What can obscure the study of DSEs to some extent is their dependence on the renormalization constants or, equivalently, on the renormalized values of the gluon and ghost dressing functions Z(μ2)=Zμ and G(0)=G0. Hence, before we proceed with the explicit numerical solutions in Sec. III B, we discuss the renormalization of the Yang-Mills system in detail. We note that the following discussion is independent of the two scenarios and equally applies to Scenario B.

Propagators, vertices and the coupling are related to their bare counterparts, denoted by the superscript (B), by the multiplicative renormalization constants, G(B)=ZcG,Γgh=Z˜ΓΓgh(B),Z(B)=ZAZ,Γ3g=Z3gΓ3g(B),g(B)=Zgg,Γ4g=Z4gΓ4g(B),(19)where Z˜Γ=ZgZA1/2Zc, Z3g=ZgZA3/2 and Z4g=Zg2ZA2 as a consequence of the STIs. In the Landau gauge the ghost-gluon vertex stays unrenormalized, so we can set Z˜Γ=1. Therefore, all renormalization constants can be related to ZA and Zc, Zg=1ZA1/2Zc,Z3g=ZAZc,Z4g=ZAZc2.(20)We can consistently renormalize the ghost and gluon propagators at different renormalization points: In practice it is convenient to renormalize the gluon dressing function at Q2=μ2 to Z(μ2)=Zμ, while the ghost dressing is fixed at Q2=0 to G(0)=G0. This fixes all renormalization constants and the resulting equations depend on g, Zμ and G0, see Appendix B for details: G(Q2)-1=G0-1+Σ(Q2)-Σ(0),Z(Q2)-1=Zμ-1+Π(Q2)-Π(μ2),F3g(Q2)=Z3g+M(Q2).(21)Here, Σ is the ghost self-energy, Π=Π+Π˜/Q2 the gluon self-energy in Eq. (14), and M are the vertex diagrams. The renormalization constants are dynamically determined by Zc=G0-1-Σ(0),ZA=Zμ-1-Π(μ2),(22)which fixes Z3g and Z4g from Eq. (20) so that no further renormalization of the vertices is required.

In practical solutions of the Yang-Mills DSEs one keeps g and Zμ fixed; the value G0 of the ghost dressing at vanishing momentum then distinguishes the scaling and decoupling solutions. Any finite G0 corresponds to a decoupling solution and the limit G0→∞ to the scaling solution. This is, however, not completely satisfactory from the viewpoint of renormalization as sketched in Fig. 6. Zμ and G0 should only multiplicatively renormalize the propagators, which would not lead to physically different solutions; in a double-logarithmic plot, a renormalization only induces vertical shifts in both functions. Similarly, the value of the coupling g should only set the scale, and when plotted over a logarithmic momentum scale it would only induce horizontal shifts in both functions. But how is it then possible to obtain a family of different decoupling solutions?

To this end, let us redefine the renormalized propagators by dividing out their values at the respective renormalization scales, G(Q2)→G(Q2)G0,Z(Q2)→Z(Q2)Zμ,(23)with the same redefinition for all other quantities that renormalize like G and Z, i.e., the renormalization constants Zc→ZcG0,ZA→ZAZμ(24)and the three- and four gluon vertices F3g→F3gZμG0,F4g→F4gZμG02.(25)This is always possible since the dressing functions are only fixed up to multiplicative renormalization. For the scaling solution, G0 becomes infinite. However, throughout this work as well as in most of the literature, the scaling case is defined as the limit of the decoupling solutions for G0→∞. Then, all redefinitions are well defined.

By inspection of the DSEs, see Appendix B, one finds that g, Zμ and G0 no longer appear individually in the equations but only in the combination α≔g24πZμG02.(26)Henceforth, we call α the coupling parameter, which can take any positive value. Note that α should not be confused with the strong coupling g2/(4π); it has the μ running of the gluon propagator and will later be related to the mass parameter. The Eqs. (21)–(22) with redefined quantities take the form G(Q2)-1=1+α[Σ(Q2)-Σ(0)],Z(Q2)-1=1+α[Π(Q2)-Π(μ2)],F3g(Q2)=Z3g+αM(Q2),Zc=1-αΣ(0),ZA=1-αΠ(μ2).(27)Here we have extracted the factor α from the redefined self-energies, which do not explicitly depend on it, apart from the two-loop terms. Thus, all renormalization constants have disappeared and only the coupling α remains.

With Eq. (26) it becomes clear that it is technically not G0 that distinguishes the decoupling solutions but α: If g and Zμ are kept fixed, changing G0 is equivalent to changing α, but equivalently we could have kept any other two variables fixed and changed the third variable. Thus, any finite α produces a decoupling solution and α→∞ corresponds to the scaling solution.

It is also convenient to remove the explicit dependence on the renormalization scale μ. To do so, we introduce a dimensionless variable x, x=Q2βμ2,(28)which is always possible since the momentum scale still carries arbitrary units and only rescales the dressing functions according to Fig. 6. If we perform the same operation for the loop momenta inside the integrals, then the dependence on μ2 drops out from the equations and moves to the cutoff of the integrals. The mass parameter β disappears from Π˜(Q2) as well, which in the redefined system becomes Π˜(Q2)Q2→1x.(29)In turn, β appears in the subtraction point since Q2=μ2 entails x=1/β. To give a concrete example, in the redefined system the ghost self-energy from Eq. (A19) in Appendix A reads Σ(x)=-Nc2π2∫0Lduu2∫-11dz(1-z2)32G(u+)Z(u-)u+u-2.(30)Here, u=k2/(βμ2), u±=u+x/4±uxz, and the μ2 dependence has moved into the cutoff L.

Finally, we redefine the ghost dressing function, and every quantity that renormalizes with it, once more by G(x)→αG(x).(31)As a consequence, the coupling α no longer appears in front of the self-energies but only in the ghost DSE. Then the renormalized equations are given by G(x)-1=1α+Σ(x)-Σ(0),Z(x)-1=1+Π(x)-Π(1β),F3g(x)=Z3g+M(x),Zc=1α-Σ(0),ZA=1-Π(1β)(32)with Π(x)=Π(x)+1/x. This is equivalent to the original equations but now the only explicit parameters are α and β. The subtraction point is no longer arbitrary but tied to the mass parameter β. In particular, the massless limit β→0 corresponds to a subtraction at x→∞. Here it is also obvious why α→∞ corresponds to the scaling solution, because α only appears in the ghost DSE and nowhere else, with G(0)=α. Note that we could have equally absorbed the coupling in the gluon dressing by setting Z(x)→αZ(x). This would lead to G(0)=1 and Z(1/β)=α but for 0<α<∞ the solutions are always identical up to multiplicative factors.

At this point, the scale x in all equations above is still arbitrary and given in internal units. A quantity that is invariant under renormalization is the running coupling α¯(x)=Z(x)G(x)2,(33)which remains finite even for α→∞. Here the parameter α is absorbed in the ghost dressing by Eq. (31). In dynamical QCD, the scale should be fixed to experiment by setting the value of α¯(x) at some momentum scale, which sets the scale in GeV units. In Yang-Mills theory, the usual procedure is to set the scale by comparing with quenched lattice QCD.

At asymptotically large momenta, the three- and four-gluon vertex dressings satisfy F3g(x)→G(x)Z(x),F4g(x)→G(x)2Z(x).(34)Hence, we can define equivalent “running couplings” by multiplying α¯(x) with powers of the renormalization-group invariants Z(x)F3g(x)/G(x) and Z(x)F4g(x)/G(x)2, e.g., α¯3g(x)=Z(x)3F3g(x)2,α¯4g(x)=Z(x)2F4g(x).(35)At large momenta these couplings are all identical but in the IR they can be different. In our calculations we set the four-gluon vertex to F4g(x)=G(x)2/Z(x), so the second relation in (34) is trivially satisfied. The deviation from the first relation will serve as a measure of the truncation error, which we discuss below.

We proceed with the numerical solution of the DSEs (32) in the (α,β) plane, where α is the coupling and β the mass parameter as explained above. The self-energy contributions for the ghost, gluon and three-gluon vertex DSEs are worked out in Appendix A. For the gluon DSE in Scenario A, we only need to keep the terms contributing to Π(Q2), whereas those for Π˜(Q2) are replaced by the constant mass term. This also means that the tadpole diagram drops out completely since it contributes to Π˜(Q2) only. For the same reason we also do not need to worry about quadratic divergences; all self-energy diagrams are only logarithmically divergent and these divergences are removed by the subtraction.

Figure 7 shows the results for the running coupling α¯(x), the ghost dressing G(x), the gluon dressing Z(x) and the three-gluon vertex dressing F3g(x) in Setup 3, i.e., with a back-coupled three-gluon vertex and including the two-loop terms in the gluon DSE. The calculations were performed at a fixed value β=0.001. The variation with β will be discussed in Sec. III C. The different curves correspond to a variation of the coupling parameter α over its full range. The running coupling is invariant under renormalization but the propagator and vertex dressing functions are not, so we plot the bare dressing functions according to Eq. (19) for better visibility, i.e., G(B)=ZcG,Z(B)=ZAZ,F3g(B)=F3gZ3g.(36)With increasing but finite α, the ghost dressing function in Fig. 7 rises in the IR and saturates at a finite value. The gluon dressing behaves like Z(x)∝x in the IR and develops a maximum at intermediate momenta. As a consequence, also the running coupling α¯(x)=Z(x)G(x)2 vanishes like Z(x)∝x in the IR and develops a maximum, however at a different value of x. The three-gluon vertex becomes increasingly smaller in the IR, where it eventually crosses zero and diverges logarithmically. The orange bands in Fig. 7 mark the onset of the “QCD-like” decoupling solutions around α≲1, as we discuss below.

For α→∞, the curves eventually converge to the scaling solution, where the dressing functions in the IR behave like G(x)∝x-κ,Z(x)∝x2κ,F3g(x)∝x-3κ.(37)In this way, the scaling solution is the envelope of the decoupling solutions and produces a finite running coupling α¯(x) which becomes constant in the IR. Thus, the emergence of a family of decoupling solutions is simply a consequence of varying the coupling parameter α, and the scaling solution is the limiting value for α→∞.

The coupled DSEs provide us with an internal way to quantify the truncation error. In practice the equations do not converge unless we modify the renormalization constant Z3g in Eq. (20) by a parameter c, Z3g=cZAZcwith0<c<1.(38)This is equivalent to an additional renormalization of the three-gluon vertex in Eq. (21) of the form F3g(Q2)=Fμ+M(Q2)-M(μ2),Z3g=Fμ-M(μ2)≠ZAZc.(39)Note that this additional renormalization would not be necessary if the STIs were preserved. Since we are working in different truncations defined by the Setups 1, 2 and 3, the STIs are no longer satisfied, but the effect can be compensated by introducing the parameter c≠1. Note however that the STIs constitute a further system of functional relations, and in nontrivial nonperturbative approximations such as the present truncations of the vertex expansions it is impossible to satisfy all sets of functional relations simultaneously; see e.g., [7,8,39,42] for respective discussions. From Eqs. (33)–(35), the parameter c2 translates to the ratio α¯3g(x)/α¯(x) evaluated in the UV. This reflects the consistency of the running couplings in the UV, which is known to be important for the quantitative accuracy of the solutions.

It turns out that for each truncation there is a value cmax where the anomalous dimensions of the ghost and gluon propagators reach their physical values. In each iteration step we employ fits of the form G(x→∞)=agh(lnx)γgh,Z(x→∞)=agl(lnx)γgl,(40)where the coefficients and exponents are free fit parameters. Thus, they are not forced to reproduce the anomalous dimensions of Yang-Mills theory given by γgh=944,γgl=1322,2γgh+γgl=1.(41)Figure 8 shows the resulting UV powers, plotted as a function of c. The β value is fixed, and we scan the full α range. For α→0, the anomalous dimensions vanish and for α→∞ they converge to the scaling solution. The largest c value where we obtain convergent solutions is marked by the vertical dashed lines. To the right of these lines we extrapolated each curve by a cubic fit with free fit parameters. The orange lines denote the physical values in Eq. (41). One can see that above a certain value of α, which marks the onset of the decoupling solutions, the curves show a substantial curvature and extrapolate to their physical values approximately at the same point cmax≈0.96…0.97. This suggests to identify cmax with the physical point in the truncation.

The rightmost plot in Fig. 8 shows the maximum value of the gluon dressing function Zmax=maxZ(B)(x) as a function of c. Close to cmax, there is a strong curvature which even appears to tend to infinity for the scaling solution. The horizontal orange band corresponds to the region of lattice results for Zmax, see below. The extrapolated values are compatible with this band, although the extrapolation induces a large uncertainty.

The results in Setups 1 and 2 are qualitatively similar to those in Fig. 8, except with a different cmax: (i)

Setup 1: cmax≈0.59…0.61,

Close to the respective value cmax, the dressing functions in Fig. 7 are very similar in all setups. In double-logarithmic plots like those for α¯(x), G(x) and F3g(x) (recall that asinh F grows logarithmically for large |F|) they are hardly distinguishable, except that in Setup 1 the three-gluon vertex stays at tree-level. The quantity that is most sensitive to the truncations is the bump of the gluon dressing function whose height defines Zmax. In Setup 2 the bump is slightly larger and narrower than in Setup 3 because the two-loop terms in the gluon self-energy are positive (cf. Fig. 4) and therefore flatten Z(x).

In any case, these observations imply that the gluon dressing function may well have only reached a fraction of its true height. The largest possible value c=0.95 we can reach in Setup 3 corresponds to the results in Fig. 7. The same observation holds true, although less pronounced, for the ghost dressing at intermediate momenta. This makes it difficult to compare with lattice results since we should match the results at c=cmax but the functions can change substantially in its vicinity. In addition, the lattice data are only determined up to scale setting (which amounts to horizontal shifts of the curves on a logarithmic scale) and multiplicative renormalization. Finally, we must identify the value of α that should be compared to the lattice-type decoupling solution.

In practice we construct simple parametrizations for the lattice data in Ref. [34], Glat(x)=11+1.9x0.7+2x2+0.38(1-e-x)ln(1+x)γgh,Zlat(x)=6.3x(1+x)2.2+6xe-2.1x+x2(0.3+x)(0.6+x)3.1(1-e-3x)ln(1+x)γgl.(42)The fits in (42) implement the one-loop running with γgh, γgl from Eq. (41), together with Glat(0)=1 and Zlat(x→0)∝x. They are determined up to normalization and rescaling and must be matched to the UV running of the DSE results. The behavior of the UV coefficient agh in Eq. (40) is analogous to Fig. 8. The curves converge approximately to the same value at cmax independently of the coupling α, from where we can match the ghost dressing with the lattice. Because this changes Glat(0) and our G(x) satisfies G(0)=α, with this we identify α∼2.5 as the value corresponding to the lattice decoupling solutions. Matching the UV running of the gluon together with the condition Z(1/β)=1 then fixes the gluon dressing function and the scale.

From the resulting curves in Fig. 9 one can see that there is a substantial gap between the lattice data and the DSE results obtained with c=0.95. However, we find a similar behavior when we apply naive cubic extrapolations like those in Fig. 8 to G(x) and Z(x) over the whole range in x: The extrapolated ghost dressing at c=0.97 reproduces the lattice data, and the extrapolated gluon dressing matches the height of the lattice data but its bump is somewhat shifted.

To check the self-consistency of our comparison, we also solved the ghost DSE for G(x) in Eq. (32), with Σ(x) from Eq. (30), using a fixed gluon input. In this case the standalone ghost DSE converges without problems, since the gluon propagator is just an input function. For that purpose, we generated a family of curves for Z(x) which interpolate between the DSE and lattice results (shown in the right panel of Fig. 9). The corresponding results for G(x) from the ghost DSE are shown in the left panel. One can see that the DSE solution with the lattice gluon input reproduces the lattice ghost without difficulties. The ghost DSE only depends on α. Solving it for different values of α, we find that the quantity ∑x|G(x)-Glat(x)| is minimized for α≈2.5 in agreement with our findings above. This also confirms that the only missing ingredient of the ghost DSE, the dressing of the ghost-gluon vertex, can only have a minor effect.

The problem of not being able to reach the physical value cmax in the present truncation appears to be related to the back-coupling of the three-gluon vertex and, by extension, to the four-gluon vertex which appears in its respective DSE. Our results are stable when changing the number of grid points or using different grids, but achieving convergence close to cmax becomes increasingly difficult even with relaxation and Newton methods (cf. Appendix C). We easily obtain large and narrow bumps for Z(x) when we employ model ansätze for F3g(x), but this comes at the expense of additional parameters and scales. This would obscure the discussion of the results in view of the gluon mass gap, and we refrained from doing so. We can also reduce the gap between the lattice and DSE results for Z(x) by identifying the lattice decoupling solution with DSE solutions at different values of α, but in those cases we can no longer simultaneously describe the ghost dressing function.

Concerning the four-gluon vertex ansatz, we employed F4g(x)=G(x)2/Z(x) but when we use instead F4g(x)=Z4g=const the results are very similar. However, in the vicinity of cmax the shape of F4g(x) especially in the mid-momentum region appears to play an increasingly prominent role in (de-)stabilizing the solutions. In Fig. 3 one can see that the four-gluon vertex appears in the gluon DSE only in the sunset diagram, which turns out to be negligibly small, so it mainly enters indirectly through the swordfish diagram in the three-gluon vertex. The general four-gluon vertex involves many tensors, including three momentum-independent ones which may lead to additional effects [64,69]. For functional computations aiming at quantitative precision in Yang-Mills theory, which also implement a dynamically back-coupled four-gluon vertex, we refer to Refs. [42,64]; e.g., in [64] the deviation from the STI was quantified to be on the subpercent level.

In summary, our analysis implies that the lattice results can be matched to the DSE results for a particular value of the coupling parameter α. In the next section we will see that this generalizes to a particular line of constant physics in the (α,β) plane.

The results discussed so far were obtained at a fixed value of the mass parameter β=0.001. The remaining question in Scenario A is how the respective results and findings change with β. For modest changes with larger β, we find that the plots in Fig. 7 are essentially unchanged except they may be shifted horizontally in logx. The overall scale is arbitrary and we are free to rescale the solutions, and after rescaling they are almost identical. Thus, in this domain the mass parameter β rescales the solutions but does not change any physics. This is in contradistinction to a variation of the coupling parameter α, which distinguishes physically different solutions.

For quantifying these observations, we show the lines of constant physics in the (α,β) plane in Fig. 10. For their determination we extracted the maximum of the running coupling α¯(x) for each parameter set (α,β). Then, we rescaled x such that the maximum always appears at the same point x=x0. Accordingly, a given line is defined by the same value α¯(x0). This amounts to interpolating the dressing functions at fixed value of β=0.001 and finding the value of α for a given α¯(x0). Along each line, all functions are identical up to rescalings, which can be seen in Fig. 11: every curve therein is a superposition of 30 curves along a given line and these curves are practically indistinguishable.

In Fig. 10 one can see that for sufficiently large β the lines of constant physics become indeed horizontal. On the other hand, if they were horizontal for all values of β this would be incompatible with physical expectations: In the massless theory with β=0, the coupling α should not produce physically different solutions but only rescale the system. In other words, in the massless limit the line of constant physics should be the vertical axis in Fig. 10. These different forms can only be reconciled with each other if the lines bend toward the origin in the (α,β) plane. Indeed, this is what we find for very small values of β, as is visible in Fig. 10.

The behavior in Figs. 10 and 11 is completely analogous in Setups 1 and 2 and thus appears to be a general feature. In practice we cannot reach β=0 for the reasons discussed in Appendix C. 1/β is the subtraction point in the gluon DSE, and the shape of the gluon self-energy defines a calculable window for β. For increasing values of α or c, the gluon dressing becomes increasingly narrow and this window shrinks. The smallest value we can reach is typically β∼10-6. The results in the left panel of Fig. 10 were obtained with c=0.92; in the right panel one can see that the effect becomes more pronounced when c is increased toward its practical limit c=0.95, i.e., the curves become steeper. However, for c≳0.92 we can no longer cover the full range in β.

The interpretation of our results is sketched in Fig. 12. The two parameters (α,β) in the Yang-Mills system form a combination λ0(α,β) that acts along the lines of constant physics and only rescales the system without changing any physics. The second combination λ1(α,β) is orthogonal to these lines and distinguishes the physically different solutions, i.e., only λ1 is an active parameter. In other words, in the presence of a mass term the coupling parameter α no longer rescales the system but distinguishes the physically different decoupling solutions, with the scaling solution as their endpoint.

In the limit β→0, the shape of the lines implies that λ0(α,0)=α, i.e., the coupling α only rescales the solutions while not changing the physics. However, this means that the scaling solution obtained at β≠0 and α→∞ is identical to the scaling solution at β=0 with arbitrary α, and it is the only solution that remains in the massless limit. This implies that in the present Scenario A the only physical solution is the scaling solution, as the mass parameter β is merely an artifact that must be sent to zero. The scaling solution is therefore a parameter-free, intrinsic property of the system. Moreover, since the scaling solution as well as the whole family of decoupling solutions implement confinement via the gluon mass gap [9,10], this implies confinement in the Landau gauge.

While this seems theoretically appealing, there is a conceptual problem that can be traced back to the determination of the IR exponent κ discussed around Eq. (17). We have replaced Π˜(Q2) by a constant mass term of the form (18), which is “removed” in the end by sending β→0. As β only appears in the term β/x, we could absorb it into the scale and thus sending β→0 is equivalent to renormalizing the equations at x=1/β→∞. This entails, however, that the mass term 1/x cannot truly be removed as shown in Fig. 13, which is the decomposition of the gluon self-energy Π(x) in Scenario A. We emphasize that its analogue in Fig. 4 corresponds to Scenario B, which we discuss below. For fixed β and α→∞ the ghost loop becomes large, but even for the scaling solution at α→∞ the term 1/x is present and dominates the IR behavior. Had we not absorbed β into the scale, for β→0 the same would happen but all curves would be pushed to the left on a logarithmic scale, so that the rise with 1/x in the IR eventually drops out of the numerical grid and is shifted to logx→-∞. Therefore, the effect of the mass term is never truly switched off.

The same rescaling effect explains the apparent discrepancy between the solutions for Z(x) at α→0 from Eqs. (27) and (32), where the first set of equations entails Z(x)→1 and the other Z(x)→1/(1+1/x-β). In the former case, the structure effects are also pushed to logx→-∞, so the renormalized theory never becomes a free massless theory for α→0.

Because the 1/x term drives the IR properties, in this case the scaling solution at x→0 with G(x)∝x-κ,Z(x)∝x2κ,F3g(x)∝x-3κ(43)has an IR exponent κ=1/2. This is also what we found analytically in Fig. 5 when sending the parameter λ to zero. The ambiguity for λ=0 reappears in the numerical solution in the following way: Since the mass term in Eq. (18) is arbitrary as long as we send β→0 in the end, we are free to replace it with any other ansatz of the form Π˜(Q2)Q2=g24πG02(βμ2Q2)2κ,(44)where κ is a free parameter, as long as we remove this term again in the end by sending β→0. After the redefinitions from Sec. III A this becomes Π˜(x)x→1x2κ,(45)which replaces the 1/x term in the DSEs (32). Its effect is again negligible in the mid- and large-momentum regions, so it does not change the behavior of the solutions, but as long as κ≥1/2 it still dominates the IR and leads to a scaling solution with IR exponent κ. We checked this numerically and found that by dialling κ we can indeed generate any scaling solution we want, where the lines of constant physics behave qualitatively in the same way as before.

In other words, in Scenario A the limit β→0 is not unique, which is a manifestation of the 0/0 problem in the analytic IR analysis and leads to a “family of scaling solutions” depending on the arbitrary value of the IR exponent κ. Because this ambiguity does not seem very appealing, in the next section we revisit the initial problem and discuss a possible alternative scenario.

The first and more practical issue is that a dynamical Δ0 term potentially comes with quadratic divergences. While Δ0 vanishes in perturbation theory using dimensional regularization, a hard cutoff interferes with gauge invariance so that Δ0 has an intrinsic “artifact” admixture that must be removed. Once the quadratic divergences are subtracted in a fully consistent way, the remainder (if it is nonzero) must be a dynamical, nonperturbative effect.

Different methods have been employed to remove quadratic divergences in the sum Π=ΔT+Δ0/Q2, see [90] for an overview. One is an explicit subtraction at the level of the integrands; in that case one analyzes their UV behavior and subtracts the terms producing the quadratic divergences. This has been employed in what we call Setup 1 [15] but it becomes cumbersome when the three-gluon vertex is back-coupled dynamically or two-loop terms are included. Another method is to subtract the quadratic divergences numerically by fitting Π to 1/Q2 terms [91]. A consistent combination of both is typically done in the fRG, where the consistency follows from the (trivial) RG consistency of the fRG setup [8,42].

Having individual access to ΔT and Δ0, a simpler method is to subtract Δ0 directly, i.e., Δ0(Q2)→Δ0(Q2)-Δ0(Q02)+g24πG02βμ2.(46)Since Δ0(Q02) is a constant, the arbitrariness in the choice of Q02 is absorbed by adding another arbitrary constant proportional to βμ2, again with a prefactor that ensures the correct renormalization. This leads to a similar form of the equations as in Scenario A, where we added a constant mass term with mass parameter β. After the redefinitions leading to Eq. (32), the self-energy becomes (47)Π(x)=ΔT(x)+1+σ(x)x,(47a)with σ(x) given by σ(x)=Δ0(x)-Δ0(x0).(47b)This procedure is equivalent to the one in Ref. [64], where a term ∝1/x is subtracted from the self-energy. In this way, the only difference between the Scenarios A and B is the term σ(x). Note that we can no longer change the power of the 1/x term like we did in Eq. (45) in Scenario A, because the subtraction must be done in its numerator. As a consequence, the IR exponent κ is no longer arbitrary but a dynamical result with a definite value depending on the truncation. Because σ(x)/x is only active in the IR, x0 should not be too large to ensure numerical stability; in practice we choose x0=1. The arbitrariness of x0 is then absorbed in the change of β which determines the subtraction point x=1/β.

The numerical results in Scenario B do not require a separate discussion, because the only change in the equations is the addition of the term σ(x) in Eq. (47a), which only affects the IR region. Thus, in the mid- and high-momentum regions, and also at low momenta as long as α is not too large, all previous statements remain intact. The shape of the solutions in Fig. 7, the behavior of the UV exponents in Fig. 8, and the values of cmax for the different setups all remain approximately the same in Scenario B. The only noticeable difference appears in the IR, where σ(x) is active.

Figure 14 shows the gluon self-energy contributions in this case. These are the same plots as in Fig. 4 except now we separate the 1/x and σ(x)/x contributions explicitly. For the decoupling solutions, σ(0) is a constant. The remaining self-energy terms in ΔT(x) are either constant or diverge at best logarithmically in the IR (see Appendix A), so that Π(x) and thus Z(x)-1 are dominated by the terms ∝1/x for x→0. These terms generate a mass gap such that the (dimensionless) gluon propagator at the origin reads D(0)=Z(x)x|x→0=11+σ(x).(48)After reinstating dimensions through x=Q2/(βμ2), the right-hand side (rhs) is divided by the (arbitrary) factor βμ2, from where one can extract the gluon mass scale according to Eq. (3).

When approaching the scaling solution for α→∞, σ(x→0) grows and eventually diverges with a power x1-2κ stemming from its ghost loop contribution. At the same time the ghost loop contribution to ΔT(x) also diverges with x-2κ in the scaling limit. Taken together, both ΔT(x) and σ(x)/x contribute with the same power to the IR divergence of Π(x)∝x-2κ, which for κ>1/2 beats the divergence from the mass term 1/x as is visible in the right panel of Fig. 14.

As a result, the dressing functions in the IR scale with G(x)∝x-κ,Z(x)∝x2κ,F3g(x)∝x-3κ.(49)The IR exponent κ can be read off from any of the dressing functions in the scaling limit. This is shown in the left panel of Fig. 15 for the ghost dressing, where the line 1/xκ is the envelope of the decoupling solutions. In practice we determine κ from the ghost self-energy by fitting it to Σ(x)=a+bxκ for x→0, where a, b and κ are free fit parameters depending on {α,β,c}, i.e., also for the decoupling solutions. From Eq. (32) we deduce that the ghost dressing function in the IR is well approximated by G(x)-1⟶x→01α+bxκ,(50)so that scaling at x→0 is achieved for α→∞ whereas for each decoupling solution G(0)=const. For sufficiently large α we may read off the scaling exponent κ≈0.58, which is compatible with contemporary DSE [63] and fRG results [42]. Note however that the approximate value undershoots the analytic scaling κ=0.595 that is obtained in the present approximation for α→∞. In the right panel of Fig. 15 one can see that κ is almost independent of c, i.e., the fact that we cannot exhaust the full range up to cmax≈0.96…0.97 is irrelevant for its determination.

When we solve the DSEs in the (α,β) plane we find the same behavior as in Scenario A. One can identify lines of constant physics, which for β→0 bend toward the origin as shown in Fig. 16. The interpretation is thus the same: One combination of α and β rescales the solutions and another one distinguishes the decoupling solutions, with the scaling solution as their endpoint.

The crucial difference to Scenario A, however, is the fact that β is no longer an artificial mass parameter that must be removed in the end by sending β→0, because now it arises from the subtraction of quadratic divergences. Therefore, any value for β is equally acceptable, which also seems more appealing considering the conceptual problems with the limit β=0 encountered in Scenario A. As a consequence, there is no longer a criterion that discriminates between the different solutions. Instead of an ambiguity in the IR exponent κ, one is left with the ambiguity between the scaling and decoupling solutions, which for β≠0 are distinguished by the parameter α. This raises the question: Can one find another criterion that allows for a selection of solutions?

The fundamental question in Scenario B is how Π˜(Q2) can vanish, given that the dynamical term Δ0(Q2)≠0 now contributes to it. As we already mentioned, this implies a nonvanishing ΔL(Q2) for the consistency relation ΔL=-Δ0/Q2 to hold. To this end, let us reexamine the assumptions we made. The crucial (implicit) assumption in our approximation so far is the use of the ghost-gluon vertex at tree-level. When introducing Scenario B in Sec. II, we already emphasized that the ghost-gluon vertex has two tensor structures, cf. Eq. (11): Γghμ(p,Q)=-igfabc[(1+A)pμ+BQμ].(51)The momentum pμ is the outgoing ghost momentum and Qμ is the incoming gluon momentum. The corresponding dressing functions are A(p2,p·Q,Q2) and B(p2,p·Q,Q2). Accordingly, the tensor structure missing in the present approximation produces to a solely longitudinal contribution to the inverse gluon propagator which belongs to the ΔL part.

The ghost-gluon vertex is UV-finite in Landau gauge and does not need to be renormalized. The function A is known to be small [56,64], which is why a tree-level vertex is a good approximation. However, little is known about the function B apart from the limit pμ=Qμ where the incoming ghost momentum vanishes, in which case the vertex becomes bare [84] and thus one has B(p2,p2,p2)=-A(p2,p2,p2).

The Lorentz tensor Qμ attached to B is purely longitudinal and vanishes whenever the gluon leg is contracted with a transverse gluon propagator in Landau gauge, i.e., for all internal gluon lines. As such, it does not contribute to the ghost self-energy in Fig. 3, but it does contribute to the ghost loop in the gluon DSE. Because the term is longitudinal, it drops out from the transverse projection Π=ΔT+Δ0/Q2 in Eq. (12). Hence, the full transverse dressing is not affected by the nonclassical dressing B in the ghost-gluon vertex (modulo small effects from A≠0). However, the longitudinal projection picks up an extra term and generalizes to Π˜(Q2)=Δ0(Q2)+Q2ΔL(Q2),(52)where Δ0 is the sum of all Π˜-like self-energy contributions listed in Appendix A (up to A≠0 effects). The longitudinal dressing ΔL can be worked out in analogy to Eq. (A21): ΔL(Q2)=-g2Nc2∫kG(k+2)G(k-2)k+2k-2B(k+2,k+·Q,Q2),(53)where k±=k±Q/2. The resulting DSEs read Z(Q2)-1=ZA+Π(Q2),L(Q2)-1=1+ξΠ˜(Q2)Q2.(54)

However, Eq. (52) allows us to eliminate the completely longitudinal term Π˜(Q2) as required by the STIs. For Π˜=0 we must have ΔL(Q2)=!-Δ0(Q2)Q2.(55)For decoupling solutions, Δ0(0) is a constant. Thus, Eq. (55) can only hold for all Q2 if the longitudinal ghost-gluon dressing B diverges like 1/Q2 for Q2→0: the ghost-gluon vertex must exhibit a longitudinal massless pole. In this case Π˜ vanishes, and we arrive at the purely transverse self-energy Πμν(Q)=Π(Q2)(Q2δμν-QμQν)(56)with Π=ΔT+Δ0/Q2. The equation for Z(Q2) is unchanged and B only serves to eliminate Π˜(Q2).