Magnetohydrodynamics and charged currents in heavy ion collisions
NNuclear Physics A 00 (2018) 1–4
NuclearPhysics A
Magnetohydrodynamics and charged currents in heavy ion collisions
Umut G ¨ursoy a , Dmitri Kharzeev b , Krishna Rajagopal c a Institute for Theoretical Physics, Utrecht University Leuvenlaan 4, 3584 CE Utrecht, The Netherlands b Department of Physics and Astronomy, Stony Brook University, New York 11794, USADepartment of Physics, Brookhaven National Laboratory, Upton, New York 11973, USA c Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139.
Abstract
The hot QCD matter produced in any heavy ion collision with a nonzero impact parameter is produced within a strong magneticfield. We study the imprint the magnetic fields produced in non-central heavy ion collisions leave on the azimuthal distributionsand correlations of the produced charged hadrons. The magnetic field is time-dependent and the medium is expanding, whichleads to the induction of charged currents due to the combination of Faraday and Hall e ff ects. We find that these currents resultin a charge-dependent directed flow v that is odd in rapidity and odd under charge exchange. It can be detected by measuringcorrelations between the directed flow of charged hadrons at di ff erent rapidities, (cid:104) v ± ( y ) v ± ( y ) (cid:105) . Keywords:
Magnetohydrodynamics, Quark Gluon Plasma, Heavy Ion Collisions
1. Introduction
Strong magnetic fields (cid:126) B are produced in all non-central heavy ion collisions (i.e. those with nonzero impactparameter b ) by the charged “spectators” (i.e. the nucleons from the incident nuclei that “miss”, flying past each otherrather than colliding). Indeed, estimates obtained via application of the Biot-Savart law to heavy ion collisions with b = e | (cid:126) B | / m π ≈ / c after a RHIC collision with √ s =
200 AGeV and e | (cid:126) B | / m π ≈ √ s = .
76 ATeV [1, 2, 3, 4, 5, 6, 7]. In recent years there has beenmuch interest in consequences of these enormous magnetic fields present early in the collision that are observable inthe final state hadrons produced by the collision, see for example, [1, 8, 9, 10].In Ref. [11] we analyze what are surely the simplest and most direct e ff ects of magnetic fields in heavy ioncollisions, and quite likely also their largest e ff ects, namely the induction of electric currents carried by the chargedquarks and antiquarks in the quark-gluon plasma (QGP) and, later, by the charged hadrons. The source of thesecharged currents is twofold. Firstly, the magnitude of (cid:126) B varies in time, decreasing as the charged spectators fly awayalong the beam direction, receding from the QGP produced in the collision. The changing (cid:126) B results in an electricfield due to Faraday’s law, and this in turn produces an electric current in the conducting medium. Secondly, becausethe conducting medium, i.e. the QGP, has a significant initial longitudinal expansion velocity (cid:126) u parallel to the beamdirection and therefore perpendicular to (cid:126) B , the Lorentz force results in an electric current perpendicular to both thevelocity and (cid:126) B , akin to the classical Hall e ff ect. Fig. 1 serves to orient the reader as to the directions of (cid:126) B and (cid:126) u ,and the electric currents induced by the Faraday and Hall e ff ects. The net electric current is the sum of that due toFaraday and that due to Hall. If the Faraday e ff ect is stronger than the Hall e ff ect, that current will result in directedflow of positively charged particles in the directions shown in Fig. 1 and directed flow of negatively charged particlesin the opposite direction. Our goal in Ref. [11] is to make an estimate of the order of magnitude of the resultingcharge-dependent v in the final state pions. We make various simplifying assumptions, explained below.1 a r X i v : . [ h e p - ph ] A ug ¨ursoy et al. / Nuclear Physics A 00 (2018) 1–4 Bz x η <0, v <0 η >0, v <0 η >0, v >0 η <0, v >0 J Hall J Faraday J Faraday J Hall u uuu
Total Directed Flow Total Directed Flow y
Figure 1. Schematic illustration of how the magnetic field (cid:126) B in a heavy ion collision results in a directed flow, v , of electric charge. The collisionoccurs in the z -direction, meaning that the longitudinal expansion velocity (cid:126) u of the conducting QGP that is produced in the collision points in the + z ( − z ) direction at positive (negative) z . We take the impact parameter vector to point in the x direction, choosing the nucleus moving towardpositive (negative) z to be located at negative (positive) x , which is to say taking the magnetic field (cid:126) B to point in the + y direction. The directionof the electric currents due to the Faraday and Hall e ff ects is shown, as is the direction of the directed flow of positive charge (dashed) in the casewhere the Faraday e ff ect is on balance stronger than the Hall e ff ect. In order to obtain the velocity (cid:126) v associated with the charged currents due to the electromagnetic field, we firstcalculate the magnetic and electric fields themselves, (cid:126) B and (cid:126) E , by solving Maxwell’s equations in the center-of-massframe (the frame illustrated in Fig. 1). The electromagnetic field produced by a single point-like charge moving with avelocity (cid:126)β in a medium with constant (our first simplifying assumption) conductivity σ can be calculated analytically[11]. The total field is obtained by integrating this over the entire distribution of all the protons in the two collidingnuclei. We also make the simplifying assumption that the protons in a nucleus are uniformly distributed within asphere of radius R , with the centers of the spheres located at x = ± b / y = + z and − z directions. For the participants we use the empirical distribution [1, 12]. We find [11] that, as other authors haveshown previously [2, 3, 4, 5, 6, 7], the presence of the conducting medium delays the decrease in the magnetic field.To model the expanding medium we use the analytic solution to relativistic viscous hydrodynamics for a conformalfluid with the shear viscosity to entropy density ratio given by η/ s = / (4 π ) found by Gubser in 2010 [13]. Thesolution describes a finite size plasma produced in a central collision that is obtained from conformal hydrodynamicsby demanding boost invariance along the beam (i.e. z ) direction, rotational invariance around z , and two specialconformal invariances perpendicular to z . As demonstrated in [11], we can choose parameters such that Gubser’ssolution yields a reasonable facsimile of the pion and proton transverse momentum spectra observed in RHIC andLHC collisions with 20 −
30% centrality, corresponding to collisions with a mean impact parameter between 7 and8 fm, see e.g. [14, 15]. We denote the velocity of Gubser’s solution as (cid:126) u .Given the electromagnetic field and the velocity of the medium (cid:126) u in the center-of-mass frame, we then determinethe total velocity (cid:126) V of the charged particles (u and d quarks) as follows. Making the assumption | (cid:126) V − (cid:126) u | / | (cid:126) u | (cid:28) (cid:126) u (cid:48) = (cid:126) E (cid:48) and (cid:126) B (cid:48) are non-vanishing. We then solve the equation of motion for a charged fluid element with mass m in this frame, using theLorentz force law and requiring stationary currents: m d (cid:126) v (cid:48) dt = q (cid:126) v (cid:48) × (cid:126) B (cid:48) + q (cid:126) E (cid:48) − µ m (cid:126) v (cid:48) = , (1)2 ¨ursoy et al. / Nuclear Physics A 00 (2018) 1–4 - - - Y - - v - - - Y - - v Figure 2. Directed flow v for positively charged pions (solid curves) and negatively charged pions (dashed curves) in our calculation withparameters chosen to give a reasonable facsimile of 20-30% centrality heavy ion collisions at the LHC (left figure) and at RHIC (right figure) at p T = .
25 (green), 0.5 (blue) and 1 GeV (red). Here we are only plotting the charge-dependent contribution to the directed flow v that originatesfrom the presence of the magnetic field in the collision and that is caused by the Faraday and Hall e ff ects. This charge-dependent contribution to v must be added to the, presumably larger, charge-independent v . where the last term describes the drag force on a fluid element with mass m on which some external (in this caseelectromagnetic) force is being exerted, with µ being the drag coe ffi cient. The nonrelativistic form of (1) is justifiedby the aforementioned assumption. For the purpose of our order-of-magnitude estimate, we use the N = µ m = . T , for a ’t Hooft coupling λ = π and T =
255 MeV. Finally, we boost back to theoriginal center-of-mass frame to obtain the total velocity (cid:126) V .The theoretical estimations we make here are based on the basic assumption that the electromagnetic interactionscan be treated classically. We checked this by comparing the total magnetic energy in the medium to the energy of asingle photon with wavelength comparable to the size of the medium and showing that the former is larger roughly bya factor that varies from ∼ ∼
50 as τ increases from 0.3 fm to 0.8 fm.
2. Results
We apply the standard prescription to obtain the hadron spectra from a hydrodynamic flow, that is here givenby Gubser’s solution, assuming sudden freezeout when the fluid cools to a specified freezeout temperature T f , wasdeveloped by Cooper and Frye [19]. We shall take T f =
130 MeV for heavy ion collisions at both the LHC and RHIC.The hadron spectrum for particles of species i with mass m i will depend on transverse momentum p T , momentumspace rapidity Y and the azimuthal angle in momentum space φ p . To establish notation, note that the dependence ofthe hadron spectrum on φ p can be expanded as S i ≡ p d N i d p = d N i p T dYd p T d φ p = v (cid:16) + v cos( φ p − π ) + v cos 2 φ p + · · · (cid:17) , (2)where in general the v n will depend on Y and p T .Once we obtained the electromagnetic field, fixed the parameters of the hydrodynamic flow and calculated the totalvelocity V ± µ as explained in the previous section, we can finally use the freezeout procedure to calculate the hadronspectra, including electromagnetic e ff ects. Figure 2 shows v for positively and negatively charged pions as a functionof momentum-space rapidity Y at transverse momenta p T = .
5, 1, and 2 GeV. We have chosen the initial magneticfield created by the spectators with beam rapidity ± Y = ± ± Y = ± . σ = .
023 fm − and the drag parameter µ m in (1) as above and we have set thefreezeout temperature to T f =
130 MeV. We see in Fig. 1 that if the current induced by Faraday’s law is greater thanthat induced by the Hall e ff ect, we expect v > Y > Y < v < Y > Y <
0. Comparing to Fig. 2, we observe thatthis is indeed the pattern for pions with p T = ff ects, the e ff ect of Faraday on pions with p T = ff ect of Hall. However, the e ff ects of Hall3 ¨ursoy et al. / Nuclear Physics A 00 (2018) 1–4 and Faraday on pions with smaller p T and small Y are comparable in magnitude, for example with the Hall e ff ect justlarger for p T = .
25 and | Y | < .
2, resulting in a reversal in the sign of v in this kinematic range at LHC. We observethat the Faraday e ff ect is dominant for pions at RHIC even for p T as low as 0.25 GeV. Directed flow for the protonsand anti-protons at the LHC and RHIC can be calculated in a similar fashion and the result can be found in [11].
3. Observables, and a look ahead
Our estimates of the magnitude of the charge-dependent directed flow of pions in heavy ion collisions at the LHCand RHIC, and their dependence on Y and p T , can be found in Fig. 2. The e ff ect is small. What makes it distinctiveis that it is opposite in sign for positively and negatively charged particles of the same mass, and that for any speciesit is odd in rapidity. Detecting the e ff ect directly by measuring the directed flow of positively and negatively chargedparticles, which we shall denote by v + and v − , is possible in principle but is likely to be prohibitively di ffi cult in practice[11]. Instead, It would be advantageous to define correlation observables that, first of all, involve taking ensembleaverages of suitably chosen di ff erences rather than just of v + or v − and that, second of all, do not require knowledgeof the direction of the magnetic field. To isolate the charge-dependent directed flow that we are after, it is helpful todefine the asymmetries between the directed flows for positive and negative hadrons A i j ( Y , Y ) ≡ v i ( Y ) − v j ( Y ),where i , j are + or -. Even if the direction of the magnetic field is not reconstructed, one can still study the correlationfunctions defined by C i j , kl ( Y , Y ) ≡ (cid:104) A i j ( Y , Y ) A kl ( Y , Y ) (cid:105) . (3)These correlation functions are quadratic in the directed flow, and so are not sensitive to the direction of (cid:126) B and thesign of v in a given event. However, they still carry the requisite information about dynamical charge-dependentcorrelations induced by the magnetic field. Analogous correlations functions have been measured with high preci-sion [20, 21].The challenge to experimentalists is to measure these correlators, or others that are also defined so as to separatethe desired e ff ects from charge-independent backgrounds. If this is possible, one may use comparisons between dataand the nontrivial p T - and Y -dependence of results like those that we have obtained in Fig. 2 to extract a wealth ofinformation, for example about the strength of the initial magnetic field and about the magnitude of the electricalconductivity of the plasma. Acknowledgements.
We are grateful to Sergei Voloshin for helpful suggestions. This work was supported byDOE grants de-sc0011090, DE-FG-88ER40388 and DE-AC02- 98CH10886 and is a part of the D-ITP consortium.
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