Of course there are relations between the correlation functions of a quantum field theory-not, necessarily, of the form stated, however. They can be worked out by calculating what happens to the expression of a correlation function, upon performing a change of variables in the path integral. One example of such relations is given by the Slavnov-Taylor identities, but it isn't the only one.

@Stam Nicolis:

Can you pls elaborate your explanation?  What other kind of relations exist other than 

Slavnov-Taylor identities?  can you provide a simple example how exactly these

relations can be calculated in path intergral?

Any change of variables in the path integral leads to relations among the correlation functions. If this change of variables  leaves the action invariant, they are called Slavnov-Taylor identities, when the transformation is a non-abelian gauge transformation. Such relations, in general, are, sometimes, called Schwinger-Dyson equations and are the foundation of the Monte Carlo renormalization group.  

@Stam Nicolis:

I have come across a gauge condition in which the gluon propagator remains singular, hence pertubation can't be done.  But it seems to be resolving a non perturbative issue- gribov ambiguity with strong evidences of the confinement. 

What would you comment on the "usefulness" of the gauge?  

A gauge condition is useful in order to resolve the overcounting of configurations when taking into account off-shell states. That's all. Once this is shown, the focus is on gauge-invariant quantities. So these are the issues that are relevant when discussing a gauge condition-it's like choosing a coordinate system in the space of field configurations-some choices are more convenient than others and singularities must be resolved but, apart from that, one choice is as good as any other, provided one can get through to quantities that shouldn't depend on this choice. The information provided is much too vague to say anything more. 

The gauge condition is given in the link below:

10.1103/PhysRevD.91.085028

The paper can, also, be found in http://arxiv.org/pdf/1408.1309v1.pdf . The gluon propagator obtained, cf. eq.(7), decreases like 1/p^2 at large p, so is OK-it's the ghosts, deduced from the gauge-fixing condition, that are non-conventional. However it's not obvious whether what they claim as physical is,  indeed, gauge invariant, i.e. independent of the gauge-fixing condition they've imposed. They *assume* some condensation mechanism at work, that, however, involves ghosts, i.e. unphysical degrees of freedom, so it's not obvious that the consequences can be trusted to have physical effects. A mass term for the gluons would break gauge invariance and, if it's used, as here, as a gauge fixing condition, this means that the mass can't be physical, any more than the coefficient of any gauge fixing term is. Indeed, they find that the mass term is purely imaginary; but this, still, doesn't mean anything, by itself and seems strange, in view of the fact that they show, apparently, that the Lagrangian is hermitian.  The framework is that of the classical Lagrangian, they don't calculate any quantum corrections, either perturbatively, or on the lattice, in any approximation, e.g. mean-field.  They've just studied some consequences of their gauge-fixing condition, under the assumption that the ghosts behave in a certain way. It's by no means mandatory that the ghosts should do so, however.