Cubic color charge correlator in a proton made of three quarks and a gluon

The three point correlation function of color charge densities is evaluated explicitly in light cone gauge for a proton on the light cone. This includes both $C$-conjugation even and odd contributions. We account for perturbative corrections to the three-quark light cone wave function due to the emission of an internal gluon which is not required to be soft. We verify the Ward identity as well as the cancellation of UV divergences in the sum of all diagrams so that the correlator is independent of the renormalization scale. It does, however, exhibit the well known soft and collinear singularities. The expressions derived here provide the $C$-odd contribution to the initial conditions for high-energy evolution of the dipole scattering amplitude to small $x$. Finally, we also present a numerical model estimate of the impact parameter dependence of quantum color charge three-point correlations in the proton at moderately small $x$.


I. Introduction 2
II. Correlator of three color charge operators, ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ 3 A. UV divergent diagrams 4 B. Finite diagrams 1. Coupling at least once to the gluon 2. Coupling only to quarks III.The correlator in impact parameter space IV.Summary

I. INTRODUCTION
In this paper we present explicit expressions for all diagrams which determine the cubic light cone gauge color charge correlator ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ in a proton.The proton is approximated by a non-perturbative three-quark Fock state, plus a perturbative gluon.This is in continuation of ref. [1] where we derived analogous expressions for ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 )⟩, and of ref. [2] where we presented numerical results for the quadratic correlator (see also Ref. [3] for a first phenomenological application).The cubic correlator quantifies corrections to Gaussian color charge fluctuations in the proton, and it provides a contribution to the scattering matrix of a dipole which is odd under C-conjugation.
The S-matrix for eikonal scattering of a quark -antiquark dipole off the fields in the target proton can be expressed as1 [4,5] Here, ⃗ b is the impact parameter vector while ⃗ r denotes the transverse separation of quark and anti-quark.The operators U (U † ) are (anti-)path ordered Wilson lines of the field in covariant gauge, representing the eikonal scattering of the quarks at transverse coordinate ⃗ x: C-conjugation transforms the generators of the fundamental representation t a → −(t a ) T .The S-matrix can be separated into its real part which at high energy is dominated by C-conjugation even two-gluon exchange, and its imaginary part which starts out as C-odd three gluon exchange: Thus, the fact that the imaginary part of S(⃗ r, ⃗ b) is non-zero is due to the existence of a color singlet three gluon (t-channel) exchange with negative C-parity in QCD [6][7][8][9][10][11]. Recently, the TOTEM and D0 collaborations have presented evidence for a difference in p−p vs. p− p elastic scattering cross sections at a CM energy of √ s ≃ 2 TeV, and low momentum transfer |t| < 1 GeV 2 [12,13] (also see ref. [14]).However, our focus here is on cubic color charge correlations in the semi-hard regime, which is related to the C and P odd contribution to the dipole scattering amplitude.
A key limitation for quantitative predictions in the energy regime of the Electron-Ion Collider (EIC) [44][45][46][47] is the crude knowledge of the initial condition for the evolution equations at moderately small x.Deriving the next-to-leading order (NLO) expressions for O(⃗ r, ⃗ b) due to one gluon emission corrections in a proton target at x ∼ 0.01 − 0.1 is the main purpose of this paper.The corresponding expressions at leading order (LO) have been published in refs.[34,48,49].The latter paper also provides numerical estimates of cubic color charge correlators and of O(⃗ r, ⃗ b) at LO, i.e. in the valence quark regime.Bartels and Motyka [50] have also calculated the proton impact factor for t-channel three gluon exchange, which agrees with the LO expressions for ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ given in refs.[34,48,49], and soft gluon emission corrections to proton-proton scattering at high energy.
The initial condition for the small-x evolution [26][27][28] of Im S, which we derive here, depends not only on the impact parameter and the dipole vectors but also on their relative angle, and on the light-cone momentum fraction x in the target.In fact, the BK equation in its standard formulation evolves the wave function of the dipole projectile, and the evolution "time" is then related to the minus component of the momentum of the gluon in the proton target [51][52][53].Ducloué et al. have reformulated [52] BK evolution at NLO in terms of the target rapidity (or Bjorken-x).They obtained an evolution equation which is non-local in rapidity and which depends explicitly on the gluon's plus momentum fraction x = k + g /P + .Therefore, it is important to determine the dependence of the initial scattering amplitude not only on impact parameter ⃗ b and dipole size ⃗ r but also its dependence on x.
The amplitude for C-odd three gluon exchange is related to the correlator of C-odd color charge fluctuations [34,48,49]  q1,q2,q3 Here, ⃗ K = −(⃗ q 1 + ⃗ q 2 + ⃗ q 3 ) is the (transverse) momentum transfer given ⃗ P = 0 for the incoming proton, and q is shorthand for d2 q/(2π) 2 .We denote the C-odd part of the correlator of three color charges as Note that G − 3 (⃗ q 1 , ⃗ q 2 , ⃗ q 3 ) from eq. ( 5) is given by the correlator of three covariant-gauge color charge densities.However, in the weak field limit, a computation in light-cone gauge is applicable.
The above correlator is symmetric under a simultaneous sign flip of all three gluon momenta, and so −iIm S(⃗ r, ⃗ b) = −iO(⃗ r, ⃗ b) is imaginary 3 .Also, it vanishes quadratically in any of the transverse momentum arguments so that −iO(⃗ r, ⃗ b) is free of infrared divergences.The light-cone gauge color charge density operator in the eikonal "shock wave limit" is given by ρ a ( ⃗ k) = ρ a qu ( ⃗ k) + ρ a gl ( ⃗ k) with [1] ρ a qu ( ⃗ k) = g i,j,σ (t a ) ij dx q d 2 q 16π 3 x q b † iσ (x q , ⃗ q) b jσ (x q , ⃗ k + ⃗ q) , ρ a gl ( ⃗ k) = g λbc (T a ) bc dx g d 2 q 16π 3 x g a † bλ (x g , ⃗ q) a cλ (x g , ⃗ q + ⃗ k) .
Here a † , a and b † , b denote creation and annihilation operators for gluons and quarks, respectively.
In the next section II we compute all contributions to ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 ) in a proton on the light cone in lightcone gauge; specifically we consider the NLO correction due to the emission or exchange of a gluon which is not required to be soft.In sec.III we describe the Fourier transform of the correlator to impact parameter space, and present a numerical model estimate.A brief summary is presented in sec.IV.Appendix A summarizes the Fock state description of the proton on the light front used throughout this paper, appendix B shows the cancellation of UV divergences in the sum of all diagrams for ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 ) , and in appendix C we check the vanishing of this correlator when ⃗ q 1 → 0 or ⃗ q 3 → 0.
In this section we compute the correlator of three color charge operators ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ where ρ a (⃗ q) = ρ a gl (⃗ q) + ρ a qu (⃗ q).This expectation value is defined as the matrix element of the product of three color charge operators between the incoming (|P ⟩) and outgoing (⟨K|) proton states, stripped of the delta functions expressing conservation of L.C. and transverse momentum: The structure of the proton state assumed in this work is explained briefly in appendix A.
In general, this correlator has both even and odd components under C-parity which transforms (t a ) ij → −(t a ) ji 4 and (T a ) bc → −(T a ) cb = (T a ) bc . 5Note that the following expressions apply when the number of colors N c = 3.We shall use the shorthand notation ⃗ q = ⃗ q 1 + ⃗ q 2 + ⃗ q 3 = ⃗ P − ⃗ K and ⃗ q ij = ⃗ q i + ⃗ q j in the following expressions 6 .Their corresponding diagrams are shown in the figures.We label them as fig.1(q 3 q 2 g), for example, corresponding to a diagram of the type shown in fig. 1 (i.e., a gluon exchange across the operator insertion by a quark with itself, with at least one of the probes attached to that internal gluon) where the first probe gluon (momentum ⃗ q 1 , color a) couples to the internal gluon, the second probe gluon (momentum ⃗ q 2 , color b) couples to the second quark, and the third probe (momentum ⃗ q 3 , color c) couples to the third quark.
a, q 1 b, q 2 c, q 3 FIG.1: UV divergent diagrams (propagators of external probes to be amputated) for ⟨ρ a (⃗ q1) ρ b (⃗ q2) ρ c (⃗ q3)⟩ where at least one of the probes attaches to the gluon in the proton.The cut is located at the insertion of the three color charge operators.
We begin with the UV divergent diagrams where a quark exchanges a gluon with itself.The diagrams where one or more of the probes attach to the gluon are shown in fig. 1.
To prepare, we first list the matrix elements of one, two, and three ρ gl (⃗ q) between one-gluon states: The matrix elements of ρ qu (⃗ q) between one-quark states are similar, with T a → t a .
With this we obtain fig. 1 with a symmetry factor of 3.Here the integration measures [dk i ] and [dx i ] and the proton valence quark wave function Ψ qqq are defined in Appendix A 1. The Lorentz invariant gluon phase space measure dk g is given in Appendix.A 2, the phase space integral is calculated in Ref. [1], and the result is also included in Appendix.B. Since tr ) bc it follows that this contribution is even under C conjugation.The remaining integral over the longitudinal and transverse momenta of the emitted gluon can be decomposed into a finite function and a UV divergent part [1].We verify the cancellation of the UV divergences in appendix B.
where (D c ) ab = d cab .Performing the traces, the color factor becomes ).The first term corresponds to a C-odd contribution while the second one is even under C.The symmetry factor for this diagram is 3.
The diagram where instead the second gluon attaches to quark 1 (not shown) is given by fig and the one where the first gluon attaches to quark 1: These diagrams come with a symmetry factor of 3 since the "active" quark may just as well be quark 2 or quark 3.
Continuing with the diagram where the third probe attaches to quark 2: The symmetry factor is 6 because the third gluon probe may also attach to quark 3.
Once again there are analogous diagrams (not shown) where the second or the first probe attaches to quark 2 (or to quark 3): Their symmetry factors are 6.
We continue with the diagrams where two of the probes attach to quarks.
The symmetry factor is 3.The SU(N c ) relation f abc t a t b = i 2 N c t c is useful for evaluating the color factor for this diagram.Performing the trace, tr t a t b t c = 1 4 (d abc + if abc ), separates the C-odd contribution proportional to d abc from the C-even contribution proportional to if abc .fig. 1(q The symmetry factor is 3.
The symmetry factor is 3.
The symmetry factor is 6, to include the contribution where the third gluon attaches to quark 3.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.
b, q 2 a, q 1 c, q 3 K + , P + , b, q 2 a, q 1 c, q 3 FIG.2: Two of the diagrams for ⟨ρ a qu (⃗ q1) ρ b qu (⃗ q2) ρ c qu (⃗ q3)⟩ in the three-quark Fock state which involve the quark wave function renormalization factor Z 1/2 q (x) ("virtual corrections").The cut is located at the insertion of the three color charge operators.
The final set of UV divergent diagrams is shown in fig. 3. Here, quark 1 exchanges a gluon with itself across the insertion of the three charge operators while the three gluon probes attach in all possible ways to the three quarks.fig.3(q with a symmetry factor of 3. FIG. 3: Final set of UV divergent "real emission" diagrams for ⟨ρ a (⃗ q1) ρ b (⃗ q2) ρ c (⃗ q3)⟩ where all three gluon probes attach to quarks in the proton.The cut is located at the insertion of the three color charge operators.
with a symmetry factor of 6.
with a symmetry factor of 6.
with a symmetry factor of 6.
with a symmetry factor of 6.
with a symmetry factor of 6.

fig. 3(q
with a symmetry factor of 6.
with a symmetry factor of 6.
with a symmetry factor of 6. fig.
with a symmetry factor of 6.
with a symmetry factor of 6. fig.
with a symmetry factor of 6. fig.
with a symmetry factor of 6.
with a symmetry factor of 6.

B. Finite diagrams
1. Coupling at least once to the gluon a, q 1 b, q 2 c, q 3 FIG.4: A sample of UV finite diagrams for ⟨ρ a (⃗ q1) ρ b (⃗ q2) ρ c (⃗ q3)⟩ where at least two of the probes attach to the gluon in the proton.The cut is located at the insertion of the three color charge operators.
We now proceed to the UV-finite contributions.To write the following expressions in more compact form we introduce the integral operator with z 1 = x g /x 1 and z 2 = x g /(x 2 + x g ).We begin with the diagrams shown in fig. 4 where two distinct quarks exchange a gluon across the operator insertion, and where two of the probes attach to that gluon.
The symmetry factor is 6 which includes a factor of 2 for interchanging the gluon emission and absorption vertices between quarks 1 and 2 (so that in |P ⟩ the gluon couples to quark 2, and in ⟨K| it couples to quark 1).

fig. 4(q
The symmetry factor 6 includes a factor of 2 due to the contribution from fig. 4(q 2 gg) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
fig. 4(gq The symmetry factor of 6 includes a factor of 2 due to the contribution of diagram fig.4(gq 2 g) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
Again, the symmetry factor for this diagram is 6 which includes the contribution of diagram fig.4(ggq 2 ) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6; this includes the contribution from fig. 4(q 1 gg) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6.
The symmetry factor is 6.
The symmetry factor is 6.

fig. 4(ggq
The symmetry factor is 6. a, q 1 b, q 2 c, q 3 FIG.5: A sample of UV finite diagrams for ⟨ρ a (⃗ q1) ρ b (⃗ q2) ρ c (⃗ q3)⟩ where one of the probes attaches to the gluon in the proton.The cut is located at the insertion of the three color charge operators.
We continue with the diagrams shown in fig. 5 where two distinct quarks exchange a gluon across the operator insertion, and where one of the probes attaches to that gluon.
The symmetry factor is 6 (this includes the contribution from diagram fig.5(q 2 q 2 g) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged).

fig. 5(q
Again, the symmetry factor is 6 (this includes the contribution from diagram fig.5(q 2 gq 2 ) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged).

fig. 5(gq
Again, the symmetry factor is 6 (this includes the contribution from diagram fig.5(gq 2 q 2 ) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged).
The symmetry factor is 6, including the contribution from diagram fig.5(q 1 q 2 g) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6, including the contribution from diagram fig.5(q 1 gq 2 ) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
fig. 5(gq The symmetry factor is 6, including the contribution from diagram fig.5(gq 1 q 2 ) (not shown) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6, including the contribution from diagram fig.5(q 2 q 1 g) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6, including the contribution from diagram fig.5(q 2 gq 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
fig. 5(gq The symmetry factor is 6, including the contribution from diagram fig.5(gq 2 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6, including the contribution from diagram fig.5(q 1 q 1 g) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6, including the contribution from diagram fig.5(q 1 gq 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor is 6, including the contribution from diagram fig.5(gq 1 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 3 q 2 g) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 3 gq 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(gq 3 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 2 q 3 g) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 2 gq 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(gq 2 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 3 q 1 g) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 3 gq 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(gq 3 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 1 q 3 g) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.5(q 1 gq 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.

fig. 5(gq
The symmetry factor of 6 includes the contribution from diagram fig.5(gq 1 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.

Coupling only to quarks
Here we consider the diagrams where all three gluon probes couple exclusively to quarks in the proton.We begin with the matrix element of ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 ) in the |qqqg⟩ Fock state.A few examples are shown in fig.6.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 2 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 2 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 2 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 1 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 1 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 1 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 3 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 3 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 3 q 2 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 2 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 2 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 2 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 1 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 1 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 1 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 3 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 3 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 3 q 1 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 2 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 2 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 2 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 1 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 1 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 1 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 2 q 3 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 1 q 3 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The symmetry factor of 6 includes the contribution from diagram fig.6(q 3 q 3 q 3 ) with the gluon emission and absorption vertices between quarks 1 and 2 interchanged.
The final set of diagrams corresponds to the virtual corrections where two quarks exchange a gluon on either side of the insertion of the ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 ) operator.All the symmetry factors are 6 × 1 2 = 3; the factor of 1 2 corrects for double counting of internal gluon exchanges as mentioned at the end of appendix A 2.
Fig. 7 shows a small subset of these diagrams.To write them in compact form we introduce the integral operator We shall also include right away the contribution from the diagram where the gluon exchange occurs on the other side of the insertion from quark 2 to quark 1.With this, In closing this section we note that the subset of diagrams where all three gluons attach to the same quark line is proportional to the UV finite, O(g 2 ) correction to the electromagnetic form factor of the proton, i.e. to the matrix element ⟨ρ em (⃗ q)⟩.Hence, for ⃗ q 2 → 0 this must vanish.Indeed, in this limit the sum of eqs.(88,101,114,116,129,142), multiplied by their respective symmetry factors, is zero.[The finite parts of the UV divergent diagrams eqs.(36,44) vanish individually when ⃗ q 2 → 0, see appendix B.]

III. THE CORRELATOR IN IMPACT PARAMETER SPACE
The vanishing of ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ when ⃗ q 1 = 0 or ⃗ q 3 = 0 leads to a sum rule in impact parameter space.Let us first separate C-odd and even contributions via Introducing the total momentum transfer ⃗ K = −(⃗ q 1 + ⃗ q 2 + ⃗ q 3 ), where we assume ⃗ P = 0 for the incoming proton, and the relative momenta ⃗ ∆ 12 = ⃗ q 1 − ⃗ q 2 , ⃗ ∆ 23 = ⃗ q 2 − ⃗ q 3 , we can Fourier transform these correlators to impact parameter space: These functions satisfy the sum rule We proceed to show a numerical estimate for G − 3 (b) for ⃗ ∆ 12 = ⃗ ∆ 23 = 0, normalized according to G − 3 (⃗ q 1 , ⃗ q 2 , ⃗ q 3 ) = 4d abc ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩/g 3 .Here, ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ is given by the sum of all diagrams computed above.For the numerical results we employ the "harmonic oscillator" three-quark model wave function Ψ qqq (x i , ⃗ k i ) by Brodsky and Schlumpf [55,56] used also in Ref. [2], which assumes a Gaussian momentum distribution of quarks in two transverse dimensions, with a specific x-dependent width.Also, the magnitude of the NLO correction depends on the value of the coupling for which we use α s = 0.2; and on the collinear regulator (see app.B) which we take as 0.2 GeV.Our result is shown in fig.8.At x = 0.1 the correction to the LO result [49] is numerically small; similar behavior was observed for the correlator of two color charge operators in ref. [2].This is an important check of the perturbative expansion about a three-quark Fock state.With decreasing x the NLO correction grows substantially.Fig. 8 also shows that the b-dependence of G − 3 does not follow a positive definite 1-body "parton density distribution", or a proton thickness function, respectively, at small b.Rather, n-body quantum correlations of color charge depend non-trivially on impact parameter, reflecting in a change of sign of G − 3 at b ≃ 0.15 fm.Lastly, we note that the generic magnitude of G − 3 , for impact parameters in the perturbative region, is similar to that of the correlator of two color charge operators G 2 shown in ref [2].Hence, for realistic values of the coupling there are substantial corrections to Gaussian color charge fluctuations in the proton at moderately small x.

IV. SUMMARY
We have computed the diagrams for the light cone gauge color charge correlator ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ in a proton made of three quarks and a perturbative gluon which is not required to carry a small light-cone momentum.This correlator provides the leading correction to Gaussian color charge fluctuations in the proton.It is independent of the renormalization scale since UV divergences cancel, but like the correlator of two color charge operators [1] it also exhibits logarithmic collinear and soft singularities.
These results may be used to obtain a more realistic picture of correlations in the proton at moderate x > ∼ 0.01.In particular, there exist contributions where one can not "pair up" the transverse momenta of two of the three exchanged gluons, i.e.where the three probes hit the target proton at three different impact parameters.Furthermore, our explicit expressions could be used as initial conditions (in the weak field / dilute regime) for small-x evolution, in particular for impact parameter dependent evolution [57][58][59][60][61] with a contribution that is odd under C and P .
The expressions for ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ have a form similar to those for ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 )⟩ obtained previously [1].Therefore, they can be evaluated numerically using the same code developed for the two-point charge correlator [2].We have provided first numerical estimates for the C-odd part G − 3 of ⟨ρ a (⃗ q 1 ) ρ b (⃗ q 2 ) ρ c (⃗ q 3 )⟩ in sec.III.We find that the NLO correction due to the |qqqg⟩ Fock state of the proton is numerically small at x = 0.1 but that it increases rapidly as x → 0.01.Also, the dependence of the three-charge correlator on impact parameter b is rather non-trivial, and its sign changes indicate b-dependent transitions in the nature of the quantum correlations of color charge in the proton.This result, together with published results on the ⟨ρ a ρ b ⟩ correlator [2], provides guidance for phenomenological models of color charge correlations in the proton (at x ∼ 0.01 − 0.1).
The latter form is used to regularize ultraviolet (UV) divergences by integrating over the momenta of all particles in D dimensions.Here, an arbitrary scale µ 2 has been introduced so that the transverse integrals preserve their natural dimensions.The quark wave function renormalization coefficient Z q can be calculated from the normalization requirement Now we replace each quark state in Eq. (A1) by the perturbative expansion in Eq. (A9).This yields We have extracted the common factor 2g(t a ) jii from ψ q→qg by defining ψ q→qg (p i ; ).The latter involves the quark helicities and the gluon polarization.Also note that C q (p + i ) ∼ O(g 2 ) while ψ q→qg ∼ O(g), and that terms of order O(g 3 ) and higher must be dropped.Finally, the integration over the plus momentum of the gluon extends up to the plus momentum of the parent quark; for example, k + g < p + 1 in the first line, and so on.
We also need to add to the r.h.s. of eq.(A12) the O(g 2 ) contributions where one quark emits a gluon which is then absorbed by a second (distinct) quark.For example, if the first quark emits and the second quark absorbs the gluon, that gives the contribution Here, the integration over k + g extends up to min(p + 1 , P + − p + 2 ).There are analogous contributions corresponding to gluon emission from quark 2 and absorption by quark 1 as well as from other pairings.Since we sum over all permutations of emitter and absorber, to avoid double counting, we should either multiply the above expression by 1 2 or else include this factor in the symmetry factors of the corresponding diagrams.We choose the latter option so that all symmetry factors for the fig.7 type diagrams in sec.II B 2 are 3. = 2π 3 C q (x 1 ) where ⃗ l, ⃗ l 1 are 2d transverse momenta (l + = l + 1 = 0), C q (x) = 1 − Z q (x) is the O(g 2 ) correction to the quark wave function renormalization factor, and F is a UV finite function satisfying F (0, 0; x/x 1 , m 2 ) → 0. Note that the UV divergent part does not depend on the momenta ⃗ l, ⃗ l 1 .The parameter m 2 is an infrared regulator for the DGLAP collinear singularity [68][69][70][71], and x is a cutoff for the soft singularity.
We begin with terms which involve Ψ * qqq (x 1 , ⃗ k 1 + x 1 ⃗ q − ⃗ q 12 ; x 2 , ⃗ k 2 + x 2 ⃗ q − ⃗ q 3 ; x 3 , ⃗ k 3 + x 3 ⃗ q).These are eq.( 17), the fourth line in eq. ( 35) times −C q (x 1 ), and eq.(37): First we verify that the subset of UV divergent diagrams satisfies the Ward identity, i.e. that their finite parts cancel when ⃗ q 1 → 0 or ⃗ q 3 → 0. We only demonstrate here the case ⃗ q 3 → 0 but we have checked the symmetry of these diagrams under ⃗ q 1 ↔ ⃗ q 3 .For the purpose of more compact expressions we will split off the pre-"factor" from the following expressions.

FIG. 8 :
FIG.8: Numerical model estimate of the impact parameter dependence of the C-odd part of the correlator of three color charge operators in the proton.

+ 6 • − 1 4 (+ 6 • 2C F − 1 2 − N c 2 tr t a t c t b + 6 •+ 6 •+ 6 • 2C F − 1 2 − N c 2 tr t a t b t c + 6 • (C F − N c 2 )+ 6 •
T c ) ba − N c tr t a t c t b + 1 4 tr T b T c (D a − T a ) (C F − N c 2 ) tr t a t b t c + (C F − 1) tr t a t c t b 2C ) ab − N c tr t a t b t c + 1 4 tr T a T c (D b − T b ) tr t a t c t b + (C F − 1) tr t a t b t c 2C 1 2 tr t a t b t c − 3 • C F − N c + 1 2 tr t a t b t c + t a t c t b − 3 • 2C F − N c + 1 2 tr t a t c t b t b t c + t a t c t b − 3 • 2C F − N c + 1 2 tr t a t b t c − 3 • 2C F − N c + 1 2 tr t a t c t b a c − 3 • 2C F (2 − N c ) tr t a t b t c = 0.