How Do Standard Model Symmetries Influence Theoretical Physics?

[Bobhawke]

1. The Standard model is an SU(3)xSU(2)xU(1) symmetric theory. To me this means that if you choose any 3 members of the groups and act on the Lagrangian, it is invariant. However, not all terms in the Lagrangian have something for a group member to act on, for example terms that don't involve anything with colour charge aren't acted on by SU(3). Then such a term would only have SU(2)xU(1) symmetry. Do we just say that this term has the full symmetry, but there is nothing for the SU(3) part to act on?

2. In a chirally symmetric theory with (say) 2 massless fermions, the L and R handed parts of the Dirac equation can be separated leaving us with a U(2)xU(2) symmetry of the Dirac part. But now what is the symmetry of the bosonic part involving the field strength tensor? It isn't U(2)xU(2), it is the original SU(2) symmetry. How can the theory then said to be invariant under U(2)xU(2)

3. Finally, what are the symmetry transformations associated with the following conserved quantities:
baryon number
lepton number
strangeness
charmness
topness
bottomness

and why aren't there conserved upness and downness numbers?

 

[malawi_glenn]

we had a discussion recently on discrete, vs symmetries and gauge symmetries :-)

strangeness is not conserved in weak interactions, but in strong. The strong interaction is isospin independet, the quarks transforms in the flavour group SU(6)_flavor and the strong interaction is invariant under flavour group transformations (IF the quarks have the same mass, the quark masses makes isospin violation even in the strong interaction, such as in certain meson decays)

https://www.physicsforums.com/showthread.php?t=296758

 

[blechman]

1. The Standard model is an SU(3)xSU(2)xU(1) symmetric theory. To me this means that if you choose any 3 members of the groups and act on the Lagrangian, it is invariant. However, not all terms in the Lagrangian have something for a group member to act on, for example terms that don't involve anything with colour charge aren't acted on by SU(3). Then such a term would only have SU(2)xU(1) symmetry. Do we just say that this term has the full symmetry, but there is nothing for the SU(3) part to act on?

This is all okay if you allow for the identity element in your "any 3 members of the groups" and remember that terms in the Lagrangian that "don't have anything to act on" actual have Kroneker deltas, so that the action of the SU(3) group element on the leptons is to "do nothing!"

2. In a chirally symmetric theory with (say) 2 massless fermions, the L and R handed parts of the Dirac equation can be separated leaving us with a U(2)xU(2) symmetry of the Dirac part. But now what is the symmetry of the bosonic part involving the field strength tensor? It isn't U(2)xU(2), it is the original SU(2) symmetry. How can the theory then said to be invariant under U(2)xU(2)

wha?! what "U(2)" are you talking about? In a globally chiral symmetric theory, there is no field strength! What happens in the EW sector of the standard model is that:

$$SU(2)_L \times SU(2)_R \rightarrow SU(2)_D$$

and it is this diagonal subgroup that is gauged. This is deep stuff, and this forum is probably not the best place to discuss it. If you want to learn more, check out H. Georgi's textbook "Weak Interactions" Chapters 3-5. It's available on his website for free. Google his name for the link.

This book is the BIBLE of theoretical particle physicists, at least where I come from.

3. Finally, what are the symmetry transformations associated with the following conserved quantities:
baryon number
lepton number
strangeness
charmness
topness
bottomness

and why aren't there conserved upness and downness numbers?

well, check out the post malawi_glenn linked to. As to your last question: "up" and "down" quarks are both effectively massless, so that they look the same, and this is what we normally call "SU(2) isospin". To the extent that the strange quark is massless too, then "strangeness" is no longer a conserved number, and instead you should use "SU(3) isospin" (purists probably don't want me to call this isospin, but I'm not sure what else to call it). Same for "charmness", etc.

Practically speaking, SU(2) describes nature pretty well; you might be able to get away with SU(3); but SU(4) and higher do a terrible job, since in no sense is the charm quark massless!

 

[Bobhawke]

1. But then you have to choose the identity of SU(3). The whole point of a symmetry group is that you can choose any elemtent of it and the Lagrangian will be invariant.

2. I am talking about $$U(2)_L X U(2)_R$$, the U(2)'s that are symmetries of the left and right handed parts of the Dirac equation after they have been separated in the massless limit.

Thank you for the information about the book by Georgi, I will definitely check it out.

 

[hamster143]

Suppose if you have a function $$f(x,y,z,t) = \sqrt(x^2+y^2) + \sqrt(z^2+t^2)$$. Both terms are invariant under the U(1) rotation in xy plane. The first term is invariant because $$x'^2+y'^2 = x^2 + y^2$$. The second term is invariant because it does not depend on x and y.

 

[hamster143]

check out H. Georgi's textbook "Weak Interactions" Chapters 3-5. It's available on his website for free. Google his name for the link.

This book is the BIBLE of theoretical particle physicists, at least where I come from.

I'll guess that you come from Harvard? It's very hard for that book to be the BIBLE anywhere else, seeing how it was published in 1984 and has not been in print since then.

http://www.people.fas.harvard.edu/~hgeorgi

 

[malawi_glenn]

1. But then you have to choose the identity of SU(3). The whole point of a symmetry group is that you can choose any elemtent of it and the Lagrangian will be invariant.

The SU(3)_c group acts on the quark-fields, not on the leptonic fields.

The SU(3)xSU(2)xU(1) symmetry is just a convenient way to express the standard model lagrangian, which has both quark and lepton fields (and higgs fields)

the lepton fields are invariant under SU(3) transformation by definition. lepton fields are two elements, SU(3) is 3x3 matrix, who are you going to combine a 3x3 matrix with a row vector with 2 elements? As blechman said, it has nothing to act on there.

since we have no GUT, one must tread L_qcd and L_ew separately

 

[blechman]

I'll guess that you come from Harvard? It's very hard for that book to be the BIBLE anywhere else, seeing how it was published in 1984 and has not been in print since then.

http://www.people.fas.harvard.edu/~hgeorgi

Georgi had a gazillion students, many of whom also had a gazillion students, propagating the "Georgi Legend (TM)" across the High-Energy Theory world! Georgi is both my grandfather and great-grandfather!

But seriously, the book has been out of print, but revised by him and turned into lecture notes, and made publically available. So don't be concerned that it looks old: it's actually quite up-to-date (the version on the web is). And it is an excellent text - but of course, it is for the advanced grad students and higher, not for those who just read Brian Greene's book and want to learn more.

Anyway, sorry, I didn't mean to hijack the thread.

 

[Haelfix]

Georgis book is definitely the bible for weak interaction particle physics. Just like Greiner is the bible for QCD stuff.

 

[malawi_glenn]

Georgi's book will come in new print at Dover publishment this spring, I will get it just since you guy's says it is that good! :-)

 

[blechman]

really?! i would just get it off the internet before it becomes illegal to do so! also, if dover's republishing it, make sure it's the UPDATED copy and not the early-80's version.

 

[hamster143]

Georgi had a gazillion students, many of whom also had a gazillion students, propagating the "Georgi Legend (TM)" across the High-Energy Theory world! Georgi is both my grandfather and great-grandfather!

But seriously, the book has been out of print, but revised by him and turned into lecture notes, and made publically available. So don't be concerned that it looks old: it's actually quite up-to-date (the version on the web is). And it is an excellent text - but of course, it is for the advanced grad students and higher, not for those who just read Brian Greene's book and want to learn more.

I'm not implying that the book is out of date - merely that it's impossible to find in paper form. It has zero reviews on amazon, for example. Our science library has one copy that is checked out. Compare with such veritable theoretical high energy physics bibles as Peskin & Schroeder or Weinberg, which you can even occasionally find in Borders.

When I was in grad school, I took classes from at least two former Georgi's students.