By Jan Smit
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Additional info for Introduction to quantum fields on a lattice : 'a robust mate'
53). 49). For small λ, γ4 ∝ λ, whereas for λ → ∞, γ4 → −2n4 /(n + 2). As usual, one expects that disconnected diagrams cancel out in expressions for the vertex functions, and that the two-point function, α β Gαβ xy = φx φy , can be expressed as a sum of connected diagrams.
The equation for the stationary points now reads (µ2 + λϕ2 )ϕα = δα0 . 4) the ground state has ϕα g pointing in the α = 0 direction, ϕα g = vδα0 , (µ2 + λv 2 )v = . 6) Consider now small fluctuations about ϕg . The equations of motion (field equations) read (−∂ 2 + µ2 + λϕ2 )ϕα = δα0 , ∂ 2 ≡ ∇2 − ∂t2 . 7) O(n) models 34 Linearizing around ϕ = ϕg , writing ϕ0 = v + σ, ϕ k = πk , k = 1, . 9) with m2σ = µ2 + 3λv 2 = 2λv 2 + /v, m2π = µ + λv = /v. 11) For µ2 > 0, v = 0 and m2σ = m2π = µ2 , whereas for µ2 < 0, v > 0 and the σ particle is heavier than the π particles.
In the continuum limit m → am, pj → apj , p4 → |ξ|−1 ap4 , Gp → a−2 ξG(p), a → 0 we obtain the Feynman propagator G(p) → −i ≡ −iGM (p). 131) where −i is meant to represent the rotation exp(−iϕ), ϕ: 0 → π/2 in the complex plane. 134) 28 Path-integral and lattice regularization without encountering the singularities at p0 = ± m2 + p2 ∓ i . Notice 3 4 that exp(ipx) is invariant under the rotation: µ=0 pµ xµ → µ=1 pµ xµ . 130) the timelike momentum is still denoted by p4 instead of p0 , because the pµ (and xµ ) in lattice units are just dummy indices denoting lattice points.
Introduction to quantum fields on a lattice : 'a robust mate' by Jan Smit