# Time-dependent observables in heavy ion collisions. Part I. Setting up the formalism

- 49 Downloads

## Abstract

We adapt the Schwinger-Keldysh formalism to study heavy-ion collisions in perturbative QCD. Employing the formalism, we calculate the two-point gluon correlation function *G* _{22} ^{ aμ,
bν} due to the lowest-order classical gluon fields in the McLerran-Venugopalan model of heavy ion collisions and observe an interesting transition from the classical fields to the quasi-particle picture at later times. Motivated by this observation, we push the formalism to higher orders in the coupling and calculate the contribution to *G* _{22} ^{ aμ,
bν} coming from the diagrams representing a single rescattering between two of the produced gluons. We assume that the two gluons go on mass shell both before and after the rescattering. The result of our calculation depends on which region of integration over the proper time of the rescattering *τ*_{Z} gives the correct correlation function at late proper time *τ* when the gluon distribution is measured. For (i) *τ*_{Z} ≫ 1*/Q*_{s} and *τ* − *τ*_{Z} ≫ 1*/Q*_{s} (with *Q*_{s} the saturation scale) we obtain the same results as from the Boltzmann equation. For (ii) *τ* − *τ*_{Z} ≫ *τ*_{Z} ≫ 1*/Q*_{s} we end up with a result very different from kinetic theory and consistent with a picture of “free-streaming” particles. Due to the approximations made, our calculation is too coarse to indicate whether the region (i) or (ii) is the correct one: to resolve this controversy, we shall present a detailed diagrammatic calculation of the rescattering correction in the *φ*^{4} theory in the second paper of this duplex.

## Keywords

Perturbative QCD Quark-Gluon Plasma## Notes

### **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.

## References

- [1]U. Heinz and R. Snellings,
*Collective flow and viscosity in relativistic heavy-ion collisions*,*Ann. Rev. Nucl. Part. Sci.***63**(2013) 123 [arXiv:1301.2826] [INSPIRE].ADSCrossRefGoogle Scholar - [2]S. Jeon and L.G. Yaffe,
*From quantum field theory to hydrodynamics: transport coefficients and effective kinetic theory*,*Phys. Rev.***D 53**(1996) 5799 [hep-ph/9512263] [INSPIRE]. - [3]F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan,
*The color glass condensate*,*Ann. Rev. Nucl. Part. Sci.***60**(2010) 463 [arXiv:1002.0333].ADSCrossRefGoogle Scholar - [4]Y.V. Kovchegov and E. Levin,
*Quantum chromodynamics at high energy*, Cambridge University Press, Cambridge, U.K. (2012).CrossRefzbMATHGoogle Scholar - [5]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]. - [6]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]. - [7]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]. - [8]Y.V. Kovchegov,
*NonAbelian Weizsäcker-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]. - [9]A. Ayala, J. Jalilian-Marian, L.D. McLerran and R. Venugopalan,
*Quantum corrections to the Weizsacker-Williams gluon distribution function at small x*,*Phys. Rev.***D 53**(1996) 458 [hep-ph/9508302] [INSPIRE]. - [10]Y.V. Kovchegov and D.H. Rischke,
*Classical gluon radiation in ultrarelativistic nucleus-nucleus collisions*,*Phys. Rev.***C 56**(1997) 1084 [hep-ph/9704201] [INSPIRE]. - [11]A. Krasnitz and R. Venugopalan,
*Nonperturbative computation of gluon minijet production in nuclear collisions at very high-energies*,*Nucl. Phys.***B 557**(1999) 237 [hep-ph/9809433] [INSPIRE]. - [12]A. Krasnitz and R. Venugopalan,
*The initial energy density of gluons produced in very high-energy nuclear collisions*,*Phys. Rev. Lett.***84**(2000) 4309 [hep-ph/9909203] [INSPIRE]. - [13]A. Krasnitz, Y. Nara and R. Venugopalan,
*Probing a color glass condensate in high energy heavy ion collisions*,*Braz. J. Phys.***33**(2003) 223.ADSCrossRefGoogle Scholar - [14]T. Lappi,
*Production of gluons in the classical field model for heavy ion collisions*,*Phys. Rev.***C 67**(2003) 054903 [hep-ph/0303076] [INSPIRE]. - [15]Y.V. Kovchegov,
*Can thermalization in heavy ion collisions be described by QCD diagrams?*,*Nucl. Phys.***A 762**(2005) 298 [hep-ph/0503038] [INSPIRE]. - [16]J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan,
*Universal attractor in a highly occupied non-Abelian plasma*,*Phys. Rev.***D 89**(2014) 114007 [arXiv:1311.3005] [INSPIRE].ADSGoogle Scholar - [17]T. Epelbaum and F. Gelis,
*Pressure isotropization in high energy heavy ion collisions*,*Phys. Rev. Lett.***111**(2013) 232301 [arXiv:1307.2214] [INSPIRE].ADSCrossRefGoogle Scholar - [18]T. Epelbaum and F. Gelis,
*Fluctuations of the initial color fields in high energy heavy ion collisions*,*Phys. Rev.***D 88**(2013) 085015 [arXiv:1307.1765] [INSPIRE].ADSGoogle Scholar - [19]J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan,
*Basin of attraction for turbulent thermalization and the range of validity of classical-statistical simulations*,*JHEP***05**(2014) 054 [arXiv:1312.5216] [INSPIRE].ADSCrossRefGoogle Scholar - [20]T. Epelbaum, F. Gelis and B. Wu,
*Nonrenormalizability of the classical statistical approximation*,*Phys. Rev.***D 90**(2014) 065029 [arXiv:1402.0115] [INSPIRE].ADSGoogle Scholar - [21]T. Epelbaum, F. Gelis, N. Tanji and B. Wu,
*Properties of the Boltzmann equation in the classical approximation*,*Phys. Rev.***D 90**(2014) 125032 [arXiv:1409.0701] [INSPIRE].ADSGoogle Scholar - [22]L. Kadanoff and G. Baym,
*Quantum Statistical Mechanics*, W.A. Benjamin Inc., New York U.S.A. (1962).Google Scholar - [23]K.-c. Chou, Z.-b. Su, B.-l. Hao and L. Yu,
*Equilibrium and nonequilibrium formalisms made unified*,*Phys. Rept.***118**(1985) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [24]E. Calzetta and B.L. Hu,
*Nonequilibrium quantum fields: closed time path effective action, Wigner function and Boltzmann equation*,*Phys. Rev.***D 37**(1988) 2878 [INSPIRE].ADSMathSciNetGoogle Scholar - [25]J.-P. Blaizot and E. Iancu,
*The quark gluon plasma: collective dynamics and hard thermal loops*,*Phys. Rept.***359**(2002) 355 [hep-ph/0101103] [INSPIRE]. - [26]P.B. Arnold, G.D. Moore and L.G. Yaffe,
*Effective kinetic theory for high temperature gauge theories*,*JHEP***01**(2003) 030 [hep-ph/0209353] [INSPIRE]. - [27]R. Baier, A.H. Mueller, D. Schiff and D.T. Son,
*‘Bottom up’ thermalization in heavy ion collisions*,*Phys. Lett.***B 502**(2001) 51 [hep-ph/0009237] [INSPIRE]. - [28]A. Kurkela and Y. Zhu,
*Isotropization and hydrodynamization in weakly coupled heavy-ion collisions*,*Phys. Rev. Lett.***115**(2015) 182301 [arXiv:1506.06647] [INSPIRE].ADSCrossRefGoogle Scholar - [29]A.H. Mueller and D.T. Son,
*On the equivalence between the Boltzmann equation and classical field theory at large occupation numbers*,*Phys. Lett.***B 582**(2004) 279 [hep-ph/0212198] [INSPIRE]. - [30]J.S. Schwinger,
*Brownian motion of a quantum oscillator*,*J. Math. Phys.***2**(1961) 407 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [31]L.V. Keldysh,
*Diagram technique for nonequilibrium processes*,*Zh. Eksp. Teor. Fiz.***47**(1964) 1515 [INSPIRE].Google Scholar - [32]N.P. Landsman and C.G. van Weert,
*Real and imaginary time field theory at finite temperature and density*,*Phys. Rept.***145**(1987) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [33]M.L. Bellac,
*Thermal field theory*, Cambridge University Press, Cambridge U.K. (2011).Google Scholar - [34]J.M. Cornwall, R. Jackiw and E. Tomboulis,
*Effective action for composite operators*,*Phys. Rev.***D 10**(1974) 2428 [INSPIRE].ADSzbMATHGoogle Scholar - [35]J. Berges,
*N-particle irreducible effective action techniques for gauge theories*,*Phys. Rev.***D 70**(2004) 105010 [hep-ph/0401172] [INSPIRE]. - [36]J. Berges,
*Introduction to nonequilibrium quantum field theory*,*AIP Conf. Proc.***739**(2005) 3 [hep-ph/0409233] [INSPIRE]. - [37]M.E. Carrington, G. Kunstatter and H. Zaraket,
*2PI effective action and gauge invariance problems*,*Eur. Phys. J.***C 42**(2005) 253 [hep-ph/0309084] [INSPIRE]. - [38]F. Gelis, T. Lappi and R. Venugopalan,
*High energy factorization in nucleus-nucleus collisions*,*Phys. Rev.***D 78**(2008) 054019 [arXiv:0804.2630] [INSPIRE].ADSGoogle Scholar - [39]F. Gelis, T. Lappi and R. Venugopalan,
*High energy factorization in nucleus-nucleus collisions. II. Multigluon correlations*,*Phys. Rev.***D 78**(2008) 054020 [arXiv:0807.1306] [INSPIRE]. - [40]S. Jeon,
*Color glass condensate in Schwinger-Keldysh QCD*,*Annals Phys.***340**(2014) 119 [arXiv:1308.0263] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [41]Y.V. Kovchegov and B. Wu,
*Time-dependent observables in heavy ion collisions II: in search of pressure isotropization in the φ*^{4}*theory*, in preparation (2017).Google Scholar - [42]A.H. Mueller,
*Small x behavior and parton saturation: a QCD model*,*Nucl. Phys.***B 335**(1990) 115 [INSPIRE].ADSCrossRefGoogle Scholar - [43]Y.V. Kovchegov,
*Quantum structure of the non-Abelian Weizsacker-Williams field for a very large nucleus*,*Phys. Rev.***D 55**(1997) 5445 [hep-ph/9701229] [INSPIRE]. - [44]A.J. Niemi and G.W. Semenoff,
*Finite temperature quantum field theory in Minkowski space*,*Annals Phys.***152**(1984) 105 [INSPIRE].ADSCrossRefGoogle Scholar - [45]M.E. Peskin and D.V. Schroeder,
*An Introduction to quantum field theory*, Addison-Wesley, Reading U.S.A. (1995).Google Scholar - [46]Y.V. Kovchegov and M.D. Sievert,
*Sivers function in the quasiclassical approximation*,*Phys. Rev.***D 89**(2014) 054035 [arXiv:1310.5028] [INSPIRE].ADSGoogle Scholar - [47]Y.V. Kovchegov and M.D. Sievert,
*Calculating TMDs of a large nucleus: quasi-classical approximation and quantum evolution*,*Nucl. Phys.***B 903**(2016) 164 [arXiv:1505.01176] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [48]
- [49]K. Fukushima,
*Initial fields and instability in the classical model of the heavy-ion collision*,*Phys. Rev.***C 76**(2007) 021902 [*Erratum ibid.***C 77**(2007) 029901] [arXiv:0711.2634] [INSPIRE]. - [50]A.H. Mueller,
*The Boltzmann equation for gluons at early times after a heavy ion collision*,*Phys. Lett.***B 475**(2000) 220 [hep-ph/9909388] [INSPIRE]. - [51]J.D. Bjorken,
*Highly relativistic nucleus-nucleus collisions: the central rapidity region*,*Phys. Rev.***D 27**(1983) 140 [INSPIRE].ADSGoogle Scholar - [52]Y.V. Kovchegov and H. Weigert,
*Collinear singularities and running coupling corrections to gluon production in CGC*,*Nucl. Phys.***A 807**(2008) 158 [arXiv:0712.3732] [INSPIRE].ADSCrossRefGoogle Scholar - [53]V. Mathieu, A.H. Mueller and D.N. Triantafyllopoulos,
*The Boltzmann equation in classical Yang-Mills theory*,*Eur. Phys. J.***C 74**(2014) 2873 [arXiv:1403.1184] [INSPIRE].ADSCrossRefGoogle Scholar - [54]T. Epelbaum, F. Gelis and B. Wu,
*Lattice worldline representation of correlators in a background field*,*JHEP***06**(2015) 148 [arXiv:1503.05333] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [55]T. Epelbaum, F. Gelis and B. Wu,
*From lattice quantum electrodynamics to the distribution of the algebraic areas enclosed by random walks on Z*^{2}, arXiv:1504.00314 [INSPIRE].