The three-gluon vertex was studied in various approaches, ranging from lattice simulations [58–65] to effective models using a massive gluon propagator [49] to functional equations [22,44,45,51,66–70]. From these studies, the nonperturbative properties for spacelike momenta are quite well understood. In particular, a suppression at intermediate momenta is seen, and the dressing of the tree-level tensor structure even becomes negative. However, the zero crossing is at rather low momenta which are difficult to reach in lattice calculations.

Here we use a model for the three-gluon vertex which also includes a term that restores the correct renormalization group behavior of the gluon propagator. Note that this is only necessary because we discarded the two-loop terms in the gluon propagator DSE. For more elaborate truncations, the correct renormalization group behavior is obtained automatically; see Refs. [22,54,71]. The vertex model is restricted to the tree-level tensor and parametrized as ΓμνρAAA,abc(p,q,r)=igfabcΓμνρAAA,(0)(p,q,r)C˜AAA(p2,q2,r2),(6)where ΓμνρAAA,(0)(p,q,r) is the Lorentz tensor of the tree-level vertex.

We tested various forms for C˜AAA(p2,q2,r2) which are detailed in Appendix C. As the results for the gluon and ghost propagators are qualitatively very similar for these models, we choose one representative for illustration in plots. This model reads C˜1AAA(x,y,z)=1Z1G(p¯2)2-2a/δ-4aZ(p¯2)2+2a.(7)Z1 is the renormalization constant of the three-gluon vertex, δ=-9/44 is the anomalous dimension of the ghost propagator and a is a parameter that determines the IR behavior of the model. Note that in the UV a drops out due to the scaling relation 1+2δ+γ=0 of the anomalous dimensions δ and γ of the ghost and gluon propagators, respectively. The vertex model is a reparametrization of the model introduced in Ref. [47] without the IR part and corresponds to a Bose-symmetrized version of the model from Ref. [55]. The non-Bose-symmetric version is recovered by replacing the dressings as G(p¯2)2→G(y)G(z) and Z(p¯2)2→Z(y)Z(z). This version was used in Ref. [28] and also tested here. Again, though, we did not find any qualitative differences.

The IR behavior of this model is determined by the parameter a. Originally, a=3δ was used which makes the expression IR finite for the scaling solution. Here, we solve for a decoupling solution for which C˜1AAA(x,y,z) is IR divergent with a=3δ.³

Functional equations allow for a family of solutions called decoupling solutions [44,72–74]. Their end point is called the scaling solution [75,76] for which the dressing functions obey simple power laws in the IR [44,74,77,78].

In this model, the analytic behavior of the three-gluon vertex is completely determined by the analytic structure of the propagator dressing functions. As we found that for the given truncation the gluon propagator seems to have a singular point in the complex plane, we wanted to remove its influence on the vertex. The corresponding models are discussed in Appendix C. While these adaptations do have quantitative effects, all qualitative aspects remain the same. Thus, we conclude that our qualitative results are robust against changes within a large class of three-gluon vertex models, but we stress that our analysis only holds for this class.

The propagator DSEs are renormalized via a momentum-subtraction scheme. For the ghost propagator DSE this leads to G(p2)-1=G(s)-1+ΣG(p2)-ΣG(s).(8)G(s) denotes the ghost dressing function at an (infrared) subtraction scale s and ΣG(p2) is the ghost self-energy. The value chosen for G(s) selects a particular solution from a one-parameter family of possible ones; for details see Ref. [74]. To be able to perform the subtraction ray by ray we need to analytically continue the value of G(s) from ray to ray. We do so using the Cauchy-Riemann condition; see Eq. (10) below and Appendix A for details.

For the gluon propagator DSE, the subtraction point needs to be chosen at large momenta for numerical stability. The hard UV cutoff employed in our calculations breaks gauge covariance and leads to additional quadratic divergences in the gluon propagator DSE. Several methods to remove them exist; see Ref. [79] and references therein for details. Such a procedure can entail ambiguities and should thus be considered as part of the model input. However, a unique subtraction might be possible with more elaborate truncations [22]. Since the present truncation scheme is constructed for qualitative tests, we use a simple but effective subtraction scheme that modifies the integrand of the gluon loop appropriately [55,80]. We also tested other methods. For one, instead of subtracting the overall divergence in the gluon loop, we split the subtraction between ghost and gluon loops [47]. For another one, we used a second renormalization condition [71,81] which already proved very useful elsewhere [22]. It turns out that the second renormalization condition can be chosen such that both procedures lead to almost similar results.

Finally, it remains to set the physical scale of our results. We do so by matching the gluon dressing function to corresponding lattice results [59] in the region above 1 GeV. Thus, we inherit the scale setting used on the lattice.

In this section we first present results for what we call the “baseline setup.” It is defined by fixed values for the renormalization conditions, the subtraction method of quadratic divergences in the gluon loop and the three-gluon vertex model from Eq. (7) with a=3δ. More details including computational parameters are explained in Appendix A. Variations and tests of this setup are presented in the subsequent subsections.

The gluon dressing function is shown in Fig. 5 as a function of the radial and angular parts of the complex variable x=p2=p˜2eiθ. In the first quadrant, the calculation works without problems. However, in the second quadrant, the bump in the gluon dressing function at p˜2≈0.5 starts to rise appreciably, however, without becoming singular. Beyond a certain value of θ, it flattens again and remains finite until the negative squared momentum axis. This behavior was seen before in a slightly different setup and with less precise numerics [28] and was interpreted as the gluon being regular at complex momenta. However, with the improved numerical treatment followed in this work, we clearly observe oscillations in the solution, which may hint towards numerical artifacts. These signals and the strong rise could also indicate that we may have hit a singular point beyond which the ray technique is no longer applicable. We tried to take this finding into account by modifying the integration path appropriately (cf. the discussion in Sec. II), but then we loose the advantage of the ray technique that we do not need to know the dressing function in unknown regions, and we did not succeed in improving the results in this way. Thus, from this plot we concluded that our results probably cannot be trusted beyond that point and that we may have hit a singularity. We corroborated this conclusion further using various tests that will be detailed below in Sec. IV B.

Our results for the ghost dressing function in the baseline setup are shown in Fig. 6. In contrast to the gluon propagator, we do not see any drastic changes in the dressing functions. The real part of the ghost is smooth throughout the complex momentum plane, whereas the imaginary part develops a negative bump at the same scale at which the gluon rises drastically. However, since the ghost and the gluon equations are directly coupled, the results for the ghost dressing should only be considered trustworthy up to the location of the potentially singular point in the gluon dressing function.

The reason for the appearance of the potential singularity is not clear. In particular, it cannot be understood in simple terms similar to the Cutkosky rules [82,83] which allow to determine the position of a branch cut from the masses of the propagators in a Feynman diagram. In the language of contour deformation, a branch cut in the external momentum arises when the integration path cannot be deformed continuously for two values of p2; see Ref. [36] for details. This is, however, not the case here.

Since the origin of the potential singularity thus seems to be dynamic, we tried to vary our setup to investigate the influence on the existence and position of the singular point. We attempted the following variations of the baseline setup: (1)

We tried several modifications of the three-gluon vertex model. The corresponding expressions are listed in Appendix C. In addition, we varied the parameter a in the model between 8δ≈-1.61 and 3δ≈-0.61. The effect of this is discussed below.

Most variations did not lead to a qualitative change of our results and we do not discuss them further. Different values for G(0) only led to small quantitative changes in the position of the singularity. A larger effect was observed from the employed three-gluon vertex model. In particular, the position of the potentially singular point in the complex momentum plane for the gluon propagator depends on the details of the vertex model. Varying the parameter a, cf. Fig. 7, we observed that the point moves closer to the real axis (and at the same time closer to the origin) when the parameter a was lowered. Lowering a as far as a=8δ≈-1.61, we were able to move the potential singularity almost onto the real axis. This, however, happens at the expense that the gluon dressing function at real spacelike momenta becomes unrealistically flat. We therefore did not lower a further.

A propagator can be calculated in the complete complex plane from the spectral density ρ(s) via D(p2)=∫0∞dsρ(s)p2+s(9)if no poles at complex momenta exist; see also Appendix B. The inverse task of extracting the spectral density from the propagator given at Euclidean momenta is an ill-posed inverse problem [84]. This is reflected in the necessity of some form of bias in these methods and a large sensitivity of the results to the precision of the input. The direct calculation performed here, on the other hand, does not have these intrinsic problems. Rather, the main challenges are of a numeric nature; see Appendix A. In particular, the global nature of analyticity can be problematic, viz., the analytic properties of a function are encoded in the behavior of the function on any region of the complex plane. Hence, the propagation of errors has to be under control and it is important to check that the numeric calculation does not interfere with analyticity. In this section, we describe several possibilities for such checks and apply them to our results.

One direct possibility for such a check is provided by Cauchy’s integral formula. We use it to reconstruct the propagators for real and spacelike momenta from any ray and monitor the quality of the reconstruction. The details for this procedure are listed in Appendix B.

We performed this reconstruction on all rays used for the calculation of the gluon propagator and ghost dressing function. For the baseline setup, the reconstructed functions are shown in Fig. 8. For guidance, we also plot the Euclidean result (in red at θ=0). The reconstruction works very well in the first quadrant (θ≤π/2) and some way into the second quadrant until it fails on the ray marked with a thick black line. Beyond this ray, it fails completely. Again, this indicates the appearance of a singularity in the complex plane at or around the ray in black.

Another possibility for a check uses the Cauchy-Riemann equations which relate the real and imaginary parts of analytic functions. In polar coordinates they can be written as ∂f(x)∂r=1ir∂f(x)∂θ(10)where x=reiθ. We can use this to test the analyticity of our results by monitoring ΔCR[f](x)=∂f(x)∂r-1ir∂f(x)∂θ.(11)We approximate the derivatives by finite differences which leads to its numeric deviations on its own. The results are shown in Fig. 9. Again, we find small deviations for the ghost and gluon propagators (cf. the scale of the z axis) up to the singular point in the second quadrant beyond which the Cauchy-Riemann equations are clearly no longer fulfilled. The two ridges are artifacts from splitting the grid for the dressing functions as discussed in Appendix A.

Having collected ample evidence that our results are analytic until we hit a potential singularity in the second quadrant, we turn the situation around and test the reliability of extrapolating the results from the spacelike axis using the analytic continuation from a Padé approximant. Specifically, we use the Schlessinger point method [85] which provides a pointwise exact description of the data. As input, we use a random subset of momentum points on the positive and real momentum axis. Note that this test has been performed already in Ref. [19] for one of the truncations that we also used here. There, indeed a singularity in the second quadrant was found. Here, we check whether this finding persists for all truncations considered in this work and, even more important, whether the results from the Schlessinger point method agree quantitatively with the explicit results from solving the DSEs in the complex momentum plane.

We compare the direct calculation and the Schlessinger point method by plotting the ratio ΔSPM[f](x)=f(x)fSPM(x),(12)where fSPM(x) is the function as obtained from the Schlessinger point method. This is shown in Fig. 10. The plots confirm that up to the potentially singular point the two methods agree quite well. Next, we compare the positions of the singular points found by the two methods for the gluon propagator. To this end, we follow a simple procedure. For a given subset of input momentum points on the real and positive momentum axis, we calculate the pole positions analytically from the coefficients of the Schlessinger point method. Poles with small residues are discarded as artifacts. For a final estimate of pole positions, we sample several subsets of the Euclidean data. This method is not as elaborate as the one introduced in Ref. [19], where the sample of input points is optimized based on the quality of the reconstruction from the spectral function, but quite effective for the present purpose. For the baseline setup, we find that the position of the pole indeed agrees very well with our expectation from the rise of the dressing function, with the location of the breakdown of reconstruction and with the region of the breakdown of the Cauchy-Riemann test. We indicate the location of the pole extracted from the Schlessinger method by a thick black line in Figs. 5, 6, 8, 9 and 10.

Thus, the combined evidence of all methods clearly points towards a singularity in the gluon dressing function at complex momenta. We observed the same quantitative agreement between the pole location predicted by the Schlessinger method and the explicit results from the ray method for all truncations studied in this work.