Chiral Lagrangian symmetry

In summary, the global flavour symmetry of the Lagrangian is broken into axial and vector parts. SU(2)XSU(2) has 9 vec generators, three of which are axial. To see all of this, one would need to have a more detailed understanding of group theory.
  • #1
LAHLH
409
1
Hi,

If I have the Lagrangian [itex] L=i\chi^{\dagger\alpha i}\bar{\sigma}^{\mu}(D_{\mu})_{\alpha}^{\beta}\chi_{\beta i}+i\xi^{\dagger}_{\bar{i}\alpha}\bar{\sigma}^{\mu}(\bar{D}_{\mu})^{\alpha}_{\beta}\xi^{\beta i}-1/4 F^{a\mu\nu}F_{\mu\nu}^{a} [/itex] where [itex]\alpha,\beta [/itex] are colour indices, and i=1,2 is a flavour index (the Lagrangian is for two massless quarks, approximating u,d quarks only), and spinor indices are supressed. chi and xi are both LH Weyl fiels. See Srednicki ch83 for more details, available free online.

Then it's obvious that this Lagrangian has global flavour symmetry [itex] \chi_{\alpha i}\to L_{i}^{j}\chi_{\alpha j} [/itex], [itex],\xi^{\alpha\bar{i}}\to (R*)^{\bar{i}}_{\bar{j}} \xi ^{\alpha\bar{j}} [/itex], where L and R* are constant unitary matrices and the c.c. of R just a notational convention. So we have [itex]U(2)_L \times U(2)_R [/itex] sym.

Then I can see that if we set [itex]L=R*=e^{i\alpha}I [/itex] , equivalent to [itex] \Psi\to e^{-i\alpha\gamma_5}\Psi [/itex] in terms of Dirac field then there is an anomaly in this axial U(1) sym, so I presume we just exclude this? then left over is the non-anomlous symmetry. Srednicki says this is [itex]SU(2)_L \times SU(2)_R \times U(1)_V [/itex], why is this the case? how has excluding this anomlous axial U(1) symmetry reduced [itex] U(2)_L\times U(2)_R [/itex] TO [itex]SU(2)_L\times SU(2)_R\times U(1)_V [/itex]?

thanks for any pointers
 
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  • #2
U(2) = SU(2) x U(1), maybe mod Z2. It's been a little while. You have the U(1)_L and U(1)_R. Then you define U(1)_V = U(1)_L + U(1)_R and U(1)_A = U(1)_L - U(1)_R.
 
  • #3
chrispb said:
U(2) = SU(2) x U(1), maybe mod Z2. It's been a little while. You have the U(1)_L and U(1)_R. Then you define U(1)_V = U(1)_L + U(1)_R and U(1)_A = U(1)_L - U(1)_R.

I see, thank you. Don't suppose you know a good reference to read up about this in particular, my group theory is a bit rusty at the moment..
 
  • #4
I learned most of my group theory from Dresselhaus and Tinkham's Group Theory books, Georgi's Lie Algebras in Particle Physics (available online for free, though not as related to this issue in particular) and Fecko's Differential Geometry and Lie Groups for Physicists. I especially like the last book.
 
  • #5
I was wondering if anyone could help me understand in more detail how SU(2)XSU(2) breaks into axial and vector parts? Also, am I correct in thinking since SU(2) has 3 generators su(2)xsu(2) has 9? I believe three of these are axial generators? since when they are broken you get three pseudogoldstone pions? So 6 vec gens? how does one see all this, if true?
 

1. What is chiral Lagrangian symmetry?

Chiral Lagrangian symmetry is a type of symmetry in theoretical physics that describes the behavior of particles with a specific handedness, or chirality. It is based on the idea that certain physical laws should remain unchanged when particles are transformed from left-handed to right-handed and vice versa.

2. How does chiral Lagrangian symmetry affect particle interactions?

Chiral Lagrangian symmetry plays a crucial role in understanding and predicting the interactions between particles. It allows for the classification of particles based on their chiral properties and helps to explain why certain interactions occur and others do not.

3. Is chiral Lagrangian symmetry a fundamental law of nature?

Chiral Lagrangian symmetry is not considered a fundamental law of nature, but rather a consequence of deeper, more fundamental symmetries such as the Standard Model of particle physics. However, it is an important concept in understanding the behavior of particles at low energies.

4. How is chiral Lagrangian symmetry related to the strong nuclear force?

Chiral Lagrangian symmetry is closely connected to the strong nuclear force, which is responsible for holding the nuclei of atoms together. The strong force exhibits chiral symmetry in the sense that it is invariant under transformations from left-handed to right-handed particles.

5. Are there any experimental tests for chiral Lagrangian symmetry?

There have been various experimental tests conducted to verify chiral Lagrangian symmetry, particularly in studies of particle interactions and decays. These tests have provided strong evidence for the validity of the theory and have helped to refine our understanding of chiral symmetry in particle physics.

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