page_title: Renormalization
section_title: Divergences in quantum electrodynamics
hierarchical_section_title: Renormalization / Divergences in quantum electrodynamics
caption_attribution_description: A penguin diagram as originally drawn by John Ellis
context_page_description: Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of these quantities to compensate for effects of their self-interactions. But even if no infinities arose in loop diagrams in quantum field theory, it could be shown that it would be necessary to renormalize the mass and fields appearing in the original Lagrangian.
  For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons, positrons, and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles shows that the electron-system behaves as if it had a different mass and charge than initially postulated. Renormalization, in this example, mathematically replaces the initially postulated mass and charge of an electron with the experimentally observed mass and charge.
context_section_description: When developing quantum electrodynamics in the 1930s, Max Born, Werner Heisenberg, Pascual Jordan, and Paul Dirac discovered that in perturbative  corrections  many integrals were divergent (see The problem of infinities).
  One way of describing the perturbation theory corrections' divergences was discovered in 1947–49 by Hans Kramers, Hans Bethe, 
  Julian Schwinger, Richard Feynman, and Shin'ichiro Tomonaga, and systematized by Freeman Dyson in 1949. The divergences appear in radiative corrections  involving Feynman diagrams with closed loops of virtual particles in them.
  While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy–momentum relation for the observed mass of that particle (that is,  is not necessarily the squared mass of the particle in that process, e.g. for a photon it could be nonzero). Such a particle is called off-shell. When there is a loop, the momentum of the particles involved in the loop is not uniquely determined by the energies and momenta of incoming and outgoing particles. A variation in the energy of one particle in the loop can be balanced by an equal and opposite change in the energy of another particle in the loop, without affecting the incoming and outgoing particles. Thus many variations are possible. So to find the amplitude for the loop process, one must integrate over all possible combinations of energy and momentum that could travel around the loop.
  These integrals are often divergent, that is, they give infinite answers.  The divergences that are significant are the "ultraviolet" (UV) ones. An ultraviolet divergence can be described as one that comes from
  the region in the integral where all particles in the loop have large energies and momenta,
  very short wavelengths and high-frequencies fluctuations of the fields, in the path integral for the field,
  very short proper-time between particle emission and absorption, if the loop is thought of as a sum over particle paths.
  So these divergences are short-distance, short-time phenomena.
  Shown in the pictures at the right margin, there are exactly three one-loop divergent loop diagrams in quantum electrodynamics:
  (a) A photon creates a virtual electron–positron pair, which then annihilates. This is a vacuum polarization diagram.
  (b) An electron quickly emits and reabsorbs a virtual photon, called a self-energy.
  (c) An electron emits a photon, emits a second photon, and reabsorbs the first. This process is shown in the section below in figure 2, and it is called a vertex renormalization. The Feynman diagram for this is also called a “penguin diagram” due to its shape remotely resembling a penguin (with the initial and final state electrons as the arms and legs, the second photon as the body and the first looping photon as the head).
  The three divergences correspond to the three parameters in the theory under consideration:
  The field normalization Z.
  The mass of the electron.
  The charge of the electron.
  The second class of divergence called an infrared divergence, is due to massless particles, like the photon. Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero. For photons, these divergences are well understood. For example, at the 1-loop order, the vertex function has both ultraviolet and infrared divergences. In contrast to the ultraviolet divergence, the infrared divergence does not require the renormalization of a parameter in the theory involved. The infrared divergence of the vertex diagram is removed by including a diagram similar to the vertex diagram with the following important difference: the photon connecting the two legs of the electron is cut and replaced by two on-shell (i.e. real) photons whose wavelengths tend to infinity; this diagram is equivalent to the bremsstrahlung process. This additional diagram must be included because there is no physical way to distinguish a zero-energy photon flowing through a
caption: (c) Example of a “penguin” diagram
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