On the Field Renormalization Constant for Unstable Particles

A recently proposed normalization condition for the imaginary part of the selfenergy of an unstable particle is shown to lead to a closed expression for the field renormalization constant Z. In turn, the exact expression for Z is necessary, in some important cases, in order to avoid power-like infrared divergences in high orders of perturbation theory. In the same examples, the width plays the rôle of an infrared cutoff and, consequently, Z is not an analytic function of the coupling constant. PACS numbers: 11.10.Gh, 11.15.Bt ∗Permanent address: Department of Physics, New York University, 4 Washington Place, New York, New York 10003, USA The unrenormalized transverse propagator of a gauge boson is of the form: D μν (s) = −iQμν s−M 0 − A(s) , (1) where Qμν = gμν − qμqν/s, qμ is the four-momentum, s = q , M0 is the bare mass, and A(s) is the unrenormalized self-energy. An analogous expression holds for a scalar boson, with −iQμν → i. The complex position of the propagator’s pole is given by s̄ = M 0 + A(s̄). (2) Combining Eqs. (1) and (2), we have D μν (s) = −iQμν s− s̄− [A(s)− A(s̄)] . (3) Parameterizing s̄ = m2 − im2Γ2, where we employ the notation of Ref. [1], and considering the real and imaginary parts of Eq. (2), we see that m2 =M 2 0 +ReA(s̄), (4) m2Γ2 = − ImA(s̄). (5) If m2 is identified with the renormalized mass, Eq. (4) tells us that the mass counterterm is given by δm2 = ReA(s̄). This is to be contrasted with the conventional mass renormalization M = M 0 +ReA(M ), (6) where M is the on-shell mass. The great theoretical advantage of using m2 and Γ2 as the basis to define mass and width is that they are intrinsically gauge-independent quantities, while M is known to be gauge dependent in next-to-next-to-leading order [1,2]. The renormalized propagator D μν (s) is obtained by dividing Eq. (1) by the field renormalization constant Z = 1− δZ. Recalling Eq. (4), one readily obtains D μν (s) = −iQμν s−m2 − S(s) + ReS(s̄)− δZ (s−m 2 2) , (7) where S(s) ≡ ZA(s). (8) Thus, the renormalized self-energy is given by S(s) = S(s)−ReS(s̄) + δZ (