A Light Cone Hamiltonian

# Initial state qqg correlations as a background for the Chiral Magnetic Effect in collision of small systems.

## Abstract

Motivated by understanding the background to Chiral Magnetic Effect in proton-nucleus collisions from first principles, we compute the three particle correlation in the projectile wave function. We extract the correlations between two quarks and one gluon in the framework of the Color Glass Condensate. This is related to the same-charge correlation of the conventional observable for the Chiral Magnetic Effect. We show that there are two different contributions to this correlation function. One contribution is rapidity-independent and as such can be identified with the pedestal; while the other displays rather strong rapidity dependence. The pedestal contribution and the rapidity-dependent contribution at large rapidity separation between the two quarks result in the negative same charge correlations, while at small rapidity separation the second contribution changes sign. We argue that the computed initial state correlations might be partially responsible for the experimentally observed signal in proton-nucleus collisions.

1

## I Introduction

Topological fluctuations in the early time Glasma state Mace et al. (2016, 2017) or thermal sphaleron transitions Arnold and McLerran (1987); Moore and Tassler (2011) during later stages of heavy-ion collisions may lead to the Chiral Magnetic Effect (CME) Kharzeev et al. (2008), the generation of the electromagnetic current along the magnetic field Skokov et al. (2009); Bzdak and Skokov (2012); McLerran and Skokov (2014). Experimentally, the associated charge separation can be measured by three particle angular average Voloshin (2004), , where is the azimuthal angle of a trigger particle defining the reaction plane and are azimuthal angles of associate particles carrying electric charge. The averaging is usually taken over a range of transverse momenta of the charged particles. The observable is also often considered as a function of relative rapidity separation between the charge particles and the multiplicity/centrality of the collision. The charge of the particles and can be either same or opposite. For more information about the experimental measurements, see Ref. Voloshin (2004); Abelev et al. (2009, 2010); Khachatryan et al. (2017); Skokov et al. (2016); Tribedy (2017).

The CME prediction for the observable can be understood as follows. In the presence of the strong magnetic field, , and initial axial charge , the CME builds an electric current along the magnetic field Kharzeev et al. (2008), which in non-central heavy-ion collisions points in the out-of-plane directions, see Refs. Skokov et al. (2009); Bzdak and Skokov (2012). The current results in the transport of the charges and subsequent formation of a dipole moment in the charge distribution, which can be described by

 dNαdϕ=N(1+2v1cos(ϕ−ψRP)+2v2cos(2[ϕ−ψRP])+2aαsin(ϕ−ψRP)+…), (1)

where is the reaction plane angle (neglecting the fluctuations, the magnetic field is perpendicular to the reaction plane), is the directed flow, is the elliptic flow and denotes the charge of the particles. The parameters describe the formation of the electric dipole . The sign of fluctuates on event by event basis rendering . Nevertheless, the parity-even fluctuations, , can still be measured in experiment. The observable suppresses the background Voloshin (2004) and is approximately equals to the fluctuations, that is

 γ≈⟨−aαaα′⟩. (2)

From this expression one can draw conclusions on the CME predictions for . For the same charges , and thus one expects ; for opposite charges , and thus . Additionally, if the backgrounds effects are negligible, the same-charge correlator should be opposite in sign but equal in magnitude to opposite-sign correlator.

In collision with heavy-ions, the first measurement of were performed at RHIC Abelev et al. (2009, 2010); it was observed that opposite-charge correlations were very close to zero or even negative, while the same-charge were negative and larger in the amplitude. The observation of close to zero opposite-charge correlations was not immediately consistent with CME, as it was predicted to have the same amplitude as same-charge. However, the observable might be potentially contaminated by large charge-independent backgrounds, that shift the values of both the same-charge and opposite-charge correlations.

In order to test CME, a few other measurements and observables were explored, for details see Ref. Kharzeev et al. (2016); Skokov et al. (2016). Nevertheless, the status of the CME in heavy-ion collisions remains inconclusive due to background correlations that may be responsible for the entirety of the observed signal Schlichting and Pratt (2011); Pratt et al. (2011); Bzdak et al. (2011, 2013).

Recently, the CMS collaboration performed measurements of the three particle correlations in proton-nucleus collisions Khachatryan et al. (2017) at = 5 TeV. The CME predicts virtually absent signal in p-A collisions due to small values of the magnetic field and its decorrelation with the event plane. However, it was observed that the differences between the same and opposite sign correlations, as functions of multiplicity and rapidity gap between the two charged particles, are of a similar magnitude in proton-nucleus and nucleus-nucleus collisions at the same multiplicities. This does not necessarily pose an immediate challenge to the CME interpretation of the charge dependent azimuthal correlations in heavy ion collisions, as the results coincide only in peripheral bins of Pb-Pb collisions, where the background effects are expected to play a dominant role Tribedy (2017).

Nevertheless, the CMS measurements make it clear that without microscopic understanding of the background contributions to the observable , any interpretation of the data will be unsatisfactory. Motivated by the data of the CMS collaboration, in this paper, we address one of the possible sources of this background; we concentrate on the same-charge correlations, which usually fall outside the scope of the conventional background models Schlichting and Pratt (2011); Hirono et al. (2014) except for the global transverse momentum conservation. We work in the framework of the Color Glass Condensate; which was successful in predicting the “ridge” correlations and is often utilized to address the systematics of the azimuthal anisotropy in the initial state, see Refs. Dumitru et al. (2011); Kovner and Lublinsky (2011, 2013); Kovchegov and Wertepny (2013); Dusling and Venugopalan (2012, 2013); Dumitru et al. (2015a); Skokov (2015); McLerran and Skokov (2017); Lappi (2015); Schenke et al. (2015); Dumitru et al. (2015b); Altinoluk et al. (2015); Lappi et al. (2016); McLerran and Skokov (2016); Schlichting and Tribedy (2016); Dusling et al. (2017); Gotsman and Levin (2017).

As was shown in Ref. Altinoluk et al. (2015) the Bose-Einstein enhancement (BSE) of soft gluons in the projectile provides the physical interpretation of the glasma-graph calculation of the “ridge” correlations. In a follow up paper the authors of Ref. Altinoluk et al. (2017) also explored the consequences of quark’s Pauli blocking in the projectile wave function. Mindful of these studies, we consider the observable and explore possible nontrivial contribution to this observable, and therefore the CME background stemming from the quantum correlations in the initial state. In practice, we consider the angular average defined as a projectile average where and are the transverse momenta of two same-charge/same-flavor quarks in the light cone wave-function of the projectile, and is the transverse momentum of the gluon. We will demonstrate that there are two distinct contributions to this quantity: the pedestal, the rapidity-independent contribution, with a negative , and the rapidity-dependent and sign changing contribution originating from Pauli blocking.

The paper is organized as follows. In Sec. II we briefly review the relevant results of Ref. Altinoluk et al. (2017) for quark-quark correlations, originating from two quark-antiquark pairs in the wave function. In Sec. III we extend this calculation to include three particle correlations, computing contribution of an additional gluon thus bringing up the relevant Fock state component to 5 particles. In this paper we limit ourselves to the calculation of correlations in the wave function of the incoming hadron, and do not attempt to calculate three particle production, which we leave for future work. Nevertheless, as demonstrated in Altinoluk et al. (2015, 2017) such initial state correlations within the CGC approach have a direct effect on production of particles, and thus can serve as a basis for qualitative understanding of the effect. In Sec. IV we discuss and summarize our findings.

## Ii Preliminaries: Quark contribution to the projectile wave-function

Let and denote quark creation and annihilation operators, while and are those of the antiquark. Additionally for gluons, we introduce and .

First we formulate, two particle, quark-antiquark, content of the light-cone wave function. This will allow us to introduce the notation we use when considering a more complicated case of two quark-two antiquark and gluon. In perturbative calculations, the quarks and antiquarks appear in the light-cone wave function of a valence charge via soft-gluon splitting or instantaneous interaction, see details in Ref. Altinoluk et al. (2017), Appendix A and the review Kogut and Soper (1970); Bjorken et al. (1971); Brodsky et al. (1998). The quark-antiquark component of the light cone wave function of a “dressed” color charge density is given by2

 |v⟩D2 = (1−g4κ4)|v⟩ (3) + g2∫dk+dαd2pd2q(2π)3 ζγδs1s2(k+,p,q,α) d†γs1(q+,q)¯d†δs2(p+,p)|v⟩,

where denotes a valence state characterised by a distribution of charge densities of valence (fast) partons. The subscript “2” in counts the perturbative order in the Yang-Mills coupling denoted as . is a constant ensuring the correct normalisation of the dressed state, are fundamental color indices, and stand for the spinor indices. The value of is irrelevant for the problem at hand. We define the longitudinal momentum fraction as

 p+=αk+,  q+=¯αk+,  ¯α=1−α, (4)

with the momentum of the parent gluon that splits into a quark and an antiquark. The splitting amplitude is given by

 ζγδs1s2(k+,p,q,α) =τaγδ∫d2k(2π)2ρa(k) ϕs1s2(k,p,q;α), (5)

where are the generators of in the fundamental representation. Here,

 ϕ=ϕ(1) + ϕ(2), (6)

with

 ϕ(1)s1s2(k,p,q;α)=−δs1s22α¯α¯αp2+αq2(2π)2δ(2)(k−p−q) (7)

and

 ϕ(2)s1s2(k,p,q;α)=1k2[¯αp2+αq2]{2α¯αk2−(¯αk⋅p+αk⋅q)+2iσ3k×p}(2π)2δ(2)(k−p−q). (8)

Thus,

 ϕs1s2(k,p,q;α)=ϕs1s2(k,p;α)(2π)2δ(2)(k−p−q) (9)

where

 ϕs1s2(k,p;α)=1k2[¯αp2+α(k−p)2]{−[¯αk⋅p+αk⋅(k−p)]+2iσ3k×p}. (10)

The term comes from the instantaneous interaction, while from the soft gluon splitting.

However, is not the state we are interested in, as it provides information about quark-antiquark content of the light cone wave function only. To probe quark-quark-gluon correlations we have to consider the two quark–two antiquark and gluon component of the dressed state, that is the state with 5 particles. We will adopt the same strategy as was used in the glasma graph calculation. That is, we focus on terms enhanced by the charge density in the wave-function; similar approach was also used in Ref. Kovchegov and Wertepny (2013). At the lowest order the relevant component of the wave function is given by

 |v⟩D5 = virtual (11) + g42∫dk+dαd2p′d2¯p′(2π)3d¯k+dβd2q′d2¯q′(2π)3ζϵιs′1s′2(k+,p′,¯p′;α)ζγδr1r2(¯k+,q′,¯q′;β) × Missing dimension or its units for \hskip × \underbracketg∫dm+(m+)1/2d2m(2π)3mim2ρa(−m)a†ai(m+,m)g|v⟩,

where we explicitly showed the part corresponding to the pair of quark-antiquark and the soft gluon. In the following section, we will use this dressed state to find the average number of two quark and a gluon âtripletsâ in the wave function.

## Iii Two-quark-gluon correlations

In this section we compute the correlations between the quarks and the gluon in the CGC wave function of the projectile. To be able to make definitive statements about correlations between produced particles this calculation has to be supplemented by the analysis of particle production, as in principle momentum distribution of produced particles is affected by momentum transfer from the target. Also scattering is not equally efficient in putting on shell all partons in the incoming wave function. In particular partons with large transverse momentum are emitted into the final state with smaller probability. Thus correlations between emitted particles are not identical to correlations between the partons in the projectile wave function. However as was observed in Ref. Altinoluk et al. (2015, 2017); Kovner et al. (2016) this change mostly affects the quantitative features preserving the qualitative pattern of the correlation. In this exploratory study we only compute the correlation in the projectile wave function and consider this to be a proxy to correlation between produced particles, at least if the transverse momenta of these particles are not too large. Already on this level, as we will demonstrate below, the calculations are non-trivial and require numerical integration.

The aim of this section is to compute the average number of two quark and a gluon “triplets” in the wave function that is formally defined, see e.g. Ref. Greiner (1998), as

 dNdp+d2pdq+d2qdm+d2m= 1(2π)6⟨D5⟨v|d†α,s1(p+,p)d†β,s2(q+,q)dβ,s2(q+,q)dα,s1(p+,p)a†fi(m+,m)afi(m+,m)|v⟩D5⟩P, (12)

i.e. first, we need to calculate the expectation value of the “number of quark-gluon triplets” in our dressed state , and then, average over the color charge densities in the projectile. For the latter we use the McLerran-Venugopalan (Gaussian) model McLerran and Venugopalan (1994a, b). This choice is somewhat restrictive and might potentially affect the result in a non universal way, especially at lower collision energies, where the odderon component becomes stronger. As it will be clear below, the observable we are computing has six powers of the charge density and thus might be sensitive to the odderon. In principle the model can be extended along the lines of Ref. Jeon and Venugopalan (2005) where the odderon is included in the averaging weight on the classical level.

A similar problem was addressed in Ref. Altinoluk et al. (2017) for two-quark correlations, see Appendix B of Ref. Altinoluk et al. (2017). We need to extend this to include a gluon. This is, thankfully quite straightforward. The extra gluon is created in the wave function independently of the quark pair from the valence charge density. The resulting expression for the correlator is, see also Fig. 1,

 dNdη1d2pdη2d2qdηgd2m = 1(2π)4g10m2∫d2kd2¯kd2ld2¯l⟨\underbracketρa(k)ρc(¯k)ρb(l)ρd(¯l)q¯qq¯q\underbracketρf(m)ρf(−m)g⟩P (13) ×{\underbrackettr(τaτb)tr(τcτd)Φ2(k,l;p)Φ2(¯k,¯l;q)A\underbracket−tr(τaτbτcτd)Φ4(k,l,¯k,¯l;p,q)B},

where , and one of are the color charge densities in the amplitude and , and the other are the color charge densities in the complex conjugate amplitude. The rapidities are defined as and , with the gluon -momentum. Note that the average number, Eq. (13), is independent of the gluon rapidity, . The functions and are defined respectively as Altinoluk et al. (2017)

 Φ2(k,l;p)≡∫10dα∫d2¯p′(2π)2∑s1s2 ϕs1,s2(k,p,¯p′;α) ϕ∗s1,s2(l,p,¯p′;α) (14)

and

 Φ4(k,l,¯k,¯l;p,q) ≡ ∑s1,s2,¯s1,¯s2∫10dαdβ(β+¯βeη1−η2)(α+¯αeη2−η1) (15) ×∫d2¯p′(2π)2d2¯q′(2π)2ϕs1s2(k,p,¯p′;α) ϕ¯s1¯s2(¯k,q,¯q′;β) ϕ∗s1¯s2(l,p,¯q′;β)ϕ∗¯s1s2(¯l,q,¯p′;α).

The integrals with respect to prime momenta represent “inclusiveness” over the antiquarks. The integrals over reduce the number of -functions to two, so that in general we can write

 Φ4(k,l,¯k,¯l;p,q) = ∑s1s2,¯s1,¯s2∫10dαdβ(β+¯βeη1−η2)(α+¯αeη2−η1) (16) × ϕs1s2(k,p;α)ϕ¯s1¯s2(¯k,q;β)ϕ∗s1¯s2(¯k−q+p,p;β)ϕ∗¯s1s2(k+q−p,q;α) × (2π)2δ(2)(¯l−k−q+p)(2π)2δ(2)(l−¯k+q−p).

Lets comment on the origin of different terms in Eq. (13). First, the particle density is proportional to : two powers of the coupling constant come from the gluon production, and the leftover originate from production of two quark-antiquark pairs, as each is proportional to owing to production of a gluon and its splitting into a quark-antiquark pair. The gluon component of Eq. (13) is trivial and proportional to , or the square of the Weizsäcker-Williams field . The quark contribution coincides with that of Ref. Altinoluk et al. (2017). It contains two distinct contributions in the curly brackets of Eq. (13): () the term proportional to corresponds to two quark loops and thus contributes with the positive sign, while the term () proportional to has one quark loop resulting in the minus sign, see Fig. 1. The latter term manifests the Pauli blocking; with the minus sign leading to the dilution of the correlation!

### The Pauli blocking term

As it is clear from Eq. (13) the correlations between the gluon and quarks originates only from the averaging over the projectile color densities. In a Gaussian model for the projectile, thus we will not consider the terms involving the contraction , since this contraction leads to uncorrelated gluon production. Additionally, we will postpone the consideration of the first term in the curly brackets of Eq. (13). This term contributes to the correlated quark production only in subleading order at large . Nevertheless compared to the second term of Eq. (13) it is enhanced by factor of 2 due to the trace over spin. Naively it is also proportional to the number of flavours . This is however not the case since we are interested in production of two quarks of a given flavour. In the real world, the ratio is not a particularly small number, and thus one should not neglect this term off hand. However, as it is clear from the definition of , this term does not depend on the rapidity separation and thus manifests itself as a pedestal in the three-particle correlation. In what follows we focus on the second term ().

In the large limit, the second contribution, proportional to , dictates that there are only 8 leading contractions for the correlated production. To understand this consider the trace . The color indices are contracted pairwise. Due to Gaussian averaging of , there are two distinct contractions: between the nearest neighbours, e.g.

 tr(τaτaτcτd)=N2c−14Ncδcd, (17)

and the contraction between two matrices separated by the other

 tr(τaτbτaτd)=−14Ncδbd. (18)

Obviously the latter is suppressed by ; and thus the corresponding contractions of the color densities will be ignored. This leaves us with 8 possible contractions: there are 4 possible ways to contract a with one of , , , ; and there are 2 possible ways to pick a neighbour to get the leading contribution.

Therefore in the leading , we get

 ⟨ρa(k)ρb(l)ρc(¯k)ρd(¯l)ρf(m)ρf(−m)⟩P ≈⟨ρa(k)ρf(m)⟩P(⟨ρb(l)ρf(−m)⟩P⟨ρc(¯k)ρd(¯l)⟩P+⟨ρd(¯l)ρf(−m)⟩P⟨ρb(l)ρc(¯k)⟩P) +⟨ρb(l)ρf(m)⟩P⟨ρc(¯k)ρf(−m)⟩P⟨ρd(¯l)ρa(k)⟩P+⟨ρc(¯k)ρf(m)⟩P⟨ρd(¯l)ρf(−m)⟩P⟨ρb(l)ρa(k)⟩P+(m→−m). (19)

The final result is symmetric with respect to the reversal of the transverse gluon vector . To simplify the equations we will keep only the terms we explicitly show, the complete expression can be constructed by symmetrizing with respect to . Using a Gaussian model for the projectile

 ⟨ρa(k)ρb(p)⟩P=(2π)2μ2(k)δabδ(2)(k+p). (20)

we obtain

Multiplying by the trace and summing with respect to the color indices we arrive at

 tr(τaτbτcτd)(2π)6⟨ρa(k)ρb(l)ρc(¯k)ρd(¯l)ρf(m)ρf(−m)⟩P=(N2c−1)24Nc ×(μ2(k)μ2(l)μ2(¯l)δ(2)(k+m)δ(2)(l−m)δ(2)(¯k+¯l)+μ2(k)μ2(¯l)μ2(l)δ(2)(k+m)δ(2)(¯l−m)δ(2)(l+¯k) +μ2(¯k)μ2(l)μ2(k)δ(2)(l+m)δ(2)(¯k−m)δ(2)(¯l+k)+μ2(¯l)μ2(¯k)μ2(k)δ(2)(¯l−m)δ(2)(¯k+m)δ(2)(l+k) +(m→−m).) (22)

Therefore the correlated piece defined by the second term of Eq. (13)

 [dNdη1d2pdη2d2qdηgd2m]Bcorr (23) =−1(2π)4g10m2∫d2kd2¯kd2ld2¯l⟨ρa(k)ρc(¯k)ρb(l)ρd(¯l)ρf(m)ρf(−m)⟩Ptr(τaτbτcτd)Φ4(k,l,¯k,¯l;p,q),

simplifies into

 [dNdη1d2pdη2d2qdηgd2m]Bcorr =−(2π)2g10μ2(m)μ2(−m)m2(N2c−1)24Nc ×∫d2lμ2(l)[Φ4(−m,m,−l,l;p,q)+Φ4(−m,l,−l,m;p,q) +Φ4(l,−m,m,−l;p,q)+Φ4(l,−l,−m,m;p,q)+(m→−m)] (24)

which eventually results in

 [dNdη1d2pdη2d2qdηgd2m]Bcorr =−(2π)6g10μ6(m)2m2(N2c−1)24Nc∫10dαdβ(β+¯βeη1−η2)(α+¯αeη2−η1) ×Tr[δ(2)(p−q)ϕ(−m,p;α)ϕ†(−m,p;α)ϕ(m,p;β)ϕ†(m,p;β) +δ(2)(p−q+2m)ϕ(−m,p;α)ϕ†(m,q;α)ϕ(m,q;β)ϕ†(−m,p;β) +δ(2)(p−q−2m)ϕ(m,p;α)ϕ†(−m,q;α)ϕ(−m,q;β)ϕ†(m,p;β) +δ(2)(p−q)ϕ(m,p;α)ϕ†(m,p;α)ϕ(−m,p;β)ϕ†(−m,p;β)] (25)

As expected, this exhibits a weakening of the correlation when the momenta of the quarks are the same (mind the minus sign in front of the integral). Another prominent feature of this expression is that it is invariant under the reversal of the gluon momentum .

The combination is proportional to the unit matrix,

 ϕ(m,p;α)ϕ†(m,p;α)=1m4(¯αp2+α(m−p)2)2{(¯αm⋅p+αm⋅(m−p))2+4(m×p)2}, (26)

while the other relevant combination is given by

 δ(2)(p−q+2m)ϕ(−m,p;α)ϕ†(m,q;α) =δ(2)(p−q+2m)m4(¯αp2+α(m+p)2)(¯α(p+2m)2+α(m+p)2) (−(¯αm⋅p−αm⋅(m+p))(¯αm⋅(p+2m)−αm⋅(m+p))−4(m×p)2 +4iσ3¯αm2 m×p). (27)

Thus

 [dNdη1d2pdη2d2qdηgd2m]Bcorr=−(2π)6g10μ6(m)2m2(N2c−1)24Nc ×Tr[δ(2)(p−q)I1(η2−η1,−m,p)I1(η1−η2,m,p)+δ(2)(p−q+2m)I2(η2−η1,m,p)I†2(η1−η2,m,p) +δ(2)(p−q−2m)I2(η2−η1,−m,p)I†2(η1−η2,−m,p)+δ(2)(p−q)I1(η2−η1,m,p)I1(η1−η2,−m,p)] (28)

with

 I1(Δη,m,p) =∫10dαα+¯αeΔηϕ(m,p,α)ϕ†(m,p,α), (29) I2(Δη,m,p) =∫10dαα+¯αeΔηϕ(−m,p,α)ϕ†(m,p+2m,α), (30)

The integrals and are matrix valued in the spin indices which are traced over in Eq. (28). Explicitly, the definitions of the integrals and read

 I1(Δη,m,p) =∫10dαα+eΔη¯α1m4(¯αp2+α(m−p)2)2{(¯αm⋅p+αm⋅(m−p))2+4(m×p)2}, (31) I2(Δη,m,p) =∫10dαα+eΔη¯α1m4(¯αp2+α(m+p)2)(¯α(p+2m)2+α(m+p)2) (−(¯αm⋅p−αm⋅(m+p))(¯αm⋅(p+2m)−αm⋅(m+p))−4(m×p)2 +4iσ3¯αm2 m×p); (32)

The integrals can be computed analytically and the key ingredients are presented in Appendix B.

Recall that our goal is to obtain the average of , that is

 γBcorr=⟨cos(ϕq+ϕp−2ϕm)⟩Bcorr=N∫d2p∫d2q∫dϕm[dNdη1d2pdη2d2qdηgd2m]Bcorrcos(ϕq+ϕp−2ϕm) (33)

Here we have fixed the magnitude of the gluon momentum, while the integral over the quark momenta and should performed inside a prescribed momentum bin. Ideally we should choose the size of the two momentum bins to be the same. This can in principle be done numerically, but would involve performing multi dimensional integrals. To get a qualitative idea of the behavior of the average we choose a simplified averaging procedure which reduces the problem to a simple two dimensional integral. We integrate with respect to the absolute value of the momentum of one of the quarks (e.g. ) from zero to infinity, while keeping the ratio of the other quark momentum to the gluon in a finite range. Defining and = we obtain

 γBcorr =−N(2π)7g10μ6(m)(N2c−1)22Nc ×∫dcpcp∫dΔϕpcos(2Δϕp)Tr[I1(η2−η1,−m,p)I1(η1−η2,m,p)] +cpcos(2Δϕp)+2cos(Δϕp)√c2p+4cpcos(Δϕp)+4Tr[I2(η2−η1,m,p)I†2(η1−η2,m,p)] (34)

The normalization includes no angular dependence; it is defined through uncorrelated production and thus is irrelevant for our qualitative study.

At large , both terms are proportional to , as was shown in Appendix B

 limΔη→∞I1(Δη,−m,p)I1(−Δη,m,p) =[(m⋅(m+p))2+4(m×p)2][(m⋅p)2+4(m×p)2]m8p4(m+p)4Δη2e−Δη (35)

and

 limΔη→∞I2(η2−η1,m,p)I†2(η1−η2,m,p) =[(m⋅(m+p))