kinetic terms in the Lagrangian, and introduce gauge covariant derivatives to ensure covariance of the Lagrangian under the respective symmetry transformations. I understand how these things are done, and how to arrive at the final leptonic Lagrangian :

 

[latex]\displaystyle{L_{lepton}=i\overline{\psi }_R\gamma ^{\mu}\left ( \partial _{\mu}+\frac{1}{2}ig_BB_{\mu} \right )\psi _R+i\overline{\psi }_L\gamma ^{\mu}\left ( \partial _{\mu}+\frac{1}{2}ig_BB_{\mu}+\frac{1}{2}ig_W\vec{\tau}\cdot \vec{W}_{\mu} \right )\psi _L}[/latex]

[latex]\displaystyle{L_{gauge}=-\frac{1}{4}f_{\mu \nu}f^{\mu \nu}-\frac{1}{8}Tr\left ( F_{\mu \nu}F^{\mu \nu} \right )}[/latex]

 

Looks horrible, but I understand the meaning of the various terms, and where they come from, so I suppose I'm ( mostly - see below ) good with this. It does show quite nicely the asymmetry of the weak interaction between left- and right-handed particles. What I am less clear about - and that is never explicitly explained in my text - is the actual status of these fields. My understanding is that these are not really to be taken as physical fields yet; instead, you perform a rotation of the B and W(3) field about the Weinberg angle, and arrive at A and Z fields, which then are physical ( their excitations are just the photon and the Z-boson ) :

 

[latex]\displaystyle{\binom{A_{\mu}}{Z_{\mu}}=R(\theta _W)\binom{B_{\mu}}{W_{\mu}^{3}}}[/latex]

 

So you basically take the original fields as purely mathematical entities, which need to be rotated by some angle ( to be determined experimentally ) in order to match physical results in the real world. The remaining two ( not yet physical ) W-fields are then combined according to electric charge, such that

 

[latex]\displaystyle{W_{\mu}^{\pm }=\frac{1}{\sqrt{2}}\left ( W_{\mu}^{1}\pm iW_{\mu}^{2} \right )}[/latex]

 

These are again physical fields, the excitations of which are just precisely the two charged W-bosons. Is this understanding correct ?

 

The next step is to break the symmetry via the usual Higgs mechanism, so that the leptons and gauge boson can acquire mass. The Higgs field in this case is a complex scalar field, and the gauge is chosen such that

 

[latex]\displaystyle{\varphi =\binom{0}{\varphi _0+\frac{h(x)}{\sqrt{2}}}}[/latex]

 

Now, this is again never really explained in my text, but I presume this gauge choice is largely arbitrary, and made simply for mathematical convenience ( what happens if I make a different choice ?? ). You then go through the usual motions - you compute the Yukawa interaction term ( and find that all neutrino terms drop out, so they are massless, while the field still couples to the electron/tau/muon ), and the gauge covariant derivative for the Higgs field ( which introduces mass terms for the W/Z gauge bosons ). Again, I understand how to do that calculation in principle, even though in practice it seems extremely tedious and complicated. By comparison with the generic quadratic mass term, and using the Weinberg angle, you can get the particle masses as functions of the coupling constants and the Higgs field vacuum.

 

Does all this sound correct, or am I way off the wall here somewhere ? I am definitely unclear on the relationship between the various fields ( B/W, A/Z ) and the Weinberg angle - my understanding is that this angle is an external parameter in the SM that needs to be determined empirically, and thus far I really have not fully grasped its meaning at all, as this is never explained at all in my text. They just basically say "now we need to perform a rotation by this angle..." without a deeper explanation - I am deducing the above from the maths only. I just need a heads-up on whether what I have written above is right. I am also somewhat confused on what the trace operator is doing in the term ;

 

[latex]-\frac{1}{8}Tr\left ( F_{\mu \nu}F^{\mu \nu} \right )[/latex]

 

The summation of the tensor indices should leave a scalar as a result, so what's with the trace ??

 

Apologies if this is elementary stuff and the answers obvious, but they aren't to me. I have to start somewhere, and unless I keep checking that my understanding is correct, I will never be able to get my head around this properly. Learning GR was child's play compared to all this QFT stuff

Edited July 22, 2016 by Markus Hanke

[Mordred]

Its been a few years since I last looked at Weinberg angle involvement on Pati-Salam. I'll have to review a bit before I answer.

 

If I remember correct though the Higgs field can be any non zero value. Then with the left/right helicity and the Weinberg mixing angle you get the values needed. (subject to off hand memory though).

 

Also no there is nothing basic about Pati-Salam....(that includes Weinberg-Salam theory)

Edited July 22, 2016 by Mordred

[Markus Hanke]

If I remember correct though the Higgs field can be any non zero value. Then with the left/right helicity and the Weinberg mixing angle you get the values needed. (subject to off hand memory though).

 

 

Ok, that makes sense.

 

 

 

Also no there is nothing basic about Pati-Salam

 

So essentially I am not the only one who finds this stuff hard-going...I'm glad to hear that

[Mordred]

Roflmao you may find SO (10) a la Pati-Salam handy. It touches on the SO (2) groups both left and right hand.

 

http://www.google.ca/url?sa=t&source=web&cd=1&ved=0ahUKEwjQ6ei66ofOAhVQwGMKHduUBbUQFggdMAA&url=http%3A%2F%2Fcds.cern.ch%2Frecord%2F546538%2Ffiles%2F0204097.pdf&usg=AFQjCNH8HRdfqO8_OjuqGnQ35xwrdMWnBA&sig2=fBInGkpTP76slqqp8styCQ

 

Im still digging on a few articles I recall

Edited July 22, 2016 by Mordred

[Mordred]

Here are some articles I was looking for. This one details the Abelion groups first then adds the non Abelion groups.

 

http://www.google.ca/url?q=https://www.theorie.physik.uni-muenchen.de/lsfrey/teaching/archiv/sose_09/rng/higgs_mechanism.pdf&sa=U&ved=0ahUKEwiAtvTzzojOAhVG_mMKHaWnAhg4ChAWCCkwBg&sig2=EL47Ra_pNKpXPuwfhVsBlw&usg=AFQjCNE8cAhfJM_PlDVnojGx2q08moVF6A

 

This one provides some greater detail.

 

http://www.google.ca/url?sa=t&source=web&cd=4&ved=0ahUKEwjp4q-arojOAhUBHmMKHXJJA2EQFggzMAM&url=http%3A%2F%2Fbolvan.ph.utexas.edu%2F~vadim%2FClasses%2F15f%2FGWS.pdf&usg=AFQjCNERKRJ8XfA8ktvOxEy8YzRI-2NZBQ&sig2=vJ51qmbImVctncPr6ysY1Q

 

Have you familiarity with the eightfold way? Some details you will need to understand the guage groups are the conservation rules in particle physics.

 

ie conservation of spin,isospin,charge,color,parity etc.

Edited July 23, 2016 by Mordred

[Markus Hanke]

Thank you all, I am going to have a bit of reading to do

A quick glance through the sources seems to indicate though that my understanding is largely in-line, but definitely still incomplete. Not too shabby for three weeks in though...