Dashen-Frautschi Fiasco

      Freeman Dyson (1923-2020).
      Photo by Heka Davis, courtesy AIP Emilio Segrč Visual Archives, Physics Today Collection.
    The first of these two achievements is the explanation of the mass difference between neutron and proton by Roger Dashen, working at the time as a graduate student under the supervision of Steve Frautschi. The neutron-proton mass difference has for thirty years been believed to be electromagnetic in origin, and it offers a splendid experimental test of any theory which tries to cover the borderline between electromagnetic and strong interactions. However, no convincing theory of the mass-difference had appeared before 1964. In this connection I exclude as unconvincing all theories, like the early theory of Feynman and Speisman, which use one arbitrary cut-off parameter to fit one experimental number. Dashen for the first time made an honest calculation without arbitrary parameters and got the right answer. His method is a beautiful marriage between old-fashioned electrodynamics and modern bootstrap techniques. He writes down the equations expressing the fact that the neutron can be considered to be a bound state of a proton with a negative pi meson, and the proton a bound state of a neutron with a positive pi meson, according to the bootstrap method. Then into these equations he puts electromagnetic perturbations, the interaction of a photon with both nucleon and pi meson, according to the Feynman rules. The calculation of the resulting mass difference is neither long nor hard to understand, and in my opinion, it will become a classic in the history of physics.

    In their paper, Dashen and Frautschi use the S-matrix method to calculate a perturbed energy level. Of course, they use approximations because they are dealing with strong interactions. If we translate what they did into the language of the Schrödinger picture, they are using the following approximation for

      (φ, δV φ),

    There are however "good" and "bad" approximations. I showed in my paper that Dashen and Frautschi use the formula

      good , δV φbad) .

    I then pointed out their infrared divergence comes from this bad approximation.

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