Question about S(3) spontaneous symmetry breaking in Peskin & Schroeder

In summary: Now he defines a variable ##\Phi = \phi_c t^c##, where ##t^c## are the generators of SU(3) which are traceless 3x3 Hermitian matrices.In summary, the author is looking for a mass matrix of bosons in a model of a multiplet of real scalar fields coupled to gauge bosons, and he gets it from the term in Lagrangian given in (20.33). He defines a variable ##\Phi = \phi_c t^c##, where ##t^c## are the generators of SU(3) which are traceless 3x3 Hermitian matrices, and finds that the invari
  • #1
Antarres
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In chapter 20 of Peskin&Schroeder about spontaneous symmetry breaking, he considers and example on page 696 of spontaneous symmetry breaking of SU(3) gauge group with generators taken in adjoint representation.

Covariant derivative is defined with:
$$D_\mu\phi_a = \partial\phi_a + gf_{abc}A^b_\mu\phi_c$$

The model considered is of a multiplet of real scalar fields coupled to gauge bosons. He's looking for mass matrix of bosons, which he gets from the term in Lagrangian given in (20.33):
$$\Delta\mathcal{L} = \frac{g^2}{2}(f_{abc}A^b_\mu\phi_c)^2$$
Now he defines a variable ##\Phi = \phi_c t^c##, where ##t^c## are the generators of SU(3) which are traceless 3x3 Hermitian matrices.
He says (equation 20.35), that we can then rewrite this term in form:
$$\Delta\mathcal{L} = -g^2tr([t^a,\Phi][t^b,\Phi])A^a_\mu A^{b\mu}$$

However, checking this trace, and substituting the definition of commutators of generators, there is a residual term that is the trace of two generators which arise from those two commutators. But that trace equals to the invariant C(r) multiplied by a Kronecker delta over the gauge indices. The invariant C(r) is equal to quadratic Casimir operator in adjoint representation of SU(N), ##C(G) = C_2(G) = N##. Therefore, a factor of 3 should arise there, and so I don't see how that factor transfers into the factor of one half in the mass term given in the first equation.

Any help would be appreciated. I suspect maybe I'm missing some kind of symmetrization step, but I'm not sure, I can't see a mistake in my calculations.
 
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  • #2
If I remember correctly, the proportionality constant in ##\Tr{t^at^b}\propto \delta^{ab}## is arbitrary, so one must specify it when defining the conventions adopted. A very used case (at least I think in particle physics, maybe not so commit in other areas) is to fix ##\Tr{t^at^b}=\frac{1}{2} \delta^{ab}##. P&S uses such convention.
With this convention, the equality follows immediately.
 
  • #3
Okay well, it is true that rescaling the generators changes this number, since it is a normalization convention. The equation is:
$$\text{tr}(t^at^b) = C(r)\delta^{ab}$$
where C(r) is a constant that is dependent on representation, and t's are generators. The representation is taken to be irreducible, and it is said that if we fix this number in one representation, then we fix it in all other representations(although I haven't tried to prove that). That means, from what I get, that this is indeed a convention, however that convention is not giving the same constant for every representation. It is conventional for fundamental representation of SU(N) to choose:
$$\text{tr}(t^at^b) = \frac{1}{2}\delta^{ab}$$
However since he mentioned that in this model we take the scalar field to transform according to the adjoint representation, I figured these generators in (20.35) would also be in adjoint representation. In adjoint representation, for the convention above, we have that:
$$\text{tr}(t^at^b) = 3\delta^{ab}$$
That is calculated in Peskin chapter 15.4. It is possible though, that when he rewrote the equation like this, he chose the generators to be in the fundamental representation, it didn't cross my mind he would do that, but I guess it's possible, since this substitution doesn't seem to have anything to do with the transformation of the fields. So maybe you're correct, I just wasn't sure that is the case.
 
  • #4
Antarres said:
Now he defines a variable ##\Phi = \phi_c t^c##, where ##t^c## are the generators of SU(3) which are traceless 3x3 Hermitian matrices.
Mmm... I think this sounds like the fundamental representation.
 
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  • #5
Yeah, you're right. Adjoint representation of SU(3) isn't three dimensional. I guess my mind slipped on that sentence. Thank you, anyways, now it all makes sense.
 
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Related to Question about S(3) spontaneous symmetry breaking in Peskin & Schroeder

1. What is S(3) spontaneous symmetry breaking?

S(3) spontaneous symmetry breaking is a concept in theoretical physics that describes the breaking of a continuous symmetry in a system. In the context of Peskin & Schroeder's book, it specifically refers to the breaking of a three-dimensional symmetry in a quantum field theory.

2. How does S(3) spontaneous symmetry breaking occur?

S(3) spontaneous symmetry breaking occurs when the ground state of a system does not exhibit the same symmetry as the underlying laws of the system. This can happen due to interactions between particles in the system, leading to the formation of a non-symmetric vacuum state.

3. What is the significance of S(3) spontaneous symmetry breaking?

S(3) spontaneous symmetry breaking is significant because it helps explain the origin of mass in the Standard Model of particle physics. The Higgs mechanism, which involves S(3) spontaneous symmetry breaking, is responsible for giving particles their mass through interactions with the Higgs field.

4. Can S(3) spontaneous symmetry breaking be observed in experiments?

Yes, S(3) spontaneous symmetry breaking can be observed indirectly through experiments such as the Large Hadron Collider (LHC) at CERN. The discovery of the Higgs boson in 2012 provided evidence for the Higgs mechanism and S(3) spontaneous symmetry breaking.

5. Are there any real-world applications of S(3) spontaneous symmetry breaking?

While S(3) spontaneous symmetry breaking has primarily been studied in the context of particle physics, it has also been applied to other fields such as condensed matter physics and cosmology. In condensed matter systems, it can help explain the formation of patterns and structures, while in cosmology, it has been used to study the early universe and the formation of galaxies.

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