How Does U(x) Relate to Pion Dynamics in Chiral Symmetry Breaking?

  • Thread starter LAHLH
  • Start date
  • Tags
    Chiral
In summary: The higher interaction terms get squished down to lower energies and eventually cancel out. So in this sense, U(x) is a field that contains the free kinetic energy terms and is most gen consistent with symmetries.
  • #1
LAHLH
409
1
Hi,

So to break the axial part of [itex] SU(2)_L \times SU(2)_R [/itex] we use the composite field [itex] \chi_{\alpha i} \xi^{\alpha \bar{j}} [/itex] (since this is Lorentz scalar, colour singlet, and has the right transformation under the flavour symmetries, only breaking the axial part etc). We assume that [itex] \langle 0| \chi_{\alpha i} \xi^{\alpha \bar{j}} |0\rangle=-v^3 \delta^{\bar{j}}_i [/itex]. (I understand roughly the mathematics of this operator does the job of breaking axial but not vector generators, but from my reading of symmetry breaking I don't really understand how we just *choose* a field in some arbitrary way like this; I thought the field had to be field involved in your Lagrangian and then you would find a continuous family of ground states of it etc)

Then my text talks about constructing a low energy effective lagrangian for the three pions (psuedo goldstone bosons) by letting [itex] |U\rangle [/itex] be a low energy state for which the expectation value of [itex] \chi_{\alpha i} \xi^{\alpha \bar{j}} [/itex] varies slowly in flavour space as a function of spacetime:
[tex] \langle U|\chi_{\alpha i} \xi^{\alpha \bar{j}}|U\rangle=-v^3 U^{\bar{j}}_i(x) [/tex]

with U(x) a spacetime dependent unitary matrix that can be written [itex] U(x)=exp\left[2i\pi^a(x)T^a/f_\pi\right] [/itex] ([itex]\pi[/itex] are pion fields, T generators, f is pion decay const)

Then why do we think of U(x) as an effective low energy field? why does specifiying it's Lagrangian to be most gen consistent with symmetries, say anything about the pions? what is really going on here? I don't see why we are forming a Lagrangian from this object that was just on the RHS of an expectation value of this composite field.
 
Physics news on Phys.org
  • #2
I'm studying this right now as well and had the same question. I think the answer has to do with the concept of "universality class" from statistical mechanics. Basically, I think the idea is that two Lagrangians that are invariant under the same symmetries will flow to the same fixed point in the IR under the renormalization group. In this case, the IR is pion physics.
 
  • #3
U is a field that - when expanded in the pion fields - contains the free kinetic energy terms + higher interaction terms which are - due to power counting - non-renormalizable. You can see that by looking at power counting of [itex]\varphi^n[/itex] theories which are 'generated' by expanding U.
 

FAQ: How Does U(x) Relate to Pion Dynamics in Chiral Symmetry Breaking?

What is chiral symmetry breaking?

Chiral symmetry breaking is a phenomenon in physics where a system that exhibits chiral symmetry at the microscopic level, breaks that symmetry at the macroscopic level. This means that while the individual parts of the system may have symmetrical properties, the system as a whole does not.

What is the importance of chiral symmetry breaking in particle physics?

Chiral symmetry breaking is important in particle physics because it helps explain why some particles have mass while others do not. The breaking of chiral symmetry is linked to the Higgs mechanism, which is responsible for giving particles their mass.

What are some examples of chiral symmetry breaking?

One example of chiral symmetry breaking is the weak interaction in particle physics. While the fundamental equations of the weak interaction are symmetrical, the observed behavior of particles under this interaction is not. Another example is the phenomenon of spontaneous symmetry breaking in condensed matter systems.

How does chiral symmetry breaking impact our understanding of the universe?

Chiral symmetry breaking is a crucial aspect of our current understanding of the universe. It helps explain the origin of mass, the behavior of subatomic particles, and the structure of matter. It also has implications for the study of dark matter and the early universe.

What research is currently being done on chiral symmetry breaking?

Scientists are continuing to study chiral symmetry breaking in various fields, including particle physics, condensed matter physics, and cosmology. Some ongoing research includes studying the effects of chiral symmetry breaking on the behavior of quarks and gluons, and investigating the potential connection between chiral symmetry breaking and the origin of the universe.

Back
Top