Equivalence between Zwanziger's horizon function and Gribov's no-pole ghost form factor

The ghost form factor entering the Gribov no-pole condition is evaluated till the third order in the gauge fields. The resulting expression turns out to coincide with Zwanziger's horizon function implementing the restriction to the Gribov region in the functional integral.


Introduction
In his seminal work [1], Gribov pointed out that the Landau gauge condition ∂ µ A a µ = 0 is plagued by the existence of gauge copies, i.e. there exist equivalent configurations A ′ µ = U A µ U −1 + iU ∂ µ U −1 which still obey the condition, ∂ µ A ′a µ = 0. As a consequence, the Landau gauge does not enable us to pick up a unique field representative for each gauge orbit 1 .
In order to get rid of the gauge copies, Gribov proposed [1] to restrict the domain of integration in the Feynman path integral to a certain region Ω, defined as the set of field configurations obeying the Landau condition and for which the Faddeev-Popov operator M ab , M ab = −(∂ 2 δ ab − gf abc A c µ ∂ µ ), is strictly positive, namely The boundary ∂Ω of the region Ω, where the first vanishing eigenvalue of the operator M ab appears, is called the first Gribov horizon. One has to note that, within the region Ω, the operator M ab is strictly positive, so that its inverse (M −1 ) ab does exist.
To restrict the domain of integration in the functional integral, Gribov worked out the so called nopole condition [1] for the ghost propagator, which is the inverse of the operator M ab , namely where the gauge field A a µ plays the role of an external classical field. Expression (2) can be represented in a functional form by means of the Faddeev-Popov ghosts According to [1], one introduces the ghost form factor σ(k, A) in momentum space as where G(k; A) is obtained by taking the Fourier transform of the trace of G ab (x, y; A), i.e. and G(x, y; A) = T r G ab (x, y; A) = Before starting the evaluation of the form factor σ(k, A) it is worth to point out that expression (4) can be obtained as the Fourier transform of the quantity where i.e.
This property 2 can be obtained from eq.(5) by performing the change of variables (z = x − y, y = y), amounting to rewrite eq.(5) as Finally, eq.(9) follows by observing that a translation of both arguments of G(z + y, y; A) by y is the same as a translation of the field configuration A a µ by y, as it can be checked term by term by looking at the expressions given in the next sections, see for example eq.(66).
As σ(k, A) turns out to be a decreasing function of the momentum k [1], Gribov required the validity of the condition which is known as the no-pole condition. From condition (11) it follows that the ghost propagator has no poles at finite values of the momentum k. Therefore, expression (4) stays always positive, meaning that the Gribov horizon ∂Ω is never crossed. The only allowed pole is at k = 0, whose meaning is that of approaching the horizon ∂Ω, where the ghost propagator is singular, due to the appearance of zero modes of the operator M ab . According to the no-pole prescription, eq.(11), the Faddeev-Popov quantization formula gets modified as [1] where S Y M is the Yang-Mills action and θ(x) stands for the step function. Making use of the integral representation it turns out that the ghost form factor σ(0, A) can be brought into the exponential of the Yang-Mills We see thus that the Yang-Mills action gets modified by the addition of the factor σ(0, A). Therefore, for the partition function Z, one writes Further, the integration over β was evaluated by a saddle point approximation [1], yielding with β * determined by the gap equation Independently, Zwanziger [3,4] implemented the restriction to the Gribov region Ω by following a different route, based on the study of the smallest eigenvalue, λ min (A), of the Faddeev-Popov operator. Relying on the equivalence between the canonical and microcanonical ensembles in the infinite volume limit, he was able to show that the restriction to the Gribov region can be achieved by adding to the Yang-Mills action a nonlocal term S h , known as the horizon function [3,4], namely The resulting partition function cut-off at the Gribov horizon turns out to be where the massive parameter γ is a dynamical parameter determined in a self-consistent way through the horizon condition [3,4] h To the first order, condition (21) reads from which one sees that, apart from a numerical coefficient, Although both Gribov's no pole condition (11) and Zwanziger's construction of the horizon function S h amount to modify the Faddeev-Popov functional measure, a discussion about the equivalence between the ghost form factor σ(0, A) and the horizon function S h has not yet been worked out. The present work aims at filling this gap. We shall evaluate the form factor σ(0, A) till the third order in the gauge fields A a µ . The resulting expression will be thus compared with that obtained by expanding the horizon function S h , hence establishing the equivalence between σ(0, A) and S h till the third order in the gauge field expansion.
The paper is organized as follows. In Sect.2 we evaluate Gribov's ghost form factor σ(0, A). In Sect.3 we expand the horizon function S h by comparing it with σ(0, A). Sect.4 is devoted to a few concluding remarks.

Evaluation of Gribov's ghost form factor
The evaluation of the ghost form factor σ(k, A) will be performed order by order in the gauge field A a µ . As we shall evaluate σ to the third order, we write where σ (1) , σ (2) , σ (3) stand, respectively, for the first, second and third order expansion of σ in powers of the gauge fields. Therefore, from the no-pole condition (11), we get Let us start thus by considering the expression of G(x, y; A) in an external background gauge field A a µ , obtained by taking the trace over the color indices of expression (3), namely In order to evaluate G(x, y; A) till the third order in the gauge field A a µ , we consider To the zeroth order approximation, it turns out that

First order
At first order we have Using we obtain Moreover, due to it follows that G (1) vanishes identically so that

Second order
Performing Wick contractions and using eq.(29), one obtains Taking the Fourier transformation of the expression above it follows where we have used the property we obtain which can be rewritten as Therefore, till the second order, for the no-pole condition we get where Owing to the transversality of the gauge field A a µ (q) we can set Thus Following [1] 3 , it turns out that so that for the Gribov no-pole form factor σ (2) , one obtains

3 • order
To the third order Performing all possible Wick contractions and proceeding as in the case of G (2) , one finds where we have defined Taking the Fouries transformation and using the transversality condition q µ A a µ (q) = 0, one gets Proceeding as in the previous case, for the Gribov ghost form factor till the third order we find where use has been made of

Expansion of the horizon function
In order to make a comparison between Gribov's ghost form factor σ(0, A) and Zwanziger's horizon function S h , we need to expand the expression till the third oder in the gauge field A a µ . To that end we evaluate the inverse of the Faddeev-Popov operator M −1 , which is equivalent to solve the problem where the Green function G ab (x, y) is evaluated as a series in the coupling constant 4 g, i.e. where Thus, at first order, we get which gives Therefore Consequently, till the third order in the gauge fields A a µ , for the horizon function we obtain Finally, moving to the Fourier space, (69) Recalling now the expression for the ghost form factor σ(0, k) till the third order, namely (70) 4 Notice that an expansion in g is equivalent to an expansion in the gauge field A a µ . and making use of it is apparent that, apart from a global factor, the expression of the Gribov ghost factor σ(0, A) coincides with that obtained by expanding the horizon function till the same order 5 , i.e.

Conclusion
In this work the equivalence between Gribov's ghost form factor σ(0, A) and Zwanziger's horizon function S h has been investigated. The form factor σ(0, A) has been evaluated till the third order in the gauge fields A a µ and proven to be equivalent with the horizon function S h , as expressed by eq.(72). Our result can be interpreted as a strong indication of the fact that Zwanziger's horizon function S h is an all orders resummation of Gribov's form factor σ(0, A).
Let us conclude by mentioning that, although being non-local, the horizon function S h can be cast in local form by means of the introduction of a suitable set of auxilairy fields. Remarkably, the resulting action turns out to be renormalizable to all orders [3,4,6,7,8,9,10].