Studying high pT momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions

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-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Studying high pT momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions © 2024 the Authors Published version Soudi, Ismail; Majumder, Abhijit Soudi, I., & Majumder, A. (2024). Studying high pT momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions. In A. Vossen, T. Mehen, M. Copeland, R. Hodges, G. Matousek, M. McEneaney, K. Parham, C. Pecar, & S. Schneider (Eds.), SPIN2023 : 25th International Symposium on Spin Physics (Article 164). Sissa. POS Proceedings of Science, 456. https://doi.org/10.22323/1.456.0164 2024 PoS(SPIN2023)164 Studying high𝒑𝑻momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions Ismail Soudi𝑎,𝑏,𝑐,∗and Abhijit Majumder𝑎 𝑎Department of Physics and Astronomy, Wayne State University, Detroit, MI 48201. 𝑏University of Jyväskylä, Department of Physics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland 𝑐Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland E-mail: [email protected],[email protected] Recent experimental results have shown that small systems such as 𝑝-𝑝and 𝑝-𝐴collisions exhibit a non-zero azimuthal anisotropy even at large 𝑝𝑇. However, no evidence of jet quenching has been observed in these collisions. We investigate the possibility that the azimuthal anisotropy of high-𝑝𝑇hadrons can be generated by the intrinsic transverse momentum of the partons in the proton. After introducing transverse momentum dependent (TMD) parton distribution and fragmentation functions, additional polarization effects are allowed. Unpolarized protons can generate transversely polarized quarks or linearly polarized gluons through a distribution known as the Boer-Mulders’ function. The fragmentation of similarly polarized partons to unpolarized hadrons is called the Collins’ function. We find that the high-𝑝𝑇azimuthal anisotropies can be obtained using these TMD distributions without modification to the angle integrated spectra. 25th International Spin Physics Symposium (SPIN 2023) 24-29 September 2023 Durham, NC, USA ∗Speaker ©Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ PoS(SPIN2023)164 Studying high-𝑝𝑇momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions Ismail Soudi 1. Introduction The study of heavy ion collisions have significantly advanced our understanding of QCD matter. It is well accepted that during high energy heavy ion collisions a new phase of matter known as the Quark-Gluon Plasma (QGP) is produced [1,2]. One of the main signatures of the QGP is anisotropic flow, which is characterized by azimuthal correlations in the final momentum distribution of the produced particles. The particle yield can be expressed as a Fourier series in the azimuthal angle 𝜙as follows 𝑑𝑁 𝑑𝜙 ∝1+2∞ ∑︁ 𝑛=1 𝑣𝑛cos(𝑛(𝜙−Ψ𝑛)) ,(1) where Ψ𝑛is the 𝑛-th order event plane angle. Anisotropies in the initial density distribution leads to large pressure gradients. Coupled with a vanishing viscosity leads to a large non-zero 𝑣2coefficient, known as elliptic flow. The creation of the QGP is corroborated by the suppression of high transverse momentum hadrons in heavy ion collisions compared to proton-proton collisions, known as jet quenching [3,4]. Since high energy jets produced in the initial hard scattering must traverse the medium before reaching the detector, they lose their energy by interacting with the medium. Recently, experimental results have observed a non-zero 𝑣2for small systems such as high multiplicity 𝑝-𝑝and 𝑝-𝐴collisions. While the 𝑣2decreases at higher transverse momentum, a sizable 𝑣2is observed even at large 𝑝𝑇≳10 GeV [5,6]. Typically, the 𝑣2at high 𝑝𝑇is attributed to final state effects such as jet-medium interactions. However, studies of jet suppression in 𝑝-𝐴 collisions have not observed any significant modification of the angle integrated high transverse momentum hadron spectra [7–9]. In these proceedings, we explore the possibility that transverse momentum dependent (TMD) parton distribution (PDF) and fragmentation functions (FF) can generate a non-zero 𝑣2at high 𝑝𝑇 without any modification to the angle integrated spectra. 2. Theoretical Framework We consider pion production in 𝑝-𝑝and 𝑝-𝐴collisions at high transverse momentum 𝑃𝑇. Following [10], the cross-section of the unpolarized processes 𝑝+𝑝→𝜋+𝑋is given by a factorized convolution of the hard partonic processes 𝑎+𝑏→𝑏+𝑐, as follows, 𝑑𝜎 𝑑𝑦𝑑2𝑃𝑇 =∫𝑑𝑥𝑎𝑑𝑥𝑏𝑑𝑧𝑑2𝑘⊥𝑎𝑑2𝑘⊥𝑏𝑑3𝑘⊥𝐶 2𝜋2𝑧3𝑠𝛿(𝒌⊥𝐶·ˆ𝑝𝑐)𝐽(𝒌⊥𝐶)Γ𝜎𝜇 (𝑥𝑎, 𝑘⊥𝑎)Γ𝛼𝜈 (𝑥𝑏, 𝑘⊥𝑏) ׈ 𝑀𝜇𝜈𝜌 ˆ 𝑀∗ 𝜎𝛼𝛽Δ𝜌𝛽 (𝑧, 𝑘⊥𝐶)𝛿(ˆ𝑠+ˆ 𝑡+ˆ𝑢),(2) where 𝐽(𝒌⊥𝐶)=(𝐸2 𝐶+√︃𝒑2 𝐶−𝒌2 ⊥𝐶)2 4(𝒑2 𝐶−𝒌2 ⊥𝐶). We denote the partonic and hadronic Madelstam variable by (ˆ𝑠, ˆ 𝑡, ˆ𝑢)and (𝑠, 𝑡, 𝑢)respectively. In these proceedings, we consider only the gluon-gluon partonic channel which dominates the cross section for pion production at the 𝑝𝑇’s and √𝑠considered. The gluon correlator projected 2 PoS(SPIN2023)164 Studying high-𝑝𝑇momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions Ismail Soudi onto the helicity basis can be written as, Γ𝜆1,𝜆2 𝑃(𝑥, 𝑘⊥)=−𝛿𝜆1,𝜆2𝑓(𝑥, 𝑘2 ⊥) +𝛿𝜆1,−𝜆2𝑘2 ⊥ 2𝑀2 𝑝 ℎ⊥(𝑥, 𝑘2 ⊥) 2𝑥,(3) where 𝑓(𝑥, 𝑘2 ⊥)is the spin-polarization independent TMD-PDFs with longitudinal momentum fraction 𝑥(𝑎,𝑏)and transverse momentum 𝑘⊥(𝑎,𝑏), relative to the 𝑧-axis defined by the incoming proton beams. The distribution of linearly polarized gluons in the proton is given by the Boer- Mulders’ function ℎ⊥ 1(𝑥, 𝑘2 ⊥)[11]. Similarly, for the fragmentation, the correlator is, Δ𝜆1,𝜆2(𝑧, 𝑘⊥)=−𝛿𝜆1,𝜆2𝐷(𝑧, 𝑘2 ⊥) +𝛿𝜆1,−𝜆2𝑘2 ⊥ 2𝑀2 𝜋 𝐻⊥(𝑧, 𝑘2 ⊥) 2/𝑧.(4) Here 𝐷(𝑧𝑐, 𝑘⊥𝐶)represents the spin-polarization independent TMD-FF for the outgoing parton (c) fragmenting to the pion (𝝅), carrying momentum 𝑧𝒑𝑐+𝒌⊥𝐶. The distribution of fragmenting 𝝅 from a linearly polarized gluon is given by the Collins’ function 𝐻⊥(𝑧, 𝑘⊥)[12]. Due to the initial transverse momentum of the hard partons, the hard scattering acquires a net transverse momentum 𝒒⊥=𝒌⊥𝑎+𝒌⊥𝑏with respect to the center of mass of the hadronic scattering. Conversely, the transverse momentum of the remaining soft partons from each hadron must be compensated by the net transverse momentum. While not all the soft partons will participate in the collisions, there will be a strong correlation between the net transverse momentum of the hard scattering and the soft hadrons. To study the azimuthal anisotropies, we will compute the Fourier coefficients of the cross section as follows 𝑣2=∫𝑑𝜙𝜋cos(2(𝜙𝒒𝑇−𝜙𝜋)) 𝑑𝜎 𝑑𝜙𝜋 ∫𝑑𝜙𝜋𝑑𝜎 𝑑𝜙𝜋 ,(5) The Boer-Mulders’ and Collins’ functions in Eqns. (3-4) flip the helicity between the matrix element and its complex conjugate. Therefore, the only allowed scattering involves two linearly polarized gluons. Due to the phases of the gluon, the contribution most relevant to the azimuthal anisotropy is the scattering involving a linearly polarized gluon in the initial and final state; we will refer to this as the Boer-Mulders’ Collins scattering (𝐵𝑀 ⊗𝐶). The combination of matrix element times complex conjugate with initial and final correlators can be expressed as, ΣBM⊗C≡ˆ 𝑀𝜇𝜈𝜌 ˆ 𝑀∗ 𝜎𝛼𝛽Δ𝜌𝛽 (𝑧, 𝑘⊥𝐶)Γ𝜎𝜇 (𝑥𝑎, 𝑘⊥𝑎)Γ𝛼𝜈 (𝑥𝑏, 𝑘⊥𝑏),(6) =𝐻⊥(1)(𝑧, 𝑘⊥𝐶)hℎ⊥(1)(𝑥𝑎, 𝑘2 ⊥𝑎)𝑓(𝑥𝑏, 𝑘2 ⊥𝑏)ˆ 𝑀1ˆ 𝑀2cos(4(𝜙𝑎𝑏 −𝜙𝑏𝑐)) +𝑓(𝑥𝑎, 𝑘2 ⊥𝑎)ℎ⊥(1)(𝑥𝑏, 𝑘2 ⊥𝑏)ˆ 𝑀1ˆ 𝑀3cos(4(𝜙𝑎𝑏 −𝜙𝑎𝑐))i,(7) where we define ℎ⊥(1)≡ (𝑘2 ⊥/2𝑀2 𝑝)ℎ⊥and 𝐻⊥(1)≡ (𝑘2 ⊥/2𝑀2 𝜋)𝐻⊥. The color and spin averaged matrix elements (times complex conjugate) can be expressed as, ˆ 𝑀1ˆ 𝑀2=𝑔4 𝑠 𝑁2 𝑁2−1 𝑡2+𝑡𝑢 +𝑢2 𝑡2,ˆ 𝑀1ˆ 𝑀3=𝑔4 𝑠 𝑁2 𝑁2−1 𝑡2+𝑡𝑢 +𝑢2 𝑢2,(8) where, using partonic momenta in spherical coordinates 𝒑𝑖=(𝑝𝑖, 𝜃𝑖, 𝜙𝑖), the phases are given by tan 𝜙𝑖 𝑗 =tan 𝜙𝑗−𝜙𝑖 2sin 𝜃𝑗+𝜃𝑖 2sin 𝜃𝑗−𝜃𝑖 2.(9) 3 PoS(SPIN2023)164 Studying high-𝑝𝑇momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions Ismail Soudi 0.01 0.1 1 1 10 p-Pb 8.16TeV 0.01 0.1 1 10 Azimuthal asymmetry: v2 Momentum: pT[GeV/c] 40% of Soffer bound, hk2 ⊥i ≤ 1GeV2fixed 20 - 40% of Soffer bound, hk2 ⊥i(x) ATLAS [EPJ C 80 (2020) 1, 73] pjet T>100 GeV pjet T>75 GeV MBT Azimuthal asymmetry: v2 Momentum: pT[GeV/c] 20 - 40% of Soffer bound, hk2 ⊥i(x) 40% of Soffer bound, hk2 ⊥i ≤ 1GeV2fixed ATLAS [PRC 96 (2017) 2, 024908] p-p 13TeV 0.01 0.1 1 1 10 p-Pb 8.16TeV 0.01 0.1 1 10 Azimuthal asymmetry: v2 Momentum: pT[GeV/c] 40% of Soffer bound, hk2 ⊥i ≤ 1GeV2fixed 20 - 40% of Soffer bound, hk2 ⊥i(x) ATLAS [EPJ C 80 (2020) 1, 73] pjet T>100 GeV pjet T>75 GeV MBT Azimuthal asymmetry: v2 Momentum: pT[GeV/c] 20 - 40% of Soffer bound, hk2 ⊥i(x) 40% of Soffer bound, hk2 ⊥i ≤ 1GeV2fixed ATLAS [PRC 96 (2017) 2, 024908] p-p 13TeV Figure 1: Azimuthal anisotropy coefficient 𝑣2as a function of the pion transverse momentum 𝑝𝑇for pp collisions at √𝑠=13 TeV (left) and pPb at 8.16 TeV (right). The solid shaded area represent the uncertainty on the momentum ⟨𝑘2 ⊥⟩ ≤ 1GeV2, while the hatched shaded area displays a different choice for the bound 0.2≤𝑏·𝐵≤0.4for 𝑥-dependent transverse momentum ⟨𝑘2 ⊥⟩1/2(𝑥). 204 406 8 Azimuthal asymmetry: v2 Momentum: pT[GeV/c] b·B= 0.4,hk2 ⊥i(x) BM ⊗C ATLAS [EPJ C 80 (2020) 1, 73] MBT pjet T>75 GeV pjet T>100 GeV 0 0.05 0.1 10 Figure 2: Decomposition of 𝑣2contributions. The filled green represents the BM ⊗Ccontribution, the blue hatched represents polarization independent contributions. 3. Results We employ a Gaussian ansatz for the transverse momentum dependence, and we use the nCTEQ parametrization [13] for the longitudinal dependence of the PDFs and leading order KKP [14] for FFs. The Boer-Mulders’ and Collins’ functions are taken to be proportional to the unpolarized PDFs and FFs respectively. The polarization independent contributions are only modified by Gaussian transverse momentum distributions that integrate out to unity when computing the angular integrated cross section. Conversely, the BM ⊗Ccontribution leads to a negligible modification of the angle integrated cross section, because the matrix element in Eq. (7) is proportional to cosine terms that are suppressed after the angular integrations. Accordingly, we find that the TMD contributions lead to minimal modification of the angle integrated cross section. On the left panel of Fig. 1, we present the azimuthal coefficient for 𝑝-𝑝collisions at 13 TeV. This analysis is limited to the leading twist calculation and is only valid at high transverse momentum (𝑃𝑇≫ ⟨𝑘2 ⊥⟩1/2), which we ensure by limiting the result to 𝑃𝑇>3GeV. The red filled area represents 4 PoS(SPIN2023)164 Studying high-𝑝𝑇momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions Ismail Soudi Azimuthal asymmetry: v3 Momentum: pT[GeV/c] BM ⊗C b·B= 0.4,hk2 ⊥ifixed ATLAS [PRC 96 (2017) 2, 024908] 0 0.02 0.04 0.06 0.08 0.1 1 10 p-Pb 5.02TeV Azimuthal asymmetry: v3 Momentum: pT[GeV/c] b·B= 0.4,hk2 ⊥ifixed b·B= 0.4,hk2 ⊥i(x) ATLAS [PRC 96 (2017) 2, 024908] 0 0.02 0.04 0.06 0.08 0.1 1 10 p-Pb 5.02TeV Azimuthal asymmetry: v4 Momentum: pT[GeV/c] BM ⊗BM BM ⊗C b·B= 0.4,hk2 ⊥ifixed ATLAS [PRC 96 (2017) 2, 024908] 0 0.01 0.02 0.03 0.04 0.05 1 10 p-Pb 5.02TeV Azimuthal asymmetry: v4 Momentum: pT[GeV/c] b·B= 0.4,hk2 ⊥ifixed b·B= 0.4,hk2 ⊥i(x) ATLAS [PRC 96 (2017) 2, 024908] 0 0.01 0.02 0.03 0.04 0.05 1 10 p-Pb 5.02TeV Figure 3: Decomposition of 𝑣3(left) and 𝑣4(right) contributions. The filled green represents the BM ⊗C contribution, filled red area represents the BM⊗BM and the blue hatched represents polarization independent contributions. our results within uncertainties on the mean transverse momentum of the Gaussian between a fixed value of 1GeV and an 𝑥-dependent ansatz [15]. We find that for 𝑝𝑇≥6GeV, the ATLAS data lies within our uncertainty band. Using the same parametrization, we compute the azimuthal anisotropy for 𝑝-𝑃𝑏 collisions at 8.16 TeV by increasing the mean transverse momentum of the Gaussian by a factor of 𝐴1/3. This 𝐴1/3enhancement is obtained from the multiple scatterings of the initial partons before the hard scattering [16–18]. While the effect of multiple scatterings is not well understood for polarized partons, we will assume the same enhancement of the mean transverse momentum. This leads to an enhancement of 𝑣2, as shown in the right panel of Fig. 1, which describes the ATLAS results remarkably well. In Fig. 2, we present the decomposition of the elliptic coefficient in contributions from spin independent and spin dependent partonic scatterings. We find that even though the spin dependent TMD distributions are suppressed, the 𝐵𝑀 ⊗𝐶contribution dominates the azimuthal anisotropy at high 𝑝𝑇. In Fig. 3, we present the decomposition of the 𝑣3and 𝑣4coefficients. We observe that at high-𝑝𝑇the 𝐵𝑀 ⊗𝐶contribution dominates the 𝑣3coefficient, while the 𝐵𝑀 ⊗𝐵𝑀 contribution dominates the 𝑣4coefficient. More experimental data is needed to understand these higher order coefficients and constrain the TMD distributions. We have presented evidence for a 𝑣2in high-𝑝𝑇hadron spectra in 𝑝-𝐴collisions without any observable modification of the angle integrated spectra, using TMD distributions. The initial transverse momenta of the partons in the proton can lead to anisotropies in the direction of the hard scattering. Moreover, intrinsic transverse momentum allows for spin dependent partonic contributions, which can lead to an enhancement of the 𝑣2. ACK: This work is supported by the US D.O.E. under grant number DE-SC0013460. IS is currently funded as a part of the European Research Council project ERC-2018-ADG-835105 YoctoLHC, and as a part of the Center of Excellence in Quark Matter of the Academy of Finland 5 PoS(SPIN2023)164 Studying high-𝑝𝑇momentum azimuthal anisotropies in unpolarized proton-proton collisions using transverse momentum dependent (TMD) parton distribution and fragmentation functions Ismail Soudi (project 346325). References [1] P. Romatschke and U. Romatschke, “Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?,” Phys. Rev. Lett., vol. 99, p. 172301, 2007. [2] D. A. Teaney, Viscous Hydrodynamics and the Quark Gluon Plasma, pp. 207–266. 2010. [3] A. Majumder and M. 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