On-shell renormalization scheme: Difference between revisions

Revision as of 20:55, 18 June 2012

Revision as of 10:13, 10 August 2012

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:<math> \langle \Omega | T(\psi(x)\bar{\psi}(0))| \Omega \rangle = \int \frac{d^4p}{(2\pi)^4}\frac{ie^{-i p\cdot x}}{p\!\!\!/-m - \Sigma(p) +i\epsilon} </math>

These corrections are often divergent because they countaincontain [[One-loop Feynman diagram|loops]].

By identifying the two expressions of the correlation function up to a certain order in <math>e</math>, the counterterms can be defined, and they are going to absorb the divergent contributions of the corrections to the fermion propagator. Thus, the renormalized quantities, such as <math>m_r</math>, will remain finite, and will be the quantities measured in experiments.

Line 33:

:<math> \langle \Omega | T(A^{\mu}(x)A^{\nu}(0))| \Omega \rangle = \int \frac{d^4q}{(2\pi)^4}\frac{-i\eta^{\mu\nu}e^{-i p\cdot x}}{q^2(1 - \Pi(q^2)) +i\epsilon} = \int \frac{d^4q}{(2\pi)^4}\frac{-iZ_3 \eta^{\mu\nu}e^{-i p\cdot x}}{q^2 +i\epsilon} </math>

The behaviour of the counterterm <math>\delta_3=Z_3-1</math> is independent of the momentum of the incoming photon <math>q</math>. To fix it, the behaviour of QED at large distances (which should help recover [[classical electrodynamics]]), iei.e. when <math>q^2\rightarrow 0</math>, is used :

:<math>\frac{-i\eta^{\mu\nu}e^{-i p\cdot x}}{q^2(1 - \Pi(q^2)) +i\epsilon}\sim\frac{-i\eta^{\mu\nu}e^{-i p\cdot x}}{q^2}</math>