Abstract
We investigate the path integral representation of the scalar propagator in a background gluon field, extending beyond the eikonal approximation by considering all gauge field components and incorporating its x− dependence. Utilizing the worldline formalism, we integrate the Schwinger proper time to express the scalar propagator in light-cone coordinates, facilitating a direct comparison with known results in the literature. The derived propagator captures the change of longitudinal momentum of the projectile within the medium. In the high-energy limit, our result simplifies to the effective gluon propagator employed in the BDMPS-Z formalism. Hence, we propose that our outcome serves as a foundational point for investigating corrections to the BDMPS-Z spectrum arising from the longitudinal momentum transfer of the radiated gluon with the medium, as well as for studying collisional energy loss phenomena. Lastly, by employing an expansion around the classical saddle point solution, we systematically derive an eikonal expansion in inverse powers of the boost parameter, encompassing corrections related to longitudinal momentum transfer and interactions of the projectile with the transverse component of the field.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.V. Gribov, E.M. Levin and M.G. Ryskin, Semihard Processes in QCD, Phys. Rept. 100 (1983) 1 [INSPIRE].
A.H. Mueller and J.-W. Qiu, Gluon Recombination and Shadowing at Small Values of x, Nucl. Phys. B 268 (1986) 427 [INSPIRE].
L.D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phys. Rev. D 49 (1994) 2233 [hep-ph/9309289] [INSPIRE].
L.D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei at small transverse momentum, Phys. Rev. D 49 (1994) 3352 [hep-ph/9311205] [INSPIRE].
L.D. McLerran and R. Venugopalan, Green’s functions in the color field of a large nucleus, Phys. Rev. D 50 (1994) 2225 [hep-ph/9402335] [INSPIRE].
Y.V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy, Oxford University Press (2013) [https://doi.org/10.1017/9781009291446] [INSPIRE].
J.-P. Blaizot, High gluon densities in heavy ion collisions, Rept. Prog. Phys. 80 (2017) 032301 [arXiv:1607.04448] [INSPIRE].
R. Baier et al., Radiative energy loss of high-energy quarks and gluons in a finite volume quark-gluon plasma, Nucl. Phys. B 483 (1997) 291 [hep-ph/9607355] [INSPIRE].
R. Baier et al., Radiative energy loss and p⊥ broadening of high-energy partons in nuclei, Nucl. Phys. B 484 (1997) 265 [hep-ph/9608322] [INSPIRE].
R. Baier, Y.L. Dokshitzer, A.H. Mueller and D. Schiff, Medium induced radiative energy loss: Equivalence between the BDMPS and Zakharov formalisms, Nucl. Phys. B 531 (1998) 403 [hep-ph/9804212] [INSPIRE].
B.G. Zakharov, Fully quantum treatment of the Landau-Pomeranchuk-Migdal effect in QED and QCD, JETP Lett. 63 (1996) 952 [hep-ph/9607440] [INSPIRE].
B.G. Zakharov, Radiative energy loss of high-energy quarks in finite size nuclear matter and quark-gluon plasma, JETP Lett. 65 (1997) 615 [hep-ph/9704255] [INSPIRE].
T. Altinoluk et al., Next-to-eikonal corrections in the CGC: gluon production and spin asymmetries in pA collisions, JHEP 07 (2014) 068 [arXiv:1404.2219] [INSPIRE].
T. Altinoluk, N. Armesto, G. Beuf and A. Moscoso, Next-to-next-to-eikonal corrections in the CGC, JHEP 01 (2016) 114 [arXiv:1505.01400] [INSPIRE].
I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to moderate x, JHEP 10 (2015) 017 [arXiv:1505.02151] [INSPIRE].
G.A. Chirilli, Sub-eikonal corrections to scattering amplitudes at high energy, JHEP 01 (2019) 118 [arXiv:1807.11435] [INSPIRE].
T. Altinoluk, G. Beuf, A. Czajka and A. Tymowska, Quarks at next-to-eikonal accuracy in the CGC: Forward quark-nucleus scattering, Phys. Rev. D 104 (2021) 014019 [arXiv:2012.03886] [INSPIRE].
T. Altinoluk and G. Beuf, Quark and scalar propagators at next-to-eikonal accuracy in the CGC through a dynamical background gluon field, Phys. Rev. D 105 (2022) 074026 [arXiv:2109.01620] [INSPIRE].
T. Altinoluk, G. Beuf, A. Czajka and A. Tymowska, DIS dijet production at next-to-eikonal accuracy in the CGC, Phys. Rev. D 107 (2023) 074016 [arXiv:2212.10484] [INSPIRE].
P. Agostini, T. Altinoluk and N. Armesto, Non-eikonal corrections to multi-particle production in the Color Glass Condensate, Eur. Phys. J. C 79 (2019) 600 [arXiv:1902.04483] [INSPIRE].
P. Agostini et al., Multiparticle production in proton-nucleus collisions beyond eikonal accuracy, Eur. Phys. J. C 82 (2022) 1001 [arXiv:2207.10472] [INSPIRE].
P. Agostini, T. Altinoluk and N. Armesto, Effect of non-eikonal corrections on azimuthal asymmetries in the Color Glass Condensate, Eur. Phys. J. C 79 (2019) 790 [arXiv:1907.03668] [INSPIRE].
P. Agostini, T. Altinoluk and N. Armesto, Finite width effects on the azimuthal asymmetry in proton-nucleus collisions in the Color Glass Condensate, Phys. Lett. B 840 (2023) 137892 [arXiv:2212.03633] [INSPIRE].
European Muon collaboration, A measurement of the Spin Asymmetry and Determination of the Structure Function g1 in Deep Inelastic Muon-Proton Scattering, Phys. Lett. B 206 (1988) 364 [INSPIRE].
G.A. Chirilli, High-energy operator product expansion at sub-eikonal level, JHEP 06 (2021) 096 [arXiv:2101.12744] [INSPIRE].
Y. Hatta et al., Gluon orbital angular momentum at small-x, Phys. Rev. D 95 (2017) 114032 [arXiv:1612.02445] [INSPIRE].
J. Jalilian-Marian, Quark jets scattering from a gluon field: from saturation to high pt, Phys. Rev. D 99 (2019) 014043 [arXiv:1809.04625] [INSPIRE].
J. Jalilian-Marian, Rapidity loss, spin, and angular asymmetries in the scattering of a quark from the color field of a proton or nucleus, Phys. Rev. D 102 (2020) 014008 [arXiv:1912.08878] [INSPIRE].
T. Altinoluk, N. Armesto and G. Beuf, Probing quark transverse momentum distributions in the color glass condensate: Quark-gluon dijets in deep inelastic scattering at next-to-eikonal accuracy, Phys. Rev. D 108 (2023) 074023 [arXiv:2303.12691] [INSPIRE].
S. Bhattacharya, R. Boussarie and Y. Hatta, Signature of the Gluon Orbital Angular Momentum, Phys. Rev. Lett. 128 (2022) 182002 [arXiv:2201.08709] [INSPIRE].
R. Boussarie and Y. Mehtar-Tani, Gauge invariance of transverse momentum dependent distributions at small x, Phys. Rev. D 103 (2021) 094012 [arXiv:2001.06449] [INSPIRE].
R. Boussarie and Y. Mehtar-Tani, A novel formulation of the unintegrated gluon distribution for DIS, Phys. Lett. B 831 (2022) 137125 [arXiv:2006.14569] [INSPIRE].
M. Li et al., Ultrarelativistic quark-nucleus scattering in a light-front Hamiltonian approach, Phys. Rev. D 101 (2020) 076016 [arXiv:2002.09757] [INSPIRE].
M. Li, T. Lappi and X. Zhao, Scattering and gluon emission in a color field: A light-front Hamiltonian approach, Phys. Rev. D 104 (2021) 056014 [arXiv:2107.02225] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Helicity Evolution at Small-x, JHEP 01 (2016) 072 [Erratum ibid. 10 (2016) 148] [arXiv:1511.06737] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Small-x asymptotics of the quark helicity distribution, Phys. Rev. Lett. 118 (2017) 052001 [arXiv:1610.06188] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Helicity Evolution at Small x: Flavor Singlet and Non-Singlet Observables, Phys. Rev. D 95 (2017) 014033 [arXiv:1610.06197] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Small-x Asymptotics of the Quark Helicity Distribution: Analytic Results, Phys. Lett. B 772 (2017) 136 [arXiv:1703.05809] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Small-x Asymptotics of the Gluon Helicity Distribution, JHEP 10 (2017) 198 [arXiv:1706.04236] [INSPIRE].
Y.V. Kovchegov and M.D. Sievert, Small-x Helicity Evolution: an Operator Treatment, Phys. Rev. D 99 (2019) 054032 [arXiv:1808.09010] [INSPIRE].
Y.V. Kovchegov and M.D. Sievert, Valence Quark Transversity at Small x, Phys. Rev. D 99 (2019) 054033 [arXiv:1808.10354] [INSPIRE].
Y.V. Kovchegov and Y. Tawabutr, Helicity at Small x: Oscillations Generated by Bringing Back the Quarks, JHEP 08 (2020) 014 [arXiv:2005.07285] [INSPIRE].
J. Borden and Y.V. Kovchegov, Analytic Solution for the Revised Helicity Evolution at Small x and Large Nc: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept, Phys. Rev. D 108 (2023) 014001 [arXiv:2304.06161] [INSPIRE].
F. Cougoulic and Y.V. Kovchegov, Helicity-dependent generalization of the JIMWLK evolution, Phys. Rev. D 100 (2019) 114020 [arXiv:1910.04268] [INSPIRE].
F. Cougoulic and Y.V. Kovchegov, Helicity-dependent extension of the McLerran-Venugopalan model, Nucl. Phys. A 1004 (2020) 122051 [arXiv:2005.14688] [INSPIRE].
F. Cougoulic, Y.V. Kovchegov, A. Tarasov and Y. Tawabutr, Quark and gluon helicity evolution at small x: revised and updated, JHEP 07 (2022) 095 [arXiv:2204.11898] [INSPIRE].
M. Li, Small x physics beyond eikonal approximation: an effective Hamiltonian approach, JHEP 07 (2023) 158 [arXiv:2304.12842] [INSPIRE].
M.J. Strassler, Field theory without Feynman diagrams: One loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].
C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].
A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scattering: Finding the triangle graph inside the box diagram in Bjorken and Regge asymptotics, Phys. Rev. D 102 (2020) 114022 [arXiv:2008.08104] [INSPIRE].
A. Tarasov and R. Venugopalan, Role of the chiral anomaly in polarized deeply inelastic scattering. II. Topological screening and transitions from emergent axionlike dynamics, Phys. Rev. D 105 (2022) 014020 [arXiv:2109.10370] [INSPIRE].
K.C. Zapp and U.A. Wiedemann, Coherent Radiative Parton Energy Loss beyond the BDMPS-Z Limit, Eur. Phys. J. C 72 (2012) 2028 [arXiv:1202.1192] [INSPIRE].
X. Feal and R.A. Vazquez, Transverse spectrum of bremsstrahlung in finite condensed media, Phys. Rev. D 99 (2019) 016002 [arXiv:1810.02645] [INSPIRE].
X. Feal and R. Vazquez, Intensity of gluon bremsstrahlung in a finite plasma, Phys. Rev. D 98 (2018) 074029 [arXiv:1811.01591] [INSPIRE].
S. Caron-Huot and C. Gale, Finite-size effects on the radiative energy loss of a fast parton in hot and dense strongly interacting matter, Phys. Rev. C 82 (2010) 064902 [arXiv:1006.2379] [INSPIRE].
C. Andres, L. Apolinário and F. Dominguez, Medium-induced gluon radiation with full resummation of multiple scatterings for realistic parton-medium interactions, JHEP 07 (2020) 114 [arXiv:2002.01517] [INSPIRE].
A.V. Sadofyev, M.D. Sievert and I. Vitev, Ab initio coupling of jets to collective flow in the opacity expansion approach, Phys. Rev. D 104 (2021) 094044 [arXiv:2104.09513] [INSPIRE].
C. Andres, F. Dominguez, A.V. Sadofyev and C.A. Salgado, Jet broadening in flowing matter: Resummation, Phys. Rev. D 106 (2022) 074023 [arXiv:2207.07141] [INSPIRE].
E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].
M. Fabbrichesi, R. Pettorino, G. Veneziano and G.A. Vilkovisky, Planckian energy scattering and surface terms in the gravitational action, Nucl. Phys. B 419 (1994) 147 [hep-th/9309037] [INSPIRE].
L. Brink, P. Di Vecchia and P.S. Howe, A Locally Supersymmetric and Reparametrization Invariant Action for the Spinning String, Phys. Lett. B 65 (1976) 471 [INSPIRE].
E.S. Fradkin and D.M. Gitman, Path integral representation for the relativistic particle propagators and BFV quantization, Phys. Rev. D 44 (1991) 3230 [INSPIRE].
I.K. Affleck, O. Alvarez and N.S. Manton, Pair Production at Strong Coupling in Weak External Fields, Nucl. Phys. B 197 (1982) 509 [INSPIRE].
X. Feal, A. Tarasov and R. Venugopalan, QED as a many-body theory of worldlines: General formalism and infrared structure, Phys. Rev. D 106 (2022) 056009 [arXiv:2206.04188] [INSPIRE].
X. Feal, A. Tarasov and R. Venugopalan, QED as a many-body theory of worldlines. II. All-order S-matrix formalism, Phys. Rev. D 107 (2023) 096021 [arXiv:2211.15712] [INSPIRE].
Y. Mehtar-Tani, Relating the description of gluon production in pA collisions and parton energy loss in AA collisions, Phys. Rev. C 75 (2007) 034908 [hep-ph/0606236] [INSPIRE].
F. Bastianelli and P. van Nieuwenhuizen, Path integrals and anomalies in curved space, Cambridge University Press (2006) [10.1017/CBO9780511535031] [INSPIRE].
Acknowledgments
We are very grateful to Xabier Feal, Andrey Tarasov, and Raju Venugopalan for reading this manuscript and their valuable comments. We thank Tolga Altinoluk and Guillaume Beuf for insightful discussions that have contributed to the development of this manuscript. Special thanks go to Néstor Armesto and Fabio Dominguez for their invaluable revision and constructive feedback on the manuscript, as well as for their inspiring and fruitful discussions. This work has been supported by Consellería de Cultura, Educacíon e Universidade of Xunta de Galicia (Spain) under the grant ED481B-2022-050. The author has received financial support from Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2019-2022, ref. ED421G-2019/05), by European Union ERDF, by the “María de Maeztu” Units of Excellence program MDM2016-0692, and by the Spanish Research State Agency under project PID2020-119632GB-I00. This work has been performed in the framework of the European Research Council project ERC-2018-ADG-835105 YoctoLHC and the MSCA RISE 823947 “Heavy ion collisions: collectivity and precision in saturation physics” (HIEIC), and has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 824093.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2307.13573
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Agostini, P. Scalar propagator in a background gluon field beyond the eikonal approximation. J. High Energ. Phys. 2023, 99 (2023). https://doi.org/10.1007/JHEP11(2023)099
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2023)099