IRrelations

As extensively discussed in Ferraro et al.(1999, 2000) a correct parameterization of the RGB characteristics as a function of the metal content of the stellar population does require the knowledge of the so-called "global" metallicity, which takes into account the iron as well as the alpha-element abundances. Indeed, the location of the RGB strongly depends on the low ionization potential [Fe+Mg+Si/H] abundance mixture (Straniero & Chieffi 1991; Salaris & Cassisi 1996) rather than [Fe/H] abundance alone. In fact, the low ionization potential elements are the main contributors to free electrons, which generate the H^- ion, the major component responsible for the continuum opacity in the RGB temperature range (3000-6000 K; Renzini 1977).

In halo/disk field stars, the average [alpha/Fe] abundance ratio shows a general enhancement of 0.3 - 0.5 dex with respect to the solar value up to [Fe/]~ -1 (see e.g. Boesgaard et al.1999; Gratton et al.2000; Carretta et al.2000; and references therein) and a linear decreasing trend toward solar [alpha/Fe] with further encreasing metallicity.
An [alpha/Fe] enhanchement is also found in the metal-poor halo GCs (see e.g. Gratton et al.2004; Sneden et al.2004; and references therein). The actual position of the knee (i.e. the metallicity at which [alpha/Fe] begins to decrease) depends on the Type Ia SN timescales, and it is also a function of the star formation rate, while the amount of the alpha-enhancement depends on the initial mass function of the progenitors of the Type II SNe (see McWilliam 1997).
High resolution optical (McWiliam & Rich 1994; Rich & McWilliam 2000; Carretta et al.2001; Zoccali et al. 2004; Lecureur et al.2006; Fulbright et al.2004,2006) and IR (Origlia et al.2002,2004,2005a,b; Melendez et al.2003; Rich & Origlia 2005) spectroscopic studies of both cluster and field stars in the Galactic bulge point toward an alpha-enhancement by a factor of 2 - 3 up to solar metallicity.
To take into account the above picture, independent sets of global metallicities and IR photometric relations have been computed according to two different scenarios of chemical enrichment (see Fig. 1). In both scenarios, the contribution of the alpha-elements enhancement has been taken into account by simply rescaling standard models to the global metallicity [M/H] according to the following relation (Salaris et al. 1993):

[M/H] = [Fe/H] + log (0.638 f_a + 0.362)

where f_a is the enhanced factor of the alpha-elements.

Equation Set in the Disk-like Enrichment scenario

Accordingly to Ferraro et al.(1999,AJ,118,1738), in the Disk-like scenario the alpha-elements enhancement factor f_a is as follows:

f_a = 10^0.28 for [Fe/H] < -1