Light quark masses beyond leading order

John F. Donoghue, Daniel Wyler
1992 Physical Review D, Particles and fields  
We describe the measurement of the light quark mass ratios when one calculates to second order in the quark masses. At this order there is an ambiguity in the meaning of the quark mass, which afBicts the past attempts to provide a model-independent measurement of the ratios. We argue that this is similar to the regularization-scheme dependence of coupling constants. We study the anomalous Ward identities and the effects of strong CP violation in an attempt to resolve the ambiguity. The
more » ... persists even with singlet fields, such as the g', but can be resolved by observing the 9 dependence of the theory. Since matrix elements of FF are related to BXQCQ/80, they are useful probes of quark masses. We give a procedure by which quark mass ratios can be measured in a model-independent way through the matrix elements (O~FF~m. ), (O~FF~r)), and (O~FF~37r), which in turn are observable in V'~V vF (t), 3n), with Vbeing t( or Y, when analyzed using a heavy-quark multipole expansion. Present data are not suScient to complete this program, but we use available results to provide the value the first error is experimental and the second is our estimate of the remaining theoretical model dependence. PACS number(s): 12.15.Ff, 11. 30.Rd, 11. 40.Fy dent, these ratios are equally well the ratios of renorrnaized masses or of bare parameters in the Lagrangian. These measurements of mass ratios will be modified if the IWe would like to emphasize that here and throughout the rest of the paper, when we describe the order of the expansion, it refers to an expansion in the mass. A perturbative expansion in the QCD coupling constant is never implied. be expanded in a series in the mass. To first order, ' the results are extremely simple [4]: m"+md m 2m, m, 2m+m " 892 1992 The American Physical Society 45 LIGHT QUARK MASSES BEYOND LEADING ORDER 893 theoretical analysis is carried out to next order, when the subtleties appear. It is the purpose of this paper to discuss the analysis of quark mass ratios at next order. There are several motivations for this work. In the first place, the quark masses are some of the basic parameters of the standard model, and it is important to document our level of understanding of them. There is in the community an almost universal acceptance of the lowest-order mass ratios of Eq. (1). This is not warranted, as we will demonstrate that sizable corrections to these ratios are allowed. A second motivation is a known ambiguity which first surfaces at second order [5] . While we will leave the precise statement of the ambiguity to the next section, it states that we can obtain the same phenomenological consequences either from a mass matrix m, or from a changed mass matrix m'" =m +X det(m)m where X is an arbitrary constant. Specifically (at second order) m""'=m"+am, m, , md =md+Am"m (~)m, ' '=m, +A, m"md . As far as phenomenology is concerned, any one in this family of mass matrices can be chosen as the primary mass matrix. There have been conAicting claims about the effect of this ambiguity [5 -7], including an interesting recent investigation by Leutwyler to resolve the ambiguity [7], and we will spend a good deal of the paper in an attempt to clarify this issue. A further motivation comes from the strong CP problem [8] . The only solution which does not require physics outside the standard model is the option with m"=O, in which case the effect of the 8 term vanishes. Put another way, the physical CP-violating parameter is 8 detm, which would vanish if detm =0. Although it has been argued in the past that this option is not viable phenomenologically, this conclusion has been questioned because of the ambiguity mentioned above [5] . We note, however, that, even if it were to be allowed phenomenologically, the "m"=O option" does not resolve the naturalness problem within the standard model as it is no more natural to set m"=O than to set 8=0. There is, in addition, a stimulating calculation by Kim, Choi, and Sze [9]. These authors consider a massless up
doi:10.1103/physrevd.45.892 pmid:10014450 fatcat:pbdxa54rmrcrrgkwmt25jvt5eq