aa r X i v : . [ nu c l - t h ] A ug FLUCTUATIONS OF ELECTROMAGNETIC FIELDS IN HEAVY IONCOLLISIONS
B.G. Zakharov L.D. Landau Institute for Theoretical Physics, GSP-1, 117940, Kosygina Str. 2, 117334 Moscow, Russia
We perform quantum calculations of fluctuations of the electromagnetic fields in AA collisions atRHIC and LHC energies. We find that in the quantum picture the field fluctuations are much smallerthan predictions of the classical Monte-Carlo simulation with the Woods-Saxon nuclear density. PACS numbers:
Non-central AA collisions at high energies can generate a very strong magnetic field perpendicular to the reactionplane [1, 2]. In this talk I present results of quantum calculations of fluctuations of the electromagnetic fields in AA collisions at RHIC and LHC energies based on the fluctuation-dissipation theorem (FDT) [3]. This issue is veryimportant in the context of the chiral magnetic effect and charge separation [1, 4, 5] in AA collisions because thefluctuations may partly destroy the correlation between the magnetic field direction and the reaction plane, andcan lead to reduction of the B -induced observables [6]. Previously the field fluctuations have been addressed byMonte-Carlo (MC) simulation with the Woods-Saxon (WS) nuclear distribution using the classical Lienard-Weichertpotentials [6–8]. But the WS nuclear distribution ignores the collective quantum dynamics of the nuclear groundstate. The classical treatment of the electromagnetic field may also be inadequate because, similarly to the van derWaals forces [9], it becomes invalid at large distances.We consider the proper time region τ ∼ . − B -induced effects inthe quark-gluon plasma (QGP). We ignore the electromagnetic fields generated by the induced currents in the QGPfireball after interaction of the colliding nuclei [10]. We consider the right moving and left moving nuclei with velocities V R = (0 , , V ) and V L = (0 , , − V ), and with the impact parameters b R = ( − b/ , ,
0) and b L = ( b/ , , z R,L = ± V t . For each nucleus the electromagnetic field is a sum of the mean field and the fluctuating field F µν = h F µν i + δF µν . (1)The mean fields h E i and h B i are given by the Lorentz transformation of the Coulomb field in the nucleus rest frame.For two colliding nuclei the mean magnetic field at r = 0 has only y -component. At t ≫ R A /γ (here γ = 1 / √ − V is the Lorentz factor, R A is the nucleus radius) in the region ρ ≪ tγ h B y ( t, ρ , z = 0) i takes a simple ρ -independentform h B y ( t, ρ , z = 0) i ≈ Zeb/γ t . (2)The contribution of each nucleus to the correlators of the electromagnetic fields in the lab-frame may be expressedvia the correlators in the nucleus rest frame. For γ ≫ h δE i δE k i = γ (cid:2) h δE i δE k i + V e il e kj h δB l δB j i (cid:3) rf , (3) h δB i δB k i = γ (cid:2) h δB i δB k i + V e il e kj h δE l δE j i (cid:3) rf , (4)where i, k are the transverse indices and the subscript rf on the right-hand side of (3), (4) indicates that the correlatorsare calculated in the nucleus rest frame.In calculations of the rest frame correlators h δE l δE j i , h δB i δB k i (hereafter we drop the subscript rf ) with the helpof the FDT we follow the formalism of [11] (formulated in the gauge δA = 0). It allows to relate the time Fouriercomponent of the vector potential correlator h δA i ( r ) δA k ( r ) i ω = 12 Z dte iωt h δA i ( t, r ) δA k (0 , r ) + δA k (0 , r ) δA i ( t, r ) i (5)and that of the retarded Green’s function D ik ( ω, r , r ) = − i Z dte iωt θ ( t ) h δA i ( t, r ) δA k (0 , r ) − δA k (0 , r ) A i ( t, r ) i . (6)In the zero temperature limit the FDT relation between (5) and (6) reads [11] h δA i ( r ) δA k ( r ) i ω = − sign( ω )Im D ik ( ω, r , r ) . (7)The time Fourier components of the electromagnetic field correlators in terms of that for the the vector potentialcorrelator (5) are given by h δE i ( r ) δE k ( r ) i ω = ω h δA i ( r ) δA k ( r ) i ω , (8) h δB i ( r ) δB k ( r ) i ω = rot (1) il rot (2) kj h δA l ( r ) δA j ( r ) i ω . (9)In the time region of interest ( t ∼ > . R A . It allows one to treat each nucleus asa point like dipole described by the dipole polarizability α ik ( ω ). The field fluctuations are described by correction tothe retarded Green’s function proportional to the dipole polarizability [11]. The retarded Green’s function coincideswith the Green’s function of Maxwell’s equation [11]. For the point like dipole at r = r A the equation for the retardedGreen’s function reads (cid:20) ∂ ∂x i ∂ l − δ il △ − δ il ω − πω α il ( ω ) δ ( r − r A ) (cid:21) D lk ( ω, r , r ′ ) = − πδ ik δ ( r − r ′ ) . (10)The correction to D ik due to α ik reads [11]∆ D ik ( ω, r , r ) = − ω D vil ( ω, r , r A ) α lm ( ω ) D vmk ( ω, r A , r ) , (11)where D vik is the vacuum Green’s function given by D vik ( ω, r , r ) = e iωr r (cid:20) − δ ik (cid:18) iωr − ω r (cid:19) + x i x k r (cid:18) iωr − ω r (cid:19)(cid:21) (12)with r = r − r .For spherical nuclei the polarizability tensor can be written as α ik ( ω ) = δ ik α ( ω ). α ( ω ) is an analytical function of ω in the upper half-plane [9]. It satisfies the relation α ∗ ( − ω ∗ ) = α ( ω ) [9]. It means that on the upper imaginary axis α ( ω ) is real. Using this fact, one can express the rest frame field correlators h δE i ( t, r ) δE k ( t, r ) i , h δB i ( t, r ) δB k ( t, r ) i via integrals of the type I n = R ∞ dξξ n e − ξ α (cid:16) iξ r (cid:17) with n = 0 − α ( ω ) reads [9] α ( ω ) = 13 X s (cid:20) |h | d | s i| ω s − ω − iδ + |h | d | s i| ω s + ω + iδ (cid:21) , (13)where d = (cid:16) eN P p r p − eZ P n r n (cid:17) /A is the dipole operator. At ω > α ( ω ) is connected withthe dipole photoabsorption cross section σ abs ( ω ) = 4 πω Im α ( ω ) . (14)For heavy nuclei the dipole strength is dominated by the giant dipole resonance (GDR) [13]. It appears as a broadpeak in σ abs at ω ∼
14 MeV. We parametrize the dipole polarizability for
Au and
Pb nuclei by a single GDRstate α ( ω ) = c (cid:20) ω − ω − i Γ / ω + ω + i Γ / (cid:21) . (15)By fitting the data on the photoabsorption cross section from [14] for Au and from [15] for
Pb we obtainedthe following values of the parameters: ω ≈ . ≈ .
38 MeV, c ≈ . − for Au, and ω ≈ . ≈ .
72 MeV, c ≈ .
93 Gev − for Pb. Fig. 1 illustrates the quality of our fit. Using these parameters wecalculated the fluctuations of the nuclear dipole moment. From (13), (15) one can obtain h | d | i = 3 π Z ∞ dω Im α ( ω ) = 6 cπ arctg (2 ω / Γ) . (16)
10 15 200200400600 10 15 20 Pb σ Photon energy [MeV]Au ab s [ m b ] (b)(a) FIG. 1: Fit of the photoabsorption cross section in the GDRregion to the experimental data for
Au and
Pb targets.The data are from Refs. [14] and [15], respectively.
Pb+Pb, s =2.76 TeVAu+Au, s =0.2 TeV t [fm] < B x > / / < B y > (b)(a) δ FIG. 2: The t -dependence of the ratio h δB x i / / h B y i at r = 0for Au+Au collisions at √ s = 0 . √ s = 2 .
76 TeV for the impact parameters b = 3,6 and 9 fm (from top to bottom). Solid lines are for quantumcalculations, dashed lines for classical MC calculations withthe WS nuclear density. This formula gives h | d | i ≈ .
91 fm and h | d | i ≈ .
02 fm for Au and
Pb, respectively. The classical MCcalculation with the WS nuclear density gives for these nuclei the values h d i ≈ .
89 fm and h d i ≈ .
39 fm . Thus,we see that the classical treatment overestimates the dipole moment squared by a factor of ∼ B x that vanishes without fluctuations. In Fig. 2 we show our quantum and classicalresults for t -dependence of the ratio h δB x i / / h B y i at x = y = 0 for several impact parameters for Au+Au collisionsat √ s = 0 . √ s = 2 .
76 TeV. This figure shows that the quantum treatment gives h δB x i / / h B y i smaller than the classical one by a factor of ∼ − ∼ −
27 for LHC.Thus, we see that in the quantum picture both for RHIC and LHC fluctuations of the direction of the magnetic fieldrelative to the reaction plane should be very small. Of course, experimentally the reaction plane itself cannot bedetermined exactly. In the event-by-event measurements the orientation of the reaction plane is extracted from theelliptic flow in the particle distribution [16, 17] (it is often called the participant plane), and it fluctuates around thereal reaction plane. Calculations of the fluctuations of the direction of the magnetic field relative to the participantplane require a joint analysis of the field fluctuations and of the fluctuations of the initial entropy deposition thatcontrol the fluctuations of the orientation of the participant plane in the hydrodynamical simulations of AA collisions.The initial entropy distribution is sensitive to the long range fluctuations of the nuclear density. Besides the nuclearfluctuations related to the GDR there are other collective nuclear modes [13] such as the giant monopole resonanceand the giant quadrupole resonance that may also be important for the participant plane fluctuations. It would be ofgreat interest to clarify the situation with the MC simulation with the WS nuclear density for these collective modes.This is of great interest for the event-by-event hydrodynamic simulations of AA collision.In summary, we have performed a quantum analysis of fluctuations of the electromagnetic field in AA collisionsat RHIC and LHC energies. Our quantum calculations show that the field fluctuations are very small. We havedemonstrated that the classical picture overestimates strongly the field fluctuations.This work is supported by the RScF grant 16-12-10151. References [1] D.E. Kharzeev, L.D. McLerran, and H.J. Warringa, Nucl. Phys. A , 227 (2008) [arXiv:0711.0950].[2] V. Skokov, A.Yu. Illarionov, and V. Toneev, Int. J. Mod. Phys. A , 5925 (2009) [arXiv:0907.1396].[3] H.B. Callen and T.A. Welton, Phys. Rev. , 34 (1951).[4] D.E. Kharzeev, Prog. Part. Nucl. Phys. , 133 (2014) [arXiv:1312.3348]. [5] D.E. Kharzeev, J. Liao, and S.A. Voloshin, Prog. Part. Nucl. Phys. , 1 (2016) [arXiv:1511.04050].[6] J. Bloczynski, X.-G. Huang, X. Zhang, and J. Liao, Phys. Lett. B , 1529 (2013) [arXiv:1209.6594].[7] A. Bzdak and V. Skokov, Phys. Lett. B , 171 (2012) [arXiv:1111.1949].[8] W.-T. Deng and X.-G. Huang, Phys. Rev. C , 044907 (2012) [arXiv:1201.5108].[9] V.B. Berestetski, E.M. Lifshits, and L.P. Pitaevski, Quantum Electrodynamics (Landau Course of Theoretical Physics Vol.4) , Oxford, Pergamon Press, 1979.[10] B.G. Zakharov, Phys. Lett. B , 262 (2014) [arXiv:1404.5047].[11] E.M. Lifshits and L.P. Pitaevski,
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