Witten’s Problems, Set Four N^{o}. 1 (solution written by D. Freed)

Problem:

(1) In the two-dimensional sigma model with target space a sphere S^{n}, for large n, we introduced a Lagrange multiplier field σ in order to quantize the theory.

(a) By using the equations of motion, identify σ as an operator in terms of the original variables.

(b) What is the dimension of the σ field and therefore what would you expect to be the large q behavior (with Euclidean q) of ⟨σ(q)σ(−q)⟩. Compare to the actual behavior.

Solution:

(a) This model has two bosonic fields: a map ϕ : ℝ^{2} → ℝ^{n+1} and a “Lagrange multiplier” σ : ℝ^{2} → ℝ. The action is

L = \frac{1}{2λ} |dϕ|^{2} − \frac{iσ}{2} (|ϕ|^{2} − 1).

It is easy to compute the classical equations of motion

|ϕ|^{2} = 1
\frac{Δ}{λ}ϕ = iσϕ.

Our Laplacian ’Δ’ is nonnegative. The first says that ϕ maps into the n-sphere S^{n} and the second that the Laplacian of ϕ is perpendicular to S^{n}. This means that ϕ is a harmonic map into S^{n}. We express σ in terms of ϕ as

(1)  σ = \frac{1}{iλ}(Δϕ, ϕ).

Up to a total derivative σ = \frac{1}{iλ}|dϕ|^{2}.

(b) According to equation (1) the σ field has dimension 2, since ϕ is a scalar field in two dimensions and so has dimension 0. We expect the behaviour of the 2-point function in position space to be

⟨σ(x)σ(0)⟩ ∼ \frac{1}{|x|^{4}} as x → 0,

or equivalently in momentum space to be

(2)  ⟨σ(q)σ(−q)⟩ ∼ q^{2} on q → ∞,

with possible logarithmic connections due to anomalous dimensions. We could integrate out the σ field first, thereby obtaining the σ-model into S^{n}, and since this theory is asymptotically free the 2-point function of |dϕ|^{2} does exhibit the behaviour (2) (up to logarithms). Instead, we integrate out ϕ first and compute the effective action for σ, at least in the large n limit.

The computation of the effective action appears in Witten’s lecture II-3:

(3)  Γ(σ̃) = \frac{n}{2}\{T r ln (Δ − iσ̃λ̃) + iσ̃\}.

where σ̃ = σ/n and λ̃ = λn. We know that we should expand around the value −iσ̃_{0}λ̃ = M^{2} for a positive constant M^{2}. The σ̃ propagator is the inverse of the second derivative of the action (3), which we evaluate at the constant σ̃_{0} = iM^{2}/λ̃:

(4)  \begin{aligned}\frac{δ^{2}Γ}{δσ̃^{2}} |σ̃_{0} & = \frac{nλ̃^{2}}{2} T r((Δ + M^{2})^{−1}δσ̃(Δ + M^{2})^{−1}δσ̃) \\ & = \frac{nλ̃^{2}}{2} ∫ dxdy G(x, y) δσ̃(y) G(y, x) δσ̃(x) \\ & = \frac{nλ̃^{2}}{2} ∫ dxdy G(x, y)^{2} δσ̃(x) δσ̃(y), \end{aligned}

where G is the kernel of the Greens function (Δ + M^{2})^{−1}. So the kernel of the inverse σ̃-propagator is \frac{nλ̃^{2}}{2} G(x, y)^{2}. In momentum space the square goes over to convolution, and so the kernel of the inverse σ̃-propagator is

(5)  K(q, −q) = \frac{nλ̃^{2}}{2} ∫ \frac{d^{2}p}{(2π)^{2}} \frac{1}{p^{2} + M^{2}} \frac{1}{(q − p)^{2} + M^{2}} .

= \frac{nλ̃^{2}}{8π^{2}} ∫_{0}^{1} dα ∫ d^{2}p \frac{1}{[p^{2} + M^{2} + α(1 − α)q^{2}]^{2}}

= \frac{nλ̃^{2}}{8π} ∫_{0}^{1} \frac{dα}{M^{2} + α(1 − α)q^{2}}

= \frac{nλ̃^{2}}{4π} \frac{ln(^{\frac{\sqrt{1+4M^{2}/q^{2}}+1}{\sqrt{1+4M^{2}/q^{2}}−1}})}{q^{2}\sqrt{1 + 4M^{2}/q^{2}}}

∼ \frac{nλ̃^{2}}{2π} \frac{ln(q^{2}/M^{2})}{q^{2}} as q → ∞.

So the σ̃-propagator behaves as

⟨σ̃(q)σ̃(−q)⟩ ∼ \frac{2π}{nλ̃^{2}} \frac{q^{2}}{ln(q^{2}/M^{2})} as q → ∞.

Since σ̃ = σ/n, we have finally

(6)  ⟨σ(q)σ(−q)⟩ ∼ \frac{2πn}{λ̃^{2}} \frac{q^{2}}{ln(q^{2}/M^{2})} as q → ∞.

Notice that the behaviour in n could have been predicted from (1), since the 2-point function of (Δϕ, ϕ) is n + 1 times the 2-point function of a single scalar field. The large q behavior is as predicted in (2), but corrected by a logarithm.

Notice that the inverse propagator (5) can be computed by diagrams. The effective action obtained by integrating our ϕ is computed by diagrams with external σ lines and internal ϕ lines. We should expand around σ = iM^{2}/λ, of course. For the quadratic term there is a single diagram, as shown, where the solid line

.......................................................................................................................................................................................................................................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

represents ϕ and the dotted line represents σ. Note the ϕ propagator is massive due to the shift in σ. This diagram is evaluated by (5).