So the Lagrangian looks like L 1 2 w 2 1 2 w 2 R a 4 2 � 4 w 2 μ 2 � 2 78 So

So the lagrangian looks like l 1 2 w 2 1 2 w 2 r a 4

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) = 0. So the Lagrangian looks like L = 1 2 ( ∂w ) 2 + 1 2 w 2 ( ∂R a 4 ) 2 - λ 4 w 2 - μ 2 λ 2 (7.8) So all the interactions for the pions are hidden in the rotation matrix and it comes in with a derivative. This means all the scattering amplitude are proportional to the momentum of the pions. If we define ζ i = π i / ( π 4 + ν ) then we have R i 4 = 2 ζ i 1 + ζ 2 , R 44 = 1 - ζ 2 1 + ζ 2 (7.9) Under the unbroken diagonal SU (2) symmetry we have ζ α × ζ but under the broken symmetry we have δ ζ = (1 - ζ 2 ) + 2 ζ ( · ζ ) (7.10) 34
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Quantum Field Theory III Lecture 7 Suppose we take μ 2 → ∞ and λ → ∞ but μ 2 fixed, then it is effectively making the Mexican hat potential steeper, and it is harder and harder to get out of the minimum. In the limit we will have σ 2 ( x ) + π 2 ( x ) -→ ν 2 (7.11) so in the above parametrization this means w ( x ) becomes a constant function. Then S = σ + i π · τ becomes ν times a unitary matrix. We can write it as S = νe i π ( x ) · τ = ν Σ( x ) (7.12) If we expand, then we will get ν Σ = ν 1 + i ν π · τ - 1 2 ν 2 ( π · τ ) 2 + . . . = ν + i π · τ - 1 2 ν π 2 + . . . (7.13) Now by symmetry the theory is invariant under Σ U Σ( x ) W . We want a low energy effective action and want to find invariant terms under the above symmetry that we can put into the action. The terms without derivatives can include Tr ΣΣ = ν 2 (7.14) But this is not very interesting because it is just a constant. Terms with derivatives can be Tr ( μ Σ)( μ Σ ) , h Tr ( μ Σ)( μ Σ ) i 2 , Tr ( μ ν Σ)( μ ν Σ ) , . . . (7.15) We can then write L eff = ν 2 4 Tr ( μ Σ)( μ Σ ) + c 1 h Tr ( μ Σ)( μ Σ ) i 2 + . . . (7.16) The only relevant term for low energies is the first term. Let’s think what are the possible vacuum for this theory, where G is broken into H . One possible vacuum is just φ 0 = (0 , 0 , 0 , ν ). Then 0 is also a valid vacuum. However H is the subgroup where φ 0 is invariant and we have 0 = φ 0 . So the space of vacua is the same as the coset space G/H , which is exactly the quotient group. Say if G = SO (4) and H = SO (3), then the space of vacua is the space of unit 4-vectors, which is just the sphere S 3 . The effect of our introducing σ is to introduce a coordinate of the 3-sphere, i.e. choosing a point of identity, and obtain a way of spatially disturbing the vauum. In general for any G/H we can put coordinates on the manifold with φ a . The Lagrangian will look like L = 1 2 g ab ( x ) μ φ a ( x ) μ φ b ( x ) + . . . (7.17) The quantity g ab can be considered as the metric over the manifold M = G/H . This is what people nowadays call the nonlinear sigma model. Let’s add a symmetry breaking term to the original sigma model Lagrangian, where c is a small positive number. Now the potential is V = λ 4 ( π + σ 2 - ν 2 ) 2 - (7.18) 35
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Quantum Field Theory III Lecture 7 The minimum of this potential is at π = 0 because we want to maximize , and the minimum is at 0 = ∂V ∂σ = λσ ( σ 2 - ν 2 ) - c, σ ν + c 2 λν 2 = w (7.19) If we ignore fermions, the equation of motion for σ is just σ = - λσ ( σ 2 + π 2 - ν 2 ) - c (7.20) and for π we have π = - λ π ( σ 2 + π 2 - ν 2 ) (7.21)
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