Eliashberg's theory  was born as a generalization of the BCS theory to explain some anomalies in the experimental data concerning lead. Subsequently, it was seen that the theory can be successfully applied to explain the experimental data of practically almost all superconducting materials [2,3], first of all low Tc phononic superconductors , then magnesium diboride [5,6], graphite intercalated compound CaC6 , iron-based superconductors [8-12]. This theory can be applied to describe particular systems such as proximized systems  and field effect junctions [14-16]. For what concerns the high Tc superconductors [17-21], their properties strongly depend on their oxygen content. It is possible to identify three different regimes: under, optimal and overdoping. While the discussion is still open as regards the underdoping regime, it is almost certain that the fundamental mechanism in the optimal and over regime is due to antiferromagnetic spin fluctuations, and especially in the over regime, the experimental data can be described satisfactorily by one band d - wave Eliashbeg's theory [22,23]. Detailed studies are present in the literature on cuprates and precisely on tunneling spectra that can be reproduced by using the framework of d - wave Eliashbeg's theory [24-26]. In this paper, we provide an extensive investigation of the consequences of a different symmetry of coupling in the two components of self-energy: the renormalization function Z(iωn) (s-wave symmetry) and the gap function ∆(iωn) (d-wave symmetry) and if some link exists between them. We focus here on physical quantities which can be evaluated in the imaginary axis formalism. Furthermore, it has been experimentally determined that, in cuprates, a link  exists between magnetic resonance energy Ω0 and critical temperature. So we will study the properties of one band d-wave Eliashbeg's theory where a fundamental role will be played by the assumption that the representative energy Ω0 of these systems is related to the critical temperature by a universal relationship  Ω0 = 5.8 kBTc. This assumption represents a very strong constraint in correlating the values of the two-electron boson coupling constants λd and λs. For each value, λs we will look for the value λd which exactly reproduces the Tc superconductor and we will study which relation exists between the d and s components of the electron boson coupling constant. Finally, we will see that this model has the particular property that the relationship between the gap and the critical temperature (
) is independent of the particular value of the critical temperature.
The one-band d-wave Eliashberg equations [23,30-35] are two coupled equations: one for the gap ∆(iωn,φ) and one for the renormalization functions Z(iωn,φ). These equations, in the imaginary axis representation (here ωn denote the Matsubara frequencies), when the Migdal theorem holds , are:
is the Heaviside function, ωc is cut-off energy and
We assume [2,23,30-35] that the electron boson spectral function
and the Coulomb pseudopotential
at the lowest order contain separated s and d -wave contributions,
as well as the self-energy functions:
We put the factor
inside the definition ∆d(ωn) because, experimentally, the peak of the density of the state is, usually, identified ∆d(ωn=0) while, as we will see, Zd(ωn) is always zero. The spectral functions α2Fs,d(Ω) are normalized in the way that
and of course, in this model the renormalization function is pure s-wave (Z(ωn,φ) = Zs((ωn)) while the gap function is pure d-wave (
). We consider just solutions of the Eliashberg equations in pure d -waveform because this is the indication of the experimental data. This means that the s component of the gap function is zero and this situation happens because, usually ,
). In the more general case, in principle, the gap function has d and s components. The renormalization function
has just the s component because the equation Zd (ωn) is a homogeneous integral equation with just the solution Zd (ωn) = 0 . For simplicity, we also assume that
the spectral functions are the difference between two Lorentzian, i.e.
, C is the normalization constant necessary to obtain
, Ω0 and γ are the peak energy and half-width, respectively. The half-width is =Ω0/2. This choice of the shape of the spectral function and the fact that
, is a good approximation of the true spectral function  connected with antiferromagnetic spin fluctuations. The same thing also happens in the case of iron pnictides . In any case, even making different choices for γ the link between λd and λs) remains the same but changes (very little) the coefficients of the linear fit. The cut-off energy is meV and the maximum quasiparticle energy is ωmax = 1100 meV. In the first approximation, we put
(if the component of the gap is zero the value of
is irrelevant). Now we fix the critical temperature and for any value, λs we seek the value λd that exactly reproduces the initial fixed critical temperature. After, via Padè approximants , we calculate the low-temperature value (T = Tc/10 K) of the gap because, in presence of a strong coupling interaction, the value ∆d(ωn=0) obtained by solving the imaginary-axis Eliashberg equations can be very different from the value ∆d obtained from the real-axis Eliashberg equations .
Results and discussions
We fix three different critical temperatures (70 K, 90 K and 110 K) and for any particular critical temperature, we choose different values λs and determine which value λd exactly reproduces the chosen critical temperature by numerical solution of Eliashberg equations. In Figure 1 we can see that the three curves λd versus λs are coincident. The inset of Figure 1 it is shown the linear fit of these results. We obtain a linear link between λd and λs
These results are general and do not depend on the particular shape of the electron-boson spectral function. If we change the shape of the electron-boson spectral function and we choose, for example,
we find that the linear link between λd and λs changes very little and becomes
. Even the introduction of a Coulomb potential different from zero, as we have verified, does not involve a substantial modification of our results. In principle, it is possible to obtain this result (the linear link between λs and λd) in a more simple but less general way. In fact, a similar conclusion relative to the linear connection between λs and λd may also be derived from the analysis of the approximate MacMillan formula for
 generalized to d-wave case :
The problem is that the MacMillan equation works just in a weak coupling regime. Now we solve, for each couple of λd and λs values, the Eliashberg equations at T = Tc/10 and after, via Pade we calculate the value of superconductive gap (the energy of the density of states peak). In Figure 2 the rates
are shown for three systems with different critical temperatures
(70 K, 90 K and 110 K). The curves are exactly coincidental. We have also studied what happens when the ratio
is equal to two as in the case of the heavy fermion 
with Tc = 2 K which could represent an extreme situation. In this case, the link remains linear and becomes
as it is possible to see in the inset of Figure 2. Finally, in the case of extremely strong coupling (
) it is possible to demonstrate in an analytical way, following the calculus of ref 26, when
, that λd ≈ λs i.e. the link remains linear.
In this article, it has been shown that one band d-wave Eliashbeg's theory presents universal aspects as the linear link between λd and λs or the values
that are independent of the particular critical temperature. These universal aspects are related to the assumption that the typical bosonic energy is correlated to the critical temperature as shown by experimental data (Ω0= 5.8 kBTc). We here proved that in a fully numerical solution of the Eliashberg equation, such linear links hold with great accuracy. A generalization and development of our results can be obtained by explicitly considering the momentum dependence of the self-energy without average on the Fermi surface as was done by Kamila A. Szewczyk, et al. . Obviously, we would include in the calculations, unlike them, as we have done now, the link, observed experimentally, between the critical temperature and the representative energy of the bosonic spectrum.