An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 11.3

Renormalization and Symmetry — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · gross-neveu-model-chiral-symmetry-breaking

Problem

11.3 The Gross-Neveu model. The Gross-Neveu model is a model in two spacetime dimensions of fermions with a discrete chiral symmetry: L=ψˉii∂ ⁣ ⁣ ⁣/ψi+12g2(ψˉiψi)2\mathcal{L} = \bar{\psi}_i i\partial\!\!\!/ \psi_i + \tfrac{1}{2}g^2(\bar{\psi}_i\psi_i)^2 with i=1,…,Ni = 1, \ldots, N. The kinetic term of two-dimensional fermions is built from matrices γμ\gamma^\mu that satisfy the two-dimensional Dirac algebra. These matrices can be 2×22 \times 2: γ0=σ2,γ1=iσ1,\gamma^0 = \sigma^2, \qquad \gamma^1 = i\sigma^1, where σi\sigma^i are Pauli sigma matrices. Define γ5=γ0γ1=σ3;\gamma^5 = \gamma^0\gamma^1 = \sigma^3; this matrix anticommutes with the γμ\gamma^\mu. (a) Show that this theory is invariant with respect to ψi→γ5ψi,\psi_i \to \gamma^5\psi_i, and that this symmetry forbids the appearance of a fermion mass. (b) Show that this theory is renormalizable in 2 dimensions (at the level of dimensional analysis). (c) Show that the functional integral for this theory can be represented in the following form: ∫Dψ ei∫d2x L=∫DψDσ  exp⁡[i∫d2x{ψˉii∂ ⁣ ⁣ ⁣/ψi−σψˉiψi−12g2σ2}],\int \mathcal{D}\psi\, e^{i\int d^2x\,\mathcal{L}} = \int \mathcal{D}\psi \mathcal{D}\sigma \;\exp\left[i\int d^2x\left\{\bar{\psi}_i i\partial\!\!\!/\psi_i - \sigma\bar{\psi}_i\psi_i - \frac{1}{2g^2}\sigma^2\right\}\right], where σ(x)\sigma(x) (not to be confused with a Pauli matrix) is a new scalar field with no kinetic energy terms. (d) Compute the leading correction to the effective potential for σ\sigma by integrating over the fermion fields ψi\psi_i. You will encounter the determinant of a Dirac operator; to evaluate this determinant, diagonalize the operator by first going to Fourier components and then diagonalizing the 2×22 \times 2 Pauli matrix associated with each Fourier mode. (Alternatively, you might just take the determinant of this 2×22 \times 2 matrix.) This 1-loop contribution requires a renormalization proportional to σ2\sigma^2 (that is, a renormalization of g2g^2). Renormalize by minimal subtraction. (e) Ignoring two-loop and higher-order contributions, minimize this potential. Show that the σ\sigma field acquires a vacuum expectation value which breaks the symmetry of part (a). Convince yourself that this result does not depend on the particular renormalization condition chosen. (f) Note that the effective potential derived in part (e) depends on gg and NN according to the form Veff(σcl)=N⋅f(g2N).V_{\text{eff}}(\sigma_{\text{cl}}) = N \cdot f(g^2 N). (The overall factor of NN is expected in a theory with NN fields.) Construct a few of the higher-order contributions to the effective potential and show that they contain additional factors of N−1N^{-1} which suppress them if we take the limit N→∞N \to \infty, (g2N)(g^2 N) fixed. In this limit, the result of part (e) is unambiguous.

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