Wilson lines - color charge densities correlators and the production of \eta^{\prime} in the CGC for pp and pA collisions

# Wilson lines - color charge densities correlators and the production of η′ in the CGC for pp and pA collisions

François Fillion-Gourdeau    Sangyong Jeon Department of Physics, McGill University, 3600 University Street, Montreal, Canada H3A 2T8
July 13, 2019
###### Abstract

We compute the inclusive differential cross section production of the pseudo-scalar meson in high-energy proton-proton () and proton-nucleus () collisions. We use an effective coupling between gluons and meson to derive a reduction formula that relates the production to a field-strength tensor correlator. For collisions we take into account saturation effects on the nucleus side by using the Color Glass Condensate formalism to evaluate this correlator. We derive new results for Wilson line - color charges correlators in the McLerran-Venugopalan model needed in the computation of production. The unintegrated parton distribution functions are used to characterize the gluon distribution inside protons. We show that in collisions, the cross section depends on the parametrization of unintegrated parton distribution functions and thus, it can be used to put constraints on these distributions. We also demonstrate that in collisions, the cross section is sensitive to saturation effects so it can be utilized to estimate the value of the saturation scale.

## I Introduction

One of the main challenges in high energy hadronic collisions is the understanding of particle production. The theoretical description of these complex phenomena involves both many-body physics and the theory of strong interactions. There are many approaches that aim toward a better comprehension of these topics. One of the most successful technique is perturbative quantum chromodynamics (pQCD) where one studies the limit where the coupling constant is small and where the usual loop expansion can be used in principle. Because QCD is an asymptotically free theory, this happens when the exchanged momenta are large compared to the QCD scale . Even in that regime however, physical observables computed using this machinery suffer from infrared divergences that spoil the naive perturbative expansion. These have to be resummed and this leads to various factorization formalisms like the collinear factorization and the -factorization. In collinear factorization, meaningful physical quantities are obtained in terms of parton distribution functions (PDF). These distributions characterize the non-perturbative (large distance) physics and have to be determined experimentally from a fit to structure functions. This formalism can be applied on system where the typical exchanged momentum is hard, which means that it satisfies the inequality given by where is the QCD scale and is the center of mass energy. Note that , which means that unless the mass of the produced particle is of the order of , the collinear factorization is valid only for very large . This can be relaxed in the -factorization formalism which considers semihard collisions, meaning that the typical exchanged momentum obeys . The resummation implemented in this approach takes care of large contributions that look like , and Collins and Ellis (1991); Catani et al. (1991); Gribov et al. (1983); Kuraev et al. (1977). This technique is used successfully to compute the production of many kinds of particles in high energy proton-proton collisions like heavy quarks Collins and Ellis (1991); Luszczak and Szczurek (2006a); Lipatov et al. (2001); Zotov et al. (2003); Jung (2002) and in a number of other processes (see Andersson et al. (2002); Andersen et al. (2004, 2006) for reviews of many applications). It is also used for predictions of Higgs boson production Luszczak and Szczurek (2006b); Lipatov and Zotov (2005). In this article, we use this formalism to compute the inclusive cross section at the RHIC energy.

When there is a nucleus involved in a collision at very high energy, there are new effects not included in the previous approaches due to the high density of gluons resulting from the emission enhancement at small- (where is the momentum fraction). These effects introduce a new scale called the saturation scale at which the probability of having interactions between gluons of the same nucleus becomes important. At this transverse momentum scale, the gluons recombine and this slows down the growth of partons at smaller . A naive estimation of shows that it depends on and the number of nucleons like Iancu et al. (2002); Iancu and Venugopalan (2003) so at small enough or large enough , the saturation scale is hard (). When the typical exchanged momentum is smaller than the saturation scale such as , saturation effects have to be taken into account even if the system is still in the perturbative regime. This can be achieved in the Color Glass Condensate (CGC) formalism which is a semi-classical effective theory where the non-linearities are dealt with by solving exactly the Yang-Mills equation of motion. This takes care of gluon recombinations and introduces the effects of saturation in observables.

In this article, we are using the CGC to compute the inclusive differential cross section of meson in collisions at the RHIC energy (). The main goal of this work is to look at the effect of saturation in production to validate the CGC approach and estimate the value of the saturation scale by comparing our predictions with experimental data. The is a pseudoscalar meson with a mass of , a decay width of and quantum numbers of Yao et al. (2006). One of the most important features of is that it couples to the QCD anomaly ’t Hooft (1976); Witten (1979). One way to implement and model this physics is by introducing an effective interaction between gluons and mesons. This was done in Atwood and Soni (1997), where the authors are proposing a vertex that couples two gluons and a meson () to explain B mesons decay (). This vertex contains a form factor that depends generally on gluons and momenta and that can be related to the wave-function. The structure of this vertex was investigated thoroughly using various techniques like the hard scattering and the running coupling approaches. Ali and Parkhomenko (2003, 2002); Muta and Yang (2000); Ahmady et al. (1998a); Agaev and Stefanis (2004); Kroll and Passek-Kumericki (2003). We use these results on the gluons- coupling to study the production mechanism based on gluon fusion.

The production in collisions at high energy was studied in Szczurek et al. (2007) where the exclusive cross section for the diffractive process is computed. In our study, we focus on the inclusive production mechanism which shares similar features with this previous analysis. The first attempt to compute production in high energy collisions was done by one of the present author in Jalilian-Marian and Jeon (2002). In this study, the collinear factorization is used to compute the cross section at RHIC by including intrinsic transverse momentum in the PDF with a Gaussian distribution. Based on physical arguments, the width of the Gaussian, which represents the typical transverse momentum of gluons inside the nucleus, is chosen to be . The authors show that the production is sensitive to the saturation scale implemented in this way. However, they acknowledge that their calculation can be improved because they use the collinear formalism outside its range of validity. The goal of this article is to revisit the production with a more rigorous approach by doing a full CGC computation that includes recombination effects more realistically.

The computation of meson production in the CGC was undertaken in the past using mostly an “hybrid” approach where the proton and the nucleus are described by the collinear factorization and the CGC respectively. In this formalism based on pQCD-like techniques, the fragmentation function of collinear factorization is convoluted with the gluon or quark cross section computed in the CGC formalism. This is suitable for well-known mesons like pions for which a wealth of experimental data have been measured and for which fragmentation functions are well-known. Pion production for collisions is computed in Dumitru et al. (2006a, b); Tuchin (2008) using this methodology. Contrary to pions, the data in high energy hadronic collisions for is scarce, so another approach is required to take care of hadronization effects and internal structure of the meson. In Fillion-Gourdeau and Jeon (2008), an effective theory is used to estimate the tensor meson production in collisions. We use a similar approach in this article where the interaction between gluons and is described by an effective theory. As discussed previously, we include these effects in an effective vertex that includes a form factor. As will be shown in this article, this can be implemented easily in the CGC formalism.

We consider only the case of and collisions at RHIC. For nucleus-nucleus () collisions, the total number of -mesons produced by semihard collisions in the first instants (for ) should be important and the saturation effects would also be present. Experimentally however, they cannot be detected because most of them decay inside the medium created by the collision. This is because the mean lifetime, which is about , is smaller than the time where the medium exists, which is from up to . Moreover, by considering and , we avoid all the complications that would result from the creation of the medium which include the understanding and modelling of the quark-gluon plasma properties. Finally, there are analytical solutions for the gauge field in and collisions, while the analytical solution in is still elusive. For these reasons, our present analysis is only applied to and collisions.

This article is organized as follows. In section II we describe the effective vertex used throughout the rest of the article. In section III, we show how to compute production in the CGC for and collisions. We start by deriving a reduction formula that relates the cross section to a correlator of field-strength tensors. This correlator is then evaluated to leading order in collisions and the result can be interpreted in terms of physical processes. We also show how the -factorized cross section for collisions can be recovered in the low density limit of cross section. In section IV, we compute the correlators appearing in the expression of the cross section using the McLerran-Venugopalan model. In section V, we evaluate numerically the cross section for and and discuss the range of validity of our computation. Sections III and IV contain a lot of technical details. The reader interested in results can jump directly to section V.

Throughout the article, we use both light-cone coordinates defined by

 p+ = p0+p3√2;p−=p0−p3√2 (1)

and Minkowski coordinates. It should be clear by the context which one is used. We also use the metric convention .

## Ii Effective Theory

The effective theory used in this article couples gluons and the meson. In momentum space, the effective vertex (where means off-shell gluon) is given by

 Vμν(M,p,q)=iF(p2,q2,M2)δabϵμνραpρqα (2)

where is the Levi-Civitta antisymmetric tensor, is the mass, and are color indices, and are gluon momenta and is the form factor. The explicit expression of the interaction vertex have been studied in a number of articles where different parametrizations of the form factor can be found Atwood and Soni (1997); Ali and Parkhomenko (2003, 2002); Muta and Yang (2000); Ahmady et al. (1998a); Agaev and Stefanis (2004); Kroll and Passek-Kumericki (2003). To get a first approximation of production and because we are mostly interested in making a comparative study between and collisions, we use a simple expression given by Szczurek et al. (2007)

 F(p2,q2,M2)=H0M4(M2−p2)(M2−q2) (3)

where . To get a better approximations of production, other parametrizations should be used. In the limit of no gluon virtualities (), the form factor is a constant that can be fixed by looking at the decay of . It is given by Atwood and Soni (1997).

The main applications of this coupling are related mostly to and decay where processes such as and are considered Atwood and Soni (1997); Kagan and Petrov (1997); Hou and Tseng (1998); Ahmady et al. (1998b); Du et al. (1998). More recently, gluon fusion was used to compute production in high energy hadronic collisions Szczurek et al. (2007); Jalilian-Marian and Jeon (2002); Jeon and Jalilian-Marian (2002) and from a thermalized medium Jeon (2002).

It is convenient for our purpose to consider the interaction Lagrangian given by

 Lint(x)=18∫d4yd4zF[(x−y)2,(x−z)2,M2]Gμνa(y)˜Ga,μν(z)η(x). (4)

that reproduces the vertex Eq. (2) in the perturbative expansion. As seen in the next section, this can then be used to derive a reduction formula. Here, is the usual field-strength tensor given by

 Gμνa(x)=∂μAνa(x)−∂νAμa(x)−gfabcAμb(x)Aνc(x) (5)

where is the gauge field of gluons and is the dual field-strength tensor. The Lagrangian is non-local because the vertex includes a form factor. It can be easily seen that contains three types of vertices, namely , and . At leading order however, only the first one is necessary and considered in this article.

## Iii Production of η′ from the CGC

In collisions at very high energy, the wave function of nuclei is dominated by gluons that have small longitudinal momenta (soft gluons) because of the emission enhancement at small-. The CGC is a semi-classical formalism that describes the dynamics of these degrees of freedom. In this approach, the hard partons, which carry most of the longitudinal momentum, and soft gluons which have small longitudinal components, are treated differently. Because the occupation number of the soft gluons is large, classical field equations can be employed to understand their dynamics. The hard partons act as sources for these classical field and are no longer interacting with the rest of the system (for reviews of CGC, see Iancu et al. (2002); Iancu and Venugopalan (2003); Venugopalan (2004)).

In this formalism, computing a physical quantity involves two main steps. The first one is to solve the Yang-Mills equation of motion

 [Dμ,Fμν(x)]=Jν(x) (6)

where the current represents random static sources localized on the light-cone Iancu et al. (2002); Iancu and Venugopalan (2003) and is the covariant derivative. The functions are color charge densities in the transverse plane of the proton and nucleus respectively. The next step is to take the average over the distribution of color charge densities in the nuclei with weight functionals . For any operator that can be related to color charge densities, this can be written as

 ⟨^O⟩=∫DρpDρAO[ρp,ρA]Wp[xp,ρp]WA[xA,ρA]. (7)

Computing the weight functional is a highly non-perturbative procedure so it usually involves approximations based on physical modelling. In the limit of a large nuclei at not too small , it can be approximated by the McLerran-Venugopalan (MV) model, which assumes that the partons are independent sources of color charge McLerran and Venugopalan (1994a, b). Within this assumption, the weight functional is a independent Gaussian distribution and the two point correlator is simply McLerran and Venugopalan (1994a, b); Iancu et al. (2002); Iancu and Venugopalan (2003)

 ⟨ρA,a(x⊥)ρA,b(y⊥)⟩=δabμ2Aδ2(x⊥−y⊥) (8)

where is the average color charge density and is the radius of the nucleus. It is assumed here that the nucleus has an infinite transverse extent with a constant charge distribution. Edge effects can be included by changing and by choosing a suitable transverse profile. Throughout this article, we only consider the constant distribution case.

Within the MV model, the weight functional does not depend on longitudinal coordinates and therefore, the model is boost invariant. This however can be relaxed by considering the quantum version of the CGC. In that theory, quantum radiative corrections become important below a certain scale . These corrections can be resummed by using a renormalization group technique which leads to the JIMWLK evolution equation Iancu et al. (2001); Ferreiro et al. (2002); Jalilian-Marian et al. (1997a, 1999, b). In the quantum CGC, the weight functionals obey this non-linear evolution equation in . Because the MV model is valid in the range , it can be used as an initial condition for the evolution at smaller . In this article however, we consider only the regime where the MV model is valid and do not consider the small- evolution although it could be done in principle.

On the proton side, the average computed with can be related to the unintegrated parton distribution function (uPDF) like

 g2⟨ρ∗p,a(p⊥)ρp,b(q⊥)⟩ = 4π2δab(N2c−1)[p⊥+q⊥2]2 (9) ×∫d2y⊥ei(p⊥−q⊥)⋅y⊥dϕ1(x,p⊥+q⊥2|y⊥)d2y⊥

where is the number of color. By construction, the uPDF obeys

 ∫d2y⊥dϕ1(x,p⊥|y⊥)d2y⊥=ϕ1(x,p⊥) (10)

and is normalized such that

 ∫μ20ϕ1,2(x,p⊥)≈xG(x,μ2) (11)

where is the collinear parton distribution function and is the factorization scale. The uPDF can be obtained from a fit to structure function and evolved to the desired value of , and using evolution equations such as the BFKL or the CCFM equations.

One important ingredient is missing for the computation of meson production cross section. We need a relation between the cross section and a correlator that can be evaluated using the CGC formalism. This is done in the next section using a reduction formula and the effective theory.

### iii.1 Reduction Formula and the Cross Section

The computation of mesons from the CGC can be calculated from a reduction formula. The starting point is the expression of the average number of produced per collisions given by where is the probability to produce particles. This can be converted to an equation in terms of creation/annihilation operators that can be evaluated in quantum field theory. This is given by Baltz et al. (2001)

 (2π)32Ekd¯nd3k=⟨0in|^a†out(k)^aout(k)|0in⟩ (12)

where is the in vacuum. Then, the standard LSZ procedure can be used to write this as Itzykson and Zuber (1980)

 (2π)32Ekd¯nd3k=1Z∫d4xd4yeik⋅(x−y)[∂2x+M2][∂2y+M2]⟨0in|^η(x)^η(y)|0in⟩ (13)

where is the wave function normalization and where we assumed the asymptotic conditions for the field operator in Heisenberg representation. It is possible to use the equation of motion of given simply by

 (∂2+M2)^η(x) = ^T(x) (14)

where we defined to rewrite the reduction formula in a convenient way. We get finally that

 (2π)32Ekd¯nd3k=⟨^T†(k)^T(k)⟩ (15)

where is the Fourier transform of evaluated at a meson on-shell momentum and where we set . The angular brackets here indicates expectation value of in the initial state.

The only assumptions used in deriving Eq. (15) are:

• There are no mesons in the initial state.

• The is produced on-shell.

The first assumption is justified by the fact that in high-energy collisions, the number of in a hadron before the collision (in the initial state) is negligible. This allows us to use the in vacuum and the fact that to simplify the reduction formula. Using the second assumption, we can treat the meson as a stable particle which can be produced on-shell and which is well described by the free spectral density that looks like . Therefore, by making this assumption, it is possible to use the asymptotic conditions described earlier. However, -mesons are resonances, so the spectral density should look rather as a Breit-Wigner function where is the decay width. These effects however are taken into account by the form factor .

Eq. (15) is the main result of this section. It relates the average number of mesons produced to a correlator of field strength tensors. This correlator can then be evaluated using any analytical or numerical methods. The average depends on the system studied. Looking at a plasma at equilibrium, it could be computed using finite temperature field theory or the AdS/CFT correspondence. These two formalisms are relevant to nucleus-nucleus collisions where a medium at equilibrium is created. We are interested here in and collisions where no such medium is formed so these techniques are not pursued in this study. Rather, we use the CGC which describes initial state and saturation effects in high-energy hadronic collisions.

Having expressed the average number of produced in terms of a correlator of field strength tensors, it is possible to compute the inclusive cross section in the CGC formalism which is given by Gelis and Venugopalan (2004a); Baltz et al. (2001)

 (2π)32Ekdσd3k = ∫d2b⊥(2π)32Ekd¯n(b⊥)d3k (16) = ∫d2b⊥∫DρpDρAT∗[ρp,ρA]T[ρp,ρA] ×Wp[xp,ρp]WA[ρA;b⊥].

In this expression, is the impact parameter. The fields are functionals of the source once the Yang-Mills equation of motion of the gauge field is solved (see Eq. (5) for the expression of the field strength tensor as a function of the gauge field).

### iii.2 Cross section in pA Collisions

In collisions, there are two saturation scales (one for the proton () and one for the nucleus ()) that satisfy . When the transverse momentum of the is small enough, the nucleus is in a saturation state while the proton is not because we have (remember that is the transverse mass of the ). In that case, the system is semi-dilute, meaning that one of the source is strong (or equivalently, the typical transverse momentum is small) and obeys while the other source is still weak Venugopalan (2004); Gelis and Venugopalan (2004b). The weak source can be used as a small parameter to solve the Yang-Mills equation perturbatively. The solution of the gauge field can be computed analytically to all orders in and to first order in in different gauges Blaizot et al. (2004a); Dumitru and McLerran (2002); Gelis and Mehtar-Tani (2006); Kovchegov and Mueller (1998). We use here the solution in the light-cone gauge of the proton Gelis and Mehtar-Tani (2006) but in Appendix C, we perform the same calculation in covariant gauge to show that our result is gauge invariant.

#### iii.2.1 Gauge Field and Power Counting

In the light-cone gauge of the proton () with a nucleus in covariant gauge moving in the negative direction, the solution of the gauge field in collisions can be separated in three parts where is the field associated with the proton (of ), is the field associated with the nuclei (of ) and is the field produced by the collision (of ) Gelis and Mehtar-Tani (2006). Note that the field is strong and satisfies . The explicit solution is given by Gelis and Mehtar-Tani (2006); Fukushima and Hidaka (2008)

 Aip,a(k) = 2πgδ(k−)kik++iϵρp,a(k⊥)k2⊥ (17) A−A,a(k) = 2πgδ(k+)ρA,a(k⊥)k2⊥ (18) AipA,a(k) = −igk2+ik+ϵ∫d2p⊥(2π)2[ki(k++iϵ)(k−+iϵ)−2pip2⊥]ρp,b(p⊥) (19) ×[Uab(k⊥−p⊥)−(2π)2δ2(k⊥−p⊥)δab]

where is a Wilson line in adjoint representation defined in Eq. (35), is the usual QCD coupling constant, is the Kronecker delta in color space and is the structure constant of the group. The component is non-zero and is related to but it does not appear in the final expression of the cross section so it is not needed in the computation of production. All the other components are zero.

The production cross section of mesons is related to a field strength tensor correlator given by

 B(k) ≡ ∫d4pd4q(2π)8F(p2,p22)F∗(q2,q22)⟨G∗μνa(p)˜G∗μνa(k−p)Gαβb(q)˜Gαβb(k−q)⟩ (20)

where . In principle, a correlator like this contains contributions from all orders in both sources. Because the proton source is weak, it is possible to simplify this considerably using a power counting argument to isolate the leading order contribution. For the sake of this power counting argument, we use which denotes terms having gauge fields and where and . At first, let us consider only the contributions from the abelian part of the field-strength tensor. The terms in these contributions have four powers of gauge field such as . Naively, one would expect the dominant contribution to come from terms that have many powers of the nucleus gauge field like and . However, these terms vanish because of the Lorentz structure. For example, a typical term in the abelian contribution would look like . When we sum on indices, this kind of term will contain at most one strong gauge field . Thus, the dominant contributions are like . Using a similar argument, it is possible to show that the non-abelian part have no leading order contribution in the sense that it is at least . The possible higher order contributions like for example also vanish because of the Lorentz structure of the correlator. This is because the typical non-abelian contributions look like and . Once the Lorentz indices are summed, the second typical term because in this gauge, . For , it contains only one strong field but it contains two powers of weak field like . Thus, when it is squared, it gives at most a contribution of .

### iii.3 Evaluation of the Correlator

Using the explicit expression of the field strength tensor in terms of gauge field and keeping only the dominant and non-zero contributions, the correlator can be written as

 B(k) = 64∫d4pd4q(2π)8F(−p2⊥,−p22,⊥)F∗(−q2⊥,−q22,⊥)(k+−p+)(k+−q+)ϵijϵklpiqk (21) ×⟨A−∗A,a(p)A−A,b(q)Aj∗s,a(k−p)Als,b(k−q)⟩

where is the Levi-Civitta antisymmetric tensor with and where we defined .

To obtain the preceding expression we make the assumption that the virtualities in the form factors are due solely to transverse momentum such as . This approximation is necessary to recover -factorization in the dilute limit as seen in section III.5.

It is convenient to separate in four different terms such as

 Bz,z′(k) = 64g2(k+)2∫dp−dq−d2p⊥d2q⊥(2π)6F(−p2⊥,−p22,⊥)F∗(−q2⊥,−q22,⊥)ϵijϵklpiqkp2⊥q2⊥ (22) ×⟨ρ∗A,a(p⊥)ρA,b(q⊥)Aj∗z,a(k−p)Alz′,b(k−q)⟩∣∣p+=q+=0

where and where we performed the integration on and using the delta functions in Eq. (18). These four terms can be evaluated explicitly by substituting the solution of gauge fields Eqs. (17), (18) and (19).

For the first term, it is a straightforward calculation to show that

 Bp,p(k) = 64g4∫d2p⊥d2q⊥d2r⊥d2s⊥(2π)8F(−p2⊥,−p22,⊥)F∗(−q2⊥,−q22,⊥) (23) ×ϵijϵklpirjqkslp2⊥q2⊥r2⊥s2⊥δbc(2π)2δ2(k⊥−p⊥−r⊥)δad(2π)2δ2(k⊥−q⊥−s⊥) ×⟨ρ∗p,d(r⊥)ρp,c(s⊥)⟩⟨ρ∗A,a(p⊥)ρA,b(q⊥)⟩.

The calculation of the second term is similar but requires some more work. By direct substitution of the expression of the gauge fields, we have

 BpA,p(k) = 64ig4(k+)∫dp−d2p⊥d2q⊥d2r⊥(2π)7F(−p2⊥,−p22,⊥)F∗(−q2⊥,−q22,⊥) (24) ×ϵijϵklpiqkql2p2⊥q2⊥q22,⊥1p22−ik+ϵ[pj2(p+2−iϵ)(p−2−iϵ)−2rjr2⊥] ×[⟨ρ∗A,a(p⊥)ρA,b(q⊥)U∗ac(p2,⊥−r⊥)⟩⟨ρ∗p,c(r⊥)ρp,b(q2,⊥)⟩ −(2π)2δ2(p2,⊥−r⊥)δac⟨ρ∗A,a(p⊥)ρA,b(q⊥)⟩⟨ρ∗p,c(r⊥)ρp,b(q2,⊥)⟩].

The integration on the longitudinal momentum can be done by looking at the analytical structure of the equation. First, we write a part of the integrand as

 1(k−p)2−ik+ϵ[pj2(k+−iϵ)(k−−p−−iϵ)−2rjr2⊥]∣∣ ∣∣p+=0= 12k+[k−−p−−(k−p)2⊥2k+−iϵ][pj2(k+−iϵ)(k−−p−−iϵ)−2rjr2⊥]. (25)

In the complex plane of , the first term in the RHS of Eq. (III.3) has two poles on the same side of the real axis and goes like when . Thus, closing the integration contour in the upper-half plane and using the residue theorem, the integration on of this term leads to a zero contribution because the contour at infinity has a zero contribution and because the contour does not enclose any singularities. For the second term of Eq. (III.3), we use the principal part () identity . The integration on the principal part is zero because the integrand does not depend on and while the delta function integration is trivial. We finally get

 BpA,p(k) = 32g4∫d2p⊥d2q⊥d2r⊥d2s⊥(2π)8F(−p2⊥,−p22,⊥)F∗(−q2⊥,−q22,⊥) (26) ×ϵijϵklpiqkrjslp2⊥q2⊥r2⊥s2⊥⟨ρ∗p,d(r⊥)ρp,c(s⊥)⟩ ×[−⟨ρ∗A,a(p⊥)ρA,b(q⊥)⟩δbc(2π)2δ2(p2,⊥−r⊥)δad(2π)2δ2(q2,⊥−s⊥) +⟨ρ∗A,a(p⊥)ρA,b(q⊥)U∗ad(p2,⊥−r⊥)⟩δbc(2π)2δ2(q2,⊥−s⊥)].

The calculations of and are similar. Going through the same steps as for , we get

 Bp,pA(k) = 32g4∫d2p⊥d2q⊥d2r⊥d2s⊥(2π)8F(−p2⊥,−p22,⊥)F∗(−q2⊥,−q22,⊥) (27) ×ϵijϵklpiqkrjslp2⊥q2⊥r2⊥s2⊥⟨ρ∗p,d(r⊥)ρp,c(s⊥)⟩ ×[−⟨ρ∗A,a(p⊥)ρA,b(q⊥)⟩δbc(2π)2δ2(p2,⊥−r⊥)δad(2π)2δ2(q2,⊥−s⊥) +⟨ρ∗A,a(p⊥)ρA,b(q⊥)Ubc(q2,⊥−s⊥)⟩δad(2π)2δ2(p2,⊥−r⊥)]

and