Renormalization

renormalization

• way of pulling out sensible answers from Feynman diagrams that explode
• there are two parts in the renormalization program
  • regularization - the divergences must be written down in some way so that they can be compared, added and subtracted
  • renormalization proper - the various divergences must be gathered together and extracted from the rest of the calculation
• set of techniques used to understand a given quantum field theory in a certain energy or length interval
  • http://en.wikipedia.org/wiki/Effective_field_theory
  • effective ~ restricted to some interval
• easiest to grasp using functional integrals

regularization

• the values of observable quantities cannot depend on the way we've chosen to take the cutoff (regularize)
• introducing momentum or distance cutoff so as to render finite the large momentum or short distance limits of correlation functions
• methods of regularization
  • momentum regularization (modify the propagator by introducing cutoff dependent mass couplings)
  • lattice regulatization (replace $R^d$ by a lattice, uses a small space cutoff)
  • dimensional regularization
• root of the problem
  • probability of creating particles of colossal energies
  • in terms of Feynman diagrams, the momentum in a loop can run away to infinity
• dimensional regularization
  • regularization scheme (especially good in QCD)
  • regard the dimension as a continuous quantity
  • coupling constant changes accordingly as the dimension changes continuously
  • we get a new parameter called regularization scheme

QED renormalization

• QED renormalization

electroweak renormalization

• spontaneous local symmetry breaking or Higgs mechanism
• mass term for gauge field is zero in the Lagranaian, but these bosons (W,Z bosons) have mass term. To resolve this, we employ Higgs mechanism

books

• Renormalization: an introduction to renormalization, the renormalization John C. Collins
• Kevin Costello, Renormalization and Effective Field Theory http://www.ams.org/bookstore-getitem/item=surv-170

expositions

• Gerard t Hooft, Reflections on the renormalization procedure for gauge theories, arXiv:1604.06257 [hep-th], April 21 2016, http://arxiv.org/abs/1604.06257
• Kasia Rejzner, Renormalization and periods in perturbative Algebraic Quantum Field Theory, http://arxiv.org/abs/1603.02748v1
• http://arxiv.org/abs/1511.03180v1
• Polyakov, A. M. ‘Kenneth Wilson in Moscow’. arXiv:1502.03502 [hep-Th, Physics:physics], 11 February 2015. http://arxiv.org/abs/1502.03502.
• Guo, Li, Sylvie Paycha, and Bin Zhang. “Counting an Infinite Number of Points: A Testing Ground for Renormalization Methods.” arXiv:1501.00429 [math-Ph], January 2, 2015. http://arxiv.org/abs/1501.00429.
• Panzer, Erik. “Renormalization, Hopf Algebras and Mellin Transforms.” arXiv:1407.4943 [hep-Th, Physics:math-Ph], July 18, 2014. http://arxiv.org/abs/1407.4943.
• Huang, Kerson. 2013. “A Critical History of Renormalization”. ArXiv e-print 1310.5533. http://arxiv.org/abs/1310.5533.

articles

• Pawlowski, Jan M., Michael M. Scherer, Richard Schmidt, and Sebastian J. Wetzel. “Physics and the Choice of Regulators in Functional Renormalisation Group Flows.” arXiv:1512.03598 [cond-Mat, Physics:hep-Th], December 11, 2015. http://arxiv.org/abs/1512.03598.
• Eröncel, Cem, and O. Teoman Turgut. ‘Exact Renormalization Group for Point Interactions’. arXiv:1412.8623 [hep-Th], 30 December 2014. http://arxiv.org/abs/1412.8623.
• Renormalization and quantum field theory R. E. Borcherds, 2010