Complex-mass renormalization in chiral effective field theory

Abstract

We consider a low-energy effective field theory of vector mesons and Goldstone bosons using the complex-mass renormalization. As an application we calculate the mass and the width of the $\rho$ meson.Comment: 7 pages, 1 fugure, REVTeX

Chiral behavior of vector meson self-energies

“…(22), our renormalized loop corrections formally start at second chiral order, with terms of O((s − (M V ) 2 ) 2 ) and O(δm ). While our results for the loop portion of the self-energies Π(s) can directly be mapped onto a dispersive representation, the power counting for the loop graphs is not straightforward, as already mentioned in the Introduction, and discussed in [22,24]. It was demonstrated in [22] that the genuine "soft-pion" part of the bubble diagram scales with the fractional power M d φ in dimensional regularization, which leads to an O(p 4 ) contribution in d → 4 space-time dimensions, and does not include the decay-threshold singularity, which could however be important phenomenologically.…”

“…This phenomenon was also observed when incorporating baryons in ChPT on the one-loop level [12]. To solve this problem, and to preserve the usual low-energy power counting scheme, a "heavy vector meson theory" was designed [18][19][20], while schemes preserving the power counting and manifest Lorentz covariance when including vector mesons were worked out some years later [21][22][23][24]. All these schemes face a problem in the resonance energy region, due to the fact that the ρ vector meson is not a stable particle under the strong interaction and can decay into two light Goldstone bosons (pions) which, by energy-momentum conservation, cannot be both of "soft" momentum (this problem does not occur for baryons since a decay into Goldstone bosons is prohibited by baryon number conservation): the imaginary part of the loop diagram which generates the decay width of the vector meson does not scale as expected from the naïve application of the low-energy counting rules to the diagram.…”

“…In the present case, however, it is in general complex, and contains relevant physics. This problem is discussed in [22,24]. While it is argued in [24,27] that the power-counting violating portion of the imaginary part can be absorbed in renormalized masses and couplings (which then become complex), without spoiling perturbative unitarity, this procedure is certainly only valid when the resonance mass is far above the decay threshold, e.g.…”

“…For example, when studying the quark mass dependence of the ρ mass, one must take care that M 3 π terms are included, which are nonanalytic in the quark masses and are generated by the ωπ sunset graphs, but not by the purely pionic loops (see e.g. [19,22,24]). In this work, we will not make use of such non-perturbative methods, and restrict ourselves to the one-loop level of perturbation theory to study the quark mass dependence of the vector meson self-energies.…”

Complex-mass scheme and effective field theory

“…Applications of the CMS in hadronic EFT can be found in Refs. [7][8][9][10][11]. Here we discuss a couple of conceptual issues and the application in the calculation of the magnetic moment of the -meson.…”