Directed flow induced by electromagnetic fields in heavy ion collisions
aa r X i v : . [ nu c l - t h ] S e p Directed flow induced by electromagnetic fields in heavyion collisions
Yifeng Sun, a , ∗ Salvatore Plumari b , a and Vincenzo Greco a , b a Laboratori Nazionali del Sud, INFN-LNS,Via S. Sofia 62, I-95123 Catania, Italy b Department of Physics and Astronomy, University of Catania,Via S. Sofia 64, 1-95125 Catania, Italy
E-mail: [email protected], [email protected],[email protected]
Strong electromagnetic fields are expected to be generated in off-central relativistic heavy ioncollisions, which can induce a splitting of the directed flow of charged particles and anti-particles( ∆ v ). Such a splitting manifests even for neutral charmed mesons pairs ( D , D ), hence being adirect probe of the formation of deconfined phase with charm quarks as degree of freedom.In the limit of large p T and weak interaction with the QGP, a formula of ∆ v ( p T , y z ) of chargedparticles and anti-particles as a function of p T and rapidity y z can be obtained, which is found tobe related to the spectra of charged particles and the integrated effect of the Lorentz force. Thisformula is expected to be valid to heavy quarks and leptons at high p T , where the modificationto their equations of motion due to the interaction with both QGP and electromagnetic fields issmall, and should have a general application. We also proposed a measurement of ∆ v ( p T , y z ) ofleptons from Z decay and its correlation to that of D mesons, which would be a strong probedetermining whether the large splitting measured in experiments has the electromagnetic origin. HardProbes20201-6 June 2020Austin, Texas ∗ Speaker © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ irected flow induced by electromagnetic fields in heavy ion collisions
Yifeng Sun
1. Introduction
A very strong magnetic field is generated in off-central relativistic heavy ion collisions, whichtriggers in the last decades intense studies of physics related to the parity and charge-parity symmetrybreaking process of QCD [1, 2]. On the other hand, there are other efforts concentrating on thedirect probe of electromagnetic fields, where the directed flow relative to the reaction plane is oneof the promising probes [3–6]. Besides these numerical studies, we tried to find a general signatureof the directed flow induced by e.m. fields in our previous work [7], and in this proceeding we willspecify more details of this work and try to convey the robustness of our conclusion. The signatureshould be applicable to heavy quarks and leptons, where the interaction with quark-gluon plasma(QGP) is not strong enough to dirty our conclusion.
2. Collective flows induced by electromagnetic fields in the limit of large p T and nointeraction with QGP We adopt the common convention on the configuration of off-central heavy ion collisions,with the center of nuclei moving in positive z direction located in positive x axis, which generatesan electromagnetic field with a huge magnetic component along the negative y direction. In thedeconfined phase, charged quarks can interact with such electromagnetic field, but also with theQGP. Though the interaction between the charged quarks with the QGP is strong, it should be muchweaker for heavy quarks ( c , b ) compared to light quarks ( u , d , s ) [8], and it becomes further weakenfor high p T heavy quarks as well as leptons of all p T . To derive a compacted formula of the directedflow v induced by electromagnetic fields, we thus first neglect the effect of the interaction withQGP, and the modification of it will be discussed in the next section.In general, electromagnetic fields move charged particles from initial momentum p i to the finalone p f = p i + ∆ , where ∆ is a distribution of the shift in the momenta of particles. In the evaluationof collective flows of charged particles relative to the reaction plane, the relevance is the shift intransverse momentum, where we can use a transition function T ( ∆ x , ∆ y , p x , p y , y z ) to represent alleffects of e.m. fields that shift one particle’s initial p x and p y at rapidity y z . Thus the distributionfunction of charged particles after the effect due to e.m. fields becomes: dNdp x dp y d y z = ∫ d ∆ x d ∆ y dp ix dp iy δ ( p x − p ix − ∆ x ) δ ( p y − p iy − ∆ y ) f ( p ix , p iy , y z ) T ( ∆ x , ∆ y , p ix , p iy , y z ) = ∫ d ∆ x d ∆ y f ( p x − ∆ x , p y − ∆ y , y z ) T ( ∆ x , ∆ y , p x − ∆ x , p y − ∆ y , y z )≈ ∫ d ∆ x d ∆ y [ f ( p x , p y , y z ) T ( ∆ x , ∆ y , p x , p y , y z ) − ∂ f T ∂ p x ∆ x − ∂ f T ∂ p y ∆ y ] = f − ( ∂ f ∆ x ∂ p x + ∂ f ∆ y ∂ p y ) , (1)where f is the initial spectra of charged particles and depends only on p T in boost-invariantapproximation, and ∆ x ( p x , p y , y z ) and ∆ y ( p x , p y , y z ) are the average shifts in the transverse momentaof a particle with initial p x , p y and y z . The approximation in Eq. (1) is valid when the shifts (muchsmaller than 1 GeV in most configurations of e.m. fields) are small compared to initial momenta.2 irected flow induced by electromagnetic fields in heavy ion collisions Yifeng SunOne can adopt a similar idea in the study of collective flows and express ∆ x and ∆ y as afunction of azimuthal angle φ in momentum space. Because the colliding system is symmetric with y ↔ − y , in momentum space one can find ∆ x ( p T , φ, y z ) = ∆ x ( p T , π − φ, y z ) and − ∆ y ( p T , φ, y z ) = ∆ y ( p T , π − φ, y z ) , and so we have: ∆ x ( p T , φ, y z ) = Õ n ( p T , y z ) cos ( n φ ) , (2) ∆ y ( p T , φ, y z ) = Õ n ( p T , y z ) sin ( n φ ) . (3)Because B z is absent even in the conducting medium [9, 10] due to colliding nuclei movingonly along z directions, one can have ∆ x = ∫ dtq ( E x − v z B y ) and ∆ y = ∫ dtq ( E y + v z B x ) with v z = tanhy z . So a n and b n depend weakly on p T , and the azimuthal angle distributions of ∆ x and ∆ y in momentum space measure directly the azimuthal angle distributions of E and B in coordinatespace. Eq. (1) can thus be further simplified to: f ′ = f − ( ∂ f∆ x ∂ p T cos ( φ ) − ∂ f∆ x ∂φ sin ( φ ) p T + ∂ f∆ y ∂ p T sin ( φ ) + ∂ f∆ y ∂φ cos ( φ ) p T ) = f − [ a + b p T f + ∂ f ∂ p T ( a + b )] − ∂ f ∂ p T ( a + b ) cos φ − Õ n = [ ∂ f ∂ p T ( a n + + b n + + a n − − b n − ) + ( n + )( a n + + b n + ) − ( n − )( a n − − b n − ) p T f ] cosn φ, (4)where we can find all collective flow coefficients from it.Eq. (4) shows that the distribution is symmetric with y ↔ − y . On the other hand, in themedium with non-zero chiral magnetic conductivity, ∆ x ( ∆ y ) is not symmetric (anti-symmetric)with y ↔ − y , because B y ( B x ) is not symmetric (anti-symmetric) with y ↔ − y and B z becomesnon-zero [10, 11]. This means that Eq. (4) should have sin ( n φ ) terms in principle. The disagreementbetween these two is that the emergence of chiral magnetic conductivity breaks parity ( y ↔ − y )symmetry [1, 2].
3. The splitting of the directed flow of charged particles
Using Eq. (4), one can obtain the directed flow in the limit of large p T [7]: v ( p T , y z ) = ∫ d φ f ′ cos φ ∫ d φ f ′ = − ∂ f ∂ p T ( a + a + b ) + a + b p T ff − a + b p T f − ∂ f ∂ p T ( a + b ) ≈ − ∂ lnf ∂ p T a ( p T , y z ) , (5)where the approximation is obtained by keeping the effect of e.m. fields in the leading order andby knowing the smallness of the quadrupole moments of e.m. fields in the overlapping region ofcolliding nuclei.It seems to be surprising that v depends on ∂ lnf ∂ p T at first sight. However, one can understandit better by studying charged particles with a spectra uniformly distributed in p T . If Lorentz forcemoves the momenta of all charged particles along positive x axis with a constant shift, one caneasily find that the distribution is exactly same as the initial one after the shift unless one looks atthe boundary of the final spectra, which means that v is still zero after this shift.3 irected flow induced by electromagnetic fields in heavy ion collisions Yifeng SunThe calculation of a ( p T , y z ) is complicated due to the complex time and spatial distributionsof E x and B y . However, we can find a compact formula by assuming eB y ( x , y , t , z ) = − B ( τ ) ρ B ( x , y ) with ρ B ( x , y ) = exp [− x σ − y σ ] . In this case a ( p T , y z ) can be evaluted from our study [7], andfinally ∆ v of positively and negatively charged particles becomes: ∆ v ( p T , y z ) ∝ −| q | ∂ lnf ∂ p T ∫ y z d χ cosh χ [ τ B ( τ ) − τ B ( τ )] , (6)with τ = τ coshy z cosh χ and τ = ( τ + Rm T / p T ) coshy z cosh χ . τ , can be treated as charged particle’s formationtime and the escape time out of the electromagnetic field, and is independent of p T at p T ≫ m .In fact, a ( p T , y z ) should be independent of p T as long as p T ≫ m for any configurations of e.m.fields, because the trajectory of particles and the Lorentz force experienced by them are same dueto the fact that the velocity of particles is independent of p T in this limit.Though Eq. (6) is obtained with a very strong assumption of e.m. fields, it may be robustat small y z , because the azimuthal angle distribution of e.m. fields is not so relevant based onour analysis of the former section, and E x and B y vary little in the overlapping region and atsmall space-time rapidity. Eq. (6) thus provides a general scaling of the d ∆ v / d y z ( p T ) of chargedparticles.In Ref [7], we studied ∆ v of charged leptons l + − l − from Z decay and D − D using thee.m. field that can reproduce the large ∆ v of D − D measured in experiments at 5.02 TeV Pb+Pbcollisions [12], and found that ∆ v ( l ) has a jump around 45 GeV / c , and its magnitude is alwayssmaller than ∆ v ( D ) . It can be understood by Eq. (6) from the peculiar spectra of leptons from thedecay of Z , and this surprising finding can be used to determine whether the measurement of ∆ v has the e.m. field origin.
4. The effect of the interaction with QGP
Heavy quarks can interact with QGP and the effect of e.m. fields may be weaken by that. Wethus adopted the standard Fokker-Planck equation with the interaction strength tuned to successfullyreproduce the nuclear modification factor R AA and elliptic flow v of D mesons [8, 13] at both RHICand LHC energies, and studied this effect numerically.We have picked up two typical configurations of e.m. fields: EM1 is the e.m. field thatcan reproduce experimental measurements of d ∆ v D / d η [12]; EM2 is the one calculated using aconstant medium conductivity in the lower limit of LQCD calculations [3–6] that is adopted by a lotof studies but with a discontinuity in the time evolution of e.m. fields. Case EM1 leads to a positive d ∆ v D / d η with its magnitude order two larger than case EM2, because the difference between τ B at the freeze-out time and the formation time of charm quarks is positive and larger for EM1. Forthe effect of the interaction with QGP, it is seen in Fig. 1 that the interaction damps d ∆ v D / d η onlyat p T < p T for these two significantly different e.m.fields. It is also seen in Fig. 1 that at high p T , d ∆ v D / d η shares a similar p T dependence for thesetwo e.m. fields even though the huge gap between their magnitude. This is because the variation of d ∆ v D / d η with p T mostly comes from ∂ lnf ∂ p T (see Eq. (6)).4 irected flow induced by electromagnetic fields in heavy ion collisions Yifeng Sun p T (GeV/c) EM1+QGP EM1 EM2+QGP ( 90) EM2 ( 90)
Pb+Pb @ 5.02 TeV, b=7.5 fm d v / d Figure 1: (Color online) p T dependence of d ∆ v D / d η of D and D for two configurations of e.m. fieldsplus with or without the interaction with QGP.
5. Conclusions
We have obtained a compact formula for the ∆ v of charged particles induced by electromagneticfields, which is shown to be applied to heavy quarks and leptons at high p T . According to our stepby step analysis, we think this formula is robust and can be used to test whether the experimentalmeasurements of ∆ v have an e.m. origin. References [1] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A (2008), 227-253[2] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D (2008), 074033[3] U. Gursoy, D. Kharzeev and K. Rajagopal, Phys. Rev. C (2014) no.5, 054905[4] S. K. Das, S. Plumari, S. Chatterjee, J. Alam and et al., Phys. Lett. B (2017), 260-264[5] U. Gursoy, D. Kharzeev, E. Marcus and et al., Phys. Rev. C (2018) no.5, 055201[6] S. Chatterjee and P. Bozek, Phys. Lett. B (2019), 134955[7] Y. Sun, S. Plumari and V. Greco, arXiv:2004.09880 [nucl-th][8] X. Dong and V. Greco, Prog. Part. Nucl. Phys. (2019), 97-141[9] K. Tuchin, Phys. Rev. C (2013) no.2, 024911[10] H. Li, X. l. Sheng and Q. Wang, Phys. Rev. C (2016) no.4, 044903[11] G. Inghirami, M. Mace, Y. Hirono and et al., Eur. Phys. J. C (2020) no.3, 293[12] S. Acharya et al. [ALICE], Phys. Rev. Lett. (2020) no.2, 022301[13] Y. Sun, G. Coci, S. K. Das and et al., Phys. Lett. B798