An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 18.1

Operator Products and Effective Vertices — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · operator-products-effective-vertices

Problem

Matrix element for proton decay. Some advanced theories of particle interactions include heavy particles XX whose couplings violate the conservation of baryon number. Integrating out these particles produces an effective interaction that allows the proton to decay to a positron and a photon or a pion. This effective interaction is most easily written using the definite-helicity components of the quark and electron fields: If uLu_L, dLd_L, uRu_R, eRe_R are two-component spinors, then this effective interaction is ΔL=2mX2ϵabcϵαβϵγδeRαuRαβuLbγdLcδ.\Delta\mathcal{L} = \frac{2}{m_X^2} \epsilon_{abc} \epsilon^{\alpha\beta} \epsilon^{\gamma\delta} e_{R\alpha} u_{R\alpha\beta} u_{Lb\gamma} d_{Lc\delta}. A typical value for the mass of the XX boson is mX=1016m_X = 10^{16} GeV. (a) Estimate, in order of magnitude, the value of the proton lifetime if the proton is allowed to decay through this interaction. (b) Show that the three-quark operator in ΔL\Delta\mathcal{L} has an anomalous dimension γ=−4g2(4π)2.\gamma = -4\frac{g^2}{(4\pi)^2}. Estimate the enhancement of the proton decay rate due to the leading QCD corrections.

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