Effective Theories and Elementary Particle Masses
The concepts of effective theory have a rich history in particle physics. The early days of effective theories have many examples, including Fermi’s theory of nucleon decay and chiral lagrangian dynamics for pion scattering. These examples are touched upo
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Effective Theories and Elementary Particle Masses
Abstract The concepts of effective theory have a rich history in particle physics. The early days of effective theories have many examples, including Fermi’s theory of nucleon decay and chiral lagrangian dynamics for pion scattering. These examples are touched upon briefly before going to the most pressing issue of today, which is the origin of elementary particle masses. The problem of mass generation is first described, where it is shown that simply writing down mass terms manifestly breaks cherished symmetries. It is then shown that spontaneous symmetry breaking cures this problem. The influence of effective field theory is then addressed, where it is shown that the smallness of neutrino masses nicely conforms with our intuition, but the weak-scale value of the Higgs boson mass is confusing. The chapter concludes with an essay describing this mystery and what the resolutions might be.
4.1 Introduction Effective theories play a central role in particle physics. Perhaps the most famous effective theory of them all is Fermi’s four-fermion interaction theory that described nucleon decay and muon decay. The theory is a “V-A theory” (vector minus axial vector interaction) and has the form: GF LV −A = − √ v¯ γ μ (1 − γ 5 ) f f¯q γμ (1 − γ 5 ) f q 2
(4.1)
where G F = 1.15 × 10−6 GeV−2 is the Fermi constant determined by experiment. These operators can then induce β decays of the neutron via the constituent quark decays d → ue¯v, and can also induce muon decay through μ → evμ v¯ e . The history behind determining the precise nature of this interaction is a fascinating one that required painstaking experiment and insightful theory (Renton 1990).
J. D. Wells, Effective Theories in Physics, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-34892-1_4, © The Author(s) 2012
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4 Effective Theories and Elementary Particle Masses
We know now that the Fermi theory is just the low-energy limit of the electroweak theory of the Standard Model.1 The Fermi constant G F that gives the strength of the four-fermion interaction is the low-energy limit of a W -boson propagator multiplied by its couplings to the two bilinear currents: lim
p 2 →0
−g 2 2 p2 − MW
=⇒
g2 GF ≡ √ 2 MW 2
(4.2)
where g is the SU (2) L gauge coupling of the Standard Model. Thus the propagator of the W -boson at very low energies compared with the W mass contracts to a point and makes an effective four-fermion interaction term governed by the Fermi Effective Theory coupling constant GF . Another place where Effective Theories are put to good use is in low-energy pion scattering theory. Pions are the lightest strongly interacting hadrons known in nature. The pions will interact with a very large number of other hadrons in the theory to mediate and alter even pure pion-pion scattering. Computing all of these interactions with the multitude of other intermediate hadrons is a daunting prospect to say the least. However, the effective lagrangian approach allows one to simplify these complicated dyna
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