Nonperturbative study of the four gluon vertex

Nonperturbative study of the four gluon vertex

D. Binosi    D. Ibañez European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) and Fondazione Bruno Kessler,
Villa Tambosi, Strada delle Tabarelle 286, I-38123 Villazzano (TN) Italy
   J. Papavassiliou Department of Theoretical Physics and IFIC, University of Valencia and CSIC, E-46100, Valencia, Spain
Abstract

In this paper we study the nonperturbative structure of the SU(3) four-gluon vertex in the Landau gauge, concentrating on contributions quadratic in the metric. We employ an approximation scheme where “one-loop” diagrams are computed using fully dressed gluon and ghost propagators, and tree-level vertices. When a suitable kinematical configuration depending on a single momentum scale is chosen, only two structures emerge: the tree-level four-gluon vertex, and a tensor orthogonal to it. A detailed numerical analysis reveals that the form factor associated with this latter tensor displays a change of sign (zero-crossing) in the deep infrared, and finally diverges logarithmically. The origin of this characteristic behavior is proven to be entirely due to the masslessness of the ghost propagators forming the corresponding ghost-loop diagram, in close analogy to a similar effect established for the three-gluon vertex. However, in the case at hand, and under the approximations employed, this particular divergence does not affect the form factor proportional to the tree-level tensor, which remains finite in the entire range of momenta, and deviates moderately from its naive tree-level value. It turns out that the kinematic configuration chosen is ideal for carrying out lattice simulations, because it eliminates from the connected Green’s function all one-particle reducible contributions, projecting out the genuine one-particle irreducible vertex. Motivated by this possibility, we discuss in detail how a hypothetical lattice measurement of this quantity would compare to the results presented here, and the potential interference from an additional tensorial structure, allowed by Bose symmetry, but not encountered within our scheme.

pacs:
12.38.Aw, 12.38.Lg, 14.70.Dj

I Introduction

Of all elementary vertices that appear in the QCD Lagrangian, the four-gluon vertex is the most poorly understood. From the point of view of continuum studies, this fact may be regarded as a consequence of the enormous proliferation of allowed tensorial structures, generated by the presence of four color and four Lorentz indices. This difficulty, in turn, complicates considerably the extraction of reliable nonperturbative information from the corresponding Schwinger-Dyson equation (SDE). In addition, even gauge-technique inspired Ansätze Salam (1963); Salam and Delbourgo (1964); Delbourgo and West (1977a, b) are extremely difficult to implement, due to the complicated structure of the Slavnov-Taylor identity that this vertex satisfies in the linear covariant () gauges (see, e.g. Binosi and Papavassiliou (2009)). Thus, the analytic studies dedicated to this vertex are very scarce, furnishing information only at the level of one-loop perturbation theory Pascual and Tarrach (1980); Gracey (2014), or involving generic constructions in the context of the pinch technique Papavassiliou (1993), or privileged quantization schemes, such as the background field method Hashimoto et al. (1994); Ahmadiniaz and Schubert (2013).

From the point of view of lattice simulations, the situation is simpler, in the sense that, to the best of our knowledge, no simulations of the four-gluon vertex have been performed, for any kinematic configuration. This is to be contrasted with the corresponding status of all other vertices, namely the quark-gluon, the ghost-gluon, and three-gluon vertex, which have been studied on the lattice, at least for some special choices of their momenta Skullerud and Kizilersu (2002); Skullerud et al. (2003); Cucchieri et al. (2004); Sternbeck (2006); Cucchieri et al. (2006, 2008).

In the present work, we carry out a preliminary nonperturbative study of the one-particle irreducible (1-PI) part of the four-gluon vertex, denoted by , motivated by recent developments in our understanding of the QCD nonperturbative dynamics of the two- and three-point sectors in the Landau gauge. Specifically, a precise nonperturbative connection between the masslessness of the ghost, the detailed shape of the gluon propagator in the deep infrared (IR), and the IR divergences observed in certain kinematic limits of the three-gluon vertex, has been put forth in Aguilar et al. (2014a) (see also Blum et al. (2014); Eichmann et al. (2014) for related contributions). This detailed study led to the conjecture that any purely gluonic -point function will display the same kind of behavior, given that ghost loops111We refer to ghost loops that exist already at the one-loop level. Ghost loops nested within gluon loops do not produce this particular effect, because the additional integrations over virtual momenta soften the IR divergence. appear in all of them (and, hence, the associated IR logarithmic divergence in ). Clearly, the confirmation of this expectation at the level of the four-gluon vertex would put our understanding of this specific IR effect on rather solid ground. In particular, it would be important to establish, even within an approximate scheme, the type of tensorial structures that will be associated with this particular divergence.

In order to simplify the calculation as much as possible without compromising its main objective, we have chosen a particularly simple configuration of the external momenta, in which a single momentum scale () appears, and the flow in the four legs is chosen to be ; this has the advantage of giving rise to loop integrals that are symmetric under the crossing of external legs thus reducing the amount of diagrams one needs to evaluate. We hasten to emphasize that the aforementioned momentum configuration has been first considered in Kellermann and Fischer (2008), in the context of the so-called “scaling” solutions Alkofer and von Smekal (2001). Instead, our analysis will be carried out using an IR finite gluon propagator and ghost dressing function , in conformity with the results obtained from a plethora of large-volume lattice simulations Cucchieri and Mendes (2007); Sternbeck et al. (2007); Bowman et al. (2007); Bogolubsky et al. (2009); Oliveira and Silva (2009); Cucchieri and Mendes (2009); Ayala et al. (2012), as well as a variety of analytic approaches Alkofer and von Smekal (2001); Szczepaniak and Swanson (2002); Maris and Roberts (2003); Szczepaniak (2004); Aguilar and Natale (2004); Fischer (2006); Kondo (2006); Braun et al. (2010); Binosi and Papavassiliou (2008a); Epple et al. (2008); Boucaud et al. (2008); Binosi and Papavassiliou (2008b); Aguilar et al. (2008); Fischer et al. (2009); Szczepaniak and Matevosyan (2010); Watson and Reinhardt (2010); Rodriguez-Quintero (2011); Campagnari and Reinhardt (2010); Pennington and Wilson (2011); Watson and Reinhardt (2012); Kondo (2011); Aguilar et al. (2011); Binosi et al. (2012). Specifically, we will consider a simplified version of the so-called “one-loop dressed” approximation, where one computes the one-loop diagrams with fully dressed gluon and ghost propagators, but with tree-level (undressed) vertices (the case with dressed ghost-vertices only is also presented).

Notice that this approach, although SDE-inspired, differs significantly from a typical SDE study, mainly because it does not involve the solution of an integral equation for the unknown form factors; instead, the form factors are simply extracted from the dressed diagrams mentioned above. In that sense, it may be thought of as a “lowest order” SDE approximation, where one simply substitutes tree-level values for all vertex form factors appearing inside diagrams. This particular method (and variations thereof) has been employed in the context of other vertices, furnishing results that compare favorably with the lattice Aguilar et al. (2014a); Aguilar et al. (2013); Aguilar et al. (2014b); of course, its effectiveness can only be justified a-posteriori (i.e., comparing with the lattice), given that there is no rigorous way of estimating the errors introduced by the omitted terms.

If one concentrates on the nonperturbative contributions that are quadratic in the metric, in the case of SU(3) only two independent tensorial structures emerge: the one associated with the tree-level four-gluon vertex (indicated by ), and a second one (denoted with ) which is totally symmetric in both Lorentz as well color indices (and therefore orthogonal in both spaces to the tree-level term). It turns out that the aforementioned divergences are entirely proportional to this latter tensorial structure, with no contribution to the tree-level tensor . Therefore, one finds that within the one-loop dressed approximation we employ, will carry all the IR divergences, whilst contains all the ultraviolet (UV) divergences, as required by the renormalizability of the theory. These findings clearly deviate from the patterns observed in the case of the three-gluon vertex, where the form factors proportional to the tree-level vertex, in addition to containing the UV divergences, were also affected by this particular IR divergence (displaying the associated “zero crossing”). In addition, the deviation of the form factor associated to the tree-level tensor from 1, namely its tree-level value, is relatively modest. In particular, when the ingredients used in its calculation are renormalized at GeV, its highest value, located at about 500 MeV, is 1.5.

The results obtained are further discussed in the specialized context of a possible future lattice simulation of the connected part of this vertex, to be denoted by . It turns out that the momentum configuration eliminates all contributions to from one-particle reducible (1-PR) graphs, thus isolating only , without any “contamination” from lower-order Green’s functions. In addition, an analysis based on Bose symmetry arguments alone, reveals that a third tensor structure, denoted by , is in principle allowed; evidently, the form factor associated with this tensor vanishes within the one-loop dressed approximation that we employ. It is likely, however, that this particular property will not persist in a complete nonperturbative computation, as the one provided by lattice simulations. Therefore, under the assumption that such a structure might eventually emerge, we describe how to express the complete set of form factors characterizing in terms of the standard lattice ratios , used in studies of the three-gluon vertex Cucchieri et al. (2006, 2008).

The article is organized as follows. In Sect. II we introduce our notation, review the relevant tensor decomposition, and recall some identities particular to the SU(3) gauge group. Next, in Sect. III we carry out the calculation of the one-loop dressed diagrams in the simplified setting where all the external momenta are set to zero. This will prove to be a very useful exercise, as it will allow to determine the tensorial structures that appear, and in particular establish that the divergent part coming from ghost loops is entirely proportional to the tensor alone. Then, in Sect. IV we carry out the calculation in the momentum configuration. After manipulating all diagrams analytically (Sect. IV.1), we evaluate numerically all the contributions obtained, using (quenched) lattice results as input for the gluon and ghost two-point sectors (Sect. IV.2). Finally, in Sect. IV.3 we show how our results can be related to quantities customarily studied on the lattice. Specifically, we prove that the special momentum configuration chosen for our study has the property of isolating the 1-PI contribution to the connected four-gluon Green’s function. Then, assuming the most general tensor decomposition of this vertex in terms of tensors allowed by Bose symmetry, we show what would be the best choice of the ratios . The paper ends with Sect. V, where we draw our conclusions, and two Appendices. In the first, we carry out a general analysis of the tensor structures (quadratic in the metric) that are allowed by Bose symmetry, paying particular attention to the case . Finally, Appendix B collects some lengthy expressions appearing in our analytical calculations.

Ii Generalities on the four-gluon vertex

The 1-PI four-gluon vertex will be denoted by the expression (all momenta entering)

(1)

At tree-level one has

(2)

where are the real and totally antisymmetric SU(N) structure constants, satisfying the normalization condition

(3)

so that the generators of the adjoint representation are given by

(4)

In Fig. 1 we show the conventions of momenta and Lorentz/color indices used throughout this paper.

Note that, due to Bose symmetry, remains unchanged under the simultaneous interchange of a set of its indices and momenta (e.g. , etc). It is elementary to verify the validity of this symmetry for the tree-level vertex .

Figure 1: The 1-PI 4-gluon vertex; we display the momenta [note that ], together with the Lorentz and color indices.

It is clear that the fully dressed is characterized, in general, by a vast proliferation of the tensorial structures (138 for general kinematics Gracey (2014)); of course, as we will see, Bose symmetry imposes restrictions on the structure of the possible form factors composing .

At the level of the rank-4 Minkowski tensors, the structures allowed are terms quadratic in the metric, linear in the metric and quadratic in the momenta, and quartic in momenta; schematically one has then the structures

(5)

At the level of the rank-4 color tensors the situation is considerably more complex, since, in addition to terms quadratic in or , the real and totally symmetric tensors will also emerge. Thus, in principle one has 15 allowed structures of the schematic type

(6)

However, these tensors are related by a set of 6 identities Pascual and Tarrach (1980), namely

(7)
(8)

and two independent permutation for each, a fact that reduces the number of required tensors down to 9.

Of course, due to practical limitations, one must restrict the present study to a considerably more reduced (but physically relevant) subset of the full Lorentz and color tensorial basis mentioned above. Specifically, as was done in Pascual and Tarrach (1980), we only consider terms quadratic in the metric tensor , namely terms proportional to , , and , neglecting terms quadratic and quartic in the momenta. Thus, a priori, for a general SU(N) gauge group, one has possible combinations. Furthermore, we will directly specialize our analysis to the case , where the additional identity

(9)

can be used, thus reducing the number of tensorial combinations down to 24.

However, it turns out that, within the one-loop dressed approximation and the kinematical configuration that we will employ (see Fig. 2 for the 18 diagrams appearing in this case), the color tensors reduce finally to the two structures appearing in the conventional one-loop calculation of this vertex (for ), namely the tree-level tensor defined in Eq. (2), and the totally symmetric (both in Minkowski and color space) tensor

(10)

In particular, notice that since the two tensors are orthogonal in both spaces

(11)

the prefactors multiplying them can be unambiguously identified222As shown in Appendix A, Bose symmetry allows an additional tensor structure to appear; the consequences of this fact will be briefly addressed in Sect. IV.3..

Let us finally point out that, in SU(3), one has the additional useful formula

(12)

where we have defined the combination

(13)

Our analysis of the four-gluon vertex will be carried out in the Landau gauge, where the study of the lower Green’s functions (such as gluon and ghost propagator, ghost-gluon vertex and three-gluon vertex) has been traditionally carried out, both in the continuum as well as on the lattice. In this particular gauge the full gluon propagator takes the form

(14)

while the ghost propagator, , and its dressing function, , are related by

(15)

Evidently, both and constitute crucial ingredients for the calculations of the four-gluon vertex that follows. It is therefore useful to briefly review some of their IR features that are most relevant to the present work. Specifically, both large-volume lattice simulations and a plethora of continuous nonperturbative studies, carried out both in SU(2) and in SU(3), converge to the conclusion that the function reaches a finite (nonvanishing) value in the IR. Moreover, the nonperturbative ghost propagator remains “massless”, and displays no IR enhancement, since its dressing function saturates in the deep IR to a finite value. As we will see in what follows, the aforementioned features have far reaching consequences for the IR behavior of the four-gluon vertex. Specifically, as happens with the tree-gluon vertex, the masslessness of the ghost-loops contributing to produces a logarithmic IR divergence. What is, however, qualitatively distinct compared to the three-gluon case, is that, at least within the approximation scheme that we employ, this particular divergence does not manifest itself in the part proportional to , but rather in the orthogonal combination .

Iii Vanishing external momenta

In this section we consider the simplest possible kinematic case, where all the momenta of the external gluons are set to zero ().

iii.1 The calculation

Since we do no consider the contribution of quark-loops (pure Yang-Mills theory), the only representation that appears in our problem is the adjoint, whose explicit realization is given in Eq. (4).

For the various integrals appearing in this calculation we will employ the standard text-book results

(16)

where has been defined in Eq. (10), and the integral measure is , with the space-time dimension333Notice that we set instead of used in Pascual and Tarrach (1980). and the ’t Hooft mass.

Figure 2: The 18 diagrams contributing to the four-gluon vertex in the one-loop dressed approximation. The fishnet diagrams carry a statistical factor of . Lorentz, color and momentum flow are as in Fig. 1.

There are two particular tensorial structures that appear in a natural way in the calculations of the graphs shown in Fig. 2, namely

(17)

Then, using the relation444Note also the particular property , which is a consequence of the antisymmetric nature of the in Eq. (4), and can be directly verified using Eq. (18)

(18)

together with Eq. (12), it is straightforward to express these structures in terms of and ,

(19)

Turning to the explicit calculation of the one-loop dressed diagrams of Fig. 2, the (six) ghost boxes give the result

(20)

which, with the aid of the formulas (19) introduced above, may be written in the simple form

(21)

Since the ghost dressing function is known to saturate in the IR, the integral above diverges logarithmically in the IR; however Eq. (21) shows that this divergence does not contribute to the structures proportional to the tree-level tensor . Even though this result has been derived in a simplified setting, it will persist within the one-loop dressed approximation employed here. Therefore, we arrive at the important conclusion that the IR divergent terms originating from the ghost loops would be completely missed, if one were to consider only the form factor proportional to the tree-level tensor .

We next consider the (three) gluon boxes; as the adjoint traces will be the same as those appearing in the ghost case above, we obtain that also the one-loop dressed gluon boxes do not contribute to the tree-level tensor structure. In particular, we get

(22)

Notice that, unlike the case of the ghost boxes treated above, the integral appearing in Eq. (22) is convergent in the IR, because the gluon propagator reaches a finite value in that limit.

We now turn to the (six) triangle diagrams. After some straightforward algebraic manipulations, one obtains

(23)

which, after using the identities (19), can be cast in the form

(24)

where

(25)

Finally, we are left with the (three) fishnet diagrams. One finds, similarly to what happens with the triangle diagrams,

(26)

The identities (19) allow us to express the result in its final form, namely

(27)

with

(28)

The results obtained are conveniently summarized in Table 1.

Diagrams Integral
0
0
Table 1: Contributions of the various class of diagrams to the four-gluon vertex in the one-loop dressed approximation with all external momenta set to zero.

iii.2 Perturbative analysis

At this point one may explore the qualitative behavior of the two contributions obtained above within a setting inspired by one-loop perturbation theory, but supplemented by a set of mass scales, which prevent the resulting expressions from diverging in the IR. Specifically, if one were to simply set and to their strict perturbative values ( and , respectively) the four integrals appearing in the second column of Table 1 reduce to a single integral, namely . At this point, it is easy to verify that, when , the total contribution proportional to vanishes, given that the sum of the coefficients appearing on the fourth column adds up to zero.

However, given that the integral is both IR and UV divergent, it is preferable to introduce a distinction between the two type of divergences. To accomplish this, we proceed as follows. Given that the (Euclidean) gluon propagator (in the Landau gauge) is known to be finite in the IR (a feature that can be self-consistently explained through the dynamical generation of an effective gluon mass), for the purposes of this simple calculation one may approximate by . This replacement makes the integrals , , and of Table 1 IR finite; of course, they still diverge logarithmically in the UV. Regarding the integral , it is known that the ghost remains nonperturbatively massless, a fact that leads to a genuine IR divergence; in order to control it, we will introduce an artificial mass scale, denoted by . Thus, the integral corresponding to will read .

Let us emphasize at this point that even though at the formal level both and serve as IR regulators, there is a profound physical difference between the two: constitutes a simplified realization of a true physical phenomenon, namely the IR saturation of the gluon propagator, while is an artificial scale, introduced as a regulator of a quantity (the ghost propagator) that is genuinely massless. Consequently, in order to recover the physically relevant (albeit simplified) limits, will be kept at some fixed nonvanishing value, while will be sent to zero.

The above considerations motivate the introduction of a particular integral, namely

(29)

where is the ’t Hooft mass, and the Euler-Mascheroni constant. Evidently, depending on the case that one considers, or .

In particular, after the replacements mentioned above, the integrals in Table 1 can be expressed in terms of as follows

(30)

where the ellipses in the last two expressions indicate linear combinations of the integrals555These latter integrals appear simply through the elementary algebraic manipulation in the numerators, and the subsequent cancellation of some of the denominators. or , which are convergent both in the IR and the UV.

At this point one may add up the corresponding contributions in the third and fourth columns of Table 1 and obtain, within this perturbative scheme, the coefficients multiplying and , to be denoted by and , respectively. Specifically, setting everywhere (but keeping in ), introducing , factoring out a to conform with the definition in Eq. (1), we find for the leading behavior

(31)

which, after the inclusion of the tree-level term, and use of Eq. (29), becomes

(32)

and

(33)

Evidently, all dependence on is contained in the coefficient multiplying , while the coefficient of is completely free of such terms, exactly as one would expect from the renormalizability of the theory. Indeed, given that the term does not appear in the original Lagrangian, a divergence of this type could not be renormalized away. Instead, the divergence proportional to will be reabsorbed in the standard way, namely through the introduction of the appropriate vertex renormalization constant, to be denoted by .

Specifically, one obtains the renormalized vertex from its unrenormalized counterpart through the condition (suppressing all indices)

(34)

Of course, the exact form of the and the resulting depend on the renormalization scheme chosen. In particular, in the minimal subtraction (MS) scheme one would simply have

(35)

which, upon multiplication with the of Eq. (32) will give (keeping up to terms of order ) the finite result

(36)

Note that coincides with the part proportional to of the corresponding expression given in (3.10) of Pascual and Tarrach (1980) (in the Landau gauge, and for ).

If one were instead to renormalize in the minimal subtraction (MOM) scheme, as is customary in lattice simulations and SDE studies, one would need to introduce a renormalization point, , and demand that at that point the value of the renormalized vertex reduces to its tree-level value. For instance, as in Pascual and Tarrach (1980), the completely symmetric choice and may be employed; then, the corresponding would read (in general)

(37)

such that (schematically)

(38)

Of course, for the case at hand, since the vertex has been computed only for vanishing momenta, one cannot implement a MOM-type scheme. However, in order to get a sense of the general trend that one might expect from a general calculation, we may assume that the subtraction point lies sufficiently far in the UV. Then, for a representative large Euclidean momentum , the qualitative behaviour of the form factor may be approximated by

(39)

so that, at one obtains

(40)

and therefore, the value of gets renormalized to

(41)

As happens typically, in the finite result the ’t Hooft scale has been replaced by the renormalization scale .

It is obvious at this point, that the above approximations require that , and, consequently, since the logarithm becomes negative, . To obtain a quantitative notion of the effect, we will use lattice-inspired values for and ; specifically, if we identify the saturation point of the gluon propagator on the lattice with , we know that, for GeV we have that MeV. Then, using that, for this particular , , we finally find

(42)

Quite interestingly, this apparent tendency of the quantum corrections to reduce the tree-level value persists in the full one-loop dressed calculation; in fact, the value quoted in Eq. (42) is fairly close to the one found in the next section.

Turning to the in Eq. (33), we notice that, when the artificial IR cutoff is taken to zero, while the physical gluon mass is kept at a nonvanishing value, the logarithm diverges to . Again, this coincides with the behavior found in the more complete calculation of the next section. Of course, the slope of the logarithm found in Eq. (33) is numerically rather suppressed when compared to the result found in the next section; however, this is to be expected, given that the function , which in Eq. (21) is raised to the fourth power, is considerably different from 1 in the IR and intermediate momenta.

Iv The special momentum configuration

Even within the one-loop dressed approximation we are employing, the calculation of the four-gluon vertex for a generic external momenta configuration (such as the one depicted in Fig. 1) is still a complex task. In addition, it is not the most expeditious way to obtain information about the IR dynamics of this vertex that could be easily contrasted with lattice simulations.

Thus, we will study a relatively simple kinematic configuration, which is obtained choosing a single momentum scale and identifying the momentum flow (see Fig. 1) with (and hence ). This kinematic configuration gives rise to loop integrals that are fully symmetric under the crossing of external legs; therefore, the crossed diagrams may be obtained from the original ones through simple permutations of the color and Lorentz indices.

As before, we will only consider terms that are quadratic in the metric . This choice, in addition to simplifying the algebraic structures considerably, corresponds precisely to the contributions that would survive on the lattice, if one were to consider the standard quantities employed in the simulations of vertices Cucchieri et al. (2006, 2008) (we will return to this point in Sect. IV.3).

iv.1 Analytical results

Consider the contribution of the ghost boxes. The aforementioned crossing property implies that the six different diagrams are proportional to the same integral. As a result, one obtains, similarly to what happens in the case,

(43)

where now

(44)

It can be easily checked that as , above reduces to the of Eq. (21); therefore we expect that the form factor will develop a (logarithmic) divergence in the deep IR.

Next, we consider the gluon boxes. The uncrossed diagram shown in Fig. 2, yields the general expression

(45)

where the integrals are not needed for the moment. Crossed diagrams are then obtained from the above expression through the replacements , and , . In addition, it turns out that the integrals and are equal666Notice that without this equality the gluon box contributions would lie outside the subset of all possible color and Lorentz tensor structures spanned by and . Moreover, observe that the realization of this equality requires shifts of the integration variable of the type ; of course, since only terms quadratic in the metric are kept, one consistently drops in the numerators terms produced by these shifts that are proportional to and carry Lorentz indices of the external legs. upon the momentum shifting , so that can be cast in the form

(46)

Thus, adding the three diagrams, one obtains

(47)

where

(48)

and the functions are reported in Eq. (93).

Figure 3: The 6 one-loop dressed triangle diagrams subdivided in two classes, containing three diagrams each, proportional to independent momentum integrals.

We next consider the triangle diagrams. In this case the six graphs can be divided in two separate classes (see Fig. 3), proportional to two independent momentum integrals, namely [class ] and [class ]. Let us then start from the first diagram of the class (see again Fig. 3); one obtains the general result

(49)

where

(50)

where again and are integrals whose explicit expression is not needed at this point.

Within this class, the remaining diagrams are then obtained through the replacements , and , . Thus, summing up all the graphs, one obtains, similarly to the zero external momentum case Eq. (24), the result

(51)

where

(52)

with the functions given in Eq. (94).

Similarly, for the class we obtain

(53)

where now

(54)

and the functions given in Eq. (95).

We are finally left with the fishnet diagrams. The uncrossed diagram of Fig. 2 yields

(55)

where