Srednicki — Quantum Field Theory · Problem 28.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · renormalization-group-two-scalar-fields

Problem

Consider the lagrangian density L=−12Zφ∂μφ∂μφ−12Zmm2φ2+Yφ−12Zχ∂μχ∂μχ−12ZMM2χ2+16Zggμ~ε/2φ3+12Zhhμ~ε/2φχ2(28.42)\mathcal{L} = -\tfrac{1}{2}Z_\varphi\partial^\mu\varphi\partial_\mu\varphi - \tfrac{1}{2}Z_m m^2\varphi^2 + Y\varphi - \tfrac{1}{2}Z_\chi\partial^\mu\chi\partial_\mu\chi - \tfrac{1}{2}Z_M M^2\chi^2 + \tfrac{1}{6}Z_g g\tilde{\mu}^{\varepsilon/2}\varphi^3 + \tfrac{1}{2}Z_h h\tilde{\mu}^{\varepsilon/2}\varphi\chi^2 \tag{28.42} in d=6−εd = 6 - \varepsilon dimensions, where φ\varphi and χ\chi are real scalar fields, and YY is adjusted to make ⟨0∣φ(x)∣0⟩=0\langle 0|\varphi(x)|0\rangle = 0. (Why is no such term needed for χ\chi?) a) Compute the one-loop contributions to each of the ZZ's in the MS‾\overline{\mathrm{MS}} renormalization scheme. b) The bare couplings are related to the renormalized ones via g0=Zφ−3/2Zggμ~ε/2 ,h0=Zφ−1Zχ−1/2Zhhμ~ε/2 .(28.43)\begin{aligned} g_0 &= Z_\varphi^{-3/2}Z_g g\tilde{\mu}^{\varepsilon/2} \,, \tag{28.43}\\ h_0 &= Z_\varphi^{-1}Z_\chi^{-1/2}Z_h h\tilde{\mu}^{\varepsilon/2} \,. \end{aligned} Define G(g,h,ε)=∑n=1∞Gn(g,h)ε−n≡ln⁡(Zφ−3/2Zg) ,H(g,h,ε)=∑n=1∞Hn(g,h)ε−n≡ln⁡(Zφ−1Zχ−1/2Zh) .(28.45)\begin{aligned} G(g,h,\varepsilon) &= \sum_{n=1}^\infty G_n(g,h)\varepsilon^{-n} \equiv \ln(Z_\varphi^{-3/2}Z_g) \,, \tag{28.45}\\ H(g,h,\varepsilon) &= \sum_{n=1}^\infty H_n(g,h)\varepsilon^{-n} \equiv \ln(Z_\varphi^{-1}Z_\chi^{-1/2}Z_h) \,. \end{aligned} By requiring g0g_0 and h0h_0 to be independent of μ\mu, and by assuming that dg/dμdg/d\mu and dh/dμdh/d\mu are finite as ε→0\varepsilon \to 0, show that μdgdμ=−12εg+12g(g∂G1∂g+h∂G1∂h) ,μdhdμ=−12εh+12h(g∂H1∂g+h∂H1∂h) .(28.47)\begin{aligned} \mu\frac{dg}{d\mu} &= -\tfrac{1}{2}\varepsilon g + \tfrac{1}{2}g\left(g\frac{\partial G_1}{\partial g} + h\frac{\partial G_1}{\partial h}\right) \,, \tag{28.47}\\ \mu\frac{dh}{d\mu} &= -\tfrac{1}{2}\varepsilon h + \tfrac{1}{2}h\left(g\frac{\partial H_1}{\partial g} + h\frac{\partial H_1}{\partial h}\right) \,. \end{aligned} c) Use your results from part (a) to compute the beta functions βg(g,h)≡lim⁡ε→0μ dg/dμ\beta_g(g,h) \equiv \lim_{\varepsilon\to 0}\mu\,dg/d\mu and βh(g,h)≡lim⁡ε→0μ dh/dμ\beta_h(g,h) \equiv \lim_{\varepsilon\to 0}\mu\,dh/d\mu. You should find terms of order g3g^3, gh2gh^2, and h3h^3 in βg\beta_g, and terms of order g2hg^2h, gh2gh^2, and h3h^3 in βh\beta_h. d) Without loss of generality, we can choose gg to be positive; hh can then be positive or negative, and the difference is physically significant. (You should understand why this is true.) For what numerical range(s) of h/gh/g are βg\beta_g and βh/h\beta_h/h both negative? Why is this an interesting question?

Related theory

Browse the problem library