Diffractive deep inelastic scattering at NLO in the dipole picture : The qq¯g contribution

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Diffractive deep inelastic scattering at NLO in the dipole picture : The qq¯g contribution © Authors, 2022 Published version Beuf, G.; Hänninen, H.; Lappi, T.; Mulian, Y.; Mäntysaari, H. Beuf, G., Hänninen, H., Lappi, T., Mulian, Y., & Mäntysaari, H. (2022). Diffractive deep inelastic scattering at NLO in the dipole picture : The qq¯g contribution. Physical Review D, 106(9), Article 094014. https://doi.org/10.1103/PhysRevD.106.094014 2022 Diffractive deep inelastic scattering at NLO in the dipole picture: The q¯ qg contribution G. Beuf ,1H. Hänninen ,2,3 T. Lappi ,2,3 Y. Mulian,4and H. Mäntysaari 2,3 1National Centre for Nuclear Research, 02-093 Warsaw, Poland 2Department of Physics, University of Jyväskylä, P.O. Box 35, 40014 University of Jyväskylä, Finland 3Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland 4Instituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia-Spain (Received 1 July 2022; accepted 23 September 2022; published 9 November 2022) We calculate the contribution from the q¯ qg state production to the diffractive cross sections in deep inelastic scattering at high energy. The obtained cross section is finite by itself and a part of the full next-toleading order result for the diffractive structure functions. We perform the calculation in exact kinematics in the eikonal limit, and show that the previously known high-Q2and large M2 Xresults for the structure functions can be extracted from our results in the appropriate limits. We furthermore discuss the steps required to obtain the full next-to-leading order results for the structure functions. DOI: 10.1103/PhysRevD.106.094014 I. INTRODUCTION In collisions with high scattering energy, one is measuring degrees of freedom of hadronic and nuclear states that only have a small fraction of the full momentum of the state, small-xdegrees of freedom. The large amount of phase space available at high collision energies leads to an exponentially cascading emission of gluons. At some point, however, this cascade must be limited by unitarity requirements on scattering amplitudes. Thus gluon mergings eventually start to be equally important, even at transverse resolution scales where a weak coupling description is appropriate. The kinematical region where these two effects balance each other is referred to as the gluon saturation regime, and understanding it is the topic of many theoretical and experimental efforts. Experimentally the saturation regime is relevant for understanding hadronic collision processes and the formation of quark gluon plasma at RHIC and the LHC. A particularly precise and clean way to access the small-xdegrees of freedom is, however, provided by high energy deep inelastic scattering (DIS), both in the HERA experiments, and at the future electron-ion collider (EIC) [1–3] and LHeC [4,5]. Theoretically, a convenient way to discuss the physics of gluon saturation is provided by the color glass condensate (CGC) [6–8] effective field theory, where the nonlinear gluon system is described as a classical color field. For the DIS process, the CGC framework naturally leads to the dipole picture [9–13]. In the dipole picture one factorizes the DIS process of a virtual photon off a hadronic target into two ingredients. First, the perturbative part of the process is the development of the photon into a partonic state, to leading order a color neutral quark-antiquark dipole. The second ingredient is the scattering of this partonic state with the gluonic target, which in the high collision energy limit can be treated as an eikonal interaction with the classical color field. With the prospect of higher luminosities and the availability of nuclear targets in future DIS experiments, there has been a systematic push in the field to improve the perturbative accuracy of the dipole picture by going to higher orders in perturbation theory. In recent years the dipole picture has been extended to NLO accuracy for the high energy BK/JIMWLK evolution [14–27] and the inclusive DIS cross section [28–40]. Exclusive or diffractive DIS is expected to be even more sensitive to gluon saturation than inclusive cross sections [3,41–43]. One way to understand this is to note that, due to the optical theorem, the total cross section is proportional to the elastic dipole-target amplitude, proportional to the gluon distribution in the target. Exclusive cross sections, on the other hand, are calculated as the square of the amplitude, and are thus much more sensitive to the large amplitudes, a signature of the saturation regime. Correspondingly, the recent work on inclusive scattering has been accompanied by several calculations of exclusive vector meson and diffractive dijet production at NLO in the dipole picture [44–53]. While these processes are an Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 106, 094014 (2022) 2470-0010=2022=106(9)=094014(33) 094014-1 Published by the American Physical Society extremely important part of the coming experimental program, they both have some drawbacks for the purpose of understanding gluon saturation. Exclusive vector meson production requires some knowledge or modeling of the bound state physics of the meson. While there are systematical ways to do this perturbatively, e.g., by a nonrelativistic QCD approach as in Ref. [48] or by using universal parton distribution amplitudes that can independently be measured in other processes [46,50,54], this unavoidably adds an additional source of uncertainty. Dijets, on the other hand, are well defined perturbative objects in a given jet algorithm, but only if the jet transverse momenta are sufficiently large. At realistic collider energies this has a tendency to push jet measurements to larger xand thus outside the saturation regime. In this paper we will focus on a process that has gathered somewhat less attention in the work to push the dipole picture to NLO accuracy; namely, inclusive diffractive DIS. Here the experimental signature is a large rapidity gap between the diffractive system (X) consisting of the photon remnants, and the target or its remnants. In the dipole picture the photon fluctuates into a variety of partonic states, which scatter off the target without exchanging color. In this sense the rapidity gap makes the process fundamentally an exclusive one, with a cross section given by the square of an amplitude. On the other hand, the measurement is inclusive in the sense that one sums over all of the different final states of the diffractive system, measuring the cross section differentially only in its total invariant mass MX. This latter inclusive aspect makes it possible to extend the perturbative description to much lower invariant masses and to lower xthan for diffractive dijets, even if the parton level final states are the same. The cross sections for such processes are expressed in terms of the diffractive structure functions FDð3Þ 2ðβ;Q 2;x PÞand FDð3Þ Lðβ;Q 2;x PÞor, equivalently, the diffractive virtual photon cross sections dσγþA→Aþn=d½PSn, which are the quantities that we will calculate here. At leading order in αs, the diffractive final state only consists of a quark-antiquark dipole. This already provides a good description of the general features of the experimental measurements at small M2 X∼Q2[41] (see also work in Refs. [55–57]). However, a strict leading order picture fails to describe the rise of the cross section towards larger MXwhere, in order to make a high invariant mass partonic state, additional gluon radiation is required. The phenomenologically most successful approach has been to use the “Wüsthoff result”[58], which includes the radiation of one extra gluon into the final state [59–64] in a large Q2 kinematical approximation. In our terminology, this treelevel gluon emission is already a part of the NLO result, being explicitly proportional to αs. In this paper we will calculate the same contributions as in the Wüsthoff result at what we call the exact kinematics in the eikonal limit. This means that the kinematics within a diffractive system is treated exactly without a large Q2approximation, while the interaction with the target is eikonal. Our result presented in this paper corresponds to a subset of the NLO results that is finite by itself, and suited for explicit numerical evaluation. The completion of the full NLO calculation requires the inclusion of virtual corrections with gluons that are not produced in the final state. We plan to return to these contributions in future work. Here we will merely outline steps that are needed to calculate these virtual contributions in our formalism. We have also here opted to calculate only the contributions where the gluon is emitted before the shockwave, not combining them with emissions after. This avoids issues with collinear and soft divergences in final state emissions, which eventually cancel against virtual corrections. This paper is structured in the following way. We will start by introducing the experimental observable, the diffractive structure function, in Sec. II and the dipole picture formulation in terms of LCPT in Sec. III. Before moving to specific diagrams we will then discuss in Sec. V the general strategy to calculate phase space integrals for 2and 3-particle final states of a fixed invariant mass MXin the context of an eikonal scattering picture where the interaction with the target happens at a fixed transverse coordinate. We will rederive the known result for the leading q¯ qcomponent of the wave function in our notations in Sec. VI. We then move to the main new result of this paper, the calculation of the q¯ qg component of the cross section in full kinematics in Sec. VII. A more detailed exposition of intermediate stages of the calculation has been presented earlier in Ref. [65]. We check in Sec. VIII that our calculation reduces to known results in the kinematical limits of large Q2(Wüsthoff [58]) and large MX(e.g., in Ref. [66]), before concluding in Sec. IX. The results in this paper cover a finite, self-contained subset of the NLO corrections to the diffractive results, generalizing earlier calculations to the full kinematics. We plan to return to the calculation of the remaining parts in a future publication, as outlined in Sec. III. II. DIFFRACTIVE STRUCTURE FUNCTIONS The diffractive cross section σD eþA→MXþpin electronnucleus (or electron-proton) DIS integrated over the squared momentum transfer tis usually expressed in terms of the diffractive structure functions FD 2and FD Ldefined as dσD eþA→MXþp dβdQ2dxP¼2πα2 em βQ4½1þð1−yÞ2FDð3Þ 2ðβ;Q 2;x PÞ −y2 1þð1−yÞ2FDð3Þ Lðβ;Q 2;x PÞ:ð1Þ The superscript (3) refers to the structure functions that depend on three variables, in this case β;Q 2, and xP G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-2 discussed above. One can also consider structure functions differentially in the squared momentum transfer t, in which case one has the structure functions FDð4Þ 2;L ðβ;Q 2;x P;tÞ.In this work we consider both t-differential and t-integrated cross sections. For simplicity we focus here on coherent diffraction that corresponds to the events where the target does not dissociate, but our results are straightforward to generalize to dissociative events in the Good-Walker [67] picture (see, e.g., Refs. [68–71]). The diffractive structure functions are related to the total diffractive cross sections in γþAscattering as xPFDð4Þ T;L ¼Q2 4π2αem Q2 β dσD γ T;LþA→MXþA dM2 Xdt;ð2Þ where Tand Lrefer to transversely and longitudinally polarized photons. The experimentally measured [72–75] total diffractive cross sections are usually reported in terms of the diffractive reduced cross section σDð3Þ rðβ;Q 2xPÞ¼FDð3Þ 2ðβ;Q 2xPÞ −y2 1þð1−yÞ2FDð3Þ Lðβ;Q 2xPÞ:ð3Þ Here the Lorentz-invariant quantities describing the kinematics are the virtuality of the photon −Q2and the fraction of the target longitudinal momentum xPcarried by the pomeron (exchanged in the scattering process) in the frame where the target has a large longitudinal momentum, defined as xP≡ðP−P0Þ·q P·q¼M2þQ2−t W2þQ2−m2 N ≈M2 XþQ2 W2þQ2:ð4Þ The invariant mass of the diffractively produced system is denoted by M2 Xand the nucleon mass by m2 N. The variable β¼Q2=ð2q·ðP−P0ÞÞ≈Q2=ðM2 XþQ2Þhas, in the frame where the target momentum is large, an interpretation as the fraction of the pomeron momentum carried by the struck quark. The four vectors P,P0are the target nucleon momenta before and after the scattering, respectively, see Fig. 1. Finally y¼ðP·qÞ=ðP·lÞis the inelasticity describing the energy transfer from the lepton with initial momentum l, and qis the photon momentum. We are working in the dipole picture, where one develops the virtual photon state in a series of partonic Fock states. Let us first consider the general case of an n-parton Fock state, for which we denote the phase space element as ½PSn, At leading order only the n¼q¯ qstate contributes, and in this work we focus on including the n¼q¯ qg contribution, which is actually the dominant component at high M2 X(small β) and at high Q2[62]. This tree-level contribution is also a necessary ingredient for the future full calculation of the diffractive structure functions at NLO accuracy. The diffractive cross section can be written as dσD γ T;LþA→MXþA dM2 X¼X nZd½PSn dσγ T;LþA→Aþn d½PSn δðM2 X−M2 nÞ; ð5Þ where M2 nis the invariant mass of the Fock state n. The cross section dσγA→Aþn d½PSnfor a production of a color-singlet state nin photon-nucleus or photon-proton scattering is expressed in terms of the scattering amplitudes (see [76], except we now normalize with a 2qþin a different place) as dσD γþA→Aþn¼2qþð2πÞδðqþ−qþ nÞY i∈F:s:n f dpijMγ→nj2:ð6Þ Here i∈F:s:n means iterating over all the particles iin the Fock state nand qþ nis the total plus momentum of the partons in this Fock state. The one particle phase space element reads f dpi≡d2pidpþ i 2pþ ið2πÞ3:ð7Þ The scattering amplitude is obtained from the matrix elements of dressed, interacting, states by leaving out a momentum conservation delta function DhF:S:njˆ S−1jγiD¼2qþð2πÞδðqþ γ−qþ nÞMγ→n;ð8Þ where in the diffractive scattering the final state F:S:n is a color singlet. At high energies, the scattering amplitudes Mγ→ncan be calculated by considering the γ→nprocess, and inserting an interaction with the shockwave (target color field) in all possible ways. At high energies the transverse coordinates of the partons are fixed when they propagate in the color field of the target and as such the interaction can be straightforwardly described in terms of Wilson lines at FIG. 1. Kinematics of inclusive diffractive DIS. DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-3 fixed transverse coordinates as discussed in more detail in Sec. IV. III. OUTLINE OF NLO CALCULATION We will in this paper compute a part of the NLO correction to the diffractive structure functions that is finite by itself. However, let us first discuss the overall structure of the full NLO contribution in terms of the contributing diagrams. This discussion will make it more clear which parts of the NLO contributions are included in our result here, and what still needs to be done to calculate the rest. We are calculating, and drawing diagrams, for an exclusive amplitude for a virtual photon-target shockwave scattering. Thus a single diagram includes both the Fock state expansion of the incoming dressed virtual photon state jγiD,and that of the outgoing dressed multiparton state DhF:s:njin the amplitude (8). In the diagrams the shockwave is represented by a blue band, with time progressing from left to right. The state furthest to the left is the asymptotic incoming state (the dressed virtual photon), which then develops into a superposition of bare parton states, corresponding to the LFWFs γ→n. The interaction with the shockwave then is given in terms of the bare states [77]. On the other side of the shockwave, furthest to the right, is the asymptotic final state DhF:s:nj, which develops into a superposition of bare states, going leftward in the figure. Thus the part of the figure to the right of the shockwave corresponds to the complex conjugate of the LFWF of the final state, in particular with energy denominators calculated with respect to the final state. At leading order, the only diagram contributing to the scattering amplitude is diagram (a) shown in Fig. 2, where the photon first splits to a q¯ qdipole, and then subsequently the quarks scatter off the target color field with no net color charge transfer to the target. In momentum space, we denote the quark and antiquark transverse momenta as p0 and p1, respectively, and in transverse coordinate space use the coordinates x0,x1. Similarly the fractions of the photon plus momentum carried by the quark and the antiquark are zi¼pþ i=qþwith z0þz1¼1. A. Radiative corrections The purpose of this paper is to calculate the gluon emission part of the next-to-leading order contributions to the diffractive structure functions, which dominates at large M2 X. In the CGC, if one integrates the transverse momentum of several final state particles without restriction, one can encounter spurious UV divergences in real higher order corrections, associated with the breakdown of the eikonal approximation. This occurs when the light-cone momentum p−scales associated with the produced system become comparable or larger than that of the target. In the case of diffractive structure functions considered here, the fixed invariant mass of the produced q¯ qg state ensures that the eikonal approximation stays valid for the whole integration range, by constraining the p−scale of the diffractive system. For that reason, no UV divergence can arise in real NLO corrections to diffractive structure functions. This is to be compared to the case of inclusive DIS [31–33] where one uses the optical theorem and thus the final state is completely fixed to be the same as the initial one. In dijet production [36,37,44,45,52], on the other hand, one typically fixes the momenta of some of the final state particles and integrates over the others. This can lead to a different pattern of cancellations between diagrams. The calculation of the loop corrections is left for future work, but for completeness we list all the relevant diagrams in the following Sec. III B. In order to calculate the n¼q¯ qg contribution to the diffractive scattering amplitude we include gluon emission contributions from both the quark and the antiquark. There can be a regular gluon emission before the shockwave shown in diagrams in Figs. 3(b) and 3(c). In light cone FIG. 2. Leading order amplitude. The blue band represents the interaction with the target color field. FIG. 3. Gluon emission before the shock. G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-4 perturbation theory there is also an instantaneous γ→q¯ qg vertex resulting in instantaneous gluon emission diagrams in Figs. 4(d) and 4(e). These are the contributions that we will calculate in detail in this paper. Similarly the gluon can be emitted after the shock, see diagrams in Figs. 5(f) and 5(g). There are several relations between these diagrams. First, since the virtual photon as a whole is color neutral, there is a destructive interference between emissions from the quark and from the antiquark. This cancels the leading small transverse momentum (collinear) divergence and serves as a useful check of the relative sign between the contributions. The contributions with emissions after the shockwave, pictured in Fig. 5, can be conveniently obtained from the corresponding ones in Fig. 3where the emission happens before by taking a specific coordinate limit, in a procedure developed in Ref. [78]. Here one first separates out from the coordinate space γ→q¯ qg wave function a piece corresponding to the final gluon emission. In terms of equations this means that one writes the γ→q¯ qg wave function obtained from diagrams in Figs. 3(b),3(c),4(d), and 4(e) in a factorized form as1 ˜ ψγ λ→q0¯q1g2¼˜ ψγ λ→q0¯q1;q0→q0g2 ˜ ψq0→q0g2 þ˜ ψγ λ→q0¯ q1;¯ q1→¯ q1g2 ˜ ψ¯ q1→¯ q1g2:ð9Þ Here ˜ ψq0→q0g2and ˜ ψ¯ q1→¯ q1g2are the 1→2particle gluon emission wave functions. Equation (9) should be understood as the definition of the remaining parts ˜ ψγ λ→q0¯ q1;q0→q0g2and ˜ ψγ λ→q0¯ q1;¯ q1→¯ q1g2. Here the notation refers to these being the parts of the wave function that are associated (e.g., by the helicity structure) with the first γ→q¯ qsplitting (γ λ→q0¯ q1), but depend on the fact that the (anti)quark will later emit a gluon (q0→q0g2, ¯ q1→¯ q1g2). Thus ˜ ψγ λ→q0¯ q1;q0→q0g2and ˜ ψγ λ→q0¯ q1;¯ q1→¯ q1g2 depend on the coordinates of all three particles in the final state. The calculation of the gluon emission diagrams after the shock wave requires the q¯ qcomponent in the dressed Dhq¯ qgjstate. This, in turn, requires the ðq¯ qg →q¯ qÞ† merging wave function, which is given by (minus) the Hermitian conjugates of the corresponding emission wave functions ˜ ψq0→q0g2and ˜ ψ¯ q1→¯ q1g2. In other words, one is here factoring out from the gluon emission before the shockwave the gluon emission wave function that appears when the gluon is emitted after. A look at the transverse coordinate space γ→q¯ qg wave function [see, e.g., Eqs. (C14) and (C19) in Ref. [33] for explicit expressions] shows that indeed the structure of the regular emission wave functions naturally factorizes like this. Using the factorized notation (9), the procedure of Ref. [78] for obtaining amplitudes for the emission-afterthe-shock contributions in Fig. 5is the following. One evaluates both the Wilson line operators and the γ→q¯ q parts of the wave functions ˜ ψγ λ→q0¯ q1;q0→q0g2,˜ ψγ λ→q0¯ q1;¯ q1→¯ q1g2 with the transverse coordinates of the gluon and its parent FIG. 4. Instantaneous diagram gluon emission wave function with emission before the cut. FIG. 5. Gluon emission after shock. Note that the intermediate transverse coordinate of the quark (antiquark) before the gluon emission is not the same as the coordinate at the cut. They are defined by x0 0≔z0x0þz2x2 z0þz2and x0 1≔z1x1þz2x2 z1þz2. 1Here ˜ ψdenotes a reduced coordinate space wave function. See Eqs. (16) and (17) for the normalization in the γ→q¯ q, γ→q¯ qg case; the convention is trivially extended to the 1→2 gluon emission case. DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-5 (anti)quark replaced by the coordinate of the parent before the emission, and changes the sign. Thus, for the emission from the quark, diagram in Fig. 3(b),one replaces x0→x0 0and x2→x0 0, in both the Wilson line operator and in ˜ ψγ λ→q0¯ q1;q0→q0g2, with the coordinate defined as x0 0≔z0x0þz2x2 z0þz2. Correspondingly, for the emission from the antiquark, diagram in Fig. 3(c), one replaces x1→x0 1and x2→x0 1,withx0 1≔z1x1þz2x2 z1þz2. It is clear that this corresponds to the Wilson line operator being evaluated at the correct coordinate for the diagrams in Fig. 5(f) and 5(g). There is no “emission after the shockwave”contribution for the instantaneous diagrams in Fig. 4(d) and 4(e). This is, however, built into the formalism of Ref. [78], since it turns out that the parts of ˜ ψγ λ→q0¯ q1;q0→q0g2,˜ ψγ λ→q0¯ q1;¯ q1→¯ q1g2corresponding to the instantaneous diagrams vanish in the coordinate limits x0;x2→x0 0and x1;x2→x0 1, respectively. One can arrive at this procedure for constructing the final state emissions in multiple ways. In Ref. [78] it is derived explicitly by looking at the expressions and noticing a relation between the energy denominators of the different diagrams. More generally, using the orthogonality of the jγiDand jq¯ qgiDstates one can derive the corresponding relation between the wavefunctions for ðq¯ qg →q¯ qÞ†, γ→q¯ q[diagrams in Fig. 5(f) and 5(g)], γ→q¯ qg [diagrams in Figs. 3(b),3(c),4(d), and 4(e)] and the process ðq¯ qg →γÞ†, corresponding to the photon crossing the shockwave first and all the emissions happening after (i.e., the bare photon state in the final Dhq¯ qgj). Since the last one does not contribute to the amplitude because the photon is color neutral, one obtains a linear relation for the contributions of the emission diagrams of Figs. 3and 5. As discussed above, one can also see this relation directly by looking at the coordinate space wave functions, using the Fourier transforms from, e.g., Appendix C of [33]. In an inclusive observable where one integrates over the momenta of the final state gluon and of its parent without any restrictions, it would be natural to always keep the initial and final state gluon emissions, Figs. 3and 5, together because they have a tendency to cancel each other in the UV region where the gluon is at the same coordinate as the emitting quark. Thus, they are often evaluated together such as in Refs. [78,79]. However, for the case of the diffractive structure function, the restriction on the diffractive system mass MXcuts out contributions of large transverse momenta. Thus it is quite natural to evaluate the contributions of the emissions before and after the shockwave separately. On the other hand, the final state gluon emissions are associated with the wave function renormalization constants for, and gluon exchanges between, the outgoing quarks. The relation to these contributions which are, in our language, a part of the q¯ qpart of the cross section is especially important for the kinematical region when the gluon becomes collinear to the quark, where corresponding IR divergences must cancel. Thus it would not be natural here to consider the diagrams with gluon emission after the shockwave, before taking into account all the loop corrections. In conclusion, for a final state with a fixed MX, the natural way to group diagrams together is different from some other observables. Since we are here leaving the NLO q¯ qcontribution overall to a future paper, we will also not calculate the final state emission contributions here. The exception to this is in Sec. VIII A, where one works in the MX→∞limit neglecting the restriction on final state momenta, and thus only the inclusion of the final state emissions allows one to get a finite result. The 3-jet cross section in diffractive DIS has been computed earlier in Ref. [44], using the shockwave formalism that should be equivalent to our result here. The calculation includes emissions both before and after the shockwave, as is appropriate for the case of a fully differential 3-jet cross section. The IR divergences associated with wave function renormalization of the outgoing quarks would appear only after integrating over the phase space of the gluon, which is not done in Ref. [44],butis done here. Checking the equivalence of the result at the final cross section level would require a significant amount of algebra which we have not performed here. However, it was found in Ref. [44] that the result is compatible with the γ→q¯ qwave function of Ref. [30], which is the starting point of our calculation, as discussed in more detail in Sec. VII. FIG. 6. Propagator and normal vertex correction diagrams calculated in [32,33]. FIG. 7. Instantaneous vertex correction diagrams calculated in [32,33]. G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-6 B. Loop corrections In addition to the radiative contributions discussed in this paper, there are also loop corrections at NLO. We will return to them in more detail in a future paper, but let us make a few remarks here. First, there are the one-loop corrections to the γ→q¯ qwave function, depicted in Figs. 6and 7. These include a quark propagator correction before the shockwave shown in diagrams in Fig. 6(h) and 6(i),and corrections to the γ→q¯ qvertex (including regular and instantaneous gluon or quark exchange) shown in diagrams in Figs. 6(j),6(k),7(l),7(m),7(n),7(o),and7(p). These oneloop wave functions have already been calculated as a part of the virtual photon NLO wave function [32,33,38–40],and the results for the loop calculations can be directly taken from these references. In diffractive scattering there are also additional diagrams that are not needed for total (inclusive) cross section and as such are not available in the literature. Now it will be necessary to add propagator correction diagrams where the gluon crosses the shockwave, diagrams in Fig. 8(q) and 8(r). These exhibit UV divergences in the limit when the gluon coordinate becomes equal to the emitting (anti)quark. Similarly to the calculation of the inclusive cross section these will have to partially cancel UV divergences in the vertex correction diagrams in Figs. 6and 7. This cancellation is the reason why the one-loop γ→q¯ qwave function alone is not sufficient to directly achieve the full NLO result. In addition to the propagator correction type diagrams, similar normal and instantaneous vertex correction diagrams in Fig. 9(s)–9(v) are also needed. In our formalism (see [31,80] for more detailed discussions, based on the seminal work of [76]) we specifically exclude diagrams containing self-energy corrections inserted on the external asymptotic particles. Thus we do not explicitly have the diagrams in Fig. 10(w) and 10(x) in our calculation. Instead, one must attach to the amplitude a wave function renormalization constant ffiffiffiffiffiffiffiffiffi Zq=¯ q p[again see Eq. (11)], which includes the same physical contribution, and is determined by the unitarity of the evolution operator. The outgoing quark and antiquark wave function renormalization constants also have UV divergences, which should cancel the rest of the UV divergences from the vertex corrections. Finally there are also diagrams with a gluon exchange in the final state [diagrams in Fig. 11(y),11(z), and 11(aa)]. Naively, one could think that these corrections correspond to a renormalization of the outgoing q¯ qstate, but they cannot be absorbed into just a constant, since the quark and antiquark actually exchange momentum in the exchange. A proper discussion of how to define the dressed q¯ qoutgoing state and treat the interactions between the outgoing particles is a major part of the discussion of the full NLO result, which we will return to in future work. IV. INITIAL AND FINAL FOCK STATES We will from now on focus on the leading order q¯ qpart and the radiative q¯ qg part of the cross section. To begin, let us define explicitly our notations and normalization for the Fock states. The Fock expansion of the incoming virtual photon state in terms of bare partonic states is jγ λðqþ;q;Q2ÞiD¼ffiffiffiffiffiffiffi Zγ λ qNon-QCD Fock states þg X q0¯q1F:s: ˜ Ψγ λ→q0¯ q1 ˜ b† 0 ˜ d† 1j0i þg X q0¯ q1g2F:s: ˜ Ψγ λ→q0¯q1g2 ˜ b† 0 ˜ d† 1 ˜ a† 2j0iþ; ð10Þ FIG. 8. Propagator correction diagrams where the gluon crosses the shockwave, but is not produced. FIG. 9. Gluon emission diagrams where the gluon crosses the shockwave, but is not produced. FIG. 10. Diagrams with propagator correction in final state. These are not included, but instead there is a wave function renormalization constant for the outgoing states. FIG. 11. Final state interaction diagrams. DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-7 where F. s. stands for “Fock states.”For the outgoing partonic states the corresponding expansions are Dh¯ q1q0j¼ ffiffiffiffiffiffi Zq p ffiffiffiffiffiffi Z¯ q ph0j ˜ d1 ˜ b0þg X q00¯ q10F:s: 0h0j ˜ d10 ˜ b00 ˜ Ψ† q0¯ q1→q00¯ q10þg X q00¯ q10g20F:s:h0j˜ a20 ˜ d10 ˜ b00 ˜ Ψ† q0¯ q1→q00¯ q10g20þ;ð11Þ Dhg2¯ q1q0j¼ ffiffiffiffiffi Zg pffiffiffiffiffiffi Zq p ffiffiffiffiffiffi Z¯q ph0j˜ a2 ˜ d1 ˜ b0þg X q00¯ q10F:s:h0j˜ d10 ˜ b00 ˜ Ψ† q0¯ q1g2→q00¯ q10þ:ð12Þ Here, we have only written out states that are needed for the full NLO cross section. In addition, marked with …, there are other states needed at higher orders and non-QCD Fock states, including the photon. In fact, for the radiative corrections that we calculate in this paper, only the leading order terms in Eqs. (11) and (12) are needed. Strictly speaking the final state wave functions are not exactly hermitian conjugates of initial state ones, but differ by the sign of the iεin the energy denominators [see Eqs. (4.2) and (4.3) of Ref. [76]]. This difference does not play a role in the calculation of this paper, but will be crucial in the calculation of the full NLO corrections to diffractive DIS, that we leave for a future publication. The notation ˜ Σdenotes the sum over the quantum numbers of each parton in the Fock state and a mixed space phase-space integration for each parton [31,65]. Here the prime 0in the sum denotes the fact that the original state is not included in the sum [76].2For brevity the transverse coordinates and flavor (f) and helicity (h) indices are not written down explicitly here. We will discuss the color structure of the final state explicitly below. The γ λ-state renormalization coefficient is Zγ λ¼1þOðe2Þand so it can be dropped in this work. The renormalization coefficients Zq;¯ q;g of the partonic states are of the order of 1þOðg2Þ, and as such do not affect the tree-level NLO corrections that we discuss in this paper. In the mixed space xiare the transverse coordinates and kþ ithe longitudinal momenta of the partons, and indices i¼0, 1 refer to the quark and the antiquark, and i¼2to the gluon. The quark, antiquark, and gluon creation and annihilation operators satisfy the (anti-)commutation relations ½aðkþ 0;x0;λ0;a 0Þ;a †ðkþ 1;x1;λ1;a 1Þ ¼ ð2kþ 0Þð2πÞδðkþ 0−kþ 1Þδð2Þðx0−x1Þδλ0;λ1δa0;a1;ð13Þ fbðkþ 0;x0;h 0;α0Þ;b †ðkþ 1;x1;h 1;α1Þg ¼ ð2kþ 0Þð2πÞδðkþ 0−kþ 1Þδð2Þðx0−x1Þδh0;h1δα0;α1;ð14Þ fdðkþ 0;x0;h 0;α0Þ;d †ðkþ 1;x1;h 1;α1Þg ¼ ð2kþ 0Þð2πÞδðkþ 0−kþ 1Þδð2Þðx0−x1Þδh0;h1δα0;α1:ð15Þ Here aiand λirefer to the gluon color and polarization, respectively. In Eq. (10) the functions ˜ Ψγ λ→q0¯ q1and ˜ Ψγ λ→q0¯ q1g2are the light front wave functions (LFWFs) describing the perturbative γ→q¯ qand γ→q¯ qg splittings. Furthermore it is convenient to factor out the overall color factor, momentum conservation and dependence on the photon transverse momentum q, and define the reduced wave functions ˜ ψγ λ→q0¯ q1and ˜ ψγ λ→q0¯ q1g2as ˜ Ψγ λ→q0¯ q1¼ð2qþÞ2πδðkþ 0þkþ 1−qþÞeiq qþ·ðkþ 0x0þkþ 1x1Þ ×1α0α1 ˜ ψγ λ→q0¯ q1;ð16Þ ˜ Ψγ λ→q0¯ q1g2¼ð2qþÞ2πδðkþ 0þkþ 1þkþ 2−qþÞ ×eiq qþ·ðkþ 0x0þkþ 1x1þkþ 2x2Þta α0α1 ˜ ψγ λ→q0¯ q1g2:ð17Þ These LFWFs are currently available in the literature. The lowest order ˜ ψγ λ→q0¯q1is a standard result [81]. Loop corrections to it, as well as tree-level wave function describing the γ→q¯ qg splitting have been recently calculated in mixed space (and ddimensions) in Refs. [30,32,38–40]; other results derived in momentum space include Refs. [82–85], and some are compatible with BFKL evolution [28,29] but not the gluon saturation regime. 2What exactly counts as the “same state”requires a more detailed discussion in the case of a two-particle state than for one particle; we plan to return to this in a future paper in the context of the full NLO calculation where this term is needed. In practice, only the diagrams including self-energy corrections on asymptotic external legs need to be excluded, since they are already taken into account thanks to the overall renormalization constants. G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-8 X h0;h1ð˜ ψγ L→q¯ 0¯q¯ 1Þ†ð˜ ψγ L→q0¯q1Þ¼2e2e2 f ð2πÞ2z3 0z3 14Q2K0ðx01 ¯ QÞK0ðx¯ 0¯ 1¯ QÞð72Þ and 1 2X Tpol λX h0;h1ð˜ ψγ λ→q¯ 0¯q¯ 1Þ†ð˜ ψγ λ→q0¯q1Þ¼e2e2 f ð2πÞ2z2 0z2 1ððz0−z1Þ2þ1Þx01 ·x¯ 0¯ 1 x01x¯ 0¯ 1 Q2K1ðx01 ¯ QÞK1ðx¯ 0¯ 1¯ QÞ:ð73Þ A more familiar form of the momentum fraction dependence can be obtained noting that ðz0−z1Þ2þ1≡ 2ðz2 0þz2 1Þ, which holds under the plus-momentum conservation z0þz1¼1. B. From wave function to diffractive cross section The total diffractive cross section at leading order can be obtained by substituting the virtual photon wave function in Eq. (40), and using the phase space integrals Ið2Þ Δand Ið2Þ MX given in Eqs. (43) and (45). Let us first consider the case where the virtual photon is longitudinally polarized. We note that although the squared wave function (72) factorizes as K0ðx01 ¯ QÞK0ðx¯ 0¯ 1¯ QÞ, the diffractive cross section cannot be written simply as a square of an amplitude, since the phase space integral Ið2Þ MXmixes the transverse coordinates in the amplitude and in the conjugate amplitude even after an integral over the total momentum transfer, see Eq. (45). In order to derive the leading order results for the q¯ q contribution to the diffractive structure functions commonly used in the literature [61], further approximations are required. In particular, we assume that (i) The invariant mass M2 Xor virtuality Q2is so large that exp ½−i2z−1 2Δ·ðr−¯ rÞ≈1(note that r2;¯ r2≲1=Q2;1=M2 X). In this case, the momentum transfer integral of Ið2Þ Δgives only δð2Þðb−¯ bÞ, see Eq. (47). This is also the case if the dipole-proton scattering amplitude depends on the center of mass of the q¯ qsystem b0¼z0x0þz1x1and not on the impact parameter b¼ðx0þx1Þ=2(see discussion in Sec. VA). (ii) The dipole-target interaction does not depend on the orientation of the dipole or that of the impact parameter, i.e., S01 ≡Sðkx0−x1k;kbkÞ≕Srb. The angular dependence is commonly neglected when the initial condition for the BK evolution is determined by fitting the HERA data [34,100–103] and in parametrizations such as IPsat [104].However,in general such a (probably weak) angular dependence should exist, see, e.g., Refs. [112–116]. Under these assumptions, the only dependence on the angle θr;¯ rbetween r≔x01 and ¯ r≔x¯ 0¯ 1is in the phase space integral Ið2Þ MX. This integral then gives Zdθr;¯ rIð2Þ MX¼Zdθr;¯ rJ0ðffiffiffiffiffiffiffiffiffi z0z1 pMXk¯ r−rkÞ ¼ 2πJ0ðffiffiffiffiffiffiffiffiffi z0z1 pMXkrkÞJ0ðffiffiffiffiffiffiffiffiffi z0z1 pMXk¯ rkÞ;ð74Þ and as such it factorizes into parts that only depend on transverse coordinates in the amplitude or in the conjugate amplitude. Integrating over tresults in a delta function forcing b¼¯ b[see Eq. (47)], and the cross section becomes dσD L;q¯ q dM2 X¼Nc e2 ð2πÞ2X f e2 fZ1 0 dz0z3 0ð1−z0Þ3Zd2bZdrrJ0ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z0ð1−z0Þ pMXrÞQK0ðr¯ QÞðSrb −1Þ2 ;ð75Þ where we have substituted the longitudinal photon wave function summed over helicities shown in Eq. (72). Note that if the impact parameter dependence factorizes from the dipole scattering amplitude, i.e., ðSrb −1Þ≡TðbÞ½Sr−1, then the impact parameter integral completely factorizes and gives Rd2bjTðbÞj2. On the other hand, for the transversely polarized photons, the leading order reduced wave function squared depends on the angle between rand ¯ ras shown in Eq. (73): X Tpol:λX h0;h1ð˜ ψγ λ→q¯ 0¯ q¯ 1Þ†ð˜ ψγ λ→q0¯ q1Þ∝r·¯ r krkk¯ rk;ð76Þ which means that the part that depends on the dipole sizes rand ¯ r—omitting the dipole amplitudes for now—reads DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-15 Zd2rZd2¯ rIð2Þ MX r·¯ r krkk¯ rk¼Zd2rZd2¯ rZd2l ð2πÞ2δðl2−z0ð1−z0ÞM2 XÞr·¯ r krkk¯ rkeil·ð¯ r−rÞ:ð77Þ Parametrizing the angles as ∠ðl;rÞ≕θand ∠ðl;¯ rÞ≕¯ θ, we have for the dot product: r·¯ r¼r¯ rðcos θcos ¯ θþsin θsin ¯ θÞ, where the sine term vanishes in the integration. Thus we are left with ZrdrdθZ¯ rd¯ rd¯ θZd2l ð2πÞ2δðl2−z0ð1−z0ÞM2 XÞcos θcos ¯ θeil¯ rcos ¯ θe−ilr cos θ ¼πZrdrZ¯ rd¯ rJ1ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z0ð1−z0Þ pMXrÞJ1ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z0ð1−z0Þ pMX¯ rÞ:ð78Þ Consequently, we find that (only) under the assumptions listed at the beginning of this subsection the diffractive cross section can be written in a factorized form independently of the photon polarization. Otherwise the transverse coordinates in the amplitude and conjugate amplitude are mixed. In the future it will be interesting to study numerically the effect of these assumptions, that were used, e.g., in Ref. [62] where a good description of the HERA diffractive structure function data was obtained. Both cross sections can now under these assumptions be expressed in terms of an auxiliary function, denoting now z0¼zwith the integral over z1is performed using the δfunction Φnðz; β;Q;bÞ¼ZdrrJnðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zð1−zÞ pMXrÞKnðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi zð1−zÞ pQrÞðSrb −1Þ2 :ð79Þ Using this, we may write the diffractive structure functions as xPFD L;q¯qðβ;x P;Q 2Þ¼NcQ4 2π3βXe2 fZd2bZ1 0 dzz 3ð1−zÞ3Q2Φ0ðz; β;Q;bÞ;ð80Þ xPFD T;q¯qðβ;x P;Q 2Þ¼NcQ4 8π3βXe2 fZd2bZ1 0 dzz 2ð1−zÞ2ðz2þð1−zÞ2ÞQ2Φ1ðz; β;Q;bÞ;ð81Þ which is in agreement with Ref. [62], once one accounts for the different normalization of the dipole amplitude and the different integration domain of z. VII. TREE-LEVEL q¯ qg CONTRIBUTION TO THE DIFFRACTIVE STRUCTURE FUNCTIONS In this section we present the main result of this paper: the tree-level calculation of diffractive q¯ qg production as a function of M2 Xand t. We consider the case where the gluon is emitted before the shockwave and the q¯ qg system then interacts with the target, corresponding to the diagrams in Figs. 3(b),3(c),4(d),and4(e). The emission-after-shock contribution could then in principle be obtained by taking the appropriate coordinate limits following the method developed in Refs. [78,107]. As discussed in Sec. III,wewill however leave it to a future publication. For simplicity we only consider the massless quark limit in this work. The diffractive q¯ qg production cross section was written in Sec. VB [see Eq. (55)] in terms of the phase space integrals Ið3Þ MXand Ið3Þ Δgiven in Eqs. (67) and (59) as dσD λ;q¯ qg dM2 Xdjtj¼4π4NcCFZ1 0 dz0 z0Z1 0 dz1 z1Z1 0 dz2 z2 δðz0þz1þz2−1ÞZx0Zx1Zx2Z¯ x0Z¯ x1Z¯ x2 Ið3Þ ΔIð3Þ MX ×X h0;h1;λ2ð˜ ψγ λ→q¯ 0¯q¯ 1g¯ 2Þ†ð˜ ψγ λ→q0¯q1g2Þ½S† ¯ 0¯ 1¯ 2−1½S012 −1:ð82Þ The only part missing from the diffractive q¯ qg production cross section is thus the calculation of the square of the tree-level wave function ˜ ψγ λ→q0¯ q1g2, with different transverse coordinates in the amplitude (xi) and in the conjugate amplitude (¯ xi), summed over helicities. The plus momentum fractions ziare external kinematical variables and therefore the same in the direct and complex conjugate amplitude. We calculate this square using the wave functions for the longitudinally and transversely polarized photons in four dimensions from Ref. [32] (see also Refs. [30,38–40]), with the modification that the factor of 2qþhas been taken out of the reduced wave functions ˜ ψ in the definition (17). The reduced LFWF for the longitudinal photon reads G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-16 ˜ ψTree γ L→q0¯ q1g2¼eefgi ð2πÞ2εj λ22QK0ðQX012Þffiffiffiffiffi z0 pffiffiffiffiffi z1 pδh1;−h0 ×z1½ð2z0þz2Þδjm −ið2h0Þz2ϵjmxm 20 x2 20−z0½ð2z1þz2Þδjm þið2h0Þz2ϵjmxm 21 x2 21;ð83Þ whereas for the transverse photon we have ˜ ψTree γ λ→q0¯ q1g2¼eefg ð2πÞ2εi λεj λ2ffiffiffiffiffi z0 pffiffiffiffiffi z1 pδh1;−h0 Q X012 K1ðQX012Þ ×z1½ð2z0þz2Þδjm −ið2h0Þz2ϵjm½ð2z1−1Þδil −ið2h0Þϵilxl 0þ2;1xm 20 x2 20 þz0½ð2z1þz2Þδjm þið2h0Þz2ϵjm½ð2z0−1Þδil þið2h0Þϵilxl 0;1þ2xm 21 x2 21 −z0z1z2 z0þz2½δij −ið2h0Þϵijþz0z1z2 z1þz2½δij þið2h0Þϵij;ð84Þ where X012,x0þ2;1 and x0;1þ2are defined as X2 012 ≔z0z1x2 01 þz0z2x2 02 þz1z2x2 12;ð85Þ x0þ2;1 ≔− z0 z0þz2 x20 þx21 ¼x01 þz2 z0þz2 x20;ð86Þ x0;1þ2≔−x20 þz1 z1þz2 x21 ¼x01 −z2 z1þz2 x21:ð87Þ The quantity Q2X2 012 corresponds to the q¯ qg formation time divided by the lifetime of the virtual photon that forms the q¯ qg system, as discussed in more detail in Ref. [30]. Configurations with large Q2X2 012 are exponentially suppressed, which enforces the restriction that the q¯ qg state must develop within a formation time that is less than the lifetime of the virtual photon. The calculation of the squared wave functions Ph0;h1;λ2ð˜ ψγ λ→q¯ 0¯ q¯ 1g¯ 2Þ†ð˜ ψγ λ→q0¯ q1g2Þis cumbersome but straightforward. More technical details are given in Ref. [65]. After a lot of algebra, we obtain the diffractive structure functions xPFDð4ÞNLO L;q¯ qg ðxBj;Q 2;β;tÞ¼4αsNcCFQ4 βX f e2 fZ1 0 dz0 z0Z1 0 dz1 z1Z1 0 dz2 z2 δðz0þz1þz2−1Þ ×Zx0Zx1Zx2Z¯ x0Z¯ x1Z¯ x2 Ið3Þ MXIð3Þ Δ4z0z1Q2K0ðQX012ÞK0ðQX¯ 0¯ 1¯ 2Þ ×z2 1ð2z0ðz0þz2Þþz2 2Þx20 x2 20 ·x¯ 2¯ 0 x2 ¯ 2¯ 0 − 1 2 x¯ 2¯ 1 x2 ¯ 2¯ 1− 1 2 x¯ 2¯ 0·x21 x2 ¯ 2¯ 0x2 21 þz2 2 2x¯ 2¯ 0·x21 x2 ¯ 2¯ 0x2 21 þx20 ·x¯ 2¯ 1 x2 20x2 ¯ 2¯ 1 þz2 0ð2z1ðz1þz2Þþz2 2Þx21 x2 21 ·x¯ 2¯ 1 x2 ¯ 2¯ 1 − 1 2 x¯ 2¯ 0 x2 ¯ 2¯ 0− 1 2 x20 ·x¯ 2¯ 1 x2 20x2 ¯ 2¯ 1þz2 2 2x¯ 2¯ 0·x21 x2 ¯ 2¯ 0x2 21 þx20 ·x¯ 2¯ 1 x2 20x2 ¯ 2¯ 1 ×½1−S† ¯ 0¯ 1¯ 2½1−S012;ð88Þ for the longitudinal structure function, and DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-17 xPFDð4ÞNLO T;q¯qg ðxBj;Q 2;β;tÞ¼2αsNcCFQ4 βX f e2 fZ1 0 dz0 z0Z1 0 dz1 z1Z1 0 dz2 z2 δðz0þz1þz2−1Þ ×Zx0Zx1Zx2Z¯ x0Z¯ x1Z¯ x2 Ið3Þ MXIð3Þ Δ z0z1Q2 X012X¯ 0¯ 1¯ 2 K1ðQX012ÞK1ðQX¯ 0¯ 1¯ 2Þ ×fϒðjðbÞj2Þ reg þϒðjðcÞj2Þ reg þϒðdÞ inst þϒðeÞ inst þϒðbÞ×ðcÞ interf g½1−S† ¯ 0¯ 1¯ 2½1−S012ð89Þ for the transverse structure function. The ϒterms of the squared virtual photon light-front wave function are ϒðjðbÞj2Þ reg ¼z2 1ð2z0ðz0þz2Þþz2 2Þð1−2z1ð1−z1ÞÞðx¯ 0þ¯ 2;¯ 1·x0þ2;1Þðx¯ 2¯ 0·x20Þ x2 ¯ 2¯ 0x2 20 −z2ð2z0þz2Þð2z1−1Þðx¯ 0þ¯ 2;¯ 1·x¯ 2¯ 0Þðx0þ2;1 ·x20Þ−ðx¯ 0þ¯ 2;¯ 1·x20Þðx0þ2;1 ·x¯ 2¯ 0Þ x2 ¯ 2¯ 0x2 20 ;ð90Þ ϒðjðcÞj2Þ reg ¼z2 0ð2z1ðz1þz2Þþz2 2Þð1−2z0ð1−z0ÞÞðx¯ 0;¯ 1þ¯ 2·x0;1þ2Þðx¯ 2¯ 1·x21Þ x2 ¯ 2¯ 1x2 21 −z2ð2z1þz2Þð2z0−1Þðx¯ 0;¯ 1þ¯ 2·x¯ 2¯ 1Þðx0;1þ2·x21Þ−ðx¯ 0;¯ 1þ¯ 2·x21Þðx0;1þ2·x¯ 2¯ 1Þ x2 ¯ 2¯ 1x2 21 ;ð91Þ ϒðdÞ inst ¼z2 0z2 1z2 2 ðz0þz2Þ2−z2 0z3 1z2 z0þz2x0þ2;1 ·x20 x2 20 þx¯ 0þ¯ 2;¯ 1·x¯ 2¯ 0 x2 ¯ 2¯ 0 þz2 0z1ðz1þz2Þ2z2 z0þz2x0;1þ2·x21 x2 21 þx¯ 0;¯ 1þ¯ 2·x¯ 2¯ 1 x2 ¯ 2¯ 1;ð92Þ ϒðeÞ inst ¼z2 0z2 1z2 2 ðz1þz2Þ2−z0z2 1ðz0þz2Þ2z2 z1þz2x0þ2;1 ·x20 x2 20 þx¯ 0þ¯ 2;¯ 1·x¯ 2¯ 0 x2 ¯ 2¯ 0 þz3 0z2 1z2 z1þz2x0;1þ2·x21 x2 21 þx¯ 0;¯ 1þ¯ 2·x¯ 2¯ 1 x2 ¯ 2¯ 1;ð93Þ ϒðbÞ×ðcÞ interf ¼−z0z1½z1ðz0þz2Þþz0ðz1þz2Þ½z0ðz0þz2Þþz1ðz1þz2Þ ×ðx¯ 0þ¯ 2;¯ 1·x0;1þ2Þðx¯ 2¯ 0·x21Þ x2 ¯ 2¯ 0x2 21 þðx¯ 0;¯ 1þ¯ 2·x0þ2;1Þðx¯ 2¯ 1·x20Þ x2 ¯ 2¯ 1x2 20 þz0z1z2ðz0−z1Þ2 ×ðx¯ 0þ¯ 2;¯ 1·x¯ 2¯ 0Þðx0;1þ2·x21Þ−ðx¯ 0þ¯ 2;¯ 1·x21Þðx0;1þ2·x¯ 2¯ 0Þ x2 ¯ 2¯ 0x2 21 þðx¯ 0;¯ 1þ¯ 2·x¯ 2¯ 1Þðx0þ2;1 ·x20Þ−ðx¯ 0;¯ 1þ¯ 2·x20Þðx0þ2;1 ·x¯ 2¯ 1Þ x2 ¯ 2¯ 1x2 20 :ð94Þ It might be elucidating to note that the above expressions satisfy the following symmetries in particle exchanges: ϒðjðcÞj2Þ reg ≡ϒðjðbÞj2Þ reg ðz0↔z1;x0↔x1;¯ x0↔¯ x1Þð95Þ ϒðeÞ inst ≡ϒðdÞ instðz0↔z1;x0↔x1;¯ x0↔¯ x1Þ;ð96Þ making their sum symmetric under the exchange of the quark and antiquark. Meanwhile the ðbÞ×ðcÞinterference term is already a sum of terms that can be obtained by this exchange, and is thus symmetric by itself. The results as expressed in Eqs. (88) and (89) are written for the most general transverse coordinate space dependence, and contain a total of six two-dimensional tranverse coordinate integrations. In practical applications, the Wilson line operators are usually available in coordinate space from solutions of the BK equation, thus one way or the other such coordinate integrations have to be present; if the cross section formula was expressed in momentum space, a similar number of transverse integrals would be needed in the Fourier-transformation of the Wilson line G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-18 operators to momentum space. More generally, with a 3-parton configuration passing through the shockwave one ultimately has to integrate over either the coordinates of 6 Wilson lines, or over 6 tranverse momenta transferred by the Wilson lines from the target to the scattering partonic system. Fully evaluating the cross section expressions numerically will be a nontrivial task. One simplification can be provided by an additional assumption of a factorized impact parameter dependence for the Wilson line operator 1−S012, an approximation that is often made in phenomenological studies. The remaining integrals are nevertheless more demanding than in the previously commonly used large Q2limit that we will discuss in Sec. VIII B. The ultimate reason is that the large Q2limit reduces the problem to an “effective adjoint dipole”with only two Wilson lines in the amplitude, instead of three. Then further kinematical approximations allow one to write the cross section in terms of the square of a single coordinate integral. In the full case the helicity sums over the produced particles, performed analytically, result in expressions that couple the coordinates in the amplitude and its complex conjugate. Thus a similar factorization is not possible. The dipole amplitude Nij ¼1−Sij satisfies the BK or JIMWLK high-energy evolution equation, where the evolution is parametrized in terms of the evolution rapidity determined from the kinematics of the process. Different versions of the evolution equations are factorized in terms of different variables, associated with either light cone energy or momentum [34], and require a consistent treatment in the cross section. We emphasize that the contribution considered in this section is one where the gluon momentum fraction z2 cannot become small, since it is limited by the finite MX.This contribution is thus not associated with the high energy evolution of the target. We will have to return to the question of evolution rapidity in more detail when addressing the full NLO diffractive structure function. These contributions to the diffractive structure functions are finite without requiring any additional cancellations with other diagrams. This is because the invariant mass of the final state is fixed, and thus ultraviolet divergences do not appear. This also meansthat the divergences in the loop contributions that we are not calculating here should cancel against each other, see discussion in Sec. III. Note that without the M2 X restriction the integration over the final state momenta would set ¯ xi→xiand an ultraviolet divergence ∼Rd2x02=x2 02 or ∼Rd2x12=x2 12 would appear. In the diffractive structure functions the corresponding structure is ∼Rd2xi2xi2=x2 i2 (with i¼0, 1) which is UV finite. The integration over the momentum transfer sets the center of mass of the q¯ qg system bto be the same in the amplitude and in the conjugate amplitude, b¼¯ b, but this does not affect the behavior of these integrals in the ultraviolet region (note that Ið3Þ MXdoes not depend on bor on ¯ b). A potential (transverse) infrared divergence is removed, for a gluon emitted before the shockwave, by the dipole amplitude part vanishing for jx02j∼jx12j→∞. For the emissions after the shockwave that we are not calculating here, these configurations are not suppressed by the Wilson line correlator, and need to cancel against the wave function renormalization of the outgoing quarks. Similarly there is no soft gluon divergence in the limit z2→0, as the invariant mass, which we keep finite, gives a lower bound for the integral z2≳1=M2 X. For a parametrically large MX our result would give a large logarithm ∼ln M2 Xfrom the lower limit of the z2integration. While such contributions could be resummed [117], they are not easily accessible at EIC energies and we will not consider this resummation further here. The BK/JIMWLK evolution of the target, on the other hand, is associated with the z2→0limit of contributions where the gluon crosses the shockwave, but is reabsorbed and not measured in the final state, which we are leaving to future work. The interpretation of the cumbersome ϒterms is actually straightforward. First, ϒðjðbÞj2Þ reg describes the contribution where the gluon is emitted by the quark in the amplitude and absorbed by the same quark in the conjugate amplitude. Similarly, ϒðjðcÞj2Þ reg corresponds to the case where the antiquark emits and absorbs the gluon. Furthermore, the instantaneous gluon emission and absorption by the quark (antiquark) is described by ϒðdÞ inst (ϒðeÞ inst). Finally, gluon emission by the quark and absorption by the antiquark (or vice versa) contributes the term ϒðbÞ×ðcÞ interf . Note that the instantaneous contribution only appears as a part of the transverse cross section. The interference between the regular and instantaneous gluon emissions is included in the terms ϒðdÞ inst and ϒðeÞ inst, with the former including the interference contributions containing the (d) diagram, and similarly the latter those of the (e) diagram. We emphasize that prior to this work the q¯ qg contribution to the diffractive cross section has only been known in approximate kinematics and for a transverse photon only (which dominates at high Q2), see discussion in Sec. VIII. For the longitudinal polarization even approximative results have been missing from the literature. The cross sections (88) and (89), that are the main results of this work, are finite and can be straightforwardly implemented in phenomenological applications. In a future work we plan to apply these results to describe the HERA diffractive structure function data. VIII. RECOVERING KNOWN LIMITS A. The large-MXlimit As a verification of our calculation, here we will extract the q¯ qg contribution for FD Tin the limit of large MX, i.e., in the limit when the emitted gluon is soft. This limit has been considered by several authors, e.g., in Refs. [66,117–121]. DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-19 To be specific, wewill use the result as it is written in Ref. [66] and also used in the phenomenological studies [61,62].The derivation simplifies considerably in the β→0limit, i.e., for final states with arbitrarily large invariant masses MXin comparison to the virtuality of the photon: M2 X≫Q2.The result of Ref. [66] for the q¯ qg contribution—including gluon emissions from the quark-antiquark dipole both before and after the scattering off the shockwave6—is, converted to our notations, xPFDðMSÞ T;q¯qg ðxP;β¼0;Q2Þ ¼αsNcCFQ2 16π5αem Zd2x0Zd2x1Zd2x2 Z1 0 dz zð1−zÞj˜ ψLO γ λ→q¯ 0¯ q¯ 1j2 ×x2 01 x2 02x2 12 ½N02 þN12 −N01 −N02N122:ð97Þ A key feature of this result is that the q¯ qg LFWF has been factorized to the leading order q¯ qLFWF, the BK kernel describing the gluon emission, and the scattering of the tripole state in terms of scatterings of daughter dipoles. Our starting point to rederive the large-MXresult is Eq. (48), with the final state momentum integrals still undone. First we note that the leading process to create high invariant mass final states is caused by the emissions of soft gluons, with z2≪1, and at the limit z2→0we have M2 X≈P2 2=z2. We consider a tintegrated cross section, whose momentum dependence in this limit reads dσD γ λ→q¯ qg dM2 X ∼Zdz2 z2Zd2p0 ð2πÞ2Zd2p1 ð2πÞ2 ×Zd2p2 ð2πÞ2eix¯ 00p0þix¯ 11p1þix¯ 22p2δp2 2 z2 −M2 X ¼δð2Þðx0−x¯ 0Þδð2Þðx1−x¯ 1ÞZdz2 ×Zd2p2 ð2πÞ2eix2¯ 2p2z2 p2 2 δz2−p2 2 M2 X ¼1 M2 X δð2Þðx0−x¯ 0Þδð2Þðx1−x¯ 1Þδð2Þðx2−x¯ 2Þ; ð98Þ where we assumed that M2 Xis dominated by the gluon light cone energy p2 2=z2. We have here integrated over z2, assuming that MXis large enough so that it is always possible to find a solution to the delta function constraint z2¼p2 2=M2 Xwith 0<z 2<1. In reality this is not possible for arbitrarily large p2. Thus our approximation has rendered the p2integral unbounded, resulting in a delta function setting x2−x¯ 2. This approximation, as discussed in Sec. III and in Sec. VII, would make the cross section UV divergent unless one also includes the diagrams with emission after the shockwave, using the procedure of subtracting from the “emission before”term the appropriate coordinate limit. Thus we will include these contributions here, unlike in the rest of this paper. At this point we can integrate over the coordinates in the conjugate amplitude using the delta functions. Starting from the equation (48) we now get dσD γ λ→q¯ qg dM2 XdxP¼NcCF ð4πÞ2Z1 0 dz0 z0Z1 0 dz1 z1 δðz0þz1−1Þ ×Zd2x0d2x1d2x2 1 M2 X ×X f;h0;h1;λ2j˜ ψγ λ→q¯ 0¯ q¯ 1g¯ 2ðS† 012 −1Þ −½emission afterj2:ð99Þ For calculating the subtraction terms correctly, we need to decompose the γ→q¯ qg wave function into gluon emission and effective γ→q¯ qparts according to Eq. (9). Here the limit of z2→0simplifies things dramatically. In the soft gluon limit the wave function is given by [note that in our normalization conventions for the reduced wave functions, see Eqs. (16) and (17), this relation is true without additional coefficients]: ˜ ψγ λ→q0¯q1g2≈ z2→0 ˜ ψLO γ λ→q0¯ q1½˜ ψq0→q0g2þ˜ ψ¯q1→¯q1g2;ð100Þ where the effective γ→q¯ qwave functions ˜ ψγ λ→q0¯ q1;q0→q0g2 and ˜ ψγ λ→q0¯ q1;¯ q1→¯ q1g2defined by Eq. (9) have become independent of the gluon emission in the limit z2→0. This simplification can be understood in several equivalent ways. In coordinate space, the argument is that for a soft emission z2→0the coordinate of the emitting particle does not change, and therefore the original γ→q¯ qsplitting is independent of the later gluon emission. For momentum space wave functions the reason is that in the limit z2→0 the gluon light cone energy blows up and thus energy denominators for gluon emission become independent of the state emitting the gluon, k− q¯ qg −k− γ≈k− g≈k− qg −k− q≈k− ¯ qg −k− ¯ q:ð101Þ This leads to a factorization of the γ→q¯ qg wave function into leading order γ→q¯ qand q→qg,¯ q→¯ qg wave functions in momentum space, which is carried over into coordinate space. We recall from Sec. III A that the 6The authors of Ref. [66] justify the contribution from the emission after the shockwave as resulting from the normalization of the photon state, which is possible in the soft gluon limit. However, this contribution is better understood as resulting from emissions after the shockwave; see discussion in Ref. [121]. The normalization condition for the photon state can only be used to obtain such contributions in the soft gluon limit where the γ→q¯ qg wave function factorizes into γ→q¯ qand gluon emission wave functions, but not in general kinematics [31,32]. G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-20 subtraction procedure consists in pulling out the gluon emission wave function and then taking the limit of the gluon coordinate being equal to its emitter in the remaining factors. In this case we can also factor out the common γ→q¯ qpart. Thus we get, restoring the coordinates for clarity, ½˜ ψγ λ→q0¯ q1g2ðx0;x1;x2ÞðS012 −1Þ−½emission after ¼˜ ψLO γ λ→q0¯ q1ðx0;x1Þf˜ ψq0→q0g2ðx0;x2Þ ×½ðS012 −1Þ−ðS012 −1Þjx2→x0 þ˜ ψ¯ q1→¯ q1g2ðx1;x2Þ½ðS012 −1Þ−ðS012 −1Þjx2→x1g: ð102Þ We now use the coordinate limits of the “tripole”(see, e.g., [32,33]) operators S012jx2→x0¼S012jx2→x1¼S01 ð103Þ and the relation (see also, e.g., [32,33]) S012 −S01 ¼Nc2 Nc2−1ðS02S21 −S01Þ≈ Nc→∞S02S21 −S01 ¼N02N21 −N02 −N21 þN01 ð104Þ with Nij ¼1−Sij to get ½˜ ψγ λ→q0¯ q1g2ðx0;x1;x2ÞðS012 −1Þ−½emission after ¼˜ ψLO γ λ→q0¯q1ðx0;x1Þf˜ ψq0→q0g2ðx0;x2Þ þ˜ ψ¯q1→¯q1g2ðx1;x2Þg½N02N21 −N02 −N21 þN01: ð105Þ We can now use the gluon emission wave function, which in our conventions [see Eq. (37) of Ref. [80] and recall the normalization of the reduced wavefunction from (16),(17)] reads ˜ ψq0→q0g2ðx0;x2Þ≈ z2→0 2gZd2p2 ð2πÞ2 p2·ε λ2 p2 2 eip2·ðx2−x0Þ ¼2gi 2πðx2−x0Þ·ε λ2 ðx2−x0Þ2;ð106Þ and similarly for the antiquark. Using this relation in Eq. (99) enables the familiar calculation of the BK kernel X λ2j˜ ψq0→q0g2ðx0;x2Þþ˜ ψ¯ q1→¯ q1g2ðx1;x2Þj2 ¼g2 π2 x2 01 x2 02x2 21 ¼4αs π x2 01 x2 02x2 21 :ð107Þ Inserting the squared wave functions from Eq. (107) and (105) into (99) one then directly recovers the result (97). B. The large-Q2limit In this section we will rederive the Wüsthoff result for the q¯ qg contribution to FD T[58,60,108], which is an approximate result at the large-Q2limit for this next-toleading order contribution. Specifically we will verify that the Wüsthoff result emerges from the NLO result calculated in exact kinematics when one takes the large-Q2limit. The Wüsthoff result for the q¯ qg contribution has been extensively used in phenomenology [60–62,64], and it has some special features we seek to understand in depth. It can be written in coordinate space in the appealing short form7: xPFDðGBWÞ T;q¯qg ðxP;β;Q 2Þ¼αsβ 8π4X f e2 fZd2bZQ2 0 dk2Z1 β dzk4ln Q2 k21−β z2 þβ z2 ×Z∞ 0 drrd˜σdip d2bðb;r;x PÞK2ðffiffiffiz pkrÞJ2ðffiffiffiffiffiffiffiffiffiffi 1−z pkrÞ2;ð108Þ while a momentum space version can be found, e.g., from Ref. [60]. An essential feature of the large-Q2structure function is the manifestation of the DGLAP splitting function for g→ q¯ qsplitting. This is associated with the DGLAP evolution of the parton distributions of the pomeron, and is written in terms of the target minus momentum fractions βand z(recall that we work in a frame where the photon has a large plus momentum). In the frame where the target has a large longitudinal momentum βcan be interpreted as the fraction of the pomeron momentum carried by the stuck parton, and it is also related to the invariant mass of the final state as β¼Q2=ðQ2þM2 XÞ. The fraction zis the fraction of the pomeron minus momentum transferred to the q¯ qsystem. On the one hand, the presence of the splitting function in the large Q2limit is unavoidable in QCD. On the other hand, since we are using light cone perturbation theory to quantize 7We use the explicit form from Ref. [62], which is written in color glass condensate formalism. Its connection to the original two-gluon exchange formulation [58,60] is discussed in Refs. [61,65]. DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-21 the projectile photon, the light cone momentum fraction ξ¼β=z does not appear spontaneously in the calculation. These kinematical variables are illustrated in Fig. 12. The three-particle phase space has been treated only approximately in the Wüsthoff result. The result is written in terms of the size rof an “effective adjoint dipole”formed by the gluon and the quark-antiquark pair. Thus the interaction with the target color field appears in the form of an adjoint representation dipole cross section ˜σdip. There is also an explicit log Q2resulting from the integral over some of the “internal”kinematics of the q¯ qpair as we will demonstrate explicitly below. The contribution corresponding to the DGLAP splitting function ½ð1−β=zÞ2þðβ=zÞ2originates from the part of phase space where emissions are strongly ordered in transverse momenta. Since wewant toobtain theDGLAP splitting function with fixed momentum fractions in k−, this implies that the kþmomentum fractions must also be strongly ordered. This is the Bjorken aligned jet [122] configuration. Thus it is more convenient to work in momentum space to make the connection to the Wüsthoff result. To make this more specific, we choose the transverse momenta for the three-particle q¯ qg final state (after the shockwave) as Pi≔pi−ziq;ð109Þ ˆ Pq¯ q≔z0P1−z1P0 z0þz1 ;ð110Þ ˆ Kg˜g≔ðz0þz1ÞP2−z2ðP0þP1Þ;ð111Þ ˆ Δ≔−P0−P1−P2;ð112Þ where piare the transverse momenta of the final state partons, and the variable ˆ Pq¯qcan be interpreted as the relative momentum between the quark and antiquark, whereas ˆ Kg˜gis the momentum of the gluon with respect to the quark-antiquark system, i.e., the “effective gluon.”The total transverse momentum transfer in the scattering process is ˆ Δ. The corresponding conjugate variables in transverse coordinate space can be determined by writing P0·x0þP1·x1þP2·x2≡ ˆ Pq¯ q·uþˆ Kg˜ g·rþ ˆ Δ·b; ð113Þ from which one obtains u¼x1−x0;ð114Þ r¼x2−z0x0þz1x1 z0þz1 ;ð115Þ b¼z0x0þz1x1þz2x2:ð116Þ Here uis the size of the q¯ qdipole, ris the distance from the gluon to the center of mass of the q¯ qsystem, i.e., the size of the effective gluonic dipole, and bis the center of mass. In the derivation of the large Q2limit of our result for FD T, Eq. (88), our starting point is the expression (82) for the diffractive cross section. The calculation then proceeds FIG. 12. Kinematics for the diffractive q¯ qg-production diagram. Left: dipole picture frame, where the probe has a large qþwhich is conserved in the interaction with the target. We show the plus and transverse momenta, with Pq¯ qthe relative transverse momentum of the q¯ qpair and Kg˜ gof the gluon-effective gluon dipole. Right: infinite target momentum frame, where one tracks the minus component of the momentum, i.e., the target momentum fraction. The zigzag line refers to the pomeron emitted from the target, and is used to illustrate a generic diffractive interaction between the virtual photon and target. The two scattering pictures are connected by the invariants of the scattering. In the GBW limit the q¯ qpair forms a hard system that is seen as pointlike by the target color field, thus M2 q¯ qand Pq¯ qare the same before and after the shockwave. In the aligned jet limit z2≪z1≪z0≈1we have M2 q¯ q≈P2 q¯ q=z1, and from the collinear factorization picture on the right: M2 q¯q¼2ðxBj=ξÞqþP−þq2¼ð1=ξ−1ÞQ2. The invariant mass of the q¯ qg system and Kg˜g, on the other hand, are affected by the interaction with the shockwave. In the dipole picture, before the shockwave, we have M2 q¯ qg ≈M2 q¯ qþK2 g˜ g=z2. The invariant mass after the shockwave is M2 X¼ð1=β−1ÞQ2≈M2 q¯ qþˆ K2 g˜ g=z2. G. BEUF et al. PHYS. REV. D 106, 094014 (2022) 094014-22 roughly as follows. First we Fourier transform the LFWFs back into momentum space where we can apply the kinematic approximations related to the leading log Q2 limit. Then we write the virtual photon LFWF in momentum space using natural relative momenta from the perspective of the physical picture that emerges at the large-Q2 limit discussed above. This enables us to apply the aligned jet limit (AJL) approximations at the LFWF level, after which we proceed to integrate over surplus degrees of freedom at the cross section level. In the aligned jet kinematics we have the strong transverse momentum ordering Q2≫P2 q¯ q≫K2 g˜g≫Δ2. The corresponding kþ momentum fraction ordering is z2≪z1≪z0or, symmetrically, z2≪z0≪z1. Since these two limits give the same contribution, we will just concentrate on the first one and in the end multiply the result by 2. The transverse Fourier transform of the LFWF into momentum space is defined as [32] ˜ ψγ T→q¯qg NLO ¼Zd2k0 ð2πÞ2Zd2k1 ð2πÞ2Zd2k2 ð2πÞ2ð2πÞ2δð2Þðk0þk1þk2−qÞeiðk0·x0þk1·x1þk2·x2Þψγ T→q¯qg NLO :ð117Þ Performing the Fourier transforms and rewriting the Fourier momenta in terms of the natural momenta using k1¼−z1 z0þz1 Kg˜ gþPq¯ q;ð118Þ k0¼−z0 z0þz1 Kg˜ g−Pq¯ q;ð119Þ the diffractive cross section from Eq. (82) becomes dσD λ;q¯qg dM2 X¼NcCF 4ð2πÞ2Z0 −∞ dtZ1 0 dz0 z0Z1 0 dz1 z1Z1 0 dz2 z2 δðz0þz1þz2−1ÞZd2ˆ Pq¯q ð2πÞ2Zd2ˆ Kg˜ g ð2πÞ2Zd2ˆ Δ ð2πÞ2 ×δð ˆ Δ2−jtjÞδˆ K2 g˜ g z2ðz0þz1Þþz0þz1 z0z1 ˆ P2 q¯ q−M2 X ×ZuZrZbZ¯ uZ¯ rZ¯ bð2πÞ6eið¯ u−uÞ· ˆ Pq¯ qeið¯ r−rÞ· ˆ Kg˜ geið¯ b−bÞ· ˆ Δ ×X h0;h1;λ2ð˜ ψγ λ→q¯ 0¯ q¯ 1g¯ 2Þ†ð˜ ψγ λ→q0¯ q1g2Þ½Sð3Þ† ¯ u¯ r¯ b−1½Sð3Þ urb −1;ð120Þ where the hatted quantities are the final state momenta. The squared virtual photon amplitude Fourier transformed to coordinate space reads ð˜ ψγ λ→q¯ 0¯ q¯ 1g¯ 2Þ†ð˜ ψγ λ→q0¯ q1g2Þ¼Zd2¯ Pq¯ q ð2πÞ2Zd2¯ Kg˜ g ð2πÞ2Zd2¯ Δ ð2πÞ2ðψγ λ→q¯ 0¯ q¯ 1g¯ 2Þ†e−i¯ u·¯ Pq¯ q−i¯ r·¯ Kg˜ g−i¯ b·¯ Δð2πÞ2δð2Þð¯ ΔÞ ×Zd2Pq¯ q ð2πÞ2Zd2Kg˜ g ð2πÞ2Zd2Δ ð2πÞ2ðψγ λ→q0¯ q1g2Þeiu·Pq¯qþir·Kg˜gþib·Δð2πÞ2δð2ÞðΔÞ:ð121Þ Now we begin the application of the aligned jet limit approximations. First, we assume that the relative transverse momentum of the parent dipole kPq¯ qkis much larger than that of either of the daughter dipoles, i.e., kPq¯ qk≫kKg˜ gk.In position space this corresponds to a configuration where the gluon is emitted far away from the parent dipole, i.e., kuk≪krk. This makes the dipole amplitude independent of the parent dipole size, and enables us to separate and perform the u-integrations in Eq. (120), which yields two delta functions Zd2u ð2πÞ2Zd2¯ u ð2πÞ2ei¯ u·ðˆ Pq¯q−¯ Pq¯qÞe−iu·ðˆ Pq¯q−Pq¯qÞ¼δð2Þðˆ Pq¯ q−¯ Pq¯ qÞδð2Þðˆ Pq¯ q−Pq¯ qÞ:ð122Þ This enables us to perform the Pq¯qand ¯ Pq¯qintegrations in Eq. (120). To perform the tintegration of (120), we separate the closely related transverse momentum transfer integrals DIFFRACTIVE DEEP INELASTIC SCATTERING AT NLO IN …PHYS. REV. D 106, 094014 (2022) 094014-23 Z0 −∞ dtZd2¯ b 2πZd2b 2πZd2ˆ Δ ð2πÞ2Zd2¯ Δ ð2πÞ2Zd2Δ ð2πÞ2δðˆ Δ2−jtjÞð2πÞ4δð2Þð¯ ΔÞδð2ÞðΔÞei¯ b·ð ˆ Δ−¯ ΔÞe−ib·ð ˆ Δ−ΔÞ ¼Z0 −∞ dtZd2¯ b 2πZd2b 2πZd2ˆ Δ ð2πÞ2ei ˆ Δ·ð¯ b−bÞδðˆ Δ2−jtjÞ ¼ Zd2¯ b 2πZd2b 2πδð2Þð¯ b−bÞ:ð123Þ An essential step that we need to take before we proceed is to consider how to connect our results written in terms of the plus-momentum fractions to the minus momentum fractions in the Wüsthoff result (108). The essential idea here is to think in terms of invariant masses of different multiparton states. To connect the frames of reference, we need to approximate the invariant masses in the aligned jet limit as M2 q¯ q¼1−ξ ξQ2¼ð1−z2Þ2P2 q¯q z0z1 ≈ P2 q¯q z1 ;ð124Þ M2 q¯ qg þQ2¼Q2þz0þz1 z0z1 P2 q¯ qþK2 g˜g z2ðz0þz1Þ ≈Q2þP2 q¯ q z1þK2 g˜ g z2¼Q2 ξþK2 g˜ g z2 ;ð125Þ where zxP¼xBj=ξis the fraction of the target momentum carried by the q¯ qsystem. Given the above relations, we can reparametrize the often occurring combination P2 q¯ qþz1Q2¼z1ðM2 q¯ qþQ2Þ¼z1Q2=ξ:ð126Þ It is useful to write the q¯ qg energy denominator in terms of the diffractive state mass M2 X¼1−β βQ2≈ ˆ P2 q¯ q z1þ ˆ K2 g˜ g z2 ;ð127Þ where ˆ Kg˜gis the gluon relative momentum after the shockwave. Recall that Pq¯ q≡ ˆ Pq¯ qis conserved in the shockwave which enables us to leverage final state information about M2 Xto simplify the γLFWF before the shockwave. We now want to eliminate z2using these variables, so that z2¼β ˆ K2 g˜g Q2ð1−zÞ:ð128Þ With this relation the three particle state invariant mass before the shockwave becomes M2 q¯ qg ¼Q2 β ˆ K2 g˜g½ðz−βÞˆ K2 g˜gþð1−zÞK2 g˜ g;ð129Þ and the “outer”(i.e., q¯ qg state) LC energy denominator in the NLO virtual photon LFWF will be M2 q¯qg þQ2¼Q2 β ˆ K2 g˜g½z ˆ K2 g˜ gþð1−zÞK2 g˜ g:ð130Þ Note the distinction that this is for the q¯ qg state before the shockwave. The momentum ˆ K2 g˜ gafter the shockwave is fixed by the final state kinematics. It, or the q¯ qg invariant mass, is not the same before (M2 q¯qg) and after (M2 X) the shockwave. The momentum argument of the wave function before the shockwave Kg˜ gwill need to be Fourier transformed into coordinate space, see Eq. (121), in order to include the interaction with the target shockwave. The momenta Kg˜ gand ¯ Kg˜ gbefore the shockwave are separate in the DA and CCA, which have to be Fourier transformed separately. 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