Abstract
The formalism of Baier-Dokshitzer-Mueller-Peigné-Schiff and Zakharov determines the modifications of parton splittings in the QCD plasma that arise from medium-induced gluon radiation. Here, we study medium-modifications of the gluon splitting into a quark-anti-quark pair in this BDMPS-Z formalism. We derive a compact path-integral formulation that resums effects from an arbitrary number of interactions with the medium to leading order in the \( 1/{N}_c^2 \) expansion. Analyses in the N = 1 opacity and the saddle point approximations reveal two phenomena: a medium-induced momentum broadening of the relative quark-anti-quark pair momentum that increases the invariant mass of quark-anti-quark pairs, and a medium-enhanced production of such pairs. We note that both effects are numerically sizeable if the average momentum transfer from the medium is comparable to the quark mass. In ultra-relativistic heavy-ion collisions, this condition is satisfied for charm quarks. We therefore focus our numerical analysis on the medium modification of \( g\to c\overline{c} \), although our derivation applies equally well to \( g\to b\overline{b} \) and to gluons splitting into light-flavoured quark-anti-quark pairs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Connors, C. Nattrass, R. Reed and S. Salur, Jet measurements in heavy ion physics, Rev. Mod. Phys. 90 (2018) 025005 [arXiv:1705.01974] [INSPIRE].
J.D. Bjorken, Energy loss of energetic partons in quark-gluon plasma: possible extinction of high pt jets in hadron-hadron collisions, FERMILAB-PUB-82-59-THY, Fermilab, Batavia, IL, U.S.A. (1982).
X.-N. Wang, M. Gyulassy and M. Plumer, The LPM effect in QCD and radiative energy loss in a quark gluon plasma, Phys. Rev. D 51 (1995) 3436 [hep-ph/9408344] [INSPIRE].
M. Gyulassy and X.-n. Wang, Multiple collisions and induced gluon bremsstrahlung in QCD, Nucl. Phys. B 420 (1994) 583 [nucl-th/9306003] [INSPIRE].
R. Baier, Y.L. Dokshitzer, A.H. Mueller, S. Peigne and D. Schiff, 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, Y.L. Dokshitzer, A.H. Mueller, S. Peigne and D. Schiff, Radiative energy loss and pT broadening of high-energy partons in nuclei, Nucl. Phys. B 484 (1997) 265 [hep-ph/9608322] [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].
S. Cao and X.-N. Wang, Jet quenching and medium response in high-energy heavy-ion collisions: a review, Rept. Prog. Phys. 84 (2021) 024301 [arXiv:2002.04028] [INSPIRE].
G.-Y. Qin and X.-N. Wang, Jet quenching in high-energy heavy-ion collisions, Int. J. Mod. Phys. E 24 (2015) 1530014 [arXiv:1511.00790] [INSPIRE].
R. Baier, Y.L. Dokshitzer, A.H. Mueller and D. Schiff, Radiative energy loss of high-energy partons traversing an expanding QCD plasma, Phys. Rev. C 58 (1998) 1706 [hep-ph/9803473] [INSPIRE].
C.A. Salgado and U.A. Wiedemann, Calculating quenching weights, Phys. Rev. D 68 (2003) 014008 [hep-ph/0302184] [INSPIRE].
U.A. Wiedemann, Gluon radiation off hard quarks in a nuclear environment: opacity expansion, Nucl. Phys. B 588 (2000) 303 [hep-ph/0005129] [INSPIRE].
U.A. Wiedemann, Jet quenching versus jet enhancement: a quantitative study of the BDMPS-Z gluon radiation spectrum, Nucl. Phys. A 690 (2001) 731 [hep-ph/0008241] [INSPIRE].
R. Baier, Y.L. Dokshitzer, A.H. Mueller and D. Schiff, On the angular dependence of the radiative gluon spectrum, Phys. Rev. C 64 (2001) 057902 [hep-ph/0105062] [INSPIRE].
B.G. Zakharov, Transverse spectra of radiation processes in-medium, JETP Lett. 70 (1999) 176 [hep-ph/9906536] [INSPIRE].
Y.L. Dokshitzer and D.E. Kharzeev, Heavy quark colorimetry of QCD matter, Phys. Lett. B 519 (2001) 199 [hep-ph/0106202] [INSPIRE].
M. Gyulassy, P. Levai and I. Vitev, Reaction operator approach to non-Abelian energy loss, Nucl. Phys. B 594 (2001) 371 [nucl-th/0006010] [INSPIRE].
X.-N. Wang and X.-f. Guo, Multiple parton scattering in nuclei: parton energy loss, Nucl. Phys. A 696 (2001) 788 [hep-ph/0102230] [INSPIRE].
J.-P. Blaizot, F. Dominguez, E. Iancu and Y. Mehtar-Tani, Medium-induced gluon branching, JHEP 01 (2013) 143 [arXiv:1209.4585] [INSPIRE].
L. Apolinário, N. Armesto, J.G. Milhano and C.A. Salgado, Medium-induced gluon radiation and colour decoherence beyond the soft approximation, JHEP 02 (2015) 119 [arXiv:1407.0599] [INSPIRE].
Y. Mehtar-Tani, Gluon bremsstrahlung in finite media beyond multiple soft scattering approximation, JHEP 07 (2019) 057 [arXiv:1903.00506] [INSPIRE].
X. Feal, C.A. Salgado and R.A. Vazquez, Jet quenching test of the QCD matter created at RHIC and the LHC needs opacity-resummed medium induced radiation, Phys. Lett. B 816 (2021) 136251 [arXiv:1911.01309] [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].
C. Andres, F. Dominguez and M. Gonzalez Martinez, From soft to hard radiation: the role of multiple scatterings in medium-induced gluon emissions, JHEP 03 (2021) 102 [arXiv:2011.06522] [INSPIRE].
J.A. Barata, Y. Mehtar-Tani, A. Soto-Ontoso and K. Tywoniuk, Medium-induced radiative kernel with the improved opacity expansion, JHEP 09 (2021) 153 [arXiv:2106.07402] [INSPIRE].
S. Schlichting and I. Soudi, Splitting rates in QCD plasmas from a nonperturbative determination of the momentum broadening kernel C(q⊥), Phys. Rev. D 105 (2022) 076002 [arXiv:2111.13731] [INSPIRE].
J.H. Isaksen, A. Takacs and K. Tywoniuk, A unified picture of medium-induced radiation, arXiv:2206.02811 [INSPIRE].
J.H. Isaksen and K. Tywoniuk, Wilson line correlators beyond the large-Nc, JHEP 21 (2020) 125 [arXiv:2107.02542] [INSPIRE].
T. Liou, A.H. Mueller and B. Wu, Radiative p⊥-broadening of high-energy quarks and gluons in QCD matter, Nucl. Phys. A 916 (2013) 102 [arXiv:1304.7677] [INSPIRE].
J.-P. Blaizot and Y. Mehtar-Tani, Renormalization of the jet-quenching parameter, Nucl. Phys. A 929 (2014) 202 [arXiv:1403.2323] [INSPIRE].
B. Wu, Radiative energy loss and radiative p⊥-broadening of high-energy partons in QCD matter, JHEP 12 (2014) 081 [arXiv:1408.5459] [INSPIRE].
N. Armesto, H. Ma, Y. Mehtar-Tani, C.A. Salgado and K. Tywoniuk, Coherence effects and broadening in medium-induced QCD radiation off a massive qq¯ antenna, JHEP 01 (2012) 109 [arXiv:1110.4343] [INSPIRE].
J. Casalderrey-Solana and E. Iancu, Interference effects in medium-induced gluon radiation, JHEP 08 (2011) 015 [arXiv:1105.1760] [INSPIRE].
Y. Mehtar-Tani, C.A. Salgado and K. Tywoniuk, The radiation pattern of a QCD antenna in a dense medium, JHEP 10 (2012) 197 [arXiv:1205.5739] [INSPIRE].
J.a. Barata, F. Domínguez, C.A. Salgado and V. Vila, A modified in-medium evolution equation with color coherence, JHEP 05 (2021) 148 [arXiv:2101.12135] [INSPIRE].
P. Arnold and S. Iqbal, The LPM effect in sequential bremsstrahlung, JHEP 04 (2015) 070 [arXiv:1501.04964] [INSPIRE].
P. Arnold, T. Gorda and S. Iqbal, The LPM effect in sequential bremsstrahlung: nearly complete results for QCD, JHEP 11 (2020) 053 [arXiv:2007.15018] [INSPIRE].
K. Zapp, J. Stachel and U.A. Wiedemann, A local Monte Carlo implementation of the non-Abelian Landau-Pomerantschuk-Migdal effect, Phys. Rev. Lett. 103 (2009) 152302 [arXiv:0812.3888] [INSPIRE].
K.C. Zapp, F. Krauss and U.A. Wiedemann, A perturbative framework for jet quenching, JHEP 03 (2013) 080 [arXiv:1212.1599] [INSPIRE].
K.C. Zapp, JEWEL 2.0.0: directions for use, Eur. Phys. J. C 74 (2014) 2762 [arXiv:1311.0048] [INSPIRE].
N. Armesto, L. Cunqueiro and C.A. Salgado, Q-PYTHIA: a medium-modified implementation of final state radiation, Eur. Phys. J. C 63 (2009) 679 [arXiv:0907.1014] [INSPIRE].
B. Schenke, C. Gale and S. Jeon, MARTINI: an event generator for relativistic heavy-ion collisions, Phys. Rev. C 80 (2009) 054913 [arXiv:0909.2037] [INSPIRE].
P. Caucal, E. Iancu, A.H. Mueller and G. Soyez, A new pQCD based Monte Carlo event generator for jets in the quark-gluon plasma, PoS HardProbes2018 (2019) 028 [arXiv:1812.05393] [INSPIRE].
J.H. Putschke et al., The JETSCAPE framework, arXiv:1903.07706 [INSPIRE].
P. Caucal, E. Iancu, A.H. Mueller and G. Soyez, Vacuum-like jet fragmentation in a dense QCD medium, Phys. Rev. Lett. 120 (2018) 232001 [arXiv:1801.09703] [INSPIRE].
P. Caucal, E. Iancu and G. Soyez, Deciphering the zg distribution in ultrarelativistic heavy ion collisions, JHEP 10 (2019) 273 [arXiv:1907.04866] [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].
Z.-B. Kang, F. Ringer and I. Vitev, Effective field theory approach to open heavy flavor production in heavy-ion collisions, JHEP 03 (2017) 146 [arXiv:1610.02043] [INSPIRE].
M.D. Sievert, I. Vitev and B. Yoon, A complete set of in-medium splitting functions to any order in opacity, Phys. Lett. B 795 (2019) 502 [arXiv:1903.06170] [INSPIRE].
W. Ke and I. Vitev, Searching for QGP droplets with high-pT hadrons and heavy flavor, arXiv:2204.00634 [INSPIRE].
F. Domínguez, J.G. Milhano, C.A. Salgado, K. Tywoniuk and V. Vila, Mapping collinear in-medium parton splittings, Eur. Phys. J. C 80 (2020) 11 [arXiv:1907.03653] [INSPIRE].
M. Cacciari, S. Frixione, N. Houdeau, M.L. Mangano, P. Nason and G. Ridolfi, Theoretical predictions for charm and bottom production at the LHC, JHEP 10 (2012) 137 [arXiv:1205.6344] [INSPIRE].
M.L. Mangano, P. Nason and G. Ridolfi, Heavy quark correlations in hadron collisions at next-to-leading order, Nucl. Phys. B 373 (1992) 295 [INSPIRE].
R.K. Ellis and J.C. Sexton, Explicit formulae for heavy flavor production, Nucl. Phys. B 282 (1987) 642 [INSPIRE].
A. Andronic et al., Heavy-flavour and quarkonium production in the LHC era: from proton-proton to heavy-ion collisions, Eur. Phys. J. C 76 (2016) 107 [arXiv:1506.03981] [INSPIRE].
P. Ilten, N.L. Rodd, J. Thaler and M. Williams, Disentangling heavy flavor at colliders, Phys. Rev. D 96 (2017) 054019 [arXiv:1702.02947] [INSPIRE].
M. Cacciari, G.P. Salam and G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896 [arXiv:1111.6097] [INSPIRE].
A.J. Larkoski, S. Marzani, G. Soyez and J. Thaler, Soft drop, JHEP 05 (2014) 146 [arXiv:1402.2657] [INSPIRE].
C. Aidala et al., sPHENIX: an upgrade concept from the PHENIX collaboration, arXiv:1207.6378 [INSPIRE].
Z. Citron et al., Report from working group 5: future physics opportunities for high-density QCD at the LHC with heavy-ion and proton beams, CERN Yellow Rep. Monogr. 7 (2019) 1159 [arXiv:1812.06772] [INSPIRE].
D. Adamová et al., A next-generation LHC heavy-ion experiment, arXiv:1902.01211 [INSPIRE].
M. Attems et al., Medium-enhanced cc¯ radiation, arXiv:2209.13600 [INSPIRE].
N. Armesto et al., Comparison of jet quenching formalisms for a quark-gluon plasma ‘brick’, Phys. Rev. C 86 (2012) 064904 [arXiv:1106.1106] [INSPIRE].
JET collaboration, Extracting the jet transport coefficient from jet quenching in high-energy heavy-ion collisions, Phys. Rev. C 90 (2014) 014909 [arXiv:1312.5003] [INSPIRE].
H. Liu, K. Rajagopal and U.A. Wiedemann, Calculating the jet quenching parameter from AdS/CFT, Phys. Rev. Lett. 97 (2006) 182301 [hep-ph/0605178] [INSPIRE].
J. Casalderrey-Solana and D. Teaney, Heavy quark diffusion in strongly coupled N = 4 Yang-Mills, Phys. Rev. D 74 (2006) 085012 [hep-ph/0605199] [INSPIRE].
F. D’Eramo, H. Liu and K. Rajagopal, Transverse momentum broadening and the jet quenching parameter, redux, Phys. Rev. D 84 (2011) 065015 [arXiv:1006.1367] [INSPIRE].
S. Caron-Huot, O(g) plasma effects in jet quenching, Phys. Rev. D 79 (2009) 065039 [arXiv:0811.1603] [INSPIRE].
M. Panero, K. Rummukainen and A. Schäfer, Lattice study of the jet quenching parameter, Phys. Rev. Lett. 112 (2014) 162001 [arXiv:1307.5850] [INSPIRE].
G.D. Moore, S. Schlichting, N. Schlusser and I. Soudi, Non-perturbative determination of collisional broadening and medium induced radiation in QCD plasmas, JHEP 10 (2021) 059 [arXiv:2105.01679] [INSPIRE].
Y.-T. Chien, A. Emerman, Z.-B. Kang, G. Ovanesyan and I. Vitev, Jet quenching from QCD evolution, Phys. Rev. D 93 (2016) 074030 [arXiv:1509.02936] [INSPIRE].
E. Bianchi, J. Elledge, A. Kumar, A. Majumder, G.-Y. Qin and C. Shen, The x and Q2 dependence of \( \hat{q} \), quasi-particles and the JET puzzle, arXiv:1702.00481 [INSPIRE].
C. Andrés, N. Armesto, M. Luzum, C.A. Salgado and P. Zurita, Energy versus centrality dependence of the jet quenching parameter \( \hat{q} \) at RHIC and LHC: a new puzzle?, Eur. Phys. J. C 76 (2016) 475 [arXiv:1606.04837] [INSPIRE].
J. Noronha-Hostler, B. Betz, J. Noronha and M. Gyulassy, Event-by-event hydrodynamics + jet energy loss: a solution to the RAA ⨂ v2 puzzle, Phys. Rev. Lett. 116 (2016) 252301 [arXiv:1602.03788] [INSPIRE].
J. Casalderrey-Solana, D.C. Gulhan, J.G. Milhano, D. Pablos and K. Rajagopal, A hybrid strong/weak coupling approach to jet quenching, JHEP 10 (2014) 019 [arXiv:1405.3864] [INSPIRE].
D. Zigic, I. Salom, J. Auvinen, M. Djordjevic and M. Djordjevic, DREENA-B framework: first predictions of RAA and v2 within dynamical energy loss formalism in evolving QCD medium, Phys. Lett. B 791 (2019) 236 [arXiv:1805.04786] [INSPIRE].
C. Andres, N. Armesto, H. Niemi, R. Paatelainen and C.A. Salgado, Jet quenching as a probe of the initial stages in heavy-ion collisions, Phys. Lett. B 803 (2020) 135318 [arXiv:1902.03231] [INSPIRE].
A. Huss, A. Kurkela, A. Mazeliauskas, R. Paatelainen, W. van der Schee and U.A. Wiedemann, Predicting parton energy loss in small collision systems, Phys. Rev. C 103 (2021) 054903 [arXiv:2007.13758] [INSPIRE].
JETSCAPE collaboration, Determining the jet transport coefficient qˆ from inclusive hadron suppression measurements using Bayesian parameter estimation, Phys. Rev. C 104 (2021) 024905 [arXiv:2102.11337] [INSPIRE].
P.B. Arnold, Simple formula for high-energy gluon bremsstrahlung in a finite, expanding medium, Phys. Rev. D 79 (2009) 065025 [arXiv:0808.2767] [INSPIRE].
S. Platzer and M. Sjodahl, The Sudakov veto algorithm reloaded, Eur. Phys. J. Plus 127 (2012) 26 [arXiv:1108.6180] [INSPIRE].
L. Lönnblad, Fooling around with the Sudakov veto algorithm, Eur. Phys. J. C 73 (2013) 2350 [arXiv:1211.7204] [INSPIRE].
R. Kleiss and R. Verheyen, Competing Sudakov veto algorithms, Eur. Phys. J. C 76 (2016) 359 [arXiv:1605.09246] [INSPIRE].
R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics, Cambridge University Press (2011) [INSPIRE].
M. Bahr et al., Herwig++ physics and manual, Eur. Phys. J. C 58 (2008) 639 [arXiv:0803.0883] [INSPIRE].
S. Höche and S. Prestel, The midpoint between dipole and parton showers, Eur. Phys. J. C 75 (2015) 461 [arXiv:1506.05057] [INSPIRE].
Particle Data Group collaboration, Review of particle physics, PTEP 2020 (2020) 083C01 [INSPIRE].
A.H. Mueller and P. Nason, Heavy particle content in QCD jets, Phys. Lett. B 157 (1985) 226 [INSPIRE].
A.H. Mueller and P. Nason, Heavy particle content in QCD jets, Nucl. Phys. B 266 (1986) 265 [INSPIRE].
K. Kastella, G.F. Sterman and J. Milana, Scattering in nuclei and QCD, Phys. Rev. D 39 (1989) 2586 [INSPIRE].
A. Kovner and U.A. Wiedemann, Gluon radiation and parton energy loss, hep-ph/0304151 [INSPIRE].
N. Armesto, C.A. Salgado and U.A. Wiedemann, Medium induced gluon radiation off massive quarks fills the dead cone, Phys. Rev. D 69 (2004) 114003 [hep-ph/0312106] [INSPIRE].
Y.V. Kovchegov and L.D. McLerran, Diffractive structure function in a quasiclassical approximation, Phys. Rev. D 60 (1999) 054025 [hep-ph/9903246] [INSPIRE].
T. Lappi and R. Paatelainen, The one loop gluon emission light cone wave function, Annals Phys. 379 (2017) 34 [arXiv:1611.00497] [INSPIRE].
H.E. Haber, Useful relations among the generators in the defining and adjoint representations of SU(N), SciPost Phys. Lect. Notes 21 (2021) 1 [arXiv:1912.13302] [INSPIRE].
J. Jalilian-Marian and Y.V. Kovchegov, Inclusive two-gluon and valence quark-gluon production in DIS and pA, Phys. Rev. D 70 (2004) 114017 [hep-ph/0405266] [INSPIRE].
L. Apolinário, J.G. Milhano, G.P. Salam and C.A. Salgado, Probing the time structure of the quark-gluon plasma with top quarks, Phys. Rev. Lett. 120 (2018) 232301 [arXiv:1711.03105] [INSPIRE].
L.S. Schulman, Techniques and applications of path integration, Dover Publications (1981) [INSPIRE].
G. Moliere, Theorie der Streuung schneller geladener Teilchen II Mehrfach-und Vielfachstreuung (in German), Z. Naturforsch. A 3 (1948) 78.
F. D’Eramo, K. Rajagopal and Y. Yin, Molière scattering in quark-gluon plasma: finding point-like scatterers in a liquid, JHEP 01 (2019) 172 [arXiv:1808.03250] [INSPIRE].
B.G. Zakharov, On validity of the eikonal approximation for calculation of the probability of ultrarelativistic positronium penetration through matter, Yad. Fiz. 46 (1987) 148 [INSPIRE].
Y.V. Kovchegov, Non-Abelian Weizsacker-Williams field and a two-dimensional effective color charge density for a very large nucleus, Phys. Rev. D 54 (1996) 5463 [hep-ph/9605446] [INSPIRE].
J. Jalilian-Marian, A. Kovner, L.D. McLerran and H. Weigert, The intrinsic glue distribution at very small x, Phys. Rev. D 55 (1997) 5414 [hep-ph/9606337] [INSPIRE].
R. Baier, A. Kovner and U.A. Wiedemann, Saturation and parton level Cronin effect: enhancement versus suppression of gluon production in p-A and A-A collisions, Phys. Rev. D 68 (2003) 054009 [hep-ph/0305265] [INSPIRE].
A. Kurkela and U.A. Wiedemann, Picturing perturbative parton cascades in QCD matter, Phys. Lett. B 740 (2015) 172 [arXiv:1407.0293] [INSPIRE].
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: 2203.11241
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
Attems, M., Brewer, J., Innocenti, G.M. et al. The medium-modified \( g\to c\overline{c} \) splitting function in the BDMPS-Z formalism. J. High Energ. Phys. 2023, 80 (2023). https://doi.org/10.1007/JHEP01(2023)080
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2023)080