Quark-gluon vertex from Nf=2 lattice QCD

We study the quark-gluon vertex in the limit of vanishing gluon momentum using lattice QCD with 2 flavors of O(a) improved Wilson fermions, for several lattice spacings and quark masses. We find that all three form factors in this kinematics have a significant infrared strength, and that both the leading form factor $\lambda_1$, multiplying the tree-level vertex structure, and the scalar, chiral symmetry breaking form factor $\lambda_3$ are significantly enhanced in the infrared compared to the quenched (Nf=0) case. These enhancements are orders of magnitude larger than predicted by one-loop perturbation theory. We find only a weak dependence on the lattice spacing and quark mass.


I. INTRODUCTION
The quark-gluon vertex is one of the basic building blocks of the strong interaction. It encodes the fundamental interactions of the quarks and gluons, and can be used to define a nonperturbative running coupling. In addition, it is an essential ingredient in functional approaches to nonperturbative quantum chromodynamics (QCD), such as Dyson-Schwinger equations (DSEs) and the functional renormalisation group (FRG). In particular, the DSE for the quark propagator contains the quark-gluon vertex, and the amount of dynamical chiral symmetry breaking is highly sensitive to the details of this vertex, see Ref. [1] for the case of QED.
Traditionally, many studies of hadron phenomenology in the DSE framework have been carried out using the rainbow-ladder truncation, where the quark-gluon vertex is approximated by its tree-level structure, multiplied by an effective coupling which is taken to depend only on the gluon momentum. While this approach has been successful in describing a range of properties of pseudoscalar and vector mesons [2][3][4][5], it has failed to provide a satisfactory description of other quantities including scalar and axial-vector mesons and the chiral transition temperature [6][7][8]. It should also be noted that this approach fails to satisfy the Slavnov-Taylor identities which encode the gauge invariance of QCD.
The quark-gluon vertex has previously been studied on the lattice in the quenched approximation in a series of papers [43][44][45][46]. Apart from the uncontrolled systematic uncertainty of the quenched approximation, these studies have only been carried out at a single lattice spacing and volume and hence do not allow for a controlled approach to the physical (continuum and infinite-volume) limit. In this paper we take the first steps towards rectifying this by computing the quark-gluon vertex on state-of-the-art lattices with N f = 2 light dynamical quarks with different masses and for several lattice spacings, and compare with equivalent results in the quenched approximation (N f = 0). Some preliminary findings were presented in [47,48]. We will conduct this study in the soft gluon kinematics where the gluon momentum is vanishing, and hence both quark momenta are equal.
The structure of this paper is as follows. In Section II we present our methods for computing the vertex, including the notation and vertex decomposition (Sec. II A), lattice simulation details (Sec. II B), procedure for extracting form factors (Sec. II C) and tree-level correction of lattice data (Sec. II D). Our results are presented in Section III, while in Section IV we summarise our findings and outline prospects for future work.

A. Quark-Gluon Vertex
Our description of the quark-gluon vertex follows the notation used in [43-45, 49, 50], which we briefly summarise here.
The proper quark-gluon vertex, (Λ a µ ) ij βρ = t a ij (Λ µ ) βρ = t a ij (−ig 0 Γ µ ) βρ , is depicted in Fig. 1 with p the outgoing quark momentum, q the outgoing gluon momentum and k = p + q the incoming quark momentum where t a ij is the group generator. The quark-gluon vertex can be described in terms of 12 spin amplitudes which would be formed using the three vectors p µ , q µ , γ µ , and four Dirac scalars, I, p, q and p q. Reorganising these spin amplitudes to form a tensor basis, the full one-particle irreducible function Λ µ F (p, q, k) can be decomposed into non-transverse and transverse parts, where λ i and L µ i (p, k, q) are non-transverse form factors and basis vectors [51], and τ i and T µ i (p, k, q) are transverse form factors and basis vectors [52] respectively. By definition the transverse part is such that q · T i (p, k, q) = 0. In terms of the incoming and outgoing quark momenta the longitudinal basis can be written as The non-transverse part of the vertex satisfies the Slavnov-Taylor identity, which encodes the gauge invariance of the system. Through these Slavnov-Taylor identities the non-transverse part of the vertex is related to the inverse quark propagator, S −1 , the ghost-quark scattering kernel, H, and the ghost dressing function, G h . However the transverse part remains unconstrained by the Slavnov-Taylor identity, i.e., q µ Λ µ T (p, q) = 0 and Λ µ T (p, p) = 0. The transverse basis tensors [52] are given by where σ µν = 1 2 [γ µ , γ ν ]. Since in this paper we will only consider the soft gluon kinematics where the gluon momentum q = 0 and k = p, these transverse tensors do not contribute here. The full vertex structure in general kinematics will be considered in a future paper.

B. Lattice simulation details
In this study we use the same action and formulation for the quark propagator as in Ref. [16], and refer to that paper and references therein for further details. We employ the Wilson action for the gauge sector and the Sheikholeslami-Wohlert (clover) fermion action [53], where S W is the ordinary (unimproved) Wilson fermion action. The improvement coefficient c SW has been determined nonperturbatively. In order for the quark propagator and quark-gluon vertex to be fully O(a) improved, it is also necessary to improve the quark propagator [54], and for this purpose we use the O(a) improved "rotated" quark propagator, where the double brackets · denote averaging over gauge fields only, while S(x, y; U ) is the quark propagator evaluated on a single gauge configuration U . We use the tree-level values for the coefficients b q and c q : b q = c q = 1/4. We note that this propagator differs from the improved propagator used in Refs. [43-45, 49, 50]. The quark-gluon vertex is then determined by where the unamputated vertex V is given by In Landau gauge it is not possible to implement Eq. (7) fully since the inverse gluon propagator, D −1 µν , does not exist for q = 0. Instead we compute the transverse projected vertex In the present work we use the same gauge ensembles as in [16], which are a subset of the gauge ensembles generated by the Regensburg QCD (RQCD) collaboration (see, e.g., [55][56][57]). These ensembles have N f = 2 dynamical quarks with pion masses in the range 280-420 MeV. Three values of the lattice spacing, a ≈ 0.081 fm, a ≈ 0.071 fm and a ≈ 0.060 fm, are used. Most of the calculations have been carried out on a lattice volume of 32 3 × 64, but we have also used a 64 4 lattice to check finite volume effects for one of the parameter choices (m π ≈ 290 MeV).
In addition, we have produced a quenched ensemble with lattice spacing matching that of ensembles L07 and H07, i.e., with a = 0.07 fm, but with a larger valence quark mass m π ≈ 1000MeV.
The parameters used are listed in Table I. In order to more easily refer to the different ensembles when presenting the results, we have labelled them such that the first letter refers to the pion mass [heavy (H), 420 MeV, or light (L), 290 MeV] and the following two digits refer to the lattice spacing. The quenched ensemble is labelled Q07.  I: Lattice parameters used in this study. The lattice spacings a and pion masses m π and critical hopping parameters used to obtain the subtracted bare quark mass m q = 1/(2κ) − 1/(2κ c ) are all taken from Ref. [56]. The quenched ensemble Q07 was produced specifically for this study. N src denotes the number of different point sources per configuration used to produce the quark propagators.
For the gauge fixing we used an over-relaxation algorithm which iteratively maximizes the Landau-gauge functional with U g xµ = g x U xµ g † x+μ and g x ∈ SU (3). As stopping criterion we used where A xµ ≡ 1 2iag0 (U g xµ − U g † xµ )| traceless and ∇ µ A xµ ≡ µ (A xµ − A x−μ,µ ), as usual.

C. Form factor extraction
In the soft gluon kinematics (q µ = 0, k µ = p µ ), the continuum vertex is given by the three non-transverse form factors λ 1 , λ 2 andλ 3 , [58] We can extract these individual form factors by taking appropriate contractions and traces: where Tr 4 ≡ Tr 4 denotes the trace over spin indices. We note that Eq. (14) implies that λ 1 and λ 2 are coupled, but we can disentangle them by contracting it with the tensors δ µν and p µ p ν and solving the coupled equations. This yields the following expressions, In practice, expressions such as Eqs (15) and (16) contain multiple terms that can lead to numerically poorly determined form factors. It may therefore be desirable to sacrifice covariance and restrict ourselves to specific momentum configurations in order to obtain simple expressions involving only a single term. For instance, with the choices of (a) ν = µ and p µ = 0 and (b) ν = µ in Eq. (14) we obtain the following non-covariant, single term expressions for the continuum λ 1 and λ 2 form factors respectively, which we will make use of in this analysis: The quark-gluon vertex computed on the lattice using Eq. (7) is not renormalised. To renormalise the vertex, we impose a momentum subtraction scheme whereby the leading form factor λ 1 takes on its tree-level value at a given renormalisation scale µ, This fixes the renormalisation constant Z 1 such that Γ lat µ (p, q, k) = Z 1 Γ R µ (p, q, k), which in turn determines the renormalisation of all the form factors.

D. Tree-level correction
In order to determine the lattice equivalents of the continuum expressions (18), (19) and (17) for the form factors λ 1 , λ 2 andλ 3 , and to estimate and reduce their lattice artefacts, we need to compute the structure of the lattice tree-level vertex. Here we will outline the main results, while the details are given in Appendix A.
The tree-level quark-gluon vertex associated with the Sheikholeslami-Wohlert action without color and group indices is given by [59,60] for the case of the 'unimproved' propagator S 0 (x, y) = ψ(x)ψ(y) . The improved vertex obtained using the rotated propagator of Eq. (6) is given at tree level by where S (0) 0 (p) is the tree-level unimproved dimensionless Wilson quark propagator, while S (0) R (p) is the level expression for the improved propagator of Eq. (6). In the soft gluon kinematics with q = 0, p = k, this reduces to where we have defined the lattice momentum variables We note in particular that there are two separate tensor structures appearing in the tree-level vertex (23) which in the continuum become equal to L 2,µ , and likewise for L 3,µ . In Eqs (29) and (30) below, these are associated with separate respectively. It should be noted that all of these are proportional to the lattice spacing a and hence vanish in the naive continuum limit, and likewise that the deviation between λ (0) 1 and the continuum, tree-level value of 1 vanishes in the same limit. However, their magnitude can be large at finite lattice spacing.
Due to asymptotic freedom, we expect the nonperturbative vertex to approach its tree-level form at large momentum, and hence be dominated by lattice artefacts. We attempt to reduce these by dividing the lattice λ 1 by its tree-level form, while subtracting off the tree-level expression from the raw data for λ 2 and λ 3 .
Making a comparison between Eq. (23) and the continuum vertex Eq. (12), we end up with the tree-level corrected, lattice equivalents of the expressions (18), (19), (17), where the lattice tree-level form factors are given by and the common prefactor F (p) is given by These (λ 1 , λ 2 , λ 3 ) are the complete form factors required to determine the quark-gluon vertex in the soft gluon limit, and Eqs (25,26,27) define the exact procedures to calculate them on the lattice. In interpreting the lattice data for the various quark-gluon vertex form factors, knowledge of the corresponding perturbative expressions will be useful. Therefore, for completeness we report on the results for the Landau gauge oneloop Euclidean-space form factors λ 1,2,3 in the soft gluon kinematics in the momentum subtraction (MOM) scheme, where µ is the renormalisation scale, see Fig. 2: C A and C F are the eigenvalues of the quadratic Casimir operator in the fundamental representation and adjoint representation, respectively: . Note that λ 3 is proportional to the quark mass and a non-vanishing λ 3 , as is observed in lattice simulations, is an indication of chiral symmetry breaking.
The one-loop expressions are plotted in Figure 3, for three different values of the quark mass. The value of the form factors in the limit p → 0 is also shown in the graph. We note that all three form factors are enhanced in the infrared, and that this enhancement increases with decreasing quark mass for all three. This has also been observed in other kinematics [29]. In the case of λ 3 , however, we note that this mass ordering only occurs in the very far infrared, p 15 MeV, and that the mass ordering is the opposite for p 50 MeV, which includes all momenta available for the lattice volumes considered in this study.

III. RESULTS
In the following, we will present our results for the quark-gluon vertex in such a way as to carefully trace the effects of tree-level correction (Sec. III A), unquenching (Sec. III B), mass (Sec. III C), volume (Sec. III D) and lattice spacing (Sec. III E) on each of the three form factors.  Due to asymptotic freedom, we expect that λ 1 (p) → 1 (plus logarithmic corrections) at large p, and we can see that the tree-level corrections have the effect of bringing the lattice data closer to this continuum-like form for all cases and reducing the violations of rotational symmetry compared to the uncorrected data. In some cases, (notably for the largest quark mass m π = 422 MeV), we see indications that significant lattice artefacts remain even after tree-level correction and, therefore, careful study of the lattice spacing dependence will be necessary to determine the large-momentum behaviour of the vertex. For the larger volume (64 3 × 64), the fluctuations of λ 1 are quite large, Fig. 4c. In all cases the tree-level correction is small below 1 GeV but becomes pronounced for p 1 GeV (pa 0.4).
Using the same lattice ensembles as in Fig. 4, the effect of the tree-level corrections on the λ 2 , are shown in Fig. 5. We see that λ 2 tends to zero as p increases, in contrast to λ 1 , which approaches a value close to 1 -in line with expectations from asymptotic freedom in both cases. On the larger volume 64 3 × 64 lattice, the tree-level correction does not appear to have any effect on the λ 2 lattice artifacts within the uncertainties of the data. However, the different vertical scales complicate the comparison with the other figures. The response of the λ 2 to tree-level corrections is noticeable for all momenta. The data in Fig. 5 also show that λ 2 is strongly enhanced at low momenta, although the volume and mass dependence of λ 2 are difficult to disentangle. The data also suggest that λ 2 is enhanced for dynamical simulations relative to the quenched case.
In Fig. 6 the effect of the tree-level corrections on λ 3 are shown. They are pronounced at large momentum of around 2 GeV onwards, and have almost no effect for momenta less than 2 GeV. The corrections pushes λ 3 upwards in the ultraviolet, and λ 3 does not vanish at higher momenta as one would expect. This appears to be the case for all ensembles. We conclude that in the case of λ 3 , the tree-level correction does not work satisfactorily for pa 1.5, corresponding to momenta greater than ∼ 4 GeV, and we will discard our results for pa 1.5 (p 4 GeV) as unreliable in the absence of a careful, controlled continuum extrapolation.
The results for all our ensembles are collated in Figs. 7. This figure shows only minor variations with the quark mass and lattice spacing, with indications that decreasing the quark mass leads to a slightly stronger enhancement in the infrared, while decreasing the lattice spacing may have a small effect in the same direction. Indeed, for all form factors we see that their magnitude is largest for the H06 ensemble (m π = 426 MeV, a = 0.06 fm). We will discuss the quark mass and lattice spacing dependence in more detail in sections III C and III E.
The most striking feature of λ 2 in Fig. 7b is that this form factor appears to diverge more strongly in the infrared as we approach the continuum limit. Our results for λ 3 are summarised in Fig. 7c. We find that the magnitude of λ 3 in the infrared increases with increasing quark mass, which is expected since according to perturbation theory this form factor breaks chiral symmetry and is proportional to the quark mass, Eq. (34). Our result also suggest that its infrared magnitude increases as we approach the continuum limit.
From now on we will only show tree-level corrected data. All data shown from here on have been renormalised at 2 GeV, and to make the presentation clearer we have averaged data with nearby values of momentum, with a momentum bin size a∆p = 0.05.

B. Quenched versus dynamical
We now turn to a comparison of results from our quenched (N f = 0) and dynamical (N f = 2) ensembles at a = 0.07 fm. Since the valence quark mass in the quenched case is quite large, we use the H07 (m π = 422 MeV) ensemble in this comparison. The results are shown in Fig. 8. The one-loop perturbative expression (32), evaluated at α = 0.3 ≈ α MS (2 GeV) [61,62], m q = 17 MeV and with µ = 2 GeV, is also plotted. We have used the subtracted bare quark mass rather than the renormalised mass in this calculation, but this makes little difference as varying m q even by a factor 2 has a negligible effect for the range of momenta we are considering here. We will now discuss each form factor in turn. λ 1 : Although we find that the qualitative behaviour is the same in both quenched and unquenched cases, we see that with dynamical fermions the vertex is more strongly infrared enhanced than in the quenched case, as seen in Fig. 8a. However, a more detailed comparison, with different valence quark masses, will be required to disentangle the effects of dynamical fermions and valence quark masses. Even at one-loop order the perturbative contribution to λ 1 shows IR enhancement, and at one loop level in Landau gauge this is purely due to the nonabelian contribution.  λ 2 : This form factor has a large uncertainty in the IR and as the momentum increases its value approaches zero and the uncertainties become small (see Fig. 8b). Here we clearly see that the dynamical quarks only have a small effect on this form factor, leading at most to a slight enhancement. We also note that our lattice results λ 2 are orders of magnitude larger than the one-loop perturbative expression, which only exhibits a very small increase in the deep infrared. Like in λ 1 , the entire contribution in Landau gauge to λ 2 at one-loop level is purely non-abelian. λ 3 : As was the case for λ 1 , we find that inclusion of dynamical fermions leads to a significant enhancement of this form factor in the infrared, however above 1 GeV the difference between dynamical and quenched fermions diminishes. At lower momentum this form factor is negative, and as the momentum increases it approaches zero. We find a zero crossing between 1 and 2 GeV. We note that at this point, the uncertainties associated with the tree-level correction are not yet significant, so it appears unlikely that this zero crossing is entirely due to lattice artefacts. The one-loop expression has contributions from both abelian and non-abelian diagrams, and the combined effect is positive in the IR region, in contrast to what is found from the lattice data. Since we expect the perturbative behaviour to be reproduced at large momentum, this suggests that λ 3 should approach zero from above in the ultraviolet, and hence have a zero crossing in the intermediate-momentum region. Fig. 9 shows results for all three form factors for the H07 (dynamical) and Q07 (quenched) ensembles, along with the abelian Ball-Chiu vertex [51] and the one-loop perturbative contributions to the quark-gluon vertex. The Ball-Chiu vertex is calculated from the quark propagator using the expressions The quark wave function A(p 2 ) = 1/Z(p 2 ) and the mass function M (p 2 ) = B(p 2 )/A(p 2 ) are calculated using the N f = 2 lattice results from Ref. [16]. We observe a hierarchy in the infrared enhancement of λ 1 where the largest enhancement is for the dynamical lattice ensembles, followed by the quenched lattice data, the Ball-Chiu vertex and the one-loop perturbative expression in that order. The same hierarchy is found for λ 2 , while for λ 3 the quenched and dynamical lattice results and the Ball-Chiu are roughly equal. More work will be required to disentangle the effects of explicit and spontaneous chiral  symmetry breaking and quark dynamics in this form factor. We will study the quark mass dependence of λ 3 further in section III C. The IR hierarchy casts light on the reliability of the various approximations when they are used to calculate nonperturbative quantities. These results combined with the Slavnov-Taylor identity (3) show that the abelian Ball-Chiu vertex is not able to saturate the identity and, therefore, the quark-ghost kernel H in Eq. (3) has sizeable contributions in the infrared region.

C. Mass dependence
In Fig. 10 we show results from the L07 (m π = 295 MeV) and H07 (m π = 422 MeV) ensembles, which differ only by the quark mass. We also show the corresponding Ball-Chiu vertices and one-loop order perturbative results.
For all form factors we see that the lighter quark mass gives rise to a larger infrared enhancement. This is particularly interesting in the case of λ 3 , where the one-loop expression (34) increases with increasing quark mass in this momentum region. However, it should be noted that the one-loop expression also shows a stronger enhancement with decreasing quark mass (albeit with the opposite sign) in the deep infrared as shown in Fig. 3c. We also note that the abelian Ball-Chiu vertex exhibits the same effect, and that the same quark mass dependence was observed for λ 3 in the quenched case [44]. We therefore consider this mass dependence to be robust.
In the case of λ 1 and λ 2 , although the one-loop perturbative contribution for the lighter and heavier masses look the same, on closer inspection one notes that in the deep IR they differ from each other following the same trend, namely that they are more enhanced the smaller the quark mass (see Fig. 3).

D. Volume dependence
In Fig. 11 we compare the results for two different volumes with the same quark mass and lattice spacing, a = 0.07fm, m π = 295 MeV on the 64 4 and 32 3 × 64 lattices respectively.
In all cases, the uncertainties for the larger volume are so large that we are not able to draw any definitive conclusions; however, we do not see any evidence of a significant finite volume effect for λ 1 or λ 3 . For λ 2 our results suggest that this form factor is more strongly enhanced in the infrared as the volume increases; however, the results for the two volumes remain consistent with each other within the uncertainties.

E. Lattice spacing
In Fig. 12 we compare results for different lattice spacings, keeping the quark mass constant. For comparison, in Fig. 12 the plots on the left show results for the H06 and H07 ensembles with m π ≈ 420 MeV, while the plots on the right are for the L07 and L08 ensembles, with m π ≈ 290 MeV.
For both quark masses we observe that all form factors have a larger infrared enhancement for the smaller lattice spacing. The effect is slightly larger for the heavier masses than for the lighter mass.
At large momentum we see that all form factors move closer to their continuum tree-level value (1 for λ 1 and 0 for λ 2 and λ 3 as the lattice spacing is reduced, suggesting that the expected perturbative behaviour will be reproduced in the continuum limit.

F. All form factors
Finally, in Fig. 13 we show all form factors as function of momentum up to 4 GeV for all lattice ensembles generated for this study. λ 1 is dimensionless but λ 2 and λ 3 have dimensions of p −2 and p −1 , respectively. In order to measure the relative strength of λ 2 and λ 3 in comparison to λ 1 , all form factors are multiplied by the appropriate lattice momenta. The lattice momentum K(p) is used here to reduce lattice artefacts and ensure the continuum limit is approached more rapidly; note that all momentum variables are equivalent in the continuum limit and that no variable is intrinsically preferred over any other away from the continuum limit. For all ensembles, regardless of quark mass, volume and lattice spacing, we find that λ 1 has the biggest contribution to the quark-gluon vertex. The second and third in the hierarchy are the λ 2 and λ 3 form factors, respectively. There is no IR divergence in the soft gluon kinematics and hence we expect p 2 λ 2 and pλ 3 to vanish in the deep infrared. Lattice calculations of λ 2 and λ 3 in the deep infrared region are numerically extremely challenging; nevertheless in all the plots in Fig. 13 we see that their contributions approach zero as p → 0.

IV. CONCLUSIONS AND OUTLOOK
We have performed the first ever study of the quark-gluon vertex in Landau gauge lattice QCD with N f = 2 dynamical fermions. The study has been carried out in the soft gluon limit (gluon momentum q = 0) which gives access to three out of the non-transverse form factors, (λ 1 , λ 2 , λ 3 ). The quark-gluon vertex is an essential element of QCD, yet its complete non-perturbative form is still unknown. This calculation may be used to calibrate any approximation made for the quark-gluon vertex. Furthermore it can be used to validate proposed non-perturbative models of the quark-gluon vertex.
We find that the leading form factor λ 1 is significantly enhanced in the infrared, and that this enhancement is stronger than in the quenched approximation and increases as the chiral limit is approached. The indications are that the enhancement is further strengthened as the continuum limit is approached. No significant finite volume effects are found.
The subleading vector form factor λ 2 also exhibits an infrared strength commensurate with, if somewhat smaller than λ 1 . This form factor has only a mild dependence on the number of quark flavors, but the enhancement appears to increase as the continuum, infinite-volume and chiral limits are approached.
The infrared strength of the scalar form factor λ 3 is considerably larger than in the quenched approximation and also appears to increase as the continuum limit is approached. No significant volume effect is found, but the strength is slightly increased for smaller quark masses. The latter is in contrast to naive expectations as this form factor violates chiral symmetry and is therefore sensitive to the explicit chiral symmetry breaking from the quark mass, but suggests that it is primarily governed by the dynamical chiral symmetry breaking.
In all three cases, the non-perturbative effects are found to be orders of magnitude larger than the corresponding one-loop perturbative contributions to these form factors. In all form factors the lightest masses cause the larger IR  enhancement and it decreases as the mass increases. The dimensionless λ 2 and λ 3 form factors approach zero in the infrared, indicating no kinematic singularities in the soft gluon kinematics.
In the future we plan to extend this study to general kinematics which will allow us to determine all form factors of the transverse-projected vertex. It is also of interest to compute the vertex in general covariant gauges. Apart from addressing gauge dependence, this would also allow us to disentangle the longitudinal and transverse components, which is not possible in Landau gauge.

(A19)
The general form of the tree-level clover-rotated vertex (A19) is highly complicated. However, it simplifies in the soft-gluon kinematics which is the case we are studying in this paper. In this kinematics the gluon momentum q = 0 and both the quark momenta are equal, k = p. In this case the 'unimproved' tree-level vertex (A6) becomes Λ (0) 0,ν (p, 0, p) = −ig 0 γ ν C ν (2p) − i 2 aQ ν (2p) I = −ig 0 γ ν C ν (p) − i aK ν (p) I , and the tree-level clover-rotated vertex (A19) reduces to Λ (0) R,µ (p, 0, p) = (−ig 0 )F (p) 2ic q a K(p) + 1 − c 2 q a 2 K 2 (p) Λ a(0) 0,µ (p, 0) 2ic q a K(p) + 1 − c 2 q a 2 K 2 (p) , where the common prefactor F (p) is given by (1 + b q am) (1 + am/2) 2 1 1 + c 2 q a 2 K 2 (p) Making a comparison between this expression and the continuum vertex Eq. (12), we can identify the tree-level expressions for the form factors in the soft-gluon kinematics: We note in particular that there are two separate tensor structures appearing in the tree-level vertex (A23) which in the continuum become equal to L 2,µ , and likewise for L 3,µ . In Eqs (A25) and (A26) these are associated with separate form factors λ Although Eq. (A29) is based on the covariant expression (17), we will in practice not sum over µ in Eq. (A29), just as there is no sum over µ and α in Eqs (A27) and (A28). Combining this with the tree-level form factors, Eqs (A24),