Stronger C-odd color charge correlations in the proton at higher energy

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/ Stronger C-odd color charge correlations in the proton at higher energy © Authors 2023 Published version Dumitru, Adrian; Mäntysaari, Heikki; Paatelainen, Risto Dumitru, A., Mäntysaari, H., & Paatelainen, R. (2023). Stronger C-odd color charge correlations in the proton at higher energy. Physical Review D, 107(1), Article L011501. https://doi.org/10.1103/PhysRevD.107.L011501 2023 Stronger C-odd color charge correlations in the proton at higher energy Adrian Dumitru * Department of Natural Sciences, Baruch College, CUNY, 17 Lexington Avenue, New York, New York 10010, USA and The Graduate School and University Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016, USA Heikki Mäntysaari † Department of Physics, University of Jyväskylä, P.O. Box 35, 40014 University of Jyväskylä, Finland and Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland Risto Paatelainen ‡ Helsinki Institute of Physics and Department of Physics, FI-00014 University of Helsinki, Finland (Received 19 October 2022; accepted 7 December 2022; published 9 January 2023) The nonforward eikonal scattering matrix for dipole-proton scattering at high-energy obtains an imaginary part due to a C-odd three gluon exchange. We present numerical estimates for the perturbative odderon amplitude as a function of dipole size, impact parameter, their relative azimuthal angle, and lightcone momentum cutoff x. The proton is approximated as ψqqqjqqqiþψqqqgjqqqgi, where ψqqq is a nonperturbative three-quark model wave function while the gluon emission is computed in light-cone perturbation theory. We find that the odderon amplitude increases as xdecreases from 0.1 to 0.01. At yet lower x, the reversal of this energy dependence would reflect the onset of universal small-xrenormalization group evolution. DOI: 10.1103/PhysRevD.107.L011501 I. INTRODUCTION The S-matrix for high-energy eikonal scattering of a quark-antiquark dipole off the proton is [1–4] Sð x;  yÞ¼ 1 NchtrUð xÞU†ð yÞi:ð1Þ Below we shall also use the impact parameter  b¼ð xþ  yÞ=2and dipole (transverse) vectors r¼ y−  xwhere r points from the antiquark to the quark. The hi brackets denote the matrix element between the incoming proton state jPþ;  P¼0iand the outgoing state hPþ;  Kj, where  Kdenotes the proton transverse momentum. Our sign convention for the coupling in the covariant derivative, Dμ¼∂μþigAa μta, follows Ref. [5]. Hence, the path ordered exponential of the field in covariant gauge (Wilson line) which represents the eikonal scattering of the quark is Uð xÞ¼Pe−ig Rdx−Aþaðx−; xÞta:ð2Þ Our convention for the Wilson line and for the dipole S-matrix agrees with Ref. [6]. Others such as Ref. [7] define Sð x;  yÞwith U↔U†; however, they also take r¼ x−  y, so in all, the sign for the imaginary part of the S-matrix is the same. Indeed, our focus here is on the imaginary part Oðr;  bÞof the S-matrix, the so-called “b-dependent odderon,”which starts out in perturbation theory as C-odd three gluon exchange. This amplitude is odd under C-conjugation, i.e., exchange of quark and antiquark. The relation of various odderon amplitudes to generalized transverse momentum-dependent parton distributions has been elucidated in Refs. [7–12]. The C-odd three gluon exchange couples to cubic color charge fluctuations in the proton [13], *[email protected] †[email protected] ‡[email protected] 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 107, L011501 (2023) Letter 2470-0010=2023=107(1)=L011501(6) L011501-1 Published by the American Physical Society ImSðr;  bÞ¼Oðr;  bÞ¼− 5 18 g61 2 1 3Z q1;q2;q3>qmin 1 q2 1 1 q2 2 1 q2 3 sinð  b· KÞG− 3ð q1; q2; q3Þ"X i¼1;2;3sinr· qiþ1 2 r· K −sinr· q0 iþ1 2 r· K0−sin1 2 r· Kþsin1 2 r· K0#:ð3Þ We have written Oðr;  bÞin a form which is more suitable for numerical integration, in particular the amplitude vanishes already at the integrand level when r⊥  band different momenta  qiappear in a symmetric form. The sign of Oðr;  bÞdiffers from Ref. [13] because here we employ the more common convention r¼ y−  xrather than r¼ x−  y. Here the parameter g¼ffiffiffiffiffiffiffiffiffiffi 4παs pis the strong coupling constant,  K¼−ð q1þ q2þ q3Þis the transverse momentum transfer given  P¼0for the incoming proton, and Rqis shorthand for Rd2q=ð2πÞ2. Also, the transverse momentum vectors  q0 icorrespond to sign-flipped components along  b. We have also introduced a low momentum cutoff qmin for numerical stability; no significant dependence on this cutoff was observed when qmin <0.1GeV, except in regions where Oðr;  bÞhas a very small magnitude. The actual numerical results shown in this paper are obtained using qmin ¼0.03 GeV. We denote the C-odd part of the light-cone gauge correlator of three color charge operators as hρað q1Þρbð q2Þρcð q3ÞiC¼−≡1 4dabcg3G− 3ð q1; q2; q3Þ:ð4Þ Here, ρacorresponds to the plus component of the color current, integrated over x−. In terms of creation and annihilation operators for quarks and gluons, it reads [14] ρað  kÞ¼gX i;j;σðtaÞij Zdxqd2q 16π3xq b† iσðxq; qÞbjσðxq;  kþ qÞ þgX λbc ðTaÞbc Zdxgd2q 16π3xg a† bλðxg; qÞacλðxg;  kþ qÞ: ð5Þ Reference [15] evaluated G− 3ð q1; q2; q3Þfor a nonperturbative three-quark light-cone constituent quark model [16,17]. This model provides realistic one-particle longitudinal and transverse momentum distributions and also encodes momentum correlations. We refer to this threequark light-cone wave function as the leading-order (LO) approximation. The diagrams corresponding to corrections to the impact factor due to the perturbative emission of a gluon have been computed in Ref. [18]; they are too numerous to be listed again here. This will be referred to as the next-to-leading order (NLO) approximation. The purpose of this paper is to present numerical results for Oðr;  bÞfrom this approach, which together with analogous results for the real part of Sðr;  bÞ[14,19] provide a complete set of initial conditions for small-xevolution of the dipole S-matrix. The questions we address here are about the overall magnitude of the three gluon exchange amplitude, and its dependence on r¼jrj, b¼j  bj, their relative angle θ, and on the cutoff xon the parton light-cone momentum which appears in G− 3. The nonvanishing imaginary part of the S-matrix can be probed, for example, via charge asymmetries in diffractive electroproduction of a πþπ−pair [20,21], exclusive production of a pseudoscalar meson [22–26] in deeply inelastic scattering (DIS) or ultraperipheral proton-nucleus collisions, and lepton-meson azimuthal angle correlations in exclusive processes [27], as well as in exclusive production of a vector meson in pþpscattering [28] via “pomeron-odderon fusion.” Finally, it is also our goal to provide numerical estimates for initial conditions for small-xQCD evolution of the (hard) odderon Oðr;  bÞ[29–31]. Their crude knowledge, see e.g., Refs. [7,31], is a key limitation for quantitative predictions of the observables mentioned above in the energy regime of the Electron-Ion Collider (EIC) [32–34]. II. RESULTS The results presented here apply when the C-odd exchange can be described by the exchange of three gluons, i.e., in the perturbative regime. This should be the case when the scattered dipole is small and/or when the momentum transfer (conjugate to the impact parameter) is large. Furthermore, since we only consider the jqqqiand jqqqgiFock states of the proton, we restrict to x≳0.01. The results shown below have been obtained with αs¼0.2; note that, aside from the overall α3 sprefactor in Eq. (3), the NLO contribution to G− 3, too, depends on the coupling, see Ref. [18]. Note also that the coupling does not run at this order as the perturbative one gluon emission corrections are OðαsÞ. The nonperturbative three-quark wave function for the proton used in the numerical analysis is the “harmonic oscillator”wave function of Ref. [17]. It has been used previously in Refs. [18,19] for estimates of the real part of the S-matrix. The parameters of the wave function are constrained by the proton radius, the anomalous magnetic moment, and the axial coupling of the proton and the DUMITRU, MÄNTYSAARI, and PAATELAINEN PHYS. REV. D 107, L011501 (2023) L011501-2 neutron. Given these constraints, color charge correlators are not very sensitive to the particular model of the threequark wave function [19]. Also, following Ref. [19], here we evaluate all diagrams for the three gluon exchange with a collinear regulator of mcol ¼0.2GeV; this is consistent with the typical quark transverse momentum in the wave function of Refs. [16,17]. At the level of accuracy that we achieved in evaluating Eq. (3), we found that the angular dependence of the odderon amplitude is well approximated by Oðr;  bÞ¼a1ðr; bÞcos θþa3ðr; bÞcos 3θ;ð6Þ where θis the azimuthal angle made by  band r.We typically find that the magnitude of a3is much smaller than that of a1except in the vicinity of a sign change of a1ðr; bÞ where Oðr;  bÞis small. The angular dependence of the odderon amplitude at r¼b¼0.3fm is shown in Fig. 1. The amplitude obtained from the leading order calculation, where the dependence on the parton momentum fraction cutoff xis negligible, is compared to the result of the NLO computation at x¼0.1,x¼0.03, and x¼0.01. These results show the correction due to the perturbative gluon for different values of x.Atx¼0.1this correction is moderate, visible mostly for (anti)parallel rand  b, as the phase space for gluon emission is restricted. Note that the odderon amplitude vanishes exactly when θ¼0as can be seen from Eq. (3). For smaller x, although the qualitative angular dependence remains the same, we observe a considerable increase of the odderon amplitude jOðr;  bÞj. To further demonstrate the role of the NLO corrections on the odderon amplitude, we show in Figs. 2and 3the dominant a1coefficient as a function of impact parameter (Fig. 2) and dipole size (Fig. 3). The next-to-leading order amplitudes computed at different longitudinal momentum fraction cutoffs xare compared with the leading order result. The odderon amplitude is parity odd and so it vanishes at b¼0. It increases with impact parameter and peaks at bslightly less than 0.2 fm, for a dipole size r¼0.3fm, followed by a smooth falloff toward large b. The peak at b≲0.2fm is seen at much smaller scales than the transverse size ffiffiffiffiffiffiffiffiffi hb2i p≃0.6fm associated with the real part of the S-matrix extracted from fits to Hadron-Electron Ring Accelerator (HERA) data on exclusive J=Ψproduction in DIS [35]. The peak position depends weakly on r but remains at b≲0.3fm for all dipole sizes r≲0.8fm considered here. Again we notice that the qualitative shape of a1ðbÞis preserved by the NLO correction. However, while this correction is moderate at x¼0.1, it increases strongly with decreasing x. FIG. 1. Angular dependence of Oðr;  bÞat various xand r¼b¼0.3fm, which is predominantly ∼ ˆ r·ˆ b. The coefficients (scaled by 100) are a1¼0.16,a3¼−0.0063 at x¼0.01, a1¼0.10,a3¼−0.0030 at x¼0.03, and a1¼0.063,a3¼ −0.0035 at x¼0.1. For comparison, at leading order the fitted coefficients are a1¼0.040 and a3¼−0.0040. The error bars show the estimated uncertainty of the numerical Monte Carlo integration. FIG. 2. Impact parameter dependence of the odderon amplitude modulation coefficient a1defined in Eq. (6). FIG. 3. Dipole size dependence of a1at b¼0.3fm and various x. STRONGER C-ODD COLOR CHARGE CORRELATIONS IN THE …PHYS. REV. D 107, L011501 (2023) L011501-3 Figure 3shows the expected rapid increase of a1with dipole size rat fixed b. It levels off at about r≃0.7fm and then decreases again toward larger rwhere the dipole grows as large as the proton and a perturbative calculation loses validity. This behavior is qualitatively similar to the one obtained for the real part of the S-matrix in a similar calculation in Ref. [19]. These results are not particularly sensitive to the collinear cutoff: using mcol ¼0.3GeV instead of 0.2 GeV results in 5% (20%) larger scattering amplitude at small (large) r. It is interesting to compare the typical magnitude of the odderon exchange amplitude obtained here to parametrizations commonly employed in the literature as initial conditions at x≃0.01 for small-xevolution. Figure 4 of Ref. [31], for example, depicts odderon amplitudes which reach maximum values of ≈0.15 and 0.4, respectively. The initial “spin-dependent odderon”amplitude of Refs. [7,11] coincides with the first model of Ref. [31]. The maximal (over angle θand dipole size r) value for the odderon that we obtain at x≳0.01 is about 5×10−3for αs¼0.2 used in this work. On the other hand, the quasiclassical odderon amplitude derived for a large nucleus, Eq. (56) of Ref. [36] (also see [8,29,37]), if applied to a proton (at r¼2b¼0.7fm) with Gaussian transverse “profile function”[35], is smaller than our result by about one order of magnitude. Finally, we illustrate the dominant a1modulation coefficient at NLO as a function of both rand bin Fig. 4for x¼0.1and in Fig. 5for x¼0.03. Aside from the increasing magnitude, there is no clear qualitative change in the shape of the odderon amplitude. At large bthe a1 coefficient also changes sign, which is visible in these figures. In the Supplemental Material [38] we provide tables for the a1and a3coefficients (which are interpolated when generating Figs. 4and 5) as functions of rand bat x¼0.1, 0.03, and x¼0.01 and for comparison also for the LO three-quark proton wave function. III. DISCUSSION We have presented for the first time an estimate for the perturbative, C-odd, dipole-proton three gluon exchange amplitude Oðr;  bÞat moderately small longitudinal momentum fraction xwhere the target proton includes a perturbative gluon on top of a nonperturbative three-quark Fock state. This is a necessary input for the perturbative small-xevolution of the odderon. We find that Oðr;  bÞ increases when the jqqqgiFock state is added as the number of diagrams increases by an order of magnitude. Once the proton contains a sufficient number of color charges, the average dipole S-matrix at rapidity Y¼ log x0=x will be given by an average over the configurations of Aþin the proton, SYð x;  yÞ¼ZDAþWY½Aþ1 Nc trUð xÞU†ð yÞ:ð7Þ Here WY½Aþis the weight functional at evolution rapidity Y, and x0is the longitudinal momentum fraction at the initial condition. A small step toward lower xallows for the emission of an additional soft gluon, resulting in a small change of WY½Aþ, i.e., the small-xrenormalization group (RG) flow [39–51]. For weak scattering the averagevalueof 1−Sis small and the evolution of the imaginary part Ois given by [29–31] ∂YOð x;  yÞ¼αsNc 2π2Zd2zð x−  yÞ2 ð x−zÞ2ðz−  yÞ2 ×½Oð x; zÞþOðz;  yÞ−Oð x;  yÞ:ð8Þ For small rthe first two terms largely cancel, leaving the negative virtual correction and a decreasing odderon amplitudewith decreasing x. (Forasymptotically small xtheabove evolution equation leads to [29] the energy-independent FIG. 4. Odderon modulation coefficient a1as a function of r and bat x¼0.1calculated at NLO accuracy. FIG. 5. Odderon modulation coefficient a1as a function of r and bat x¼0.03 calculated at NLO accuracy. Note that the color scheme is different than in Fig. 4. DUMITRU, MÄNTYSAARI, and PAATELAINEN PHYS. REV. D 107, L011501 (2023) L011501-4 Bartels-Lipatov-Vacca odderon [52].) The observation of such behavior would indicate the onset of the universal flow predicted by the small-xRG. Our analysis provides a lower bound on the number of prepopulated Fock states. The angular dependence of the odderon amplitude is found to be well described by cos ϕr  b, with a small correction proportional to cos 3ϕr  bwhich is significant only in the region where Oðr;  bÞis very small. The small magnitude of the perturbative odderon amplitude obtained here indicates that high luminosities available e.g., at the EIC are necessary to access the odderon experimentally. For example, Ref. [26] obtained dσ=dt≃40 fb=GeV2for exclusive ηcproduction in DIS at low Q2,jtj¼1.5GeV2, x¼0.1, in the LO approximation with αs¼0.35.We intend to compute cross sections for various physical processes from our dipole S-matrix in the future. ACKNOWLEDGMENTS We thank Y. Hatta, A. Kovner, and V. Skokov for useful comments. A. D. acknowledges support by the DOE Office of Nuclear Physics through Award No. DESC0002307, and The City University of New York for PSC-CUNY Research Grant No. 65079-00 53. This work was supported by the Academy of Finland, the Centre of Excellence in Quark Matter, and Projects No. 338263 and No. 346567 (H. M), and Projects No. 347499 and No. 353772 (R. P). H. M. is also supported under the European Union’s Horizon 2020 research and innovation program by the European Research Council (ERC, Grant Agreement No. ERC-2018-ADG835105 YoctoLHC) and by the STRONG-2020 project (Grant Agreement No. 824093). The content of this article does not reflect the official opinion of the European Union and responsibility for the information and views expressed therein lies entirely with the authors. Computing resources from CSC—IT Center for Science in Espoo, Finland and from the Finnish Grid and Cloud Infrastructure (persistent identifier urn:nbn:fi: research-infras-2016072533)wereusedin this work. 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DUMITRU, MÄNTYSAARI, and PAATELAINEN PHYS. REV. D 107, L011501 (2023) L011501-6