On-shell renormalization scheme: Difference between revisions

Revision as of 20:27, 17 July 2013

Revision as of 05:46, 2 November 2013

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==Fermion propagator in the interacting theory==

Knowing the different [[propagator (Quantum Theory)|propagators]]s is the basis for being able to calculate [[Feynman diagram]]s which are useful tools to predict, for example, the result of scattering experiments. In a theory where the only field is the [[Fermionic field|Dirac field]], the Feynman propagator reads

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

where <math>T</math> is the [[Time ordered#Time ordering|time-ordering operator]], <math>|0\rangle</math> the vacuum in the non interacting theory, <math>\psi(x)</math> and <math>\bar{\psi}(x)</math> the Dirac field and its Dirac adjoint, and where the left-hand handsideside of the equation is the [[Correlation function (quantum field theory)|two-point correlation function]] of the Dirac field.

In a new theory, the Dirac field can interact with another field, for example with the electromagnetic field in quantum electrodynamics, and the strength of the interaction is measured by a parameter, in the case of QED it is the bare electron charge, <math>e</math>. The general form of the propagator should remain unchanged, meaning that if <math>|\Omega\rangle</math> now represents the vacuum in the interacting theory, the two-point correlation function would now read

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:<math> \langle \Omega | T(\psi(x)\bar{\psi}(0))| \Omega \rangle = \int \frac{d^4q}{(2\pi)^4}\frac{i Z_2 e^{-i p\cdot x}}{p\!\!\!/-m_r+i\epsilon} </math>

Two new quantities have been introduced. First the renormalized mass <math>m_r</math> has been defined as the pole in the Fourier transform of the Feynman propagator. This is the main prescription of the on-shell renormalization scheme (there is then no need to introduce other mass scales like in the minimal substractionsubtraction scheme). The quantity <math>Z_2</math> represents the new strength of the Dirac field. As the interaction is turned down to zero by letting <math>e\rightarrow 0</math>, these new parameters should tend to a value so as to recover the propagator of the free fermion, namely <math>m_r\rightarrow m</math> and <math>Z_2\rightarrow 1</math>.

This means that <math>m_r</math> and <math>Z_2</math> can be defined as a series in <math>e</math> if this parameter is small enough (in the unit system where <math>\hbar=c=1</math>, <math>e=\sqrt{4\pi\alpha}\simeq 0.3</math>, where <math>\alpha</math> is the [[fine-structure constant]]). Thus these parameters can be expressed as