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The No-Pole Condition in Landau gauge: Properties of the Gribov Ghost Form-Factor and a Constraint on the 2d Gluon Propagator

We study the Landau-gauge Gribov ghost form-factor sigma(p^2) for SU(N) Yang-Mills theories in the d-dimensional case. We find a qualitatively different behavior for d=3,4 w.r.t. d=2. In particular, considering any (sufficiently regular) gluon propagator D(p^2) and the one-loop-corrected ghost propagator G(p^2), we prove in the 2d case that sigma(p^2) blows up in the infrared limit p ->0 as -D(0)\ln(p^2). Thus, for d=2, the no-pole condition \sigma(p^2)<1 (for p^2>0) can be satisfied only if D(0) = 0. On the contrary, in d=3 and 4, sigma(p^2) is finite also if D(0)>0. The same results are obtained by evaluating G(p^2) explicitly at one loop, using fitting forms for D(p^2) that describe well the numerical data of D(p^2) in d=2,3,4 in the SU(2) case. These evaluations also show that, if one considers the coupling constant g^2 as a free parameter, G(p^2) admits a one-parameter family of behaviors (labelled by g^2), in agreement with Boucaud et al. In this case the condition sigma(0)<= 1 implies g^2<= g^2_c, where g^2_c is a 'critical' value. Moreover, a free-like G(p^2) in the infrared limit is obtained for any value of g^2<g^2_c, while for g^2 = g^2_c one finds an infrared-enhanced G(p^2). Finally, we analyze the Dyson-Schwinger equation (DSE) for sigma(p^2) and show that, for infrared-finite ghost-gluon vertices, one can bound sigma(p^2). Using these bounds we find again that only in the d=2 case does one need to impose D(0) = 0 in order to satisfy the no-pole condition. The d=2 result is also supported by an analysis of the DSE using a spectral representation for G(p^2). Thus, if the no-pole condition is imposed, solving the d=2 DSE cannot lead to a massive behavior for D(p^2). These results apply to any Gribov copy inside the so-called first Gribov horizon, i.e. the 2d result D(0) = 0 is not affected by Gribov noise. These findings are also in agreement with lattice data.


I. INTRODUCTION
Green functions of Yang-Mills theories are gauge-dependent quantities. They can, however, be used as a starting point for the evaluation of hadronic observables (see for example [1][2][3][4]). Thus, the study of the infrared (IR) behavior of propagators and vertices is an important step in our understanding of QCD. In particular, the confinement mechanism for color charges [5] could reveal itself in the IR behavior of (some of) these Green functions. This IR behavior should also be relevant for the description of the deconfinement transition and of the deconfined phase of QCD. Indeed, at high temperature color charges are expected to be Debye-screened and the (electric and magnetic) screening masses should be related to the IR behavior of the gluon propagator (see for example [6][7][8]).
Among the gauge-fixing conditions employed in studies of Yang-Mills Green functions, a very popular choice is the Landau gauge, which in momentum space reads p µ A b µ (p) = 0. From the continuum perspective this gauge has various important properties, including its renormalizability, various associated nonrenormalization theorems [9] and a ghost-antighost symmetry [1]. In the past few years many analytic studies of Green functions in Landau gauge have focused on the solution of the Yang-Mills Dyson-Schwinger equations (DSEs), which are the exact quantum equations of motion of the theory (see for example [1,2,10,11]). Since the DSEs are an infinite set of coupled equations, any attempt of solving them requires a truncation scheme. Then, some Green functions (usually the gluon and the ghost propagators) are obtained self-consistently from the considered equations, while all the other Green functions entering the equations are given as an input. For the coupled DSEs of gluon and ghost propagators two solutions have been extensively analyzed (see for example Chapter 10 in Ref. [5] and Ref. [12] for recent short reviews). The scaling solution [1,[13][14][15][16] finds in d = 2, 3 and 4 an IR-enhanced ghost propagator G(p 2 ) and a vanishing gluon propagator D(p 2 ) at zero momentum. 1 In particular, the IR behavior of the two propagators should be given respectively by G(p 2 ) ∼ (p 2 ) −κG−1 and by D(p 2 ) ∼ (p 2 ) 2κD +(2−d)/2 with κ G = κ D ≈ 0.2(d − 1) [14,16]. On the other hand, the massive solution [18][19][20][21][22][23][24][25][26] gives (for d = 3 and 4) a free-like ghost propagator in the IR limit, i.e. κ G = 0, and a massive behavior for the gluon propagator, 2 that is, D(0) > 0 and κ D = (d − 2)/4.
The existence of two different types of solution for the coupled gluon and ghost DSEs is now understood as due to the use of different auxiliary boundary conditions. 3 These conditions can be given in terms of the value of the ghost dressing function F (p 2 ) = p 2 G(p 2 ) at a given momentum scale p [22,31]. In particular, if one considers p = 0, it is clear that 1/F (0) = 0 gives an IR-enhanced ghost propagator G(p 2 ) while 1/F (0) > 0 yields a free-like behavior for G(p 2 ) at small momenta. As stressed in Ref. [22] (see also the discussion in Section 4.2.2 of Ref. [12]), the scaling condition 1/F (0) = 0 relies on a particular cancellation in the ghost DSE which, in turn, implies a specific "critical" value g 2 c for the coupling constant g 2 [12,32]. Thus, at least from the mathematical point of view, there is a oneparameter family of solutions for the gluon and ghost DSEs, labelled by g 2 or, equivalently, by 1/F (0): in the case g 2 = g 2 c one recovers the scaling solution while, for all cases g 2 < g 2 c , the solution is a massive one. 4 In 4d, the SU(3) physical value of the coupling seems to select 5 the massive solution [21,34].
At this point we should recall that, when considering gauge-dependent quantities in non-Abelian gauge theories, one has to deal with the existence of Gribov copies [35] (see also Ref. [36] for a recent review). Indeed, for compact non-Abelian Lie groups defined on the 4-sphere [37] or on the 4-torus [38], it is impossible to find a continuous choice of one (and only one) connection A µ (x) on each gauge orbit. The effect of Gribov copies is not seen in perturbation theory [35], i.e. the usual Faddeev-Popov-quantization procedure is correct at the perturbative level. However, these copies could be relevant at the non-perturbative level, i.e. in studies of the IR properties of Yang-Mills theories.
Different approaches have been proposed in order to quantize a Yang-Mills theory while taking into account the existence of Gribov copies (see for example [35,[39][40][41][42][43][44][45]). The one usually considered, both in the continuum and on the lattice, is based on restricting the functional integration to a subspace of the hyperplane of transverse configurations. The original proposal, made by Gribov [35], was based on the observation that the Landau gauge condition ∂ µ A b µ (x) = 0 allows for (infinitesimally) gauge-equivalent configurations if the Landau-gauge Faddeev-Popov operator M bc (x, y) = −δ(x − y)∂ µ D bc µ has zero modes. (Here D bc µ is the usual covariant derivative.) Indeed, since an infinitesimal gauge transformation δω(x) gives A b µ (x) → A b µ (x) + D bc µ ω c (x), it is clear that the exclusion (in the path-integral measure) of the zero modes of M bc (x, y) implies that gauge copies connected by such infinitesimal gauge transformations are ignored in the computation of expectation values. In order to exclude these zero modes, Gribov considered a stronger condition by requiring that the functional integration be restricted to the region Ω of gauge configurations A b µ (x) defined as This set, known as the (first) Gribov region, clearly includes the vacuum configuration A b µ (x) = 0, for which the Faddeev-Popov operator is given by −δ(x − y)δ bc ∂ 2 µ . The region Ω can also be defined (see for example [46,47]) as the whole set of local minima 6 . Since usually each orbit allows for more than one local minimum of E[A], it is clear that the region Ω is not free of Gribov copies. On the contrary, in the interior of the so-called fundamental modular region Λ, given by the set of the absolute minima of the functional E[A], no Gribov copies occur [43,48].
The characterization of the fundamental modular region Λ, i.e. finding the absolute minima of the energy functional E[A], is a problem similar to the determination of the ground state of a spin glass system [49,50]. Thus, a local formulation of a Yang-Mills theory, with the functional measure delimited to Λ, is not available, whereas a practical way of restricting the physical configuration space to the region Ω was introduced by Gribov [35]. To this end, he required that the ghost dressing function F (p 2 ) cannot have a pole at finite nonzero momenta. After setting this condition can be written as σ(p 2 ) < 1 for where σ(p 2 ) is the so-called Gribov ghost form-factor [35]. Indeed, since the ghost propagator is given by i.e. it is related to the inverse of the Faddeev-Popov matrix M bc (x, y), the above inequality -known as the no-pole condition -should be equivalent to the restriction of the functional integration to the Gribov region Ω, i.e. to the condition M bc (x, y) > 0. From the discussion above, it is clear that both scaling and massive solutions of DSEs satisfy the no-pole condition, i.e. 1/F (p 2 ) = 1 − σ(p 2 ) > 0 for p 2 > 0. Indeed, in the scaling case [1,[13][14][15][16], this condition [together with the condition 1/F (0) = 0] is imposed from the beginning to the solution of the DSEs. On the contrary, for the massive solution, the no-pole condition is either verified a posteriori, as in Ref. [20], or used [together with the condition 1/F (0) > 0] as an input for the solution of the DSEs, as in Refs. [19,25,26]. In particular, in Ref. [26], the value of 1/F (0) is fixed using lattice data.
The restriction to the first Gribov region Ω is also always implemented in lattice numerical simulations of Green functions in Landau gauge by (numerically) finding local minima of the functional E[A]. Results obtained using very large lattice volumes [51][52][53][54] (see also Chapter 10 in Ref. [5], Section 3 in Ref. [12] and Ref. [55] for recent short reviews) have shown that in d = 3 and 4 the gluon propagator D(p 2 ) is finite and nonzero in the limit p → 0 while the ghost propagator G(p 2 ) behaves as 1/p 2 . On the contrary, for d = 2 the lattice data [56][57][58][59] are in quantitative agreement with the scaling solution and one finds κ D = κ G ≈ 0.2.
Since the region Ω is not free of Gribov copies, their (possible) influence on the numerical evaluation of gluon and ghost propagators has been studied by various groups [61][62][63][64][65][66]. It has been found that these effects are usually observable only the IR limit and that any attempt to restrict the functional integration to the fundamental modular region Λ gives a stronger suppression at small momenta for both propagators, i.e. reducing the value of D(0) and increasing that of 1/F (0). More recently, it has been suggested [67][68][69][70] that the one-parameter family of solutions obtained for the gluon and ghost DSEs should be related 7 to Gribov-copy effects and that the value of 1/F (0) could be used as a gauge-fixing parameter. This analysis finds indeed IR-enhanced ghost propagators (and sometimes a disconcerting over-scaling 8 ). On the other hand, the gluon propagator still shows a finite nonzero value at zero momentum, that is, D(0) > 0. Moreover, this approach does not explain why the numerical results found in d = 2 are different from those obtained in d = 3 and 4, even though Gribov copies inside the first Gribov region Ω are clearly present in any space-time dimension d > 1.
From the analytical point of view, following Gribov's approach, Zwanziger modified the usual Yang-Mills action in order to restrict the path integral to the first Gribov region Ω [71] Although this restriction is obtained using a non-local term, the Gribov-Zwanziger (GZ) action 9 can be written as a local action and it is proven [72][73][74] to be renormalizable. At tree level the GZ gluon propagator is given by D(p 2 ) = p 2 /(p 4 + λ 4 ), where λ is a parameter with mass-dimension 1. At the same time, the ghost propagator is given by G(p 2 ) ∼ 1/p 4 . Thus, as in the scaling solution of the gluon and ghost DSEs, the gluon propagator is null at zero momentum 10 and the ghost propagator is IR-enhanced [42]. These tree-level results, also in agreement with the original work by Gribov [35], have been confirmed by one-loop calculations in the three-and four-dimensional cases [77][78][79][80][81].
More recently, the GZ action has been modified by considering (for d = 3 and 4) dimension-two condensates [82][83][84][85]. The corresponding action, called the Refined Gribov-Zwanziger (RGZ) action, still imposes the restriction of the functional integration to the region Ω and it is renormalizable. However, the RGZ action allows for a finite nonzero value of D(0) and for a free-like ghost propagator G(p 2 ) in the IR limit. Thus, nonzero values for these dimension-two condensates yield for the gluon and ghost propagators an IR behavior in agreement with the massive solution of the gluon and ghost DSEs. 11 Indeed, the RGZ tree-level gluon propagator describes well the numerical data in the SU(2) case [59,60], for d = 3 and 4, and in the SU(3) case [88] with d = 4. It is also interesting to note that the fitting values for the dimension-two condensates are very similar for the SU(2) and SU(3) gauge groups in the four-dimensional case.
As stressed above, the restriction of the functional integration to the first Gribov region Ω and the no-pole condition (3) are key ingredients in the study of the IR sector of Yang-Mills theories in Landau gauge. However, to our knowledge, a detailed investigation of the properties of the Gribov form-factor σ(p 2 ) as well as of the possible implications of the no-pole condition was missing up to now, although some interesting one-loop results were already presented in Refs. [84,[89][90][91]. In particular, in Appendix B.2 of [84] it was shown that, if the gluon propagator D(p 2 ) is positive, then in the 2d case the derivative ∂σ(p 2 )/∂p 2 is negative for all values of p 2 , i.e. σ(p 2 ) is largest at p 2 = 0. Also, in Ref. [89] it was proven that in the RGZ framework the form-factor σ(p 2 ) presents a logarithmic IR singularity − ln(p 2 ) for d = 2. This result precluded the use of the RGZ action in the two-dimensional case, leading to a first interpretation of the different behavior found in lattice numerical simulations for the 2d case, compared to the d = 3 and 4 cases. Similar findings have been (more recently) presented in Refs. [33,92,93].
In this work we collect some general properties of the Gribov form-factor σ(p 2 ) and we study the consequences of imposing the no-pole condition. In particular, in Section II, using the expression for σ(p 2 ) obtained from the evaluation of the ghost propagator at one loop, we prove that σ(p 2 ) attains its maximum value at p 2 = 0 for any dimension d ≥ 2. Since this expression for σ(p 2 ) depends on the gluon propagator D(p 2 ), in the same section we also investigate (for a general d-dimensional space-time) which IR behavior of the gluon propagator is necessary in order to satisfy the no-pole condition σ(p 2 ) < 1. By considering a generic (and sufficiently regular) gluon propagator D(p 2 ), we find in the d = 2 case that σ(p 2 ) is unbounded unless the gluon propagator is null at zero momentum. More exactly, we find σ(p 2 ) ∼ −D(0) ln(p 2 ) in the p → 0 limit, in agreement with [89]. This result does not apply to the d = 3 and 4 cases. Indeed, in these cases one can introduce, for all values of p 2 , simple finite upper bounds for the Gribov form-factor. In Section III we present explicit one-loop calculations for σ(p 2 ) using for the gluon propagator D(p 2 ) linear combinations of Yukawa-like propagators (with real and/or complex-conjugate poles), which have been recently used to model lattice data of the gluon propagator in the SU(2) case [59,60]. Besides confirming the results obtained in Section II, we also find that the ghost propagator admits a one-parameter family of behaviors [21] labelled by the coupling constant g 2 , considered as a free parameter. Moreover, the massive solution G(p 2 ) ∼ 1/p 2 , corresponding to σ(0) < 1, is obtained for all values of g 2 smaller than a "critical" value g 2 c . At the "critical" value g 2 c , implying σ(0) = 1, one finds an IR-enhanced ghost propagator. (As already stressed above, the case g 2 > g 2 c corresponds to σ(0) > 1 and one obtains a negative ghost propagator at small momenta.) Finally, in Section IV, we analyze the DSE for σ(p 2 ). We stress that in this case we do not try to solve the DSE but we focus only on general properties of this equation. As we will see, considering IR-finite ghost-gluon vertices, we confirm and extend the one-loop analysis of the no-pole condition presented in Section II. In particular, after introducing bounds for the Gribov form-factor, we show again for d = 2 that the gluon propagator D(p 2 ) must vanish at zero momentum in order to keep σ(p 2 ) finite. On the contrary, such a constraint does not apply in the three-and four-dimensional cases. We also present alternative evidence for the d = 2 result using a spectral representation for the ghost propagator in the DSE.
It is important to note that all our results in Sections II and IV apply irrespective of which set of Gribov copies (inside the region Ω) is considered, i.e. they are not affected by the so-called Gribov noise. We end with our Conclusion in Section V. Some technical details have been collected in four Appendices. In particular, in Appendix B we present properties of the Gauss hypergeometric function 2 F 1 (a, b; c; z) that are relevant to prove some of our results.

II. THE ONE-LOOP-CORRECTED GHOST PROPAGATOR AND THE GRIBOV FORM-FACTOR
In this Section, as well as in Section III below, we consider the one-loop-corrected Landau-gauge ghost propagator, diagrammatically represented in Figure 1. This propagator can be written [for the SU(N c ) gauge group in the d-dimensional case] as where δ ab D(q 2 ) P µν (q) is the tree-level gluon propagator [not necessarily given by D(q) = 1/q 2 ] and P µν (q) = δ µν − q µ q ν /q 2 is the usual projector onto the transverse sub-space, i.e. q µ P µν (q) = 0. We have also considered the tree-level ghost-gluon vertex igf adc p ν , where p is the outgoing ghost momentum. The indices a, d, c refer, respectively, to the incoming ghost, to the gluon and to the outcoming ghost. After using f adc f cdb = −N c δ ab , valid for the adjoint representation, we obtain where σ(p 2 ) is the momentum-dependent function Finally, we can write [as in Eq. (2)] which corresponds to the usual resummation of an infinite set of diagrams into the self-energy. Note that this resummation makes sense only when σ(k 2 ) < 1, i.e. when the no-pole condition (3) is satisfied. Clearly the function σ(p 2 ) is dimensionless and it should go to zero for p → ∞, modulo possible logarithmic corrections. Also, this function coincides with the so-called Gribov ghost form-factor [35,90,91], even though the latter is obtained in a slightly different way. 12 As discussed in the Introduction, the no-pole condition σ(p 2 ) < 1 for p 2 > 0 should be equivalent to the restriction of the path integral to the first Gribov region Ω [defined in Eq. (1)]. In this section we will derive general properties of σ(p 2 ) in d ≥ 2 space-time dimensions. In particular, as we will see below, the weaker condition σ(p 2 ) < +∞ is already sufficient to obtain a strong constraint on the IR behavior of the gluon propagator D(p 2 ) in the d = 2 case.
A. First Derivative of σ(p 2 ) for d = 2 Following Appendix B.2 of Ref. [84] one can show that, if the gluon propagator D(q 2 ) is positive, then in the 2d case and for all values of p 2 we have with σ(p 2 ) defined in Eq. (7). This implies that σ(p 2 ) attains its maximum value at p 2 = 0. To this end, we choose the positive x direction parallel to the external momentum p and write (using polar coordinates) .
The integral in dθ can be evaluated using contour integration on the unit circle and the residue theorem. This yields where dz represents the integral on the unit circle |z| = 1, we indicated with z the complex-conjugate of z = e iθ and θ(x) is the step function. This integral is also evaluated in Eqs. (A17) and (A18) in the Appendix A (for the general d-dimensional case). Considering also Eq. (B8) and Eq. (A5), with d = 2, we find 12 Note that in the Gribov ghost form-factor there is usually an extra factor 1/(N 2 c −1) [90,91] compared to our Eq. (7). However, this is due to the fact that in Eq. (5) above we considered for the Landau-gauge gluon propagator the usual expression D ab µν (q 2 ) = δ ab Pµν (q) D(q 2 ) while, in the derivation of the Gribov ghost form-factor, one usually writes A a µ (q)A a ν (−q) = ω(q 2 )Pµν (q), as in Eq. (255) of Ref. [90].
Then, by using ∂ x θ(x) = δ(x), where δ(x) is the Dirac delta function, the derivative of σ(p 2 ) with respect to p 2 yields which is clearly negative, for any value of p 2 , if D(q 2 ) is positive. We can evaluate the limit p 2 → 0 of this derivative using, for example, the trapezoidal rule 13 One arrives at the same result after writing Eq. (15) as whereD(p 2 ) is a primitive of D(p 2 ), that is,D ′ (p 2 ) = D(p 2 ) where we indicate with ′ the first derivative with respect to the variable p 2 . The limit p 2 → 0 then yields again Eq. (17). Clearly, one finds an IR singularity at p 2 = 0, unless D(0) = 0. If this condition is satisfied, using again the trapezoidal rule, we have from Eq. (17) that For a gluon propagator D(p 2 ) that is regular at small momenta, i.e. that can be expanded as D(p 2 ) ≈ D ′ (0) p 2 + D ′′ (0)p 4 /2 at small p 2 , the above limit is finite. On the other hand, if the leading IR behavior of D(p 2 ) is proportional to p 2η with 1 > η > 0, as found for example in the 2d case in Refs. [14,16,56,59], then the above limit gives a singular value, due to the non-integer-power (and non-analytic) behavior of D(p 2 ).
B. Infrared Singularity of σ(p 2 ) for d = 2 Here we prove that -for d = 2 and for any gluon propagator D(p 2 ) that goes to zero sufficiently fast at large momenta, e.g. as 1/p 2 , and that is reasonably regular at small momenta, e.g. that can be expanded at p = 0 as D(p 2 ) ≈ D(0) + B p 2η + Cp 2ξ (with ξ > η > 0 and D(0), B and C finite) 14 -the ghost form-factor (7) displays a logarithmic divergence for p → 0 proportional to D(0). Indeed, by considering Eq. (12), one obtains 13 The trapezoidal rule gives the numerical approximation which can be obtained by integrating . Thus, the trapezoidal rule is equivalent to using a linear Taylor  where x = q 2 ,D(x) is again a primitive of D(x) and M > 0 is a (finite) constant. If we indicate with H(x) the quantity in square brackets in the last line, then we have Note that for D(x) = 1/(x η + M ) we find H(x) = 1/M = D(0) and the last term in Eq. (24) is zero. Since lim x→∞ D(x) = 0 we also have that lim x→∞ H(x) = D(0) and the two logarithmic singularities for infinite Λ cancel each other. Thus, we get If D(x) ∼ 1/x at large x, it is easy to check 15 that σ(p 2 ) is null for p 2 → ∞, as expected. At the same time, in the limit p 2 → 0 we obtain where we used and H(0) = −M B is a finite constant. 16 Finally, in Appendix C we show that, under the assumptions made for the gluon propagator, 17 the last term on the r.h.s. of Eq. (26) is finite. Thus, the only IR singularity in the ghost form-factor σ(p 2 ) is proportional to −D(0) ln(p 2 ). This result is in qualitative agreement with [89]. An IR singularity plaguing the 2d calculation has also been recently obtained in Ref. [92].
An alternative (equivalent) proof 18 can be done by performing an integration by parts. Then, Eq. (12) becomes where we used the assumption D(x) ∼ 1/x at large x. Note that the above result coincides with Eq. (25) when M = 0, which implies H(x) = D(0) − D(x). A second integration by parts yields Here we used the hypothesis that D ′ (x) goes to zero sufficiently fast at large momenta, e.g. as 1/x 2 . As before, one easily sees that σ(p 2 ) is null for p 2 → ∞ (see Appendix C). At the same time, under the assumptions made for the gluon propagator D(p 2 ), in the limit p 2 → 0 we obtain and we again find 19 an IR singularity proportional to −D(0) ln(p 2 ), unless one has D(0) = 0.
Thus, in the 2d case and using a generic (sufficiently regular) gluon propagator, a null value for D(0) is a necessary condition to obtain a finite value for σ(0) at one loop. As a consequence, the condition D(0) = 0 must be imposed if one wants to satisfy the no-pole condition (3) and keep the functional integration inside the first Gribov region Ω. It is important to stress again that our proofs apply to any Gribov copy inside the first Gribov horizon, i.e. the result D(0) = 0 is not affected by the Gribov noise.

C. Properties of σ(p 2 ) in d Dimensions: Approximate Calculation
We can easily extend the result to the d-dimensional case by using for the integral in d d q the so-called y-max approximation or angular approximation (see for example [13,18,94]). The same approach allows us to show that the IR singularity −D(0) ln(p 2 ) is present only in the two-dimensional case. Indeed, by using hyperspherical coordinates (see Appendix A) and by considering the positive x 1 direction parallel to the external momentum p, we can write the d-dimensional ghost form-factor (7) as In the y-max approximation one substitutes 1/(p − q) 2 with 1/p 2 , for q 2 < p 2 , and with 1/q 2 , for p 2 < q 2 . Then, we obtain 19 In Appendix C we will prove that the integral on the r.h.s. of Eq. (33) is finite under the hypotheses made for the gluon propagator D(p 2 ). In the same Appendix we will also show how these hypotheses can be relaxed in this case.
where we used Eq. (A12). Note that, for d = 2 and using Eq. (A5) one gets the exact result (14). By repeating the argument shown in the Section II A, the proof of the inequality (34) follows directly from Eq. (38). At the same time, we can write Eq. (37) as where the integral I d (p 2 , ℓ) is defined in Eq. (B34). In Appendix B we have also shown that, for d > 2, this integral is finite when the gluon propagator D(p 2 ) is finite and nonzero at p 2 = 0. Thus, using the y-max approximation, we find that only in the 2d case the condition D(0) = 0 is necessary in order to obtain a finite value for the Gribov form-factor σ(p 2 ) for all values of p 2 .
Of course, in case of ultraviolet (UV) divergences we should regularize the integral defining σ(p 2 ), as done for example in Section III C below for the 4d case using the modified minimal subtraction (MS) scheme and dimensional regularization. One can also consider a fixed momentum µ and subtract 20 the value σ(µ 2 ) from the Gribov form-factor σ(p 2 ). Due to the use of the y-max approximation the result of the subtraction is very simple. Indeed, instead of Eq. (39) we have the relation which is valid for p 2 ≤ µ 2 as well as for µ 2 < p 2 . Then, we find again if D(x) is positive. We can also easily check that, for D(0) > 0 and d > 2, the Gribov form-factor σ(p 2 ) in Eq. (40) does not display an IR singularity.

D. Properties of σ(p 2 ) in d Dimensions: Exact Calculation
One can improve the results obtained in the previous Section by considering the formulae reported in Appendices A and B, which allow us to perform the angular integration in Eq. (35) without approximations. Indeed, we have 21 with I(p 2 , ν, d, ℓ) defined in Eq. (B31). Since ν = 1 in this case, for d ≥ 2 we can also make use of the inequalities (B36) and write Note that I d (p 2 , ∞) is the same integral obtained on the right-hand side of Eq. (39). Thus, the y-max approximation of the previous Section provides, for d = 3 and 4, an upper bound for the Gribov ghost factor. On the contrary, for d = 2, the above inequalities become equalities. At the same time, as one can see in Appendix B, the integral we do not need to impose the condition D(0) = 0 in order to attain a finite value for σ(p 2 ) in the IR limit.
By evaluating the derivative with respect to p 2 of the result (B32) we also obtain indicates the derivative with respect to the variable z of the Gauss hypergeometric function 2 F 1 (a, b; c; z) (see Appendix B). Here we used again the properties of the theta and of the Dirac delta functions and Eq. (B6). For d = 2, the last two terms in Eq. (44) are null [see Eq. (B23)] and, using the result (B8), we find again Eq. (15). In the 4d case one can use the expression (B9) for the Gauss hypergeometric function 2 F 1 (1, 1 − d/2; 1 + d/2; z). Then, from Eq. (44) -or, equivalently, by evaluating the derivative with respect to p 2 of Eq. (D1) in Appendix D -we find that where y = q/p and we have used Eq. (A5). For D(p 2 ) > 0 both terms in square brackets are negative, i.e. the derivative ∂σ(p 2 )/∂p 2 is negative for all values of the momentum p. Let us note that in the original work by Gribov [35] the same result was proven [see comment after Eq. (37) in the same reference] under the much stronger hypothesis of a gluon propagator D(q 2 ) decreasing monotonically with q 2 over the main range of integration. A similar analysis can be done in the 3d case using Eq. (B17). In order to simplify the notation we define This gives Then, after setting d = 3 in Eq. (44) and using Eq. (A5), we obtain where we also made the substitution x = q 2 . Next, the change of variable x = yp 2 in the first integral and x = p 2 /y in the second integral yield As one can see in Figure 2, the factor − [ Ψ(y) + y Ψ ′ (y) ] is negative for y ∈ [0, 1]. At the same time, from Eq. (B24) we know that Ψ ′ (y) is negative for y ≥ 0 (see also the corresponding plot in Figure 2). Thus, for a positive gluon propagator D(p 2 ), the 3d derivative ∂σ(p 2 )/∂p 2 is negative for p 2 > 0. Finally, we can consider a general d > 2 and, after suitable changes of variables (for p 2 > 0), we write Note that the dependence on p 2 is only in the global factor p d−4 and in the argument of the gluon propagator. From Appendix B we know that the derivative 2 F ′ 1 (1, 1 − d/2; 1 + d/2; x) is negative for x ∈ [0, 1] and d > 2 and that, under the same hypotheses, the expression in square brackets is positive. Thus, for a positive gluon propagator, both term in the r.h.s. of the above expression are negative and we have proven that, for any dimension d ≥ 2, the Gribov form-factor σ(p 2 ) (at one loop) is monotonically decreasing with p 2 , i.e. it gets its maximum value at p 2 = 0.

III. EVALUATION OF THE ONE-LOOP CORRECTED GHOST PROPAGATOR USING (LINEAR COMBINATIONS OF) YUKAWA-LIKE GLUON PROPAGATORS
In the previous Section we have proven that, at one-loop level and for a sufficiently regular gluon propagator D(p 2 ), the Gribov ghost form-factor σ(p 2 ) is always finite in three and four space-time dimensions while, in d = 2, one needs to impose D(0) = 0 in order to avoid an IR singularity of the type −D(0) ln(p 2 ). In this Section we present an explicit calculation of σ(p 2 ) at one loop for d = 2, 3 and 4 using, for the gluon propagator, results recently presented in Ref. [59,60] from fits to lattice data of D(p 2 ) in the SU(2) case. The expressions obtained below for the ghost propagator G(p 2 ) will be used in a subsequent work [95] to model lattice data of SU(2) ghost propagators.
In this Section, besides recovering the same results reported in Section II, we also find that the ghost propagator G(p 2 ) admits a one-parameter family of behaviors [21,34] labelled by the coupling constant g 2 , considered as a free parameter. The no-pole condition σ(0) ≤ 1 implies g 2 ≤ g 2 c , where g 2 c is a "critical" value. Moreover, for g 2 smaller than g 2 c one has σ(0) < 1 and the ghost propagator shows a free-like behavior in the IR limit, in agreement with the so-called massive solution of gluon and ghost DSEs [18][19][20][21][22][23][24][25]. On the contrary, for g 2 = g 2 c one finds σ(0) = 1 and the ghost propagator is IR enhanced [1,[13][14][15][16].

A. Yukawa-Like Gluon Propagators and Set Up
In Ref. [59,60] the SU(2) gluon propagator was fitted in 2, 3 and 4 space-time dimensions using, respectively, the functions 22 and The last two propagators are tree-level gluon propagators that arise in the study of the RGZ action [82][83][84][85]. The first one is a simple generalization of the form (55) that fits well the 2d data. Note that these three functions can be written as a linear combination of propagators of the type 1/(p 2 + ω 2 ), where ω 2 is in general a complex number. [In the 2d case we need to consider the more general form p η /(p 2 + ω 2 ) with η ≥ 0.] Thus, in order to evaluate σ(p 2 ) in Eq. (7) using the above gluon propagators D(p 2 ), we first consider the integral The evaluation of f (p, ω 2 ) can be done in three and four space-time dimensions by introducing Feynman parameters and applying the usual shift in the momentum q. The integration then yields with Since the Gamma function has the behavior Γ(x) ≈ 1/x for small x, it is clear that the first two integrals are UV finite for d < 4 while the third one is UV finite for d < 6. Below we will calculate the integral (57) for d = 3 and 4. We start from the case d = 3, where all terms are finite, and then we evaluate the integral for the case d = 4, using the MS scheme. On the contrary, as stressed above, in the 2d case one needs to evaluate the more general function with η ≥ 0. This case will be treated (in a slightly different way) in Section III D.
Most of the analytic results reported in this Section have been checked using Mathematica and/or Maple.

B. Ghost Propagator in the 3d Case
In the 3d case the residual x-and y-integrations in Eq. (57) are straightforward and give where the three square brackets highlight the contribution from the three terms in Eq. (57). Here we have only made the assumption p 2 > 0. By simplifying the above result, we find where In order to use the result (61) we need to write the gluon propagator (54) as Here ω 2 1 , ω 2 2 and ω 2 3 are the three roots of the cubic equation, with respect to the variable p 2 , obtained by setting equal to zero the denominator of Eq. (54). Thus, by combining Eqs. (7), (63) and (56) we can write for the function σ(p 2 ) in the 3d case the expression or σ(p 2 ) = g 2 N c 32 π ω 2 p 3 α g(p 2 , ω 2 1 ) + β g(p 2 , ω 2 2 ) + γ g(p 2 , ω 2 3 ) with g(p 2 , ω 2 ) given in Eq. (62). In general, the roots ω 2 1 , ω 2 2 and ω 2 3 are all real or there is one real root, for example ω 2 1 , and two complex-conjugate roots, i.e. (ω 2 2 ) * = ω 2 3 , implying also β = γ * . Since the fits in Ref. [59,60] support the latter case we write and Then, following for example [96], we find for ω 2 2 the relations and similar results for ω 2 3 . We also use the expression (see for example [97]) This allows us to write the function σ(p 2 ) only in term of real quantities, i.e.
where g(p 2 , ω 2 ) is given in Eq. (62) above. Also, we have with and One can check that σ(p 2 ) is null in the limit p → ∞. Finally, by expanding σ(p 2 ) around p 2 = 0 in Eqs. (72)-(81) we find which implies G(p 2 ) ∝ p −2 at very small momenta. However, if the constant term in the above expression is equal to 1/(g 2 N c ), yielding σ(0) = 1, then one gets in the IR limit G(p 2 ) ∝ p −4 or G(p 2 ) ∝ p −3 , depending on whether the term α R 2 + 2 (a v + b w)ω 2 1 vanishes or not. In particular, in the original GZ case, i.e. when the terms containing ω 2 1 and α are absent, we do recover the usual 1/p 4 behavior. Also note that for purely imaginary poles, i.e. when v = b = 0 (and R = w), the condition σ(0) = 1 simplifies to Clearly, for a given value of N c and with a suitable choice of g 2 , one can always set σ(0) = 1 in Eq. (82). For example, using the numerical data in the second column of Table XI of Ref. [59] and N c = 2, we find from Eq. (82) the result 23 σ(p 2 ) 2 g 2 ≈ 0.039(0.001) − 0.017(0.003) p + O p 2 (84) and we have σ(0) = 1 if g 2 ≈ 12.82. Thus, if one considers g 2 as a free parameter, then Eq. (82) gives a one-parameter family of behaviors, labeled by g 2 . For a specific value of g 2 = g 2 c we have σ(0) = 1 and one finds an IR-enhanced ghost propagator at one loop. On the contrary, for g 2 < g 2 c we obtain σ(0) < 1 and G(p 2 ) ∝ p −2 in the IR limit. Finally, for g 2 > g 2 c the no-pole condition σ(0) ≤ 1 is not satisfied, i.e. the ghost propagator is negative in the IR limit. These findings are in qualitative agreement with the DSE results obtained in Refs. [21,34]. Finally, note that at small momenta the function σ(p 2 ) in the above formula (84) is decreasing as p 2 increases, as expected from Section II D.

C. Ghost Propagator in the 4d Case
We want now to evaluate f (p 2 , ω 2 ) in Eq. (57) for d = 4. As stressed above, in this case we have to deal with UV divergences. We do the calculation in the MS renormalization scheme using dimensional regularization with d = 4 − ε.
For the first term in Eq. (57) we have where γ E is the Euler constant. Then, using the usual MS prescription, we find and the dx integration yields −1 16π 2 ln where µ is the renormalization scale. For the second term in Eq. (57), which is also divergent, we first perform the y integration exactly, obtaining The ε expansion then gives where we have already applied the MS prescription, and after integrating in dx we find 1 192p 4 ω 2 π 2 p 4 p 2 + 3ω 2 ln ω 2 p 2 + p 2 + ω 2 3 ln 23 The errors in brackets have been evaluated using a Monte Carlo analysis with 10000 samples (see Ref. [59] for details).
Finally, the third term, which is finite, yields = − 1 96p 4 ω 2 π 2 p 2 ω 2 (p 2 − 2ω 2 ) + 2 p 6 ln ω 2 p 2 + 2 p 6 + ω 6 ln By summing the three results above we ultimately find (in the MS scheme) with As shown in Ref. [59,60], in the 4d case the fit of the gluon-propagator data is done using the expression (55). Thus, in order to use the above result (93)-(96), we need to write the gluon propagator as where ω ± are the roots of the quadratic equation, with respect to the variable p 2 , obtained by setting equal to zero the denominator of Eq. (55). Then, the ghost form-factor in the MS scheme is given by and we have Note that the function σ MS (p 2 ) is real. From [59,60] we know that ω 2 ± are complex-conjugate roots, i.e. ω 2 − = (ω 2 + ) * and α − = α * + . By writing α ± = a ± ib and ω 2 ± = v ± iw we find with and Also note that, at large momenta, one gets Finally, by expanding σ MS (p 2 ) around p 2 = 0 in Eqs. (100)-(112) we obtain Thus, if σ MS (0) = 1 we have that G MS (p 2 ) ∼ 1/p 4 at small momenta (plus logarithmic corrections). Clearly, also in 4d, we obtain a one-parameter family of behaviors, labelled by the value of g 2 , and the IR-enhanced ghost propagator corresponds to the upper value of g 2 allowed by the no-pole condition (3). With the numerical values reported in the second column of Table IV of Ref. [59] and N c = 2 we find 24 σ MS (p 2 ) 2 g 2 = 0.0240(0.0007) + −0.0082(0.0003) + 0.0060(0.0002) ln p 2 p 2 (115) and the condition σ MS (0) = 1 corresponds 25 to g 2 c ≈ 20.83. Note again the negative sign of the leading order corrections at small momenta (see Section II D).

D. Ghost Propagator in the 2d Case
As stressed in Section III A above, Ref. [59] has shown that the fit of the gluon-propagator data in the 2d case can be done using the expression where c is real, α − = α * + , ω 2 − = (ω 2 + ) * and ω ± are the roots of the quadratic equation, with respect to the variable p 2 , obtained by setting equal to zero the denominator of Eq. (53). Thus, in order to evaluate the ghost form-factor σ(p 2 ) we need to consider the function f (p, ω 2 , η), defined in Eq. (59) above. To this end, we can choose again the positive x direction parallel to the external momentum p and consider polar coordinates. Then, after evaluating the angular integral we find valid both for η = 0 and for η > 0.
In the case η = 0 the momentum integration is straightforward giving Note that the second term above blows up logarithmically in the IR limit p → 0, in agreement with the result obtained in Section II B. For η > 0 the second integral in Eq. (117) can be written, after the change of variable t = ω 2 /(q 2 + ω 2 ), as where is the incomplete Beta function, which is defined for a, b > 0 [97], implying 2 > η > 0 in our case. For the first integral in Eq. (117) we cannot use directly the changes of variable v = 1/q and t = 1/(1 + ω 2 v 2 ) because we get an incomplete Beta function (122) with b < 0. In this case it is convenient to introduce a Feynman parameter (using non-integer exponents) and write where we have also done the integration in dq. After suitable changes of variables, the last formula can be written as where B (a, b) = B (1; a, b) = Γ(a) Γ(b)/Γ(a + b) is the Beta function. Thus, by summing the two results above, we find f (p, ω 2 , η) = (ω 2 ) η/2 4π η p 2 Note that for p = 0 the incomplete Beta function B ω 2 /(p 2 + ω 2 ); 1 − η/2, 1 + η/2 becomes the Beta function B 1 − η 2 , 1 + η 2 . Then, by Taylor expanding f (p, ω 2 , η) for small momenta p, we obtain yielding a constant contribution at p = 0. Using the expression (116), the ghost form-factor in the 2d case is given by with f (p 2 , ω 2 ) and f (p 2 , ω 2 , η) defined, respectively, in Eqs. (120) and (127). Of course, the function σ(p 2 ) is real. By writing α ± = a ± ib and ω 2 ± = v ± iw we get for the first two terms above where ℓ 3 (p 2 ), a 2 (p 2 ), a 3 (p 2 ) and R have already been defined in Eqs. (107), (110), (111) and (112) and As shown in Section II B, there is a logarithmic singularity ℓ 5 (p 2 ) ∼ − ln(p 2 ) at small momenta proportional to the gluon propagator at zero momentum, that is, D(0) = 2(av + bw)/R 2 . We also have where we have indicated with ℑ the imaginary part of the expression in square brackets. One can easily check that σ(p 2 ) is null at large momenta. Finally, the results (130) and (132), together with the expressions (128) and (129), allow us to evaluate the behavior of the ghost propagator at small momenta. We obtain Note that, if σ(0) = 1, one finds a ghost propagator with a behavior 1/p 2+η in the IR limit. As in 3d and in 4d we have a one parameter family of solutions labelled by the value of g 2 .
As explained in Ref. [59], the 2d data for the gluon propagator suggest the relations a = −b and v = w, implying av + bw = 0 and R 2 = 2v 2 . Then, we find Using the approximate result η ≈ 1 (see again Ref. [59]) this formula simplifies to On the contrary, for N c = 2 and with the numerical values reported in [59] -see the second column of Table XIV and, for the exponent η, the last line of Table XIII - The coefficient (av + bw)/R 2 ∝ D(0), multiplying the logarithmic IR singularity, is zero within error and we have omitted the corresponding term. Note that σ(p 2 ) decreases for increasing momenta p 2 , as proven in Section II A above. Also note that we have σ(0) = 1 for g 2 c ≈ 17.24 and in this case the ghost propagator behaves as ∼ 1/p 2.9 in the IR limit.

IV. THE GHOST PROPAGATOR BEYOND PERTURBATION THEORY
The one-loop analysis above has shown that, in the 2d case, an IR singularity −D(0) ln(p 2 ) appears in the Gribov form-factor σ(p 2 ) when p 2 → 0. Thus, one needs a null gluon propagator at zero momentum in order to satisfy the no-pole condition σ(0) ≤ 1. On the contrary, for d = 3 and 4, we found that σ(p 2 ) is finite also for D(p 2 ) > 0.
In this section we improve our analysis by considering the DSE for the ghost propagator G(p 2 ) (see for example [13,18,94]). As stressed in the Introduction, here we do not try to solve the ghost propagator DSE, but instead we concentrate on general properties of this equation for different space-time dimensions. In particular, the results obtained in Section II are confirmed by considering a generic (sufficiently regular) gluon propagator D(p 2 ) and an IR-finite ghost-gluon vertex igf adc p λ Γ λν (p, q).

A. The 2d Case
In the 2d Landau gauge the DSE for the ghost propagator is written as where s = p − q, the gluon and the ghost propagators -respectively D(p 2 ) and G(p 2 ) -are full propagators and we indicated with igf adc p λ Γ λν (p, q) the full ghost-gluon vertex. The above result implies if one uses Eq. (8). For a tree-level ghost-gluon vertex Γ λν (p, q) = δ λν and using the transversality of the gluon propagator we finally find which should be compared to the one-loop result (7). As in Section II A above, we can choose the x direction along the external momentum p obtaining (using polar coordinates) with s 2 = p 2 + q 2 − 2 p q cos(θ). This equation will be analyzed below using two different approaches. A first result can, however, be easily obtained using again the y-max approximation, as in Section II C above. This gives us where we have done the angular integration and set x = q 2 . In the limit of small momenta p 2 we then obtain In order to avoid IR singularities in the above equation we have to impose D(p 2 ) ≈ Bp 2η , i.e. the gluon propagator should be null at zero momentum. In particular, if σ(0) < 1, i.e. for a free-like ghost propagator at small momenta, it is sufficient to have η > 0. On the contrary, if the ghost propagator is IR enhanced and 1 − σ(0) ∝ x κ for small x with κ > 0, then the condition η > κ should be satisfied. Note that the predictions of the scaling solution [14][15][16], i.e. η = 0.4 and κ = 0.2, are consistent with the above inequality. The same results can also be obtained by setting p 2 = 0 directly in Eq. (141). This makes the θ integral trivial and gives Note, however, that in both cases we essentially miss the logarithmic IR singularity − ln(p 2 ) which is found below. In the first case this is probably related to the very crude y-max approximation. On the contrary, in Eq. (144), this is due to the (improper) exchange of the q integration with the p 2 → 0 limit [89].

Bounds on the Gribov Form-Factor
Since the Gribov form-factor is non-negative, we can easily construct a lower bound for the l.h.s. of Eq. (141) by writing where we use the definitions (B31), (B34) and the relations (B36). The last integral in the above equation has already been analyzed in Section II B, where it was shown that I 2 (p 2 , ∞) develops an IR singularity proportional to − ln(p 2 ) if D(0) = 0. Thus, σ(p 2 ) also is IR singular, unless D(0) = 0. One can also find an upper bound for σ(p 2 ) and check that the IR singularity is indeed only logarithmic. To this end we can notice that, if σ(0) < 1, one can write 26 where we have also used Eqs. (145) and (145) above. Therefore, the upper bound also blows up as − ln(p 2 ) in the IR limit. At the same time, if σ(0) = 1, with σ(p 2 ) ≈ 1 − cp 2κ at small momenta we find Note that the quantity in square brackets in the last integral is finite at s = 0 if the behavior of σ(p 2 ) is given by 1 − cp 2κ + O(p τ ) with τ ≥ 4κ. Moreover, this quantity goes to 1 at large momenta and its absolute value is clearly bounded from above by some positive constant M if σ(p 2 ) ∈ [0, 1]. Hence, we have For 1/2 > κ we can also use the upper bound in Eq. (B33) and write where M ′′ is a positive constant. Thus, we have again an IR singularity proportional to − ln(p 2 ) if D(0) is not zero. We conclude that σ(p 2 ) can be finite solely if D(0) = 0. Let us remark that the only hypothesis considered in this case is the IR expansion σ(p 2 ) = 1 − cp 2κ + O(p τ ) with 1 > 2κ and τ ≥ 4κ. Also note that the 2d lattice data [56] show for the ghost propagator an IR behavior in good agreement with the so-called scaling solution [14][15][16] that predicts κ = 0.2. Thus, the condition 1 > 2κ is verified in both cases. One can also note that, by considering in Eq. (139) the full ghost-gluon vertex Γ λν (p, q), instead of the tree-level one δ λν , the above results still applies for an IR-finite vertex. This hypothesis is usually adopted in DSE studies of gluon and ghost propagators [12,20,25,31] and it is confirmed by lattice data [98][99][100][101].

Analysis of the Gribov Form-Factor Using a Spectral Representation
In this section we analyze the DSE (141) in an alternative way, also avoiding the y-max approximation. To this end, let us first consider the θ-integral using contour integration. After setting z = e iθ we find where the integral dz is again taken on the unit circle |z| = 1. Clearly, besides the poles at q = k/z and at q = kz in the first denominator on the r.h.s. of the above equation, one has to consider possible divergences in the function In particular, if we assume ghost enhancement, i.e. σ(0) = 1, then f (z) is divergent at z = q/k and at z = k/q. Note that these divergences are not necessarily poles of the function f (z). Indeed, f (z) could display a branch cut in the unit disc or one passing through it. For example, the usual d = 2 DSE scaling solution has G(k 2 ) ∼ 1/(k 2 ) ν in the limit k 2 → 0, where ν is a fractional number. This behavior signals a non-analyticity for G(k 2 ) at the origin and implies a non-analyticity for the function f (z) at z = k/q or at z = q/k. Also, since the ghost is "massless" we should expect that the ghost propagator develops a branch cut along the real axis for k 2 < 0. Then, z = q/k or z = k/q would correspond to branch points of the function f (z), making quite difficult the evaluation of the contour integral in the above expression.
In order to overcome this problem, we make the hypothesis that a spectral representation for the ghost propagator can be introduced, i.e. we write 27 which reproduces the branch cut in G(k 2 ) for k 2 < 0 (see for example [102]). If we assume σ(∞) = 0 and write it is clear that the spectral density ρ(t) must satisfy the normalization condition Also note that the tree-level ghost propagator G(p 2 ) = 1/p 2 corresponds to the spectral density ρ(t) = 2δ(t), where δ(t) is the Dirac delta function. This case will be used below to recover results obtained in the one-loop analysis carried on in Sections II A and II B. In the general case, the spectral density ρ(t) is proportional to the discontinuity of the ghost propagator along the cut. 28 Considering Eqs. (141) and (152) we can write where we indicated with z the complex-conjugate of z = e iθ . Thus, using the representation (156) we can avoid dealing directly with the integral of an unknown function along the branch cut. In exchange, we have in our formulae an extra integration of the (also unknown) spectral density ρ(t). Nevertheless, as we will see below, the above equation will allow us to control the p 2 → 0 limit [at least in the case ρ(t) > 0]. To this end, let us first note that in the contour integral (156) there is a double pole at z = 0 and there are single poles at Since p, q, t ≥ 0 we have that p 2 + q 2 + t ≥ 2pq ≥ 0 and one can check that the pole z − lies within the unit disc while z + lies outside of it. Moreover, for p 2 + q 2 + t = 2pq (which implies t = 0 and p = q) the two poles coincide and we have z ± = 1. It is also easy to check that the residues, inside the unit circle, for the z-integrand are Then, using the residue theorem, we find 27 Since we are working in the d = 2 case, the theory should be UV finite and we do not need to consider renormalization factors here. 28 Note that, if G(k 2 ) has a branch cut along a curve C in the complex plane and if it goes to zero sufficiently fast at infinity, using Cauchy's theorem we could write down an integral relation similar to Eq. (152) with the variable t running over the curve −C, with z ∈ −C ⇔ −z ∈ C. Also, possible poles can be included by adding δ-functions to the spectral density ρ(t) or, equivalently, by pulling the pole terms out of the spectral integral. and we can write the ghost DSE (141) as Note that, for ρ(t) = 2δ(t) and using we find from Eq. (160) the one-loop result (11). Also note that, by Taylor expanding the integrand at p 2 = 0, one finds where we used the definition (152). As shown above [see Eq. (144)], this result can also be obtained immediately by setting p 2 = 0 in Eq. (141). However, as already pointed out below Eq. (144) and in Ref. [89], one should not exchange the q integration and the p 2 → 0 limit. Therefore, in order to properly evaluate σ(p 2 ) for small momenta p 2 , we write Eq. (162) as The first integral can be estimated using the the trapezoidal rule. We then obtain lim p 2 →0 where we used again Eq. (152). For the second integral we define and find where ′ refers to the derivative w.r.t. the x variable and we used the fact that D(x) goes to zero at large x. Note that, in the one-loop case ρ(t) = 2δ(t), we have G(x, p 2 ) = D(x)/(8π) and Eq. (171) becomes in agreement with Eqs. (28) and (30). By collecting the above results we can therefore write We can now verify that the last integral in the above expression is finite. Indeed, we have Then, for large x we find where we used the normalization condition (154), while for small x we have where we used the definition (153). Thus, the integral  to be finite. We can also check that the integral is finite and nonzero if ρ(t) is non-negative. 29 To this end let us first note that the numerator 2p 2 + t − t 2 + 4tp 2 is non-negative since 2p 2 + t ≥ t 2 + 4tp 2 when t, p 2 ≥ 0. Moreover, if we define 29 Our results cannot be easily extended to the general case of a spectral density ρ(t) that is negative for some values of t. However, they apply if one can explicitly verify that the integral in Eq. (178) is indeed finite and nonzero. Also note that we cannot simply consider the limit p → 0 of the integral (178), since the factor multiplying ρ(t) vanishes in this limit and we might erroneously conclude that the above expression is equal to zero. Indeed, already in the tree-level case, i.e. for ρ(t) = 2δ(t), one finds that the integral (178) is non-zero and equal to 2. Finally, one could also formally expand the integrand in powers of p 2 leading to where a b = Γ(a + 1) is the usual binomial coefficient. However, this series is not converging for all values of t ∈ [0, +∞).
Moreover, for n = 2 we have a term proportional to and this integral is divergent. Indeed, by (formally) setting it is clear that 2 = Φ(0, p 2 ) ≥ Φ(t, p 2 ) ≥ 0, since the quantity t − t 2 + 4tp 2 is negative for t > 0. This implies where we used again the normalization condition (154). At the same time we can write where we use the fact that the function (t + p 2 )Φ(t, p 2 )/p 2 is positive and gets its minimum value, equal to 3/2, for t/p 2 = 1/2. Then, using the definition (153) and the condition σ(p 2 ) ≥ 0, we can write From the above results we conclude that in Eq. (173) we have two possible IR singularities, i.e. the term D(0)/[1 − σ(p 2 )], if σ(0) = 1, and the term proportional to −D(p 2 ) ln(p 2 ). In both cases we need to impose the condition D(0) = 0 in order to avoid the singularity. Thus, we find again that a massive gluon propagator in the d = 2 case is not compatible with the restriction of the functional integration to the first Gribov region.

B. The 3d Case
In the 3d case we expect no UV divergences when using dimensional regularization 30 and the DSE for the Gribov form-factor is simply where we used the tree-level ghost-gluon vertex Γ λν (p, q) = δ λν and s 2 = p 2 + q 2 − 2 p q cos(φ 1 ). We can now work as in Section IV A 1 and use the results of Appendix B. In this way we obtain the upper bounds if σ(p 2 ) ≤ σ(0) < 1, and if σ(p 2 ) ≤ σ(0) = 1 with σ(p 2 ) ≈ 1 − cp 2κ + O(p τ ) for small p 2 . In the latter case we also need the conditions 1 > κ and τ ≥ 4κ. As we saw in Eq. (B35), under simple assumptions for the gluon propagator D(q 2 ), the integral I 3 (p 2 , ∞) is finite in the IR limit p → 0. Thus, in both cases the upper bound of σ(p 2 ) is also finite 31 and, in order to have a finite value for σ(0) in the 3d case we do not need to set D(0) = 0. This result also applies when an IR-finite ghost-gluon vertex is included in the ghost DSE (186). Let us also note that the scaling solution predicts in the 3d case [14][15][16] a value κ ≈ 0.4 for which the condition 1 > κ > 0 is satisfied. 30 As shown in Section III B above at one-loop level, the evaluation of the ghost propagator in 3d usually involves Gamma functions with half-integer arguments, which do not generate infinities. Indeed, for nonnegative values of n with n integer, one has [97] Γ (n + 1/2) = √ π 2 −n (2n − 1)!! and Γ (−n + 1/2) = (−2) n √ π/ (2n − 1)!! , where n!! denotes the double factorial. 31 Using the fact that σ(p 2 ) is nonnegative and Eqs. (B31) and (B36), the lower bound C. The 4d Case In 4d the DSE for σ(p 2 ) is given by (see for example [94]) where Z 3 and Z 1 are the renormalization constants for the ghost propagator and the ghost-gluon vertex respectively 32 and s 2 = p 2 + q 2 − 2 p q cos(φ 1 ). In order to eliminate these constants from the expression for σ(p 2 ) we can subtract 33 the same equation for some fixed value p 2 = µ 2 and set Z 1 = 1, using the non-renormalization of the ghost-gluon vertex in Landau gauge [103]. This gives with t 2 = µ 2 + q 2 − 2 µ q cos(φ 1 ). Clearly, considering D(q 2 ) ∼ 1/q 2 at large momenta, the UV (logarithmic) divergence of the two integrals cancels out. In order to derive upper bounds for the above expression without spoiling the cancellation of UV divergences and since we are interested in the IR limit p → 0, we can consider a momentum scale ℓ sufficiently large and write .
For small momenta p only the first integral on the r.h.s. of the above equation can produce an IR singularity. Following the analysis presented in the 3d case above we can then write if σ(p 2 ) ≤ σ(0) < 1, and if σ(p 2 ) = 1 − cp 2κ + O(p τ ) with τ ≥ 4κ and 3/2 > κ. Again, thanks to the result (B35), both upper bounds are finite in the IR limit p → 0 also for D(0) > 0. An alternative proof can be given by working directly with Eq. (191) and using dimensional regularization, i.e. considering a dimension d = 4 − ε. In this case we can write .
(197) 32 Note that we are again considering a tree-level ghost-gluon vertex Γ λν (p, q) = δ λν . 33 Again, this corresponds to a MOM scheme with the condition G(µ 2 ) = 1/µ 2 . 34 Of course, with d = 4 − ε and ε > 0, the integral in Eq. (197) is no longer dimensionless. To keep the dimensionality correct we should, as always, scale out a dimensional factor m 4−d where m is a mass scale, which could then be combined with the coupling constant g 2 , making σ(p 2 ) dimensionless also for ε > 0. This is important when evaluating the ε expansion in order to single out UV divergencies. Since here we are mainly interested in the IR behavior of σ(p 2 ), we do not keep track explicitly of all the terms depending on ε and we simply consider the coupling g 2 dimensionful.
Then, if we can show that no IR singularities occur for 2 < d ≤ 4, the UV infinity that appears for d → 4 is taken care of by the renormalization factors. In order to show that σ d (p 2 ) is IR finite we can work as done above and write or depending on the value of σ d (0). In the latter case we considered again the IR expansion σ d (p 2 ) ≈ 1 − cp 2κ + O(p τ ) and the conditions τ ≥ 4κ and 3/2 > κ . We conclude that also in the 4d case, σ(0) is finite if D(0) is also finite (but not necessarily null).
Let us remark that the IR exponent usually obtained in the scaling solution [14][15][16] is κ ≈ 0.6 in the 4d case, i.e. the condition κ < 3/2 is satisfied. Also note that when (d − 1)/2 ≤ κ the hypergeometric function is not convergent at z = 1 and we cannot use the above proofs in order to derive properties of the Gribov form-factor. However, these large values of κ imply for the ghost propagator G(p 2 ) a very strong IR enhancement with a behavior at least as singular as 1/k 5 in 4d and at least as singular as 1/k 4 for d = 3.

V. CONCLUSION
Summarizing, in this manuscript we have considered general properties of the Landau-gauge Gribov ghost formfactor σ(p 2 ) for SU(N c ) Euclidean Yang-Mills theories in d ≥ 2 space-time dimensions. This form-factor is in a one-to-one correspondence with the ghost propagator G(p 2 ) via Eq. (2). Also, σ(p 2 ) is bounded by 1 if the no-pole condition (3) is imposed, i.e. if one restricts the functional integration to the first Gribov region Ω. The main result of this work is an exact proof of qualitatively different behavior of σ(p 2 ) for d = 3, 4 with respect to d = 2. In particular, for d = 2, the gluon propagator D(p 2 ) needs to vanish at zero momentum in order to avoid in σ(p 2 ) an IR singularity proportional to −D(0) ln(p 2 ). On the contrary, for d = 3 and 4, an IR-finite ghost form-factor σ(p 2 ) is obtained also when D(0) > 0. These results were proven, in Section II, using perturbation theory at one loop and, in Section IV, by considering the DSE for the ghost propagator. Let us stress again that in DSE studies of correlation functions in minimal Landau gauge, besides using the no-pole condition, a specific boundary condition is usually imposed on the Gribov ghost form-factor at zero momentum. Here, instead, we have tried to prove general properties of the Gribov ghost form-factor σ(p 2 ) when the restriction to the first Gribov horizon is considered.
At the same time, in Section III, we have presented closed analytic expressions for the Gribov form-factor σ(p 2 ) at one loop, considering for the gluon propagator linear combinations of Yukawa-like propagators (with real and/or complex-conjugate poles). These functional forms, briefly described in Eqs. (53)-(55), were recently used to fit lattice data of the gluon propagator in the SU(2) case [59,60]. The expressions obtained for σ(p 2 ) confirm the results presented in Section II. These expressions also show that, for the ghost propagator G(p 2 ), there is a one-parameter family of behaviors [21,34] labelled by the coupling constant g 2 , when it is considered as a free parameter. The no-pole condition σ(0) ≤ 1 then implies g 2 ≤ g 2 c , where g 2 c is a "critical" value. For g 2 smaller than g 2 c one has σ(0) < 1 and the ghost propagator is a massive one. On the contrary, at the "critical" value g 2 c , i.e. for σ(0) = 1, one finds an IR-enhanced ghost propagator. As stressed in the Introduction, the physical value of the coupling is expected to select the actual value of σ(0). Present results [21,34] give σ(0) < 1 in the four-dimensional SU(3) case.
Our findings imply that a massive gluon propagator cannot be obtained in the two-dimensional case, in disagreement with some of the results presented in Ref. [16] (see their Table 2). 35 A possible massive behavior for the gluon propagator in the 2d case was also explicitly conjectured in Ref. [65] as a Gribov-copy effect. However, since our 2d result is valid for any Gribov copy inside the first Gribov region, we have shown that, at least for the 2d gluon propagator D(p 2 ) in the minimal Landau gauge, Gribov-copy effects do not alter our conclusion for the value of the gluon propagator at zero momentum, i.e. D(0) must vanish. This observation also represents an explicit counterexample to the identification of the one-parameter family of solutions for the gluon and ghost DSEs [22,31] with different Gribov copies, as suggested in [65,67,68]. where B (a, b) = Γ(a) Γ(b)/Γ(a + b) is the Beta function.
In our calculations we consider two different functions f (φ 1 ), i.e.