Impact parameter dependence of color charge correlations in the proton

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/ Impact parameter dependence of color charge correlations in the proton © A. Dumitru et al. Published version Dumitru, Adrian; Mäntysaari, Heikki; Paatelainen, Risto Dumitru, A., Mäntysaari, H., & Paatelainen, R. (2022). Impact parameter dependence of color charge correlations in the proton. In DIS 2021 : Proceedings of the 28th International Workshop on Deep-Inelastic Scattering and Related Subjects (Article 058). Stichting SciPost. SciPost Physics Proceedings. https://doi.org/10.21468/scipostphysproc.8.058 2022 SciPost Phys. Proc. 8, 058 (2022) Impact parameter dependence of color charge correlations in the proton Adrian Dumitru1,2?, Heikki Mäntysaari3,4 and Risto Paatelainen4 1Department of Natural Sciences, Baruch College, CUNY, 17 Lexington Avenue, New York, NY 10010, USA 2The Graduate School and University Center, The City University of New York, 365 Fifth Avenue, New York, NY 10016, USA 3Department of Physics, University of Jyväskylä, P.O. Box 35, 40014 University of Jyväskylä, Finland 4Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland [email protected] Proceedings for the XXVIII International Workshop on Deep-Inelastic Scattering and Related Subjects, Stony Brook University, New York, USA, 12-16 April 2021 doi:10.21468/SciPostPhysProc.8 Abstract The impact parameter dependence of color charge correlators in the proton is obtained from the light front formalism in light cone gauge. We include NLO corrections due to the |qqqg〉Fock state via light-cone perturbation theory. Near the center of the proton, the bdependence of the correlations is very different from a “transverse profile function”. The resulting t-dependence of exclusive J/Ψphotoproduction transitions from exponential to power law at |t| ≈ 1GeV2. This prediction could be tested at upcoming DIS facilities or in nucleus-proton ultraperipheral collisions (UPCs). Copyright A. Dumitru et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. Received 19-05-2021 Accepted 06-05-2022 Published 12-07-2022 Check for updates doi:10.21468/SciPostPhysProc.8.058 1 Introduction The Hamiltonian light front formalism [1]in light cone gauge provides essential insight into correlations of color charges in the proton [2]. These can be expressed as matrix elements of nonperturbative boost-invariant light cone Fock-space wave functions of the QCD Hamiltonian, and related to physical observables such as the exclusive final states measured in DIS experiments. For example, at leading order the light-cone gauge color charge correlator is related to the average quark transverse momentum vector and to the Sivers asymmetry [3]. Furthermore, in the mixed transverse momentum – transverse coordinate space representation, the color charge correlator can be related to the Wigner distribution [4–6], and to various other 058.1 SciPost Phys. Proc. 8, 058 (2022) KP KP Figure 1: Handbag (left) and cat’s ears (right) diagrams at LO. The former (latter) dominates for small (large) momentum transfer |t|. generalized parton distribution functions. This detailed information about the partonic structure of the proton can be accessed experimentally e.g. in exclusive dijet or meson production, or in vector meson – lepton azimuthal correlations in DIS, as recently argued in refs. [7–10]. The notion of color charge density fluctuations in the transverse impact parameter plane emerges naturally in high-energy (small-x) scattering. The projectile charge traverses without recoil the (color) field produced coherently by all “valence” charges in the target, and the scattering amplitude follows from a correlator of path ordered exponentials of that field [11]. The scale separation in soft coherent fields sourced by “frozen” valence charges was introduced by McLerran and Venugopalan (MV) in Refs. [12,13]. Their model was devised for a very large nucleus and describes Gaussian fluctuations of classical color charge densities. However, when the density of valence charges in the target is not very large, one would rather take the twodimensional color charge density as an operator acting on the light-cone wave function of the target [2]. In this contribution we focus on the non-trivial impact parameter dependence of color charge correlations in the proton, from which we predict a non-exponential, power-law tail of dσ/dtfor high-energy exclusive (coherent) J/Ψphotoproduction at large transverse momentum transfer |t| ≥ 1 GeV2. 2 The color charge correlator and coherent J/Ψphotoproduction at high momentum transfer The central object of consideration is the two-point color charge correlator 〈ρa(~q1)ρb(~q2)〉 ≡ δab g2G2(~q1,~q2). (1) The notation 〈···〉 denotes an expectation value between proton states, 〈K|and |P〉, stripped of the delta-functions for conservation of transverse and light-cone momentum. The insertion of the charge operators ρa(~q1),ρb(~q2)between the incoming and scattered proton states corresponds to the attachment of two static gluon probes (with amputated propagators) to the color charges in the proton, in all possible ways. The two static gluons that probe the proton structure carry transverse momenta ~q1and ~q2, and the total momentum transfer to the proton is ~ K−~ P=−(~q1+~q2). (2) From here on we choose ~ P=0 for the incoming proton. Figure 1shows “handbag” and “cat’s ears” diagrams at leading order (LO) O(g2). The former are represented by one-body operators so that the entire momentum transfer ~ Kto the 058.2 SciPost Phys. Proc. 8, 058 (2022) proton flows into a single valence quark line. Therefore, for ~q1→ −~q2(i.e. ~ K→0) there is maximal wave function overlap for the LO handbag diagram. This diagram is proportional to the electromagnetic (Dirac) form factor, i.e. to the distribution 〈ρ(~q)〉of electric charge in the proton. See, for example, Eqs. (9,10) in ref. [14]. For large momentum transfer |t| ≃ ~ K2, on the other hand, wave function overlap in the handbag diagram is highly suppressed. Here, the overlap of incoming and scattered proton is much greater for the cat’s ears diagram because the momentum transfer is shared by two (or even three, at NLO) valence quarks. Since ~ Kis the Fourier conjugate to the two-dimensional (2D) transverse coordinate vector (impact parameter) ~ b, it follows that color charge correlators near the center of the proton are dominated by diagrams where the momentum transfer is shared by multiple partons (n-body operators). NLO corrections due to the emission of a perturbative gluon by one of the quarks (plus the corresponding virtual corrections) have been computed in ref. [15]. These are suppressed by an additional factor of αsbut enhanced by a logarithm of the minimal light-cone momentum fraction xof the gluon [16–18]. Diagrams where a quark exchanges a gluon with itself across 〈K|→|P〉also involve a DGLAP [19–21]collinear logarithm. Numerically, the one-gluon emission corrections to the color charge correlator G2are small at x≃0.1 but turn into the dominant contribution at x≃0.01 [22]. To show the behavior of G2as a function of impact parameter we perform a Fourier transform w.r.t. the momentum transfer, G2(~q12,~ b) = Zd2~ K (2π)2e−i~ b·~ KG2~q12 −~ K 2,−~q12 +~ K 2. (3) Here ~q12 =~q1−~q2denotes the relative transverse momentum of the two gluon probes, the Fourier conjugate to the transverse distance ~rbetween the two gluons. For ~q12 =0, the integral of G2over the transverse impact parameter plane vanishes, Zd2~ b G2(~q12 =0,~ b) = 0 . (4) This follows from the fact that G2(~q1,~q2)satisfies a Ward identity and vanishes when either ~qi→0[15,23–25]. Figure 2shows G2versus b. Near the center of the proton we find that G2<0, i.e. “repulsive” two-body correlations dominate here. Also, comparing x=0.1 to x=0.01 we note that the NLO correction mainly affects G2at small bto strongly boost these negative two-body correlations, more so for smaller q12. However, hand-bag type contributions become more prominent with increasing q12 or b(the standard GPD limit). Generically, the large-btails of the two-point correlator G2exhibit a fall-off that resembles a transverse profile function. We may compare the above result from LCPT to the widely used IPsat model [26,27]. Here, the color charge correlator is assumed to factorize into a function of ~q12 (or of ~r, in coordinate space representation) times a Gaussian spatial profile function Tp(~ b) = 1 2πBe−b2/2B. (5) A similar factorized form of G2was proposed in ref. [28]. In such models the correlator of course does not change sign as a function of b. Also, it does not satisfy the sum rule eq. (4). 058.3 SciPost Phys. Proc. 8, 058 (2022) 0.0 0.2 0.4 0.6 0.8 b [fm] 0.15 0.10 0.05 0.00 0.05 G 2( q 12, b ) NLO, s = 0.2 x = 0.01, q 12 = 0.5GeV x = 0.01, q 12 = 1.0GeV x = 0.1, q 12 = 0.5GeV x = 0.1, q 12 = 1.0GeV 0.0 0.5 1.0 1.5 2.0 | t | [GeV2] 10 1 100 101 102 d /d t ( + p J / + p ) x = 0.01, Q 2= 0GeV2 LCPT NLO ( s = 0.25) (1 t / m 2 g )2 p fit in | t | > 1GeV2 IPsat H1 W = 52GeV × 0.75 Figure 2: Left: Color charge correlator G2(~q12,~ b)as a function of impact parameter for various relative gluon transverse momenta ~q12 [22]. The bands indicate the variation with the collinear cutoff, over the range m=0.1...0.4 GeV. Right: cross section for exclusive J/Ψphotoproduction versus squared transverse momentum transfer. We compare predictions of our light-cone perturbation theory approach and of the IPsat model at xP=0.01 (or W≈30 GeV). The bands reflect the uncertainty of the J/Ψwave function [29]. For reference, we also show scaled data by the H1 collaboration [30]taken at higher energy. From the color charge correlator we can compute the scattering amplitude of a quarkantiquark dipole as a function of its size and of impact parameter [2]: N(~r,~ b) = −g4CFZd2~ Kd2~q (2π)4 cos~ b·~ K (~q−1 2~ K)2(~q+1 2~ K)2 ×cos(~r·~q)−cos ~r·~ K 2 G2~q−1 2~ K,−~q−1 2~ K. (6) This expression applies in the two-gluon exchange approximation in the regime of weak scattering, N(~r,~ b)1 since it does not resum the Glauber-Mueller multiple scattering series. To perform such resummation, the color charge correlator would have to be transformed from light cone to covariant gauge. The dipole scattering amplitude exhibits an interesting dependence on the azimuthal angle between ~rand ~ bwhich is discussed in more detail in ref. [22]. We now employ our color charge correlator G2(~q1,~q2)obtained from light-cone perturbation theory (LCPT) to compute the cross section for γ+p→J/Ψ+pas a function of transverse momentum transfer. Our main interest is in the regime of relatively high momentum transfer where, as explained above, the quark-antiquark dipole in the photon predominantly scatters from multiple partons in the proton. The computation of dσ/dtfrom the dipole scattering amplitude is described in detail in the literature, see for example refs. [29,31,32](and ref. [33]for a first next-to-leading order calculation presented at this Workshop). Our result at xP=0.01 is shown in Fig. 2. (For reference, we also show HERA data taken at higher energy, xP≈0.0035 shifted to match the normalization of our calculation. However, our main focus here is on the t-dependence.) The IPsat model leads to an exponential drop off with |t|which of course follows from the parameterized b-dependence eq. (5). The proton structure that emerges from our LCPT computation exhibits a similar exponential fall-off up to |t| ≈ 0.8 GeV2. However, this turns into a power-law dependence at greater momentum transfer. The cross section at large |t| 058.4 SciPost Phys. Proc. 8, 058 (2022) probes color charge correlations at small b, near the center of the proton. As shown in the previous section, there our correlation function G2(~ b,~q12)is very different from a “profile function”. A description of dσ/dtin terms of a dipole form factor has been proposed previously [34] in order to account for the non-exponential slope at small transverse momentum transfer, in particular for low energies near the kinematic production threshold. Instead, here we consider the high-energy limit far above threshold and high |t| ≃ K2 T>1 GeV2, where the cross section is not expressed in terms of a form factor but in terms of a two-point correlation function of color charge densities in the proton, as explained above. 3 Conclusion In conclusion, we suggest that the impact parameter dependence of color charge correlations in the proton (at moderately small x) is very different from a “transverse profile function”, especially at small b. In particular, the two-point correlation function is negative at b→0, and changes sign at intermediate impact parameters. The non-trivial impact parameter dependence of color charge correlation functions leads to at-dependence of the cross section for exclusive J/Ψproduction which exhibits an exponential fall-off with |t|at intermediate momentum transfer, which then changes into a power-law (approximately dipole-like) fall-off beyond |t| ≈ 1 GeV2. 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