Radiative Corrections

Radiative Corrections

1. The Electron Vertex Function

   • Here we wish to calculate the vertex factor -ie\Gamma^\mu which is the sum of all amputated corrections to the Feynman diagram for electron scattering with a heavy object.
   • Form Factors
     • These arise in the expression for \Gamma^\mu when writing it in its most general form and applying the Ward identity as well as the Gordon identity.
     • The factors F_1 and F_2 are functions of q^2  = (p'-p)^2, and contain complete information about the influence of the electromagnetic field on the electron.
     • It follows from the expression of \Gamma^\mu that to lowest order F_1 = 1 and F_2 = 0.
     • The form factors also recieve corrections from the vertex correction diagram.
     • Electric Charge
       • It can be shown by calculating the Feynman amplitude for electron scattering off an electromagnetic field and comparing with the Born approximation, that F_1(0) can be identified with the electric charge in units of e.
       • Since this calculation is done in the limit \vec q = 0 only F_1 contributes.
       • Since F_1(0) = 1 already in leading order all radiative corrections to F_1(0) must vanish.
     • Anomalous Magnetic Moment
       • By letting the electron scatter of a vector potential we see that the form factors are related to the magnetic moment of the electron.
       • g-2 of the electron
         • It turns out that the form factors become the Landé g-factor in the relation between magnetic moment and spin. More precisely F_2(0) = (g-2)/2. Since F_2 is order \alpha in lowest order, QED predicts an order \alpha correction to the g-factor.
   • Evaluation
     • We evaluate to order \alpha that is we seek the term  \delta \Gamma^\mu = \gamma^\mu - \Gamma^\mu
     • Feynman Parameters
       • These come in handy when evaluating the denominators in the momentum integral stemming from the propagators in the Feynman diagram.
       • This method relies on an identity that allows us to put the factors in the denominator in one sum of parameters and factors lifted to, usually, the third power.
       • Completing the Square
         • We complete the square of momenta by introducting a parameter \ell, appropriately defined in terms of the momenta and Feynman parameters. We then have an integral in terms of the parameter \ell.
         • Wick Rotation
           • The minus sign in the Minkowski metric prevents us from doing a 4-dimensional spherical integration, but because of the location of the poles in the \ell^0 complex plane and the fact the integrand falls off at large |\ell^0|, we can simply rotate the integration contour such that \ell^0 \equiv i \ell_E^0. We then get a Euclidean metric and are able to do spherical integration.
     • Pauli-Villars Regularization
       • Since the photon propagator in the corrected vertex leads to a divergent integral, we need to regularize it. This Pauli-Villars regularization accomplishes this by replacing the propagator with the same propagator with a subtracted term. This term is the propagator of a heavy photon, i.e., the term -\Lambda^2 is added in the denominator. This leads to cancelling infinities in the integral.
       • The added photon mass is fictitious and has no physical meaning. We therefore hope that it vanishes when calculating physical observables.
     • Final Result for Form Factors
       • After regularization of the divergent integrals, we obtain an expression for the one-loop vertex correction, in a form where the expressions for the form factors are manifest. 
       • We find that the ultraviolet divergent term corrects F_1(q^2=0) which we know is wrong. Therefore, we adopt the ad hoc procedure to subtract a term such that we obtain no correction, i.e., make the substitution  \delta F_1(q^2) -> \delta F_1(q^2) - \delta F_1(0).
       • The Feynman integral for the vertex correction also contains an infrared divergence, which is cured by introducing a very small mass \mu in the photon propagator. We then get an infrared diverging expression for F_1(q^2).
       • F_2 (q^2) does not suffer from divergences, so this function can be evaluated at q^2 = 0 to yield the result  F_2 (0) = \frac{\alpha}{2\pi}. We thus get a correction to the g-2 of the electron of order \alpha.
       • Infrared Divergence of F_1 (q^2)
         • Evaluating F_1 (q^2) as far as we can we find a term proportional to a logarithm divergent in \mu^2. The proportionality factor is a positive number depending on q^2.
         • Resolution of Infrared Divergence Problem
           • By making the substitution e -> eF_1 (q^2) in the cross section we find a correcting factor which is negative and infinite. By evaluating the coefficient of the diverging logarithm, we see that the correction to the scattering cross section exactly cancels correction to the cross section for Brehmsstrahlung.
           • Since particle detectors can only detect photons down to an energy level of E_\ell, the probability that a detector does not see a photon in a scattering event, is the sum of the probability of scattering and the probability for emitting a Brehmsstrahlung photon below the threshold for detection. The two infinite corrections therefore cancel each other, and we get a sensible result.

2. Formal Developments

   • Field Strength Renormalization
     • This covers the investigation of the analytical structure of the two-point correlation function, and how far we can take the interpretation of a propagation amplitude in an interacting theory.
     • This will then lead to an evaluation of the external leg correction investigated in the electron vertex function.
     • Two-Point Correlation Function for Interacting Theory
       • This is expressed in terms of the spectral density function which is the sum over all zero-momentum states of the product 2 \pi \delta(M^2-m_\lambda^2) Z, where M is a variable of mass and Z is the field-strength renormalization related to the probability of creating a zero-momentum state from the interacting theory groundstate. This function is multiplied with the Feynman propagator for mass M and integrated over all masses M, to give the two-point correlation function.
       • Choosing Z = 1 for a single particle state and Z = 0 for all other states recovers the free field propagator.
       • The analytic structure of the Fourier-transformed T-P CF is such that a single particle state gives a pole at m^2 in the p^2 plane and multiparticle states give a branch cut.
     • The Electron Self-Energy
       • We can calculate the order-\alpha correction to the correlation function which is called the electron self-energy. The final result becomes a regularized divergent Feynman integral.
       • The final result shows that the branch cut is exactly at the threshold for creation of a final state with two particles (an electron and a photon), as was expected from the Källén-Lehman spectral representation.
       • We also need to find the expected simple pole at the physical mass m. This requires an infinte number of Feynman diagrams, but expressing the Fourier transform of the correlation function as a sum of products of 1PI's allows us to write this sum as a geometric series.
       • Taylor-expanding the numerator of the correlation function around the pole gives the renormalization factor Z_2 explicitly, and we see that find the identity  \delta F_1 (0) = \delta Z_2.
   • LSZ Reduction Formula
     • This formula relates Fourier transforms of correlation functions to S-matrix elements.
     • This is done by first computing the Fourier transform of a correlation function, look at the region in momentum space where the external particles are close to mass shell, and then identify the coefficient of the multiparticle pole. This coefficient is the S-matrix element.
     • Formulating this in terms of Feynman diagrams we see that the poles arising from each external propagator contributes a factor (\sqrt Z)^(n+2) to the exact (n+2)-point correlation function, which corrects for the omitted proportionality factors when first computing Feynman diagrams.
     • The electron field-strength renormalization also has an effect on the electron form factors. Including a factor \sqrt Z_2 for the initial and final electrons gives a LHS Z_2\Gamma^\mu in the expression with form factors. F_1 (q^2) then gets a new correction equal to \delta Z_2. With the identity  \delta Z_2 = - \delta F_1 (0) we have then justified the ad hoc replacement in the original calculation of  F_1 (q^2).
   • Charge Renormalization
     • Here we seek to calculate the vacuum polarization diagram in order to find a correction to the electromagnetic charge.