Particle Production in Strong Electromagnetic Fields in Relativistic Heavy-Ion Collisions

Abstract

I review the origin and properties of electromagnetic fields produced in heavy-ion collisions. The field strength immediately after a collision is proportional to the collision energy and reaches ~𝑚²_𝜋 at RHIC and ~10𝑚²_𝜋 at LHC. I demonstrate by explicit analytical calculation that after dropping by about one-two orders of magnitude during the first fm/c of plasma expansion, it freezes out and lasts for as long as quark-gluon plasma lives as a consequence of finite electrical conductivity of the plasma. Magnetic field breaks spherical symmetry in the direction perpendicular to the reaction plane, and therefore all kinetic coefficients are anisotropic. I examine viscosity of QGP and show that magnetic field induces azimuthal anisotropy on plasma flow even in spherically symmetric geometry. Very strong electromagnetic field has an important impact on particle production. I discuss the problem of energy loss and polarization of fast fermions due to synchrotron radiation, consider photon decay induced by magnetic field, elucidate 𝐽/𝜓 dissociation via Lorentz ionization mechanism, and examine electromagnetic radiation by plasma. I conclude that all processes in QGP are affected by strong electromagnetic field and call for experimental investigation.

1. Origin and Properties of Electromagnetic Field

1.1. Origin of Magnetic Field

We can understand the origin of magnetic field in heavy-ion collisions by considering collision of two ions of radius 𝑅 with electric charge 𝑍𝑒 (𝑒 is the magnitude of electron charge) at impact parameter 𝐛. According to the Biot and Savart law they create magnetic field that in the center-of-mass frame has magnitude 𝑏𝐵∼𝛾𝑍𝑒𝑅³ (1) and points in the direction perpendicular to the reaction plane (span by the momenta of ions). Here 𝛾=√𝑠_𝑁𝑁/2𝑚_𝑁 is the Lorentz factor. At RHIC heavy ions are collided at 200 GeV per nucleon, hence 𝛾=100. Using 𝑍=79 for gold and 𝑏∼𝑅_𝐴≈7 fm we estimate 𝑒𝐵≈𝑚²_𝜋∼10¹⁸ G. To appreciate how strong is this field, compare it with the following numbers: the strongest magnetic field created on earth in a form of electromagnetic shock wave is ~10⁷ G [1], and magnetic field of a neutron star is estimated to be 10¹⁰–10¹³ G, that of a magnetar up to 10¹⁵ G [2]. It is perhaps the strongest magnetic field that has ever existed in nature.

It has been known for a long time that classical electrodynamics breaks down at the critical (Schwinger) field strength 𝐹=𝑚²_𝑒/𝑒 [3]. In cgs units the corresponding magnetic field is 10¹³ G. Because 𝑚_𝜋/𝑚_𝑒=280, electromagnetic fields created at RHIC and LHC are well above the critical value. This offers a unique opportunity to study the super-strong electromagnetic fields in laboratory. The main challenge is to identify experimental observables that are sensitive to such fields. The problem is that nearly all observables studied in heavy-ion collisions are strongly affected both by the strong color forces acting in quark-gluon plasma (QGP) and by electromagnetic fields often producing qualitatively similar effects. An outstanding experimental problem thus is to separate the two effects. In Sections 2–7 I examine several processes strongly affected by intense magnetic fields and discuss their phenomenological significance. But first, in this section, let me derive a quantitative estimate of electromagnetic field.

Throughout this paper, the heavy-ion collision axis is denoted by 𝑧. Average magnetic field then points in the 𝑦-direction; see Figures 1 and 7. Plane 𝑥𝑧 is the reaction plane, and 𝑏 is the impact parameter.

Figure 1 

Heavy-ion collision geometry as seen along the collision axis 𝑧. Adapted from [4].

1.2. Magnetic Field in Vacuum

1.2.1. Time Dependence

To obtain a quantitative estimate of magnetic field we need to take into account a realistic distribution of protons in a nucleus. This has been first done in [5] (in the case of high-energy 𝑝𝑝 collisions, magnetic field was first estimated in [9] who also pointed out a possibility of formation of 𝑊-condensate [9, 10]). Magnetic field at point 𝐫 created by two heavy ions moving in the positive or negative 𝑧-direction can be calculated using the Liénard-Wiechert potentials as follows: 𝑒𝐄(𝑡,𝐫)=𝛼_em𝑎1−𝑣²_𝑎𝐑_𝑎𝑅³_𝑎1−𝐑_𝑎×𝐯_𝑎²/𝑅²_𝑎^3/2,𝑒𝐁(𝑡,𝐫)=𝛼_em𝑎1−𝑣²_𝑎𝐯_𝑎×𝐑_𝑎𝑅³_𝑎1−𝐑_𝑎×𝐯_𝑎²/𝑅²_𝑎^3/2, (2)(3) with 𝐑_𝑎=𝐫−𝐫_𝑎(𝑡), where sums run over all 𝑍 protons in each nucleus, their positions and velocities being 𝐫_𝑎 and 𝐯_𝑎. The magnitude of velocity 𝑣_𝑎 is determined by the collision energy √𝑠_𝑁𝑁 and the proton mass 𝑚_𝑝,𝑣²_𝑎=1−(2𝑚_𝑝/√𝑠_𝑁𝑁)². These formulas are derived in the eikonal approximation, assuming that protons travel on straight lines before and after the scattering. This is a good approximation, since baryon stopping is a small effect at high energies. Positions of protons in heavy ions can be determined by one of the standard models of the nuclear charge density 𝜌(𝐫_𝑎). Reference [5] employed the “hard sphere” model, while [4] used a bit more realistic Woods-Saxon distribution.

Numerical integration in (3) including small contribution from baryon stopping yields for magnetic field the result shown in Figure 2 as a function of the proper time 𝜏=(𝑡²−𝑧²)^1/2. Evidently, magnetic field rapidly decreases as a power of time, so that after first 3 fm it drops by more than three orders of magnitude.

(a)

(b)

Figure 2 

Magnetic field 𝐁=𝐵𝐲 (multiplied by 𝑒) at the origin 𝐫=0 produced in collision of two gold ions at beam energies (a) √𝑠𝑁𝑁=62 GeV and (b) √𝑠𝑁𝑁=200 GeV. Adapted from [5]. Note that 𝑒𝐵 is the same in Gauss and Lorentz-Heaviside units in contrast to 𝐵.

1.2.2. Event-by-Event Fluctuations in Proton Positions

Nuclear charge density 𝜌 provides only event-averaged distribution of protons. The actual distribution in a given event is different form 𝜌 implying that in a single event there is not only magnetic field along the 𝑦-direction, but also other components of electric and magnetic fields. This leads to event-by-event fluctuations of electromagnetic field [4]. Shown in Figure 3 are electric and magnetic field components at 𝑡=0 at the origin (denoted by a black dot in Figure 1) in AuAu collisions at √𝑠_𝑁𝑁=200 GeV.

(a)

(b)

Figure 3 

The mean absolute value of (a) magnetic field and (b) electric field at 𝑡=0 and 𝐫=0 as a function of impact parameter 𝑏 for AuAu collision at √𝑠𝑁𝑁=200 GeV.

Figure 3 clearly shows that although on average the only nonvanishing component of the field is 𝐵_𝑦, which is also clear from the symmetry considerations, other components are finite in each event and are of the same order of magnitude 𝐵_𝑥≈𝐸_𝑥≈𝐸_𝑦. (4) To appreciate the magnitude of electric field produced in heavy-ion collisions note that 𝐸∼𝑚²_𝜋=10²¹ V/cm. The corresponding intensity is 10³⁹ W/cm² which is instructive to compare with the power generated by the most powerful state-of-the-art lasers: 10²³ W/cm².

Electromagnetic fields created in heavy-ion collisions were also examined in more elaborated approaches in [11–13]. They yielded qualitatively similar results on electromagnetic field strength and its relaxation time.

1.3. Magnetic Field in Quark-Gluon Plasma

1.3.1. Liénard-Wiechert Potentials in Static Medium

In the previous section, I discussed electromagnetic field in vacuum. A more realistic estimate must include medium effects. Indeed, the state-of-the-art phenomenology of quark-gluon plasma (QGP) indicates that strongly interacting medium is formed at as early as 0.5 fm/c. Even before this time, strongly interacting medium exists in a form of Glasma [14, 15]. Therefore, a calculation of magnetic field must involve response of medium determined by its electrical conductivity. It has been found in the lattice calculations that the gluon contribution to electrical conductivity of static quark-gluon plasma is [16] ()𝑇𝜎=5.8±2.9𝑇_𝑐MeV, (5) where 𝑇 is plasma temperature and 𝑇_𝑐 it critical temperature. This agrees with [17] but is at odds with an earlier calculation [18]. It is not clear whether (5) adequately describes the electromagnetic response of realistic quark-gluon plasma because it neglects quark contribution and assumes that medium is static. Theoretical calculations are of little help at the temperatures of interest, since the perturbation theory is not applicable. In absence of a sensible alternative I will use (5) as a best estimate of electrical conductivity. If medium is static then 𝑇 is constant as a function of time 𝑡. The static case is considered in this section, while in the next section I consider expanding medium.

In medium, magnetic field created by a charge 𝑒 moving in 𝑧-direction with velocity 𝑣 is a solution of the following equations: 𝛁⋅𝐁=0,𝛁×𝐄=−𝜕𝐁,()𝛿(𝐛),𝜕𝑡𝛁⋅𝐄=𝑒𝛿𝑧−𝑣𝑡𝛁×𝐁=𝜕𝐄̂()𝛿(𝐛),𝜕𝑡+𝜎𝐄+𝑒𝑣𝐳𝛿𝑧−𝑣𝑡 (6)(7) where we used the Ohm’s law 𝐣=𝜎𝐄 to describe currents induced in the medium. Position of the observation point is specified by the longitudinal and transverse coordinates 𝑧 and 𝐛,𝐫=𝑧𝐳̂+𝐛. Taking curl of the second equation in (7) and substituting (6) we get ∇²𝜕𝐁=²𝐁𝜕𝑡²+𝜎𝜕𝐁[̂()𝛿(𝐛.𝜕𝑡−𝑒𝑣𝛁×𝐳𝛿𝑧−𝑣𝑡)] (8) The particular solution reads 𝐁(𝑧,𝐛,𝑡)=^𝑡₀𝑑𝑡^′∞_−∞𝑑𝑧𝑑²𝐛𝐺𝑧−𝑧^′,𝐛−𝐛^′,𝑡−𝑡^′𝑒𝑣×𝛁^′𝐳̂𝛿𝑧^′−𝑣𝑡^′𝛿𝐛^′, (9) where Green’s function 𝐺(𝑧−𝑧^′,𝐛−𝐛^′,𝑡−𝑡^′) satisfies the following equation: ∇²𝜕𝐺−²𝐺𝜕𝑡²−𝜎𝜕𝐺𝜕𝑡=−𝛿𝑧−𝑧^′𝛿𝐛−𝐛^′𝛿𝑡−𝑡^′, (10) which is solved by 𝐺𝑧−𝑧^′,𝐛−𝐛^′,𝑡−𝑡^′=𝑑²𝑘_⟂(2𝜋)²𝑒^{𝑖(𝐛−𝐛′)⋅𝐤⟂∞}_−∞𝑑𝑘_𝑧𝑒2𝜋^{𝑖𝑘𝑧(𝑧−𝑧′)}×^∞_−∞𝑑𝜔𝑒2𝜋^{−𝑖𝜔(𝑡−𝑡′)}1𝑘²_𝑧+𝑘²_⟂−𝜔²,−𝑖𝜔𝜎 (11) where 𝐤=𝑘_𝑧𝐳̂+𝐤_⟂. Plugging this into (9) and substituting for the expression in the square brackets in (9) its Fourier image, we obtain 𝐁()𝑑𝑧,𝐛,𝑡=2𝜋𝑒𝑣²𝑘_⟂(2𝜋)²𝑒^{𝑖𝐛⋅𝐤⟂∞}_−∞𝑑𝑘_𝑧𝑒2𝜋^{𝑖𝑘𝑧𝑧}×^∞_−∞𝑑𝜔𝑒2𝜋^{−𝑖𝜔𝑡}𝑖𝐤×𝐳̂𝑘²_𝑧+𝑘²_⟂−𝜔²−𝑖𝜔𝜎×𝛿𝜔−𝑘_𝑧𝑣𝑑=𝑒²𝑘_⟂(2𝜋)²𝑒^{𝑖𝐛⋅𝐤⟂}×^∞_−∞𝑑𝜔𝑒2𝜋^{−𝑖𝜔𝑡}𝑒^{𝑖𝜔𝑧/𝑣}×𝑖𝐤_⟂×𝐳̂𝜔²/𝑣²+𝑘²_⟂−𝜔².−𝑖𝜔𝜎 (12)(13) We are interested in the 𝑦-component of the field. Noting that (𝐤_⟂×𝐳̂)⋅𝐲=−𝑘_⟂cos𝜙, where 𝜙 is the azimuthal angle in the transverse plane, and integrating over 𝑑²𝑘_⟂ we derive 𝑒𝐵_𝑦=𝛼_em𝜋^∞_−∞𝑠(𝜔)𝐾₁(𝑠(𝜔)𝑏)𝑒^{𝑖𝜔(𝑧/𝑣−𝑡)}𝑑𝜔, (14) where 𝛼_em=𝑒²/4𝜋, and we introduced notation 𝑠(𝜔)=𝜔1𝑣²−𝜖(𝜔), (15) where 𝜖(𝜔) is the dielectric constant of the plasma with the following frequency dependence: 𝜖(𝜔)=1+𝑖𝜎𝜔. (16)

Equation (14) is actually valid for any functional form of 𝜖(𝜔) [19], which can be easily verified by using electric displacement 𝐃 instead of 𝐄 in (7). In this case (16) can be viewed as a low frequency expansion of 𝜖(𝜔). Magnetic field in this approximation is quasistatic. Therefore, we could have neglected the second time derivative in (8), and then keeping only the leading powers of 𝜔 we would have derived (14) with 𝑠²=𝑖𝜔𝜎. After integration over 𝜔 this gives (21). Let us take notice of the fact that neglecting the second time derivative in (8) yields diffusion equation for magnetic field in plasma.

It is instructive to compare time dependence of magnetic field created by moving charges in vacuum and in plasma. In vacuum, setting 𝜎=0 in (13) and integrating first over 𝜔 and then over 𝐤_⟂ give 𝑒𝐁=𝐲𝛼_em𝑏𝛾𝑏²+𝛾²(𝑡−𝑧)^23/2, (17) where we used 𝑣≈1. This coincides with (3) for a single proton when we take 𝐑_𝑎=𝐛+(𝑧−𝑣𝑡)𝐳̂. Consider field strength (17) at the origin 𝑧=0. At times 𝑡<𝑏/𝛾 the field is constant, while at 𝑡≫𝑏/𝛾 it decreases as 𝐵_∞∝1/𝑡³. At the time 𝑡≈𝑏 the ratio between these two is 𝐵₀𝐵_∞=1𝛾³≪1, (18) which is a very small number (~10⁻⁶ at RHIC).

In matter 𝜎>0. Let me write the modified Bessel function appearing in (14) as follows: 𝑠𝐾₁(𝑠𝑏)=^∞₀𝐽₁(𝑥𝑏)𝑥²𝑑𝑥𝑥²+𝑠². (19) Substituting (19) into (14) and using (16), we have (𝑣=1) 𝛼𝑒𝐁=^em𝜋𝐲^∞₀𝑑𝑥^∞_−∞𝐽𝑑𝜔¹(𝑥𝑏)𝑥²𝑥²𝑒−𝑖𝜔𝜎^{𝑖𝜔(𝑧−𝑡)}. (20) Closing the contour in the lower half-plane of complex 𝜔 picks a pole at 𝜔=−𝑖𝑥²/𝜎. We have 𝑒𝐁=2𝛼_em𝜎𝐲^∞₀𝑑𝑥𝑥²𝐽₁(𝑥𝑏)𝑒^{−(𝑥2/𝜎)(𝑡−𝑧)}=𝐲𝛼_em𝑏𝜎2(𝑡−𝑧)²𝑒^{−𝑏2𝜎/4(𝑡−𝑧)}. (21) At 𝑧=0 this function vanishes at 𝑡=0 and 𝑡→∞ and has maximum at the time instant 𝑡=𝑏²𝜎/8 which is much larger than 𝑏/𝛾. The value of the magnetic field at this time is 𝑒𝐵_max=32𝑒⁻²𝛼_em𝑏³𝜎. (22) (Here 𝑒 is the base of natural logarithm.) This is smaller than the maximum field in vacuum 𝐵_max𝐵₀=32𝑒⁻²𝜎𝑏𝛾 (23) but is still a huge field. We compare the two solutions (17) and (21) in Figure 4. We see that in a conducting medium magnetic field stays for a long time.

Figure 4 

Relaxation of magnetic field at 𝑧=0 in vacuum (blue), in static conducting medium at 𝜎=5.8 MeV (red) and at 𝜎=16 MeV (brown), and in the expanding medium (green). Units of 𝐵 is fm−2≈2𝑚2𝜋. 𝑏=7 fm, 𝑍=79 (gold nucleus), 𝛾=100 (RHIC).

One essential component is still missing in our arguments—time dependence of plasma properties due to its expansion. Let us now turn to this problem.

1.3.2. Magnetic Field in Expanding Medium

So far I treated quark-gluon plasma as a static medium. Expanding medium temperature and hence conductivity are functions of time. In Bjorken scenario [20], expansion is isentropic, that is, 𝑛𝑉=const, where 𝑛 is the particle number density and 𝑉 is plasma volume. Since 𝑛∼𝑇³ and at early times expansion is one-dimensional 𝑉∼𝑡 it follows that 𝑇∝𝑡^−1/3. (Eventually, we will consider the midrapidity region 𝑧=0, therefore distinction between the proper time and 𝑡 is not essential.) Equation (5) implies that 𝜎∼𝑡^−1/3. I will parameterize conductivity as follows: 𝜎(𝑡)=𝜎₀𝑡₀𝑡₀+𝑡^1/3, (24) where I took 𝑡₀≈0.5 fm to be the initial time (or longitudinal size) of plasma evolution. Suppose that plasma lives for 10 fm/c and then undergoes phase transition to hadronic gas at 𝑇_𝑐. Then employing (5) we estimate 𝜎₀≈16MeV. Let me define another parameter that I will need in the forthcoming calculation: 𝛽=4𝜎₀3𝑡₀≈43MeV.fm (25)

Magnetic field in expanding medium is still governed by (8). As was explained in the preceding subsection, time evolution of magnetic field is quasi-static, which allows me to neglect the second time derivative. Let me introduce a new “time” variable 𝜌 as follows: 𝜌=1+𝑡𝑡₀^4/3−1. (26) Field 𝐁(𝑧,𝐛,𝜌) satisfies equation ∇²𝐁=𝛽𝜕𝐁̂𝜌𝛿(𝐛,𝜕𝜌−𝑒𝑣𝛁×𝐳𝛿𝑧−𝑣𝑡) (27) where 𝑡𝜌=𝑡₀𝜌+1^3/4−1. (28) Its solution can be written as 𝐁𝑧,𝐛,𝜌=^𝜌₀𝑑𝜌^′∞_−∞×𝑑𝑑𝑧₂𝐛𝔊𝑧−𝑧^′,𝐛−𝐛^′,𝜌−𝜌^′̂𝜌𝛿(𝐛,𝑒𝑣×𝛁𝐳𝛿𝑧−𝑣𝑡) (29) in terms of Green’s function 𝔊(z−𝑧^′,𝐛−𝐛^′,𝜌−𝜌^′) satisfying ∇²𝔊−𝛽𝜕𝔊𝜕𝜌=−𝛿𝑧−𝑧^′𝛿𝐛−𝐛^′𝛿𝜌−𝜌^′. (30) To solve this equation we represent 𝔊 as three-dimensional Fourier integral with respect to the space coordinates and Laplace transform with respect to the “time” coordinate: 𝔊=𝑑𝑧,𝐛,𝜌³𝑘(2𝜋)³𝑒^{𝑖(𝐤⟂⋅𝐛+𝑘𝑧𝑧)}_𝐶𝑑𝜆𝑒2𝜋𝑖^𝜆𝜌1𝑘²_⟂+𝑘²_𝑧,+𝛽𝜆 (31) with the contour 𝐶 running parallel to the imaginary axis to the right of all integrand singularities. Now I would like to write the expression in the curly brackets in (29) also as Fourier-Laplace expansion. To this end we calculate 𝑓_𝐤,𝜆=𝑑²𝐛^∞_−∞×𝑑𝑧_∞0𝑑𝜌𝑒^{−𝑖(𝐤⟂⋅𝐛+𝑘𝑧𝑧)}𝑒^−𝜆𝜌𝛿𝜌𝛿(𝐛)=𝑧−𝑣𝑡_∞0𝑑𝜌𝑒^{−𝑖𝑘𝑧𝑣𝑡0[(𝜌+1)3/4−1]}𝑒^−𝜆𝜌. (32)(33) Therefore, ̂𝜌𝛿(𝐛𝑑𝑒𝑣𝛁×𝐳𝛿𝑧−𝑣𝑡)=𝑒𝑣³𝑘(2𝜋)³𝑒^{𝑖(𝐤⟂⋅𝐛+𝑘𝑧𝑧)}×_C𝑑𝜆𝑒2𝜋𝑖^𝜆𝜌𝑖𝐤_⟂×𝐳̂𝑓_𝐤,𝜆. (34) Substituting (31) and (34) into (29) we obtain upon integration over the volume and time as follows: 𝐁=𝑑𝑧,𝐛,𝜌³𝑘(2𝜋)³𝑒^{𝑖(𝐤⟂⋅𝐛+𝑘𝑧𝑧)}×_𝐶𝑑𝜆𝑒2𝜋𝑖^𝜆𝜌𝑒𝑣𝑖𝐤_⟂×𝐳̂𝑘²_⟂+𝑘²_𝑧𝑓+𝛽𝜆^𝐤,𝜆𝜃𝜌, (35) where 𝜃 is the step function. Taking consequent integrals over 𝜆 and 𝑘_𝑧 gives 𝐁0,𝐛,𝜌=𝑒𝑣𝛽𝑑²𝑘_⟂(2𝜋)²𝑒^{𝑖𝐤⟂⋅𝐛∞}_−∞𝑑𝑘_𝑧2𝜋𝑖𝐤_⟂×𝐳̂^𝜌₀𝑑𝜌^′𝑒^{−𝑖𝑘𝑧𝑣𝑡0[(𝜌′+1)3/4−1]}×𝑒^{−((𝑘2⟂+𝑘2𝑧)/𝛽)(𝜌−𝜌′)}=𝑒𝑣𝛽𝑑²𝑘_⟂(2𝜋)²𝑒^{𝑖𝐤⟂⋅𝐛}12𝜋𝑖𝐤_⟂×𝐳̂^𝜌₀𝑑𝜌^′𝑒^{−(𝑘2⟂/𝛽)(𝜌−𝜌′)}×𝜋𝛽𝜌−𝜌^′𝑒^{−𝑣2𝑡20𝛽[(𝜌′+1)3/4−1]2/4(𝜌−𝜌′)}. (36) Consider now 𝐵_𝑦. Integrating over azimuthal angle 𝜙 and then over 𝑘_⟂ as in (13), (14) yields 𝑒𝐵_𝑦=𝛼0,𝐛,𝜌_em𝑣𝑏𝛽^3/22√𝜋^𝜌₀𝑑𝜁𝜁^−5/2×𝑒^{−𝐛2𝛽/4𝜁}𝑒^{−𝑣2𝑡20𝛽[(𝜌−𝜁+1)3/4−1]2/4𝜁}, (37) where 𝜁=𝜌−𝜌^′.

The results of a numerical calculation of (37) are shown in Figure 4. We see that expansion of plasma tends to increase the relaxation time, although this effect is rather modest. We conclude that due to finite electrical conductivity of QGP, magnetic field essentially freezes in the plasma for as long as plasma exists. Similar phenomenon, known as skin effect, exists in good conductors placed in time-varying magnetic field: conductors expel time dependent magnetic fields from conductor volume confining them into a thin layer of width 𝛿∼𝜔^−1/2 on the surface.

1.3.3. Diffusion of Magnetic Field in QGP

The dynamics of magnetic field relaxation in conducting plasma can be understood in a simple model [21]. Suppose at some initial time 𝑡=0 magnetic field 𝐁(0,𝐫) permits the plasma. The problem is to find the time dependence of the field at 𝑡>0. In this model, the field sources turn off at 𝑡=0 and do not at all contribute to the field at 𝑡>0. Electromagnetic field is governed by the following equations: 𝛁×𝐁=𝐣,𝐣=𝜎𝐄,𝛁×𝐄=−𝜕𝐁𝜕𝑡,𝛁⋅𝐁=0, (38) that lead to the diffusion equation for 𝐁, after we neglect the second-time derivative as discussed previously as follows: ∇²𝐁=𝜎𝜕𝐁.𝜕𝑡 (39) For simplicity we treat electrical conductivity 𝜎 as constant. Initial condition at 𝑡=0 reads 𝐁(0,𝐫)=𝐁₀𝑒^{−𝐛2/𝑅2}, (40) where the Gaussian profile is chosen for illustration purposes, and 𝑅 is the nuclear radius. Solution to the problem (39), (40) is 𝐁(𝑡,𝐫)=𝑑𝑉^′𝐁0,𝐫^′G𝑡,𝐫−𝐫^′, (41) where Green’s function is 𝐺()=1𝑡,𝐫(4𝜋𝑡/𝜎)^3/2−𝐫exp².4𝑡/𝜎 (42) Integrating over the entire volume we derive 𝐁(𝑡,𝐫)=𝐁₀𝑅²𝑅²−𝐛+4𝑡/𝜎exp²𝑅².+4𝑡/𝜎 (43) It follows from (43) that as long as 𝑡≪𝑡_relax, where 𝑡_relax is a characteristic time 𝑡_relax=𝑅²𝜎/4 and magnetic field 𝐁 is approximately time independent. This estimate is the same as the one we arrived at after (21).

In summary, magnetic field in quark-gluon plasma appears to be extremely strong and slowly varying function of time for most of the plasma lifetime. At RHIC it decreases from 𝑒𝐵≈(2.5𝑚_𝜋)² right after the collision to 𝑒𝐵≈(𝑚_𝜋/4)² at 𝑡≈5 fm see Figure 4. This has a profound impact on all the processes occurring in QGP.

1.3.4. Schwinger Mechanism

Schwinger mechanism of pair production [3] is operative if electric field exceeds the critical value of 𝑚²/𝑒, where 𝑚 is mass of lightest electrically charged particle. Indeed, in order to excite a fermion out of the Dirac sea, electric force 𝑒𝐄 must do work along the path 𝑑𝓵 satisfying ^ℓ₀𝑒𝐄⋅𝑑𝓵^′>2𝑚. (44) If 𝐄=const, then 𝐸≳𝑚/ℓ𝑒. The maximal value of ℓ is the fermion Compton’s wavelength ℓ∼𝜆^-=1/𝑚 implying that the minimum (or critical) value of electric field is 𝐸_𝑐=𝑚²𝑒. (45) Notice that in stronger fields ℓ∼𝑚/𝑒𝐸<𝜆^-. Figure 3 indicates that electron-positron pairs are certainly produced at RHIC. An important question then is the role of these pairs in the electromagnetic field relaxation in plasma. There are two associated effects: (i) before 𝑒⁻𝑒⁺ pairs thermalize, they contribute to the Foucault currents; (ii) after they thermalize, their density contributes to the polarization of plasma in electric field and hence to its conductivity.

Since space dimensions of QGP are much less than 𝜆^-_𝑒=380 fm, it may seem inevitable that space dependence of electric field (in addition to its time dependence) has a significant impact on the Schwinger process in heavy-ion collisions. However, this conclusion is premature. Indeed, suppose that electric field is a slow function of coordinates. Then 𝐄(𝐫)≈𝐄(0)+𝐫⋅𝛁𝐄(0). Work done by electric field is 𝑒𝐄(^ℓ₀0)⋅𝑑𝓵^′+^ℓ₀(𝐫⋅𝛁)𝑒𝐄(0)⋅𝑑𝓵^′∼𝑒𝐸ℓ+𝑒ℓ²𝐸(0)𝜆, (46) where 𝜆 is length scale describing space variation of electric field. In order that contribution of space variation to work be negligible, the second term in the r.h.s. of (46) must satisfy 𝑒ℓ²𝐸(0)/𝜆≪𝑚. Employing the estimate ℓ∼𝑚/𝑒𝐸(0) that we obtained after (45) implies 𝑚/𝑒𝐸(0)𝜆≪1. Following [22] I define the inhomogeneity parameter 𝑚𝛾=𝜆𝑒𝐸 (47) that describes the effect of spatial variation of electric field on the pair production rate. For electrons 𝑚=0.5 MeV in QGP 𝜆∼0.5 fm at 𝑒𝐸∼𝑚²_𝜋 we have 𝛾∼0.01. Therefore, somewhat counter intuitively, electric field can be considered as spatially homogeneous. The same conclusion can be derived from results of [23]. Schwinger mechanism in spatially dependent electric fields was also discussed in [24, 25].

In view of smallness of 𝛾 one can employ the extensive literature on Schwinger effect in time-dependent spatially-homogeneous electric fields. As far as heavy-ion physics is concerned, the most comprehensive study has been done in [6, 26, 27] who developed an approach to include the effect of backreaction. They argued that time evolution of electric field can be studied in adiabatic approximation and used the kinetic approach to study the time evolution. Their results are exhibited in Figure 5. Similar results were obtained in [28]. We observe that response time of the current density of Schwinger pairs ~10⁴ fm/c is much larger than the plasma lifetime ~10 fm/c, and therefore no sizable electric current is generated.

(a)

(b)

Figure 5 

Time dependence of electric field due to the Schwinger mechanism back reaction and the corresponding electric current density of Schwinger pairs. Dimensionless time variable is defined as 𝜏=𝑡/𝜆-. For electrons 𝜆-=380 fm. Plasma undergoes phase transition at about 𝜏=1/38. Adapted from [6].

In summary, strong electric field is generated in heavy-ion collisions in every event but averages to zero in a large event ensemble. This field exceeds the critical value for electrons and light 𝑢,𝑑 quarks. However, during the plasma lifetime no significant current of Schwinger pairs is generated.

2. Flow of Quark-Gluon Plasma in Strong Magnetic Field

2.1. Azimuthal Asymmetry

Magnetic field is known to have a profound influence on kinetic properties of plasmas. Once the spherical symmetry is broken, distribution of particles in plasma is only axially symmetric with respect to the magnetic field direction. This symmetry, however, is not manifest in the plane span by magnetic field and the impact parameter vectors, namely, 𝑥𝑦-plane in Figure 1. Charged particles moving along the magnetic field direction 𝑦 are not influenced by the magnetic Lorentz force, while those moving the 𝑥𝑧-plane (i.e., the reaction plane) are affected the most. The result is azimuthally anisotropic flow of expanding plasma in the 𝑥𝑦-plane even when initial plasma geometry is completely spherically symmetric. The effect of weak magnetic field on quark-gluon plasma flow was first considered in [29] who argued that magnetic field is able to enhance the azimuthal anisotropy of produced particles up to 30%. This conclusion was reached by utilizing a solution of the magnetohydrodynamic equations in weak magnetic field.

A characteristic feature of the viscous pressure tensor in magnetic field is its azimuthal anisotropy. This anisotropy is the result of suppression of the momentum transfer in QGP in the direction perpendicular to the magnetic field. Its macroscopic manifestation is decrease of the viscous pressure tensor components in the plane perpendicular to the magnetic field, which coincides with the reaction plane in the heavy-ion phenomenology. Since Lorentz force vanishes in the direction parallel to the field, viscosity along that direction is not affected at all. In fact, the viscous pressure tensor component in the reaction plane is twice as small as the one in the field direction. As the result, transverse flow of QGP develops azimuthal anisotropy in presence of the magnetic field. Clearly, this anisotropy is completely different from the one generated by the anisotropic pressure gradients and exists even if the later is absent. In fact, because spherical symmetry in magnetic field is broken, viscous effects in plasma cannot be described by only two parameters: shear 𝜂 and bulk viscosity 𝜁. Rather the viscous pressure tensor of magnetoactive plasma is characterized by seven viscosity coefficients, among which five are shear viscosities and two are bulk ones.

2.2. Viscous Pressure in Strong Magnetic Field

2.2.1. Viscosities from Kinetic Equation

Generally, calculation of the viscosities requires knowledge of the strong interaction dynamics of the QGP components. However, in strong magnetic field these interactions can be considered as a perturbation, and viscosities can be analytically calculated using the kinetic equation [30–33]. To apply this approach to QGP in strong magnetic field we start with kinetic equation for the distribution function 𝑓 of a quark flavor of charge 𝑧𝑒 as follows: 𝑝^𝜇𝜕_𝜇𝑓=𝑧𝑒𝐵^𝜇𝜈𝜕𝑓𝜕𝑢^𝜇𝑢_𝜈+𝒞𝑓,…, (48) where 𝒞 is the collision integral and 𝐵^𝜇𝜈 is the electromagnetic tensor, which contains only magnetic field components in the laboratory frame. Ellipsis in the argument of 𝒞 indicates the distribution functions of other quark flavors and gluons (I will omit them in the following). The equilibrium distribution reads 𝑓₀=𝜌4𝜋𝑚³𝑇𝐾₂𝑒𝛽𝑚^{−𝛽𝑝⋅𝑈(𝑥)}, (49) where 𝑈(𝑥) is the macroscopic velocity of the fluid, 𝑝^𝜇=𝑚𝑢^𝜇 is particle momentum, 𝛽=1/𝑇, and 𝜌 is the mass density. Since (𝜕𝑓₀/𝜕𝑢^𝜇)∝𝑢_𝜇, the first term on the r.h.s. of (48) as well as the collision integral vanishes in equilibrium. Therefore, we can write the kinetic equation as an equation for 𝛿𝑓: 𝑝^𝜇𝜕_𝜇𝑓₀=𝑧𝑒𝐵^𝜇𝜈𝜕𝛿𝑓𝜕𝑢^𝜇𝑢_𝜈+𝒞𝛿𝑓, (50) where 𝛿𝑓 is a deviation from equilibrium. Differentiating (49) we find 𝜕_𝜇𝑓₀=−𝑓₀1𝑇𝑝^𝜆𝜕_𝜇𝑈_𝜆(𝑥). (51) Since 𝑈^𝜆=(𝛾_𝑉,𝛾_𝑉𝐕) and 𝑝^𝜆=(𝜀,𝐩)=(𝛾_𝑣𝑚,𝛾_𝑣𝑚𝐯) it follows that 𝑚𝑝⋅𝑈=√1−𝑣²1−𝑉²(1−𝐯⋅𝐕). (52) Thus, in the comoving frame 𝜕_𝜇𝑓_0𝐕=0=𝑓₀1𝑇𝑝_𝜈𝜕_𝜇𝑉^𝜈. (53) Substituting (53) in (50) yields −𝑓₀𝑇𝑝^𝜇𝑝^𝜈𝑉_𝜇𝜈=𝑧𝑒𝐵^𝜇𝜈𝜕𝛿𝑓𝜕𝑢^𝜇𝑢_𝜈+𝒞𝛿𝑓, (54) where I defined 𝑉_𝜇𝜈=12𝜕_𝜇𝑉_𝜈+𝜕_𝜈𝑉_𝜇. (55) and used 𝑢^𝜇𝑢^𝜈𝜕_𝜇𝑉_𝜈=𝑢^𝜇𝑢^𝜈𝑉_𝜇𝜈.

Since the time derivative of 𝑓₀ is irrelevant for the calculation of the viscosity I will drop it from the kinetic equation. All indices thus become the usual three-vector ones. To avoid confusion we will label them by the Greek letters from the beginning of the alphabet. Introducing 𝑏_𝛼𝛽=𝐵⁻¹𝜀_𝛼𝛽𝛾𝐵_𝛾, we cast (54) in the form 1𝑇𝑝^𝛼𝑢^𝛽𝑉_𝛼𝛽𝑓₀=−𝑧𝑒𝐵𝑏_𝛼𝛽𝑣^𝛽𝜕𝛿𝑓𝜕𝑣_𝛼1𝜀−𝒞𝛿𝑓. (56)

The viscous pressure generated by a deviation from equilibrium is given by the tensor −Π_𝛼𝛽=𝑝_𝛼𝑝_𝛽𝑑𝛿𝑓³𝑝𝜀. (57) Effectively it can be parameterized in terms of the viscosity coefficients as follows (we neglect the bulk viscosities): Π_𝛼𝛽=⁴_𝑛=0𝜂_𝑛𝑉^(𝑛)_𝛼𝛽, (58) where the linearly independent tensors 𝑉^(𝑛)_𝛼𝛽 are given by𝑉⁽⁰⁾_𝛼𝛽=3𝑏_𝛼𝑏_𝛽−𝛿_𝛼𝛽𝑏_𝛾𝑏_𝛿𝑉_𝛾𝛿−13,𝑉∇⋅𝐕_(1)𝛼𝛽=2𝑉_𝛼𝛽+𝛿_𝛼𝛽𝑉_𝛾𝛿𝑏_𝛾𝑏_𝛿−2𝑉_𝛼𝛾𝑏_𝛾𝑏_𝛽−2𝑉_𝛽𝛾𝑏_𝛾𝑏_𝛼+𝑏_𝛼𝑏_𝛽−𝛿_𝛼𝛽∇⋅𝐕+𝑏_𝛼𝑏_𝛽𝑉_𝛾𝛿𝑏_𝛾𝑏_𝛿,𝑉⁽²⁾_𝛼𝛽=2𝑉_𝛼𝛾𝑏_𝛽𝛾+𝑉_𝛽𝛾𝑏_𝛼𝛾−𝑉_𝛾𝛿𝑏_𝛼𝛾𝑏_𝛽𝑏_𝛿,𝑉⁽³⁾_𝛼𝛽=𝑉_𝛼𝛾𝑏_𝛽𝛾+𝑉_𝛽𝛾𝑏_𝛼𝛾−𝑉_𝛾𝛿𝑏_𝛼𝛿𝑏_𝛼𝛾𝑏_𝛽𝑏_𝛿−𝑉_𝛾𝛿𝑏_𝛽𝛾𝑏_𝛼𝑏_𝛿,𝑉⁽⁴⁾_𝛼𝛽=2𝑉_𝛾𝛿𝑏_𝛼𝛿𝑏_𝛼𝛾𝑏_𝛽𝑏_𝛿+𝑉_𝛾𝛿𝑏_𝛽𝛾𝑏_𝛼𝑏_𝛿. (59a)(59b)(59c)(59d)(59e)Here 𝑏_𝛼 is a unit vector in the direction of magnetic field. For the calculation of the shear viscosities 𝜂_𝑛, 𝑛=0,…,4 we can set ∇⋅𝐕=0 and 𝑉_𝛼𝛽𝑏_𝛼𝑏_𝛽=0.

Let us expand 𝛿𝑓 to the second order in velocities in terms of the tensors 𝑉^(𝑛)_𝛼𝛽 as follows: 𝛿𝑓=⁴_𝑛=0𝑔_𝑛𝑉^(𝑛)_𝛼𝛽𝑣^𝛼𝑣^𝛽. (60) Then, substituting (60) into (58) and requiring consistency of (57) and (58) yield 𝜂_𝑛2=−15𝜀𝑣⁴𝑔_𝑛𝑑³𝑝. (61) This gives the viscosities in the magnetic field in terms of the deviation of the distribution function from equilibrium. Transition to the nonrelativistic limit in (61) is achieved by the replacement 𝜀→𝑚 [30].

2.2.2. Collisionless Plasma

In strong magnetic field we can determine 𝑔_𝑛 by the method of consecutive approximations. Writing 𝛿𝑓=𝛿𝑓⁽¹⁾+𝛿𝑓⁽²⁾ and substituting into (56), we find 1𝑇𝑝^𝛼𝑣^𝛽𝑉_𝛼𝛽𝑓₀=−𝑧𝑒𝐵𝑏_𝛼𝛽𝑣^𝛽𝜕𝛿𝑓⁽¹⁾+𝛿𝑓⁽²⁾𝜕𝑣_𝛼1𝜀+𝒞𝛿𝑓⁽¹⁾. (62) Here I assumed that the deviation from equilibrium due to the strong magnetic field is much larger than due to the particle collisions. The explicit form of 𝒞 is determined by the strong interaction dynamicsbut drops off the equation in the leading oder. The first correction to the equilibrium distribution obeys the equation 1𝑇𝑝_𝛼𝑣_𝛽𝑉_𝛼𝛽𝑓₀=−𝑧𝑒𝐵𝑏_𝛼𝛽𝑣_𝛽𝜕𝛿𝑓⁽¹⁾𝜕𝑣_𝛼1𝜀. (63) Using (60), we get 𝑏_𝛼𝛽𝑣_𝛽𝜕𝛿𝑓⁽¹⁾𝜕𝑣_𝛼=2𝑏_𝛼𝛽𝑣_𝛽⁴_𝑛=0𝑔_𝑛𝑉^(𝑛)_𝛼𝛾𝑣_𝛾. (64) Substituting (64) into (63) and using (59a), (59b), (59c), (59d), and (59e) yield 𝜀𝑝𝑇𝑧𝑒𝐵^𝛼𝑣_𝛽𝑉_𝛼𝛽𝑓₀=−2𝑏_𝛽𝜈𝑣_𝛼𝑣_𝜈×𝑔₁2𝑉_𝛼𝛽−2𝑉_𝛽𝛾𝑏_𝛾𝑏_𝛼+2𝑔₂𝑉_𝛽𝛾𝑏_𝛾𝑏_𝛼+𝑔₃𝑉_𝛼𝛾𝑏_𝛽𝛾+𝑉_𝛽𝛾𝑏_𝛼𝛾−𝑉_𝛾𝛿𝑏_𝛼𝑏_𝛿+2𝑔₄𝑉_𝛾𝛿𝑏_𝛽𝛾𝑏_𝛼𝑏_𝛿, (65) where I used the following identities 𝑏_𝛼𝛽𝑏_𝛼=𝑏_𝛼𝛽𝑏_𝛽=𝑏_𝛼𝛽𝑣_𝛼𝑣_𝛽=0. Clearly, (65) is satisfied only if 𝑔₁=𝑔₂=0. Concerning the other two coefficients, we use the identities𝑏_𝛼𝛽𝑏_𝛽𝛾=𝑏_𝛾𝑏_𝛼−𝛿_𝛼𝛾𝑏²,𝜀_𝛼𝛽𝛾𝜀_𝛿𝜖𝜁=𝛿_𝛼𝛿𝛿_𝛽𝜖𝛿_𝛾𝜁−𝛿_𝛽𝜁𝛿_𝛾𝜖−𝛿_𝛼𝜖𝛿_𝛽𝛿𝛿_𝛾𝜁−𝛿_𝛽𝜁𝛿_𝛾𝛿+𝛿_𝛼𝜁𝛿_𝛽𝛿𝛿_𝛾𝜖−𝛿_𝛽𝜁𝛿_𝛾𝛿 (66a)(66b)that we substitute into (65) to derive −𝜀𝑝2𝑇𝑧𝑒𝐵^𝛼𝑣^𝛽𝑉_𝛼𝛽𝑓₀=𝑔₃2𝑉_𝛼𝛽𝑏_𝛼𝑏_𝛽−4𝑉_𝛼𝛽𝑣_𝛼𝑏_𝛽(𝐛⋅𝐯)+2𝑔₄𝑉_𝛼𝛽𝑣_𝛼𝑏_𝛽(𝐛⋅𝐯). (67) Since 𝑝_𝛼=𝜀𝑣_𝛼, we obtain 𝑔₃=𝑔₄2𝜀=−²𝑓₀.4𝑇𝑧𝑒𝐵 (68) Using (49), (68) in (61) in the comoving frame (of course 𝜂_𝑛s do not depend on the frame choice) and integrating using 3.547.9 of [34], we derive [35] 𝜂₃=𝐾₃𝛽𝑚𝐾₂𝛽𝑚𝜌𝑇2𝑧𝑒𝐵. (69) The nonrelativistic limit corresponds to 𝑚≫𝑇 in which case we get 𝜂^NR₃=𝜌𝑇.2𝑧𝑒𝐵 (70) In the opposite ultrarelativistic case 𝑚≪𝑇 (high-temperature plasma) 𝜂^UR₃=2𝑛𝑇²,𝑧𝑒𝐵 (71) where 𝑛=𝜌/𝑚 is the number density.

2.2.3. Contribution of Collisions

In the relaxation-time approximation we can write the collision integral as 𝒞𝛿𝑓=−𝜈𝛿𝑓, (72) where 𝜈 is an effective collision rate. Strong field limit means that 𝜔_𝐵≫𝜈, (73) where 𝜔_𝐵=𝑧𝑒𝐵/𝜀 is the synchrotron frequency. Whether 𝜈 itself is function of the field depends on the relation between the Larmor radius 𝑟_𝐵=𝑣_𝑇/𝜔_𝐵, where 𝑣_𝑇 is the particle velocity in the plane orthogonal to 𝐁 and the Debye radius 𝑟_𝐷. If 𝑟_𝐵≫𝑟_𝐷, (74) then the effect of the field on the collision rate 𝜈 can be neglected [30]. Assuming that (74) is satisfied, the collision rate reads 𝜈=𝑛𝑣𝜎_𝑡, (75) where 𝜎_𝑡 is the transport cross-section, which is a function of the saturation momentum 𝑄_𝑠 [36, 37]. We estimate 𝜎_𝑡∼𝛼²_𝑠/𝑄²_𝑠, with 𝑄_𝑠∼1 GeV and 𝑛=𝑃/𝑇 with pressure 𝛼²_𝑠𝑃∼1 GeV/fm³; we get 𝜈∼40MeV. Inequality (73) is well satisfied since 𝑒𝐵≃𝑚²_𝜋 [5, 11], and 𝑚 is in the range between the current and the constituent quark masses. On the other hand, applicability of the condition (74) is marginal and is very sensitive to the interaction details. In this section we assume that (74) holds in order to obtain the analytic solution. Additionally, the general condition for the applicability of the hydrodynamic approach ℓ=1/𝜈≪𝐿, where ℓ is the mean free path and 𝐿 is the plasma size is assumed to hold. Altogether we have 𝑟_𝐷≪𝑟_𝐵≪ℓ≪𝐿.

Equation for the second correction to the equilibrium distribution 𝛿𝑓⁽²⁾ follows from (62) after substitution (72) 𝑧𝑒𝐵𝜀𝑏_𝛼𝛽𝑣_𝛽𝜕𝛿𝑓⁽²⁾𝜕𝑣_𝛼=−𝜈𝛿𝑓⁽¹⁾. (76) Now, plugging𝛿𝑓⁽¹⁾=𝑔₃𝑉⁽³⁾_𝛼𝛽+𝑔₄𝑉⁽⁴⁾_𝛼𝛽𝑣_𝛼𝑣_𝛽,𝛿𝑓⁽²⁾=𝑔₁𝑉⁽¹⁾_𝛼𝛽+𝑔₂𝑉⁽²⁾_𝛼𝛽𝑣_𝛼𝑣_𝛽, (77a)(77b)into (76) yields 2𝑧𝑒𝐵𝜀𝑔₁2𝑉_𝛽𝛼𝑏_𝛼𝛾𝑣_𝛽𝑣_𝛾−2𝑉_𝛽𝛼𝑏_𝛼𝛾𝑣_𝛽𝑣_𝛾(𝐯⋅𝐛)+2𝑔₂𝑉_𝛽𝛼𝑏_𝛼𝛾𝑣_𝛽𝑣_𝛾(𝐯⋅𝐛)=−𝜈𝑔₃−2𝑉_𝛽𝛼𝑏_𝛼𝛾𝑣_𝛽𝑣_𝛾−6𝑉_𝛽𝛼𝑏_𝛼𝛾𝑣_𝛽𝑣_𝛾(𝐯⋅𝐛), (78) where I used 𝑔₄=2𝑔₃. It follows that 𝑔₁=𝑔₂4=𝜈𝛾_𝑣𝑔₃2𝜔_𝐵. (79) With the help of (80), (49), and (65) we obtain [35] 𝜂₁=𝜂₂4=85√𝜌2𝜋²𝜎_𝑡𝑇^3/2(𝑧𝑒𝐵)²𝑚^1/2𝐾_7/2𝛽𝑚𝐾₂.𝛽𝑚 (80)

2.3. Azimuthal Asymmetry of Transverse Flow: A Simple Model

To illustrate the effect of the magnetic field on the viscous flow of the electrically charged component of the quark-gluon plasma I will assume that the flow is non-relativistic and use the Navier-Stokes equations that read 𝜌𝜕𝑉_𝛼𝜕𝑡+𝑉_𝛽𝜕𝑉_𝛼𝜕𝑥_𝛽=−𝜕𝑃𝜕𝑥_𝛼+𝜕Π_𝛼𝛽𝜕𝑥_𝛽, (81) where Π_𝛼𝛽 is the viscous pressure tensor, 𝜌=𝑚𝑛 is mass density, and 𝑃 is pressure. I will additionally assume that the flow is nonturbulent and that the plasma is non-compressible. The former assumption amounts to dropping the terms nonlinear in velocity, while the later implies vanishing divergence of velocity 𝛁⋅𝐕=0. (82) Because of the approximate boost invariance of the heavy-ion collisions, we can restrict our attention to the two dimensional flow in the 𝑥𝑦-plane corresponding to the central rapidity region.

The viscous pressure tensor in vanishing magnetic field is isotropic in the 𝑥𝑦-plane and is given by Π⁰_𝛼𝛽=𝜂𝜕𝑉_𝛼𝜕𝑥_𝛽+𝜕𝑉_𝛽𝜕𝑥_𝛼𝑉=2𝜂^𝑥𝑥𝑉_𝑥𝑦𝑉_𝑦𝑥𝑉_𝑦𝑦, (83) where the superscript 0 indicates absence of the magnetic field. In the opposite case of very strong magnetic field the viscous pressure tensor has a different form (58). Neglecting all 𝜂_𝑛 with 𝑛≥1, we can write Π^∞_𝛼𝛽=𝜂₀−𝑉_𝑦𝑦002𝑉_𝑦𝑦=2𝜂₀12𝑉_𝑥𝑥00𝑉_𝑦𝑦, (84) where we also used (82). Notice that Π^∞_𝑥𝑥=(1/2)Π^∞_𝑦𝑦=(1/2)Π⁰_𝑥𝑥 indicating that the plasma flows in the direction perpendicular to the magnetic field with twice as small viscosity as in the direction of the field. The later is not affected by the field at all, because the Lorentz force vanishes in the field direction. Substituting (84) into (81) we derive the following two equations characterizing the plasma velocity in the strong magnetic field [35]: 𝜌𝜕𝑉_𝑥𝜕𝑡=−𝜕𝑃𝜕𝑥+𝜂₀𝜕²𝑉_𝑥𝜕𝑥²,𝜌𝜕𝑉_𝑦𝜕𝑡=−𝜕𝑃𝜕𝑦+2𝜂₀𝜕²𝑉_𝑦𝜕𝑦². (85) Additionally, we need to set the initial conditions 𝑉_𝑥𝑡=0=𝜑₁𝑥,𝑦,𝑉_𝑦𝑡=0=𝜑₂𝑥,𝑦. (86) The solution to the the problem (85), (86) is𝑉_𝑥=𝑥,𝑦,𝑡_∞−∞𝑑𝑥^′𝜑₁𝑥^′,𝑦𝐺_1/2𝑥−𝑥^′−1,𝑡𝜌^𝑡₀𝑑𝑡^′∞_−∞𝑑𝑥^′𝐺_1/2𝑥−𝑥^′,𝑡−𝑡^′𝜕𝑃𝑥^′,𝑦,𝑡^′𝜕𝑥^′,𝑉_𝑦=𝑥,𝑦,𝑡_∞−∞𝑑𝑦^′𝜑₂x,𝑦^′𝐺₁𝑦−𝑦^′−1,𝑡𝜌^𝑡₀𝑑𝑡^′∞_−∞𝑑𝑦^′𝐺₁𝑦−𝑦^′,𝑡−𝑡^′𝜕𝑃𝑥,𝑦^′,𝑡^′𝜕𝑦^′. (87a)(87b)Here Green’s function is given by 𝐺_𝑘=1𝑦,𝑡√4𝜋𝑎²𝑒𝑘𝑡^{−𝑦2/4𝑎2𝑘𝑡} (88) and the diffusion coefficient by 𝑎²=2𝜂₀𝜌. (89)

Suppose that the pressure is isotropic; that is, it depends on the coordinates 𝑥,𝑦 only via the radial coordinate 𝑟=𝑥²+𝑦²; accordingly we pass from the integration variables 𝑥^′ and 𝑦^′ to 𝑟^′ in (87a) and (87b) correspondingly. At later times we can expand Green’s function (88) in inverse powers of 𝑡. The first terms in the r.h.s. of (87a) and (87b) are subleading, and we obtain𝑉_𝑥1𝑥,𝑦,𝑡≈−𝜌^𝑡₀𝑑𝑠^∞_−∞1𝑑𝑟√2𝜋𝑎²𝑠𝜕𝑃(𝑟,𝑡−𝑠)1𝜕𝑟=−𝜌^𝑡₀1𝑑𝑠√2𝜋𝑎²𝑠[𝑃(𝑅(𝑠),𝑡−𝑠)−𝑃(0,𝑡−𝑠)], (90a) and by the same token 𝑉_𝑦1𝑥,𝑦,𝑡≈−𝜌^𝑡₀1𝑑𝑠√4𝜋𝑎²𝑠[𝑃(𝑅(𝑠),𝑡−𝑠)−𝑃(0,𝑡−𝑠)], (90b)where 𝑅(𝑡) denotes the boundary beyond which the density of the plasma is below the critical value. We observe that 𝑉_𝑥/𝑉_y=√2. Consequently, the azimuthal anisotropy of the hydrodynamic flow is [35] 𝑉²_𝑥−𝑉²_𝑦𝑉²_𝑥+𝑉²_𝑦=1−(1/2)()=11+1/23. (91) Since I assumed that the initial conditions and the pressure are isotropic, the azimuthal asymmetry (91) is generated exclusively by the magnetic field.

We see that at later times after the heavy-ion collision, flow velocity is proportional to 𝜂₀^−1/2, where 𝜂₀ is the finite shear viscosity coefficient; see (87a) and (87b). If the system is such that in absence of the magnetic field it were azimuthally symmetric, then the magnetic field induces azimuthal asymmetry of 1/3; see (91). The effect of the magnetic field on flow is strong and must be taken into account in phenomenological applications. Neglect of the contribution by the magnetic field leads to underestimation of the phenomenological value of viscosity extracted from the data [38–40]. In other words, the more viscous QGP in magnetic field produces the same azimuthal anisotropy as a less viscous QGP in vacuum.

A model that I considered in this section to illustrate the effect of the magnetic field on the azimuthal anisotropy of a viscous fluid flow does not take into account many important features of a realistic heavy-ion collision. To be sure, a comprehensive approach must involve numerical solution of the relativistic magnetohydrodynamic equations with a realistic geometry. A potentially important effect that I have not considered here is plasma instabilities [41, 42], which warrant further investigation.

The structure of the viscous stress tensor in very strong magnetic field (84) is general, model independent. However, as explained, the precise amount of the azimuthal anisotropy that it generates cannot be determined without taking into account many important effects. Even so, I draw the reader’s attention to the fact that analysis of [29] using quite different arguments arrived at similar conclusion. Although a more quantitive numerical calculation is certainly required before a final conclusion can be made, it looks very plausible that the QGP viscosity is significantly higher than the presently accepted value extracted without taking into account the magnetic field effect [38–40] and is perhaps closer to the value calculated using the perturbative theory [43, 44].

3. Energy Loss and Polarization due to Synchrotron Radiation

3.1. Radiation of Fast Quark in Magnetic Field

General problem of charged fermion radiation in external magnetic field was solved in [45–47]. It has important applications in collider physics; see, for example, [48, 49]. In heavy-ion phenomenology, synchrotron radiation provides one of the mechanisms of energy loss in quark-gluon plasma, which is an important probe of QGP [50, 51] (synchrotron radiation in chromo-magnetic fields was discussed in [52–54]).

A typical diagram contributing to the synchrotron radiation, that is, radiation in external magnetic field, by a quark is shown in Figure 6 [21]. This diagram is proportional to (𝑒𝐵)^𝑛, where 𝑛 is the number of external field lines. In strong field, powers of 𝑒𝐵 must be summed up, which may be accomplished by exactly solving the Dirac equation for the relativistic fermion and then calculating the matrix element for the transition 𝑞→𝑞+𝑔. Such calculation has been done in QED for some special cases including the homogeneous constant field and can be readily generalized for gluon radiation. Intensity of the radiation can be expressed via the invariant parameter 𝜒 defined as 𝜒²𝛼=−^em𝑍²_𝑞ℏ³𝑚⁶𝐹_𝜇𝜈𝑝^𝜈2=𝛼_em𝑍²_𝑞ℏ³𝑚⁶(𝐩×𝐁)², (92) where the initial quark 4-momentum is 𝑝^𝜇=(𝜀,𝐩) and 𝑍_𝑞 is the quark charge in units of the absolute value of the electron charge 𝑒. At high energies, 𝑍𝜒≈^𝑞𝐵𝜀𝐵_𝑐𝑚. (93) The regime of weak fields corresponds to 𝜒≪1, while in strong fields 𝜒≫1. In our case, 𝑒𝐵/𝑒𝐵_𝑐≈(𝑚_𝜋/𝑚_𝑢)²≫1 (at RHIC), and therefore 𝜒≫1. In terms of 𝜒, spectrum of radiated gluons of frequency 𝜔 can be written as [45] 𝑑𝐼𝑑𝜔=−𝛼_𝑠𝐶_𝐹𝑚²𝜔𝜀²×^∞_𝑥(𝜉)2Ai𝑑𝜉+𝑥+𝜔𝜀𝜒𝑥^1/2Ai^′(𝑥), (94) where 𝐼 is the intensity 𝑥=ℏ𝜔𝜀^′𝜒^2/3, (95) and 𝜀^′ is quark’s energy in the final state. Ai is the Ayri function. Equation (94) is valid under the assumption that the initial quark remains ultrarelativistic, which implies that the energy loss due to the synchrotron radiation Δ𝜀 should be small compared to the quark energy itself Δ𝜀≪𝜀.

Figure 6 

A typical diagram contributing to the synchrotron radiation by a quark.

Figure 7 

Geometry of a heavy-ion collision. 𝐩 denotes momentum of a fast quark. Two orthogonal planes are the reaction plane span by the initial heavy-ion momenta and the mid rapidity plane 𝜃=𝜋/2, which is labeled as 𝜂=0.

Energy loss by a relativistic quark per unit length is given by [48] 𝑑𝜀𝑑𝑙=−^∞₀𝑑𝜔𝑑𝐼𝑑𝜔=𝛼_𝑠𝐶_𝐹𝑚²𝜒²2^∞₀4+5𝜒𝑥^3/2+4𝜒²𝑥³1+𝜒𝑥^3/24Ai^′(𝑥)𝑥𝑑𝑥. (96) In two interesting limits, energy loss behaves quite differently. At 𝜂=𝜑=0 we have [48] 𝑑𝜀𝑑𝑙=−2𝛼_𝑠ℏ𝐶_𝐹𝑍_𝑞𝑒𝐵²𝜀²3𝑚⁴,𝜒≪1,𝑑𝜀𝑑𝑙=−0.37𝛼_𝑠ℏ^−1/3𝐶_𝐹𝑍_𝑞𝑒𝐵𝜀^2/3,𝜒≫1. (97a)(97b)In the strong field limit energy loss is independent of the quark mass, whereas in the weak field case it decreases as 𝑚⁻⁴. Since 𝜒∝ℏ, limit of 𝜒≪1 corresponds to the classical energy loss.

To apply this result to heavy-ion collisions we need to write down the invariant 𝜒 in a suitable kinematic variables. The geometry of a heavy-ion collision is depicted in Figure 7. Magnetic field 𝐁 is orthogonal to the reaction plane span by the impact parameter vector 𝐛 and the collision axis (𝑧-axis). For a quark of momentum 𝐩 we define the polar angle 𝜃 with respect to the 𝑧-axis and azimuthal angle 𝜑 with respect to the reaction plane. In this notation, 𝐁=𝐵𝐲 and 𝐩=𝑝_𝑧𝐳̂+𝑝_⟂(𝐱cos𝜑+𝐲sin𝜑), where 𝑝_⟂=|𝐩|sin𝜃≈𝜀sin𝜃. Thus, (𝐁×𝐩)²=𝐵²(𝑝²_𝑧+𝑝²_⟂cos²𝜑). Conventionally, one expresses the longitudinal momentum and energy using the rapidity 𝜂 as 𝜀=𝑚_⟂cosh𝜂 and 𝑝_𝑧=𝑚_⟂sinh𝜂, where 𝑚²_⟂=𝑚²+𝑝²_⟂. We have 𝜒²=ℏ²(𝑒𝐵)²𝑚⁶𝑝²_⟂sinh²𝜂+cos²𝜑. (98)

In Figure 8 a numerical calculation of the energy loss per unit length in a constant magnetic field using (96) and (98) is shown [21]. We see that energy loss of a 𝑢 quark with 𝑝_⟂=10 GeV is about 0.2 GeV/fm at RHIC and 1.3 GeV/fm at LHC. This corresponds to the loss of 10% and 65% of its initial energy after traveling 5 fm at RHIC and LHC, respectively. Therefore, energy loss due to the synchrotron radiation at LHC gives a phenomenologically important contribution to the total energy loss.

(a)

(b)

(c)

(d)

Figure 8 

Energy loss per unit length by quarks moving in constant external magnetic field as a function of (a) 𝑝𝑇 at RHIC at 𝜂=𝜑=0, (b) 𝑝𝑇 at LHC at 𝜂=𝜑=0, (c) azimuthal angle 𝜑 with respect to the reaction plane for 𝑝𝑇=5 GeV and 𝜂=0,1, and (d) fractional energy loss versus rapidity 𝜂 at 𝑝𝑇=5 GeV and 𝜑=0,𝜋/4.

Energy loss due to the synchrotron radiation has a very nontrivial azimuthal angle and rapidity dependence that comes from the corresponding dependence of the 𝜒-parameter (98). As can be seen in Figure 8(c), energy loss has a minimum at 𝜑=𝜋/2 that corresponds to quark’s transverse momentum 𝐩_⟂ being parallel (or antiparallel) to the field direction. It has a maximum at 𝜑=0,𝜋 when 𝐩_⟂ is perpendicular to the field direction and thus lying in the reaction plane. It is obvious from (98) that at midrapidity 𝜂=0 the azimuthal angle dependence is much stronger pronounced than in the forward/backward direction. Let me emphasize that the energy loss (96) divided by 𝑚², that is, 𝑑𝜀/(𝑑𝑙𝑚²) scales with 𝜒. In turn, 𝜒 is a function of magnetic field, quark mass, rapidity, and azimuthal angle. Therefore, all the features seen in Figure 8 follow from this scaling behavior.

3.2. Azimuthal Asymmetry of Gluon Spectrum

In magnetic field gluon spectrum is azimuthally asymmetric. It is customary to describe this asymmetry by Fourier coefficients of intensity defined as 𝐼𝜑=𝐼1+2^∞_𝑛=1𝑣_𝑛,𝑣cos𝑛𝜑_𝑛=12𝜋𝐼^𝜋−𝜋𝐼𝜑cos𝑛𝜑𝑑𝜑. (99) Azimuthal averaged intensity is 𝐼=∫^𝜋_−𝜋𝐼(𝜑)𝑑𝜑/2𝜋. In strong fields 𝜒≫1, and we can write 𝐼𝜑=0.37𝛼_𝑠𝐶_𝐹𝑚²𝜒^2/3=0.37𝛼_𝑠𝐶_𝐹𝑒𝐵𝑝_⟂^2/3sinh²𝜂+cos²𝜑^1/3. (100) We have 𝑣_𝑛=∫^𝜋_−𝜋𝐼𝜑cos𝑛𝜑𝑑𝜑∫^𝜋_−𝜋𝐼𝜑=∫𝑑𝜑_𝜋−𝜋sinh²𝜂+cos²𝜑^1/3cos𝑛𝜑𝑑𝜑∫^𝜋_−𝜋sinh²𝜂+cos²𝜑^1/3.𝑑𝜑 (101) At 𝜂=0 the Fourier coefficients 𝑤_𝑛 can be calculated analytically using formula 3.631.9 of [34] as follows: 𝑣_𝑛=𝐵(4/3,4/3)𝐵)+(),()−(𝑣((4/3𝑛/24/3𝑛/2)),if𝑛∈even;_𝑛=0,if𝑛∈odd, (102) where 𝐵 is Euler’s beta function. The corresponding numerically values of the lowest harmonics are 𝑣₂=0.25,𝑣₄=−0.071,𝑣₆=0.036,𝑣₈=−0.022,and𝑣₁₀=0.015.

3.3. Polarization of Light Quarks

Synchrotron radiation leads to polarization of electrically charged fermions, this is known as the Sokolov-Ternov effect [46]. It was applied to heavy-ion collisions in [21]. Unlike energy loss that I discussed so far, this is a purely quantum effect. It arises because the probability of the spin-flip transition depends on the orientation of the quark spin with respect to the direction of the magnetic field and on the sign of fermion’s electric charge. The spin-flip probability per unit time reads [46] 5√𝑤=3𝛼_𝑠𝐶_𝐹ℏ16𝑚²²𝜀𝑚⁵𝑍_𝑞𝑒|𝐯×𝐁|𝜀³×21−9(𝜻⋅𝐯)²−8√3𝑒15sign_𝑞(𝜻⋅𝐳̂), (103) where 𝜻 is a unit axial vector that coincides with the direction of quark spin in its rest frame and 𝐯=𝐩/𝜀 is the initial fermion velocity.

The nature of this spin-flip transition is transparent in the nonrelativistic case, where it is induced by the interaction Hamiltonian [49] as follows: ℋ=−𝝁⋅𝐁=−𝑔𝑒𝑍_𝑞ℏ2𝑚𝐬⋅𝐁. (104) It is seen that negatively charged quarks and antiquarks (e.g., 𝑢 and 𝑑) tend to align against the field, while the positively charged ones (e.g., 𝑢 and 𝑑) align parallel to the field.

Let 𝑛_↑(↓) be the number of fermions or antifermions with given momentum and spin direction parallel (antiparallel) to the field produced in a given event. At initial time 𝑡=𝑡₀ the spinasymmetry defined as 𝑛𝐴=^↑−𝑛_↓𝑛_↑+𝑛_↓ (105) vanishes. Equation (103) implies that at later times, a beam of nonpolarized fermions develops a finite asymmetry given by [46] 8√𝐴=3151−𝑒^{−(𝑡−𝑡0)/𝜏}, (106) where 8𝜏=5√3𝑚𝛼_𝑠𝐶_𝐹𝑚𝜀²𝑚²𝑍_𝑞𝑒|𝐯×𝐁|³ (107) is the characteristic time over which the maximal possible asymmetry is achieved. This time is extremely small on the scale of 𝑡₀. For example, it takes only 𝜏=6⋅10⁻⁸ fm for a 𝑑 quark of momentum 𝑝_⟂=1 GeV at 𝜂=𝜑=0 at RHIC to achieve the maximal asymmetry of 𝐴_𝑚=8/5√3=92%. Therefore, quarks and antiquarks are polarized almost instantaneously after being released from their wave functions. However, subsequent interaction with QGP and fragmentation washes out the polarization of quarks.

A more sensitive probe are leptons weakly interacting with QGP and not undergoing a fragmentation process. Thus, their polarization can present a direct experimental evidence for the existence and strength of magnetic field. In case of muons we can estimate 𝜏 by replacing 𝛼_𝑠𝐶_𝐹→𝛼_em. For muons we get 𝜏=0.3 fm/c, which is still much smaller than magnetic field life-time. Observation of such a lepton polarization asymmetry is perhaps the most definitive proof of existence of the strong magnetic field at early times after a heavy-ion collision regardless of its later time-dependence.

In summary, energy loss per unit length for a light quark with 𝑝_𝑇=15 GeV is about 0.27 GeV/fm at RHIC and 1.7 GeV/fm at LHC, which is comparable to the losses due to interaction with the plasma. Thus, the synchrotron radiation alone is able to account for quenching of jets at LHC with 𝑝_⟂ as large as 20 GeV. Synchrotron radiation is definetely one of missing pieces in the puzzle of the jet energy loss in heavy-ion collisions. Quarks and leptons are expected to be strongly polarized in plasma in the direction parallel or anti-parallel to the magnetic field depending on the sign of their electric charge.

4. Photon Decay

In this section I consider pair production by photon in external magnetic field [55], which is a cross-channel of synchrotron radiation discussed in the previous section. Specifically, we are interested to determine photon decay rate 𝑤 in the process 𝛾𝐵→𝑓𝑓𝐵, where 𝑓 stands for a charged fermion, as a function of photon’s transverse momentum 𝑘_𝑇, rapidity 𝜂, and azimuthal angle 𝜑. Origin of these photons in heavy-ion collisions will not be of interest to us in this section.

Characteristic frequency of a fermion of species 𝑎 of mass 𝑚_𝑎 and charge 𝑧_𝑎𝑒 (𝑒 is the absolute value of electron charge) moving in external magnetic field 𝐵 (in a plane perpendicular to the field direction) is ℏ𝜔_𝐵=𝑧_𝑎𝑒𝐵𝜀, (108) where 𝜀 is the fermion energy. Here—in the spirit of the adiabatic approximation—𝐵 is a slow function of time. Calculation of the photon decay probability significantly simplifies if motion of electron is quasiclassical; that is, quantization of fermion motion in the magnetic field can be neglected. This condition is fulfilled if ℏ𝜔_𝐵≪𝜀. This implies that 𝜀≫√𝑧𝑒𝐵. (109) For RHIC it is equivalent to 𝜀≫𝑚_𝜋, for LHC 𝜀≫4𝑚_𝜋.

Photon decay rate was calculated in [45] and, using the quasi-classical method, in [56]. It reads 𝑤=−_𝑎𝛼_em𝑧³_𝑎𝑒𝐵𝑚_𝑎𝜘_𝑎^∞_4/𝜘𝑎^2/32𝑥^3/2+1/𝜘_𝑎Ai^′(𝑥)𝑥^11/4𝑥^3/2−4/𝜘_𝑎^3/2, (110) where summation is over fermion species and the invariant parameter 𝜘 is defined as 𝜘²_𝑎𝛼=−^em𝑧²_𝑎ℏ³𝑚⁶_𝑎𝐹_𝜇𝜈𝑘^𝜈2=𝛼_em𝑧²_𝑎ℏ³𝑚⁶_𝑎(𝐤×𝐁)², (111) with the initial photon 4-momentum 𝑘^𝜇=(ℏ𝜔,𝐤). With notations of Figure 7, 𝐤=𝑘_𝑧𝐳̂+𝑘_⟂(𝐱cos𝜑+𝐲sin𝜑), where 𝑘_⟂=|𝐤|sin𝜃=ℏ𝜔sin𝜃. Thus, (𝐁×𝐤)²=𝐵²(𝑘²_𝑧+𝑘²_⟂cos²𝜑). Employing ℏ𝜔=𝑘_⟂cosh𝜂 and 𝑘_𝑧=𝑘_⟂sinh𝜂, we write 𝜘_𝑎=ℏ𝑧_𝑎𝑒𝐵𝑚³_𝑎𝑘_⟂sinh²𝜂+cos²𝜑. (112)

Plotted in Figure 9 is the photon decay rate (110) for RHIC and LHC. The survival probability of photons in magnetic field is 𝑃=1−𝑤Δ𝑡, where Δ𝑡 is the time spent by a photon in plasma. Estimating Δ𝑡=10 fm we determine that photon survives with probability 𝑃_RHIC≈97% at RHIC, while only 𝑃_LHC≈80% at LHC. Such strong depletion can certainly be observed in heavy-ion collisions at LHC.

(a)

(b)

Figure 9 

Decay rate of photons moving in reaction plane in magnetic field as a function of transverse momentum 𝑘𝑇: (a) at RHIC, (b) at LHC. Broken lines from bottom to top give contributions of 𝛾→𝑑𝑑,𝛾→𝑢𝑢,𝛾→𝜇+𝜇−, and 𝛾→𝑒+𝑒− channels. Upper solid line is the total rate.

Azimuthal distribution of the decay rate of photons at LHC is azimuthally asymmetric as can be seen in Figure 10 [55]. The strongest suppression is in the 𝐵 field direction, that is, in the direction orthogonal to the reaction plane. At 𝜂≳1 the 𝜑 dependence of 𝜘_𝑎 is very weak which is reflected in nearly symmetric azimuthal shape of the dashed line in Figure 10.

Figure 10 

Azimuthal distribution of the decay rate of photons at different rapidities at LHC. Only contribution of the 𝛾→𝑒+𝑒− channel is shown. Note that the rate 𝑤 vanishes at 𝜂=0 and 𝜑=𝜋/2, that is, in the direction of 𝐁. However, it happens at so small values of |𝜑−𝜋/2| that it is beyond the experimental resolution and is not shown in the figure.

To quantify the azimuthal asymmetry it is customary to expand the decay rate in Fourier series with respect to the azimuthal angle. Noting that 𝑤 is an even function of 𝜑, we have 𝑤𝜑=12𝑤₀+^∞_𝑛=1𝑤_𝑛,𝑤cos𝑛𝜑_𝑛=1𝜋^𝜋_−𝜋𝑤𝜑cos𝑛𝜑𝑑𝜑. (113) In strong fields 𝜘_𝑎≫1. For example, for 𝛾→𝜇⁺𝜇⁻ at RHIC at 𝜑=𝜂=0 and 𝑘_𝑇=1 GeV we get 𝜘_𝜇=19. Therefore, we can expand the rate (110) at large 𝜘_𝑎 as [45] 3𝑤≈^1/65Γ²(2/3)2^4/37𝜋^1/2Γ(7/6)_𝑎𝛼_em𝑒𝐵𝑧³_𝑎𝑚_𝑎𝜘_𝑎^1/3≡𝐴sinh²𝜂+cos²𝜑^1/6,𝜘_𝑎≫1. (114) At 𝜂=0 the Fourier coefficients 𝑤_𝑛 can be calculated analytically using formula 3.631.9 of [34] 𝑤_2𝑘=32^1/3𝐴𝐵)()),𝑤((5/6+𝑘,5/6−𝑘_2𝑘+1=0,𝑘=0,1,2,…, (115) where 𝐵 is the Euler’s beta function and 𝐴 is defined in (114), Substituting these expressions into (113) we find 1𝑤=2𝑤₀1−^∞_𝑘=1√𝜋Γ(−1/6)2^2/3𝐵)()).((5/6+𝑘,5/6−𝑘cos2𝑘𝜑 (116) The first few terms in this expansion read 1𝑤=2𝑤₀21−5cos2𝜙+14,55cos4𝜙−⋯ (117)

What is measured experimentally is not the decay rate, but rather the photon spectrum. This spectrum is modified by the survival probability 𝑃 which is obviously azimuthally asymmetric. To quantify this asymmetry, we write using (113) 𝑃=𝑃1+^∞_𝑘=1𝑣_2𝑘,𝑣cos2𝜑𝑘_2𝑘=−1−𝑃𝑃2𝑤_2𝑘𝑤₀, (118) where 𝑃=⟨1−𝑤Δ𝑡⟩_𝜑=1−𝑤₀Δ𝑡 is the survival probability averaged over the azimuthal angle. Since 𝑤₀Δ𝑡≪1, as can be seen in Figure 9, we can estimate using (114) and (115) 𝑣_2𝑘≈−2𝑤_2𝑘𝑤₀𝑤₀Δ𝑡=−2𝑤_2𝑘𝑤₀Δ𝑡56^2/3Γ(2/3)7𝜋_𝑎𝛼_em(𝑒𝐵)^2/3𝑧_𝑎^8/3𝑘_𝑇^1/3. (119) In particular, the “elliptic flow” coefficient is [55] 𝑣₂=Δ𝑡26^2/3Γ(2/3)7𝜋_𝑎𝛼_em(𝑒𝐵)^2/3𝑧_𝑎^8/3𝑘_𝑇^1/3. (120) For example, at 𝑘_𝑇=1 GeV and Δ𝑡∼10 fm/c one expects 𝑣₂≃2% at RHIC and 𝑣₂≃14% at LHC only due to the presence of magnetic field. We see that decay of photons in external magnetic field significantly contributes to the photon asymmetry in heavy-ion collisions along with other possible effects.

In summary, I calculated photon pair-production rate in external magnetic field created in off-central heavy-ion collisions. Photon decay leads to depletion of the photon yield by a few percent at RHIC and by as much as 20% at the LHC. The decay rate depends on the rapidity and azimuthal angle. At midrapidity the azimuthal asymmetry of the decay rate translates into asymmetric photon yield and contributes to the “elliptic flow.” Let me also quote a known result that photons polarized parallel to the field are 3/2 times more likely to decay than those polarized transversely [45]. Therefore, polarization of the final photon spectrum perpendicular to the field is a signature of existence of strong magnetic field. Finally, photon decay necessarily leads to enhancement of dilepton yield.

5. Quarkonium Dissociation in Magnetic Field

5.1. Effects of Magnetic Field on Quarkonium

Strong magnetic field created in heavy-ion collisions generates a number of remarkable effects on quarkonium production, some of which I will describe in this section. Magnetic field can be treated as static if the distance 𝜆 over which it significantly varies is much larger than the quarkonium radius. If Δ𝑡 is magnetic field life-time, then 𝜆∼𝑐Δ𝑡. For a quarkonium with binding energy 𝜀_𝑏 and radius 𝛼_𝑠/𝜀_𝑏, the quasi-static approximation applies when 𝜀_𝑏𝜆/𝛼_𝑠≫1. Estimating conservatively 𝜆∼2 fm we get for 𝐽/𝜓: 𝜀_𝑏𝜆/𝛼_𝑠≈23, which is comfortably large to justify the quasi-static approximation, where I assumed that 𝜀_𝑏 is given by its vacuum value. As temperature 𝑇 increases 𝜀_𝑏 drops. Temperature dependence of 𝜀_𝑏 is model dependent; however, it is certain that eventually it vanishes at some finite temperature 𝑇₀. Therefore, only in the close vicinity of 𝑇₀, that is, at very small binding energies, the quasi-static approximation is not applicable. I thus rely on the quasi-static approximation to calculate 𝐽/𝜓 dissociation [57, 58].

Magnetic field has a three-fold effect on quarkonium.(1)Lorentz ionization. Consider quarkonium traveling with constant velocity in magnetic field in the laboratory frame. Boosting to the quarkonium comoving frame, we find mutually orthogonal electric and magnetic fields given by (121a), (121b), and (122). In the presence of an electric field quark and antiquark have a finite probability to tunnel through the potential barrier thereby causing quarkonium dissociation. In atomic physics such a process is referred to as Lorentz ionization. In the nonrelativistic approximation, the tunneling probability is of order unity when the electric field 𝐸 in the comoving frame satisfies 𝑒𝐸≳𝑚^1/2𝜀_𝑏^3/2 (for weakly bound states), where 𝑚 is quark mass; see (144). This effect causes a significant increase in quarkonium dissociation rate; numerical calculation for 𝐽/𝜓 is shown in Figure 13.(2)Zeeman effect. Energy of a quarkonium state depends on spin 𝑆, orbital angular momentum 𝐿, and total angular momentum 𝐽. In a magnetic field these states split; the splitting energy in a weak field is Δ𝑀=(𝑒𝐵₀/2𝑚)𝑔𝐽_𝑧, where 𝐽_𝑧=−𝐽,−𝐽+1,…,𝐽 is projection of the total angular momentum on the direction of magnetic field, 𝑚 is quark mass, and 𝑔 is Landé factor depending on 𝐽, 𝐿, and 𝑆 in a well-known way; see, for example, [59]. For example, 𝐽/𝜓 with 𝑆=1,𝐿=0, and 𝐽=1 (𝑔≈2) splits into three states with 𝐽_𝑧=±1,0 and with mass difference Δ𝑀=0.15 GeV, where we used 𝑒𝐵₀=15𝑚²_𝜋. Thus, the Zeeman effect leads to the emergence of new quarkonium states in plasma.(3)Distortion of the quarkonium potential in magnetic field. This effect arises in higher-order perturbation theory and becomes important at field strengths of order 𝐵∼3𝜋𝑚²/𝑒³ [60]. This is 3𝜋/𝛼 times stronger than the critical Schwinger’s field. Therefore, this effect can be neglected at the present collider energies.

Some of the notational definitions used in this section: 𝐕 and 𝐏, are velocity and momentum of quarkonium in the lab frame; 𝑀 is its mass; 𝐩 is the momentum of quark or anti-quark in the comoving frame; 𝑚 is its mass; 𝐁₀ is the magnetic field in the lab frame, 𝐄 and 𝐁 are electric and magnetic fields in the comoving frame; 𝛾_𝐿 is the quarkonium Lorentz factor; and 𝛾 is a parameter defined in (139). I use Gauss units throughout the section; note that expressions 𝑒𝐵, 𝑒𝐸, and 𝑒𝐵₀ are the same in Gauss and Lorentz-Heaviside units.

5.2. Lorentz Ionization: Physical Picture

In this section I focus on Lorentz ionization, which is an important mechanism of 𝐽/𝜓 suppression in heavy-ion collisions [57, 58]. Before we proceed to analytical calculations it is worthwhile to discuss the physics picture in more detail in two reference frames: the quarkonium proper frame and the lab frame. In the quarkonium proper frame the potential energy of, say, antiquark (with 𝑒<0) is a sum of its potential energy in the binding potential and its energy in the electric field −𝑒𝐸𝑥, where 𝑥 is the electric field direction; see Figure 11. Since |𝑒|𝐸𝑥 becomes large and negative at large and negative 𝑥 (far away from the bound state) and because the quarkonium potential has finite radius, this region opens up for the motion of the antiquark. Thus there is a quantum mechanical probability to tunnel through the potential barrier formed on one side by the vanishing quarkonium potential and on the other by increasing absolute value of the antiquark energy in electric field. Of course the total energy of the antiquark (not counting its mass) is negative after tunneling. However, its kinetic energy grows proportionally to 𝑒𝐸𝑥 as it goes away. By picking up a light quark out of vacuum it can hadronize into a 𝐷-meson.

Figure 11 

Effective potential 𝑈(𝑥)=𝑚2+(𝑝𝑧+𝑒𝐵𝑥)2+𝑝2𝑥−𝑒𝐸𝑥−𝑚2+𝑝2𝑥+𝑝2𝑥 for 𝑝𝑥=0,𝑝𝑧=𝑚/6,and𝐵=𝑚 (except the blue line where 𝐵=0). The width of the potential barrier decreases with 𝐸 and increases with 𝐵. 1−𝜖0 corresponds to the binding energy in units of 𝑚.

If we now go to the reference frame where 𝐸=0 and there is only magnetic field 𝐵 (we can always do so since 𝐸<𝐵), then the entire process looks quite different. An energetic quarkonium travels in external magnetic field and decays into quark-antiquark pair that can later hadronize into 𝐷-mesons. This happens in spite of the fact that 𝐽/𝜓 mass is smaller than masses of two 𝐷-mesons due to additional momentum 𝑒𝐀 supplied by the magnetic field. Similarly a photon can decay into electron-positron pair in external magnetic field.

5.3. Quarkonium Ionization Rate

5.3.1. Comoving Frame

Consider a quarkonium traveling with velocity 𝐕 in constant magnetic field 𝐁₀. Let 𝐁 and 𝐄 be magnetic and electric fields in the comoving frame, and let subscripts ‖ and ⟂ denote field components parallel and perpendicular to 𝐕 correspondingly. Then,𝐸_‖=0,𝐄_⟂=𝛾_𝐿𝐕×𝐁₀,𝐵_‖=𝐁₀⋅𝐕𝑉,𝐁_⟂=𝛾_𝐿𝐕×𝐁₀×𝐕𝑉², (121a)(121b)where 𝛾_𝐿=(1−𝑉²)^−1/2. Clearly, in the comoving frame 𝐁⋅𝐄=0. If quarkonium travels at angle 𝜙 with respect to the magnetic field in the laboratory frame, then 𝐵=𝐵₀cos²𝜙1−𝛾²_𝐿+𝛾²_𝐿,𝐸=𝐵₀𝛾_𝐿𝑉sin𝜙. (122) We choose 𝑦 and 𝑥 axes of the comoving frame such that 𝐁=𝐵𝐲 and 𝐄=𝐸𝐱. A convenient gauge choice is 𝐀=−𝐵𝑥𝐳̂ and 𝜑=−𝐸𝑥. For a future reference we also define a useful dimensionless parameter 𝜌 [61]: 𝐸𝜌=𝐵=𝛾_𝐿𝑉sin𝜙cos²𝜙1−𝛾²_𝐿+𝛾²_𝐿. (123) Note that (i) 0≤𝜌≤1 because 𝐵²−𝐸²=𝐵²₀≥0 and (ii) when quarkonium moves perpendicularly to the magnetic field 𝐁₀, 𝜌=𝑉.

5.3.2. WKB Method

I assume that the force binding 𝑞 and 𝑞 into quarkonium as a short-range one, that is, (𝑀𝜀_𝑏)^1/2𝑅≪1, where 𝜀_𝑏 and 𝑀 are binding energy and mass of quarkonium, respectively, and 𝑅 is the radius of the nuclear force given by 𝑅≈(𝛼_𝑠/𝜎)^1/2, where 𝜎=1 GeV/fm is the string tension. For example, the binding energy of 𝑐 and 𝑐 in 𝐽/𝜓 in vacuum is 𝜀_𝑏=0.64GeV≪𝑀/𝑅²=𝑀𝜎/𝛼_𝑠≈3 GeV. This approximation is even better at finite temperature on account of 𝜀_𝑏 decrease. Regarding 𝐽/𝜓 as being bound by a short-range force enables us to calculate the dissociation probability 𝑤 with exponential accuracy 𝑤≈𝑒^−𝑓, independently of the precise form of the quarkonium wave function. This is especially important, since solutions of the relativistic two-body problem for quarkonium are not readily available.

It is natural to study quarkonium ionization in the comoving frame [57]. As explained in the Introduction, ionization is quantum tunneling through the potential barrier caused by the electric field 𝐄. In this subsection I employ the quasiclassical WKB approximation to calculate the quarkonium decay probability 𝑤. For the gauge choice specified in Section 5.3.1, quark energy 𝜀₀(𝜀₀<𝑚) in electromagnetic field can be written as 𝜀₀=𝑚²+(𝐩−𝑒𝐀)²=+𝑒𝜑𝑚²+𝑝_𝑧+𝑒𝐵𝑥²+𝑝²_𝑥+𝑝²_𝑦−𝑒𝐸𝑥. (124) In terms of 𝜀₀, quarkonium binding energy is 𝜀_𝑏=𝑚−𝜀₀. To simplify notations, we will set 𝑝_𝑥=0, because the quark moves constant momentum along the direction of magnetic field.

The effective potential 𝑈(𝑥)=𝜀₀(𝑥)−𝑚²+𝐩² corresponding to (124) is plotted in Figure 11. We can see that the tunneling probability is finite only if 𝐸>0. It is largest when 𝐵=0. It has been already noted before in [61–63] that the effect of the magnetic field is to stabilize the bound state. In spite of the linearly rising potential (at 𝐵>𝐸) tunneling probability is finite as the result of rearrangement of the QED vacuum in electric field.

Ionization probability of quarkonium equals its tunneling probability through the potential barrier. The later is given by the transmission coefficient 𝑤=𝑒^{−2∫𝑦10−𝑝2𝑦𝑑𝑦}≡𝑒^−𝑓. (125) In the nonrelativistic approximation one can also calculate the preexponential factor, which appears due to the deviation of the quark wave function from the quasi-classical approximation. This is discussed later in Section 5.5.2. We now proceed with the calculation of function 𝑓. Since 𝐵>𝐸 (125) can be written as [57] 𝑝²_𝑥=−𝑒²𝐵²−𝐸²𝑥−𝑥₁𝑥−𝑥₂, (126) where 𝑥_1,2=𝜀₀𝐸−𝑝_𝑧𝐵∓𝜀₀𝐸−𝑝_𝑧𝐵²−𝐵²−𝐸²−𝜀²₀+𝑚²+𝑝²_𝑧𝑒𝐵²−𝐸². (127) Define dimensionless variables 𝜖₀=𝜀₀/𝑚 and 𝑞=𝑝_𝑧/𝑚. Integration in (74) gives 𝑓𝑚²=−𝜖²₀+1+𝑞²𝜖₀𝐸−𝑞𝐵𝑒𝐵²−𝐸²−𝜖₀𝐸−𝑞𝐵²−𝐵²−𝐸²−𝜖²₀+1+𝑞²𝑒𝐵²−𝐸^23/2𝜖×ln⁰𝐸−𝑞𝐵+𝐵²−𝐸²−𝜖²₀+1+𝑞²𝜖₀𝐸−𝑞𝐵²−𝐵²−𝐸²𝜖²₀+1+𝑞². (128)𝑤=𝑒^𝑓 gives the corresponding ionization probabilities, at a given 𝑞. The largest probability corresponds to smallest 𝑓, which occurs at momentum 𝑞_𝑚 determined by [62] 𝜕𝑓𝑞_𝑚𝜕𝑞_𝑚=0. (129) Using (128) and parameter 𝜌 defined in (123), we find [57] 𝜌𝜖₀−𝜌𝑞_𝑚1−𝜌²𝜖ln⁰𝜌−𝑞_𝑚+1−𝜌²−𝜖²₀+1+𝑞²_𝑚𝜖₀−𝜌𝑞_𝑚²−1+𝜌²=−𝜖²₀+1+𝑞²_𝑚1−𝜌². (130) This is an implicit equation for the extremal momentum 𝑞_𝑚=𝑞_𝑚(𝜖₀,𝜌). Substituting 𝑞_𝑚 into (128) one obtains 𝑓=𝑓(𝜖₀,𝜌), which by means of (125) yields the ionization probability. The quasi-classical approximation that we employed in this section is valid in as much as 𝑓(𝑞_𝑚)≫1.

In order to compare with the results obtained in [62] using the imaginary time method, we can rewrite (130) in terms of an auxiliary parameter 𝜏₀ as𝜏₀=1−𝜌²−𝜖²₀+1+𝑞²_𝑚𝜌𝜖₀−𝜌𝑞_𝑚,tanh𝜏₀𝜏₀𝜖=𝜌⁰−𝜌𝑞_𝑚𝜖₀𝜌−𝑞_𝑚. (131a)(131b) Taking advantage of these equations, (128) can be cast into a more compact form as follows 𝑓_𝑚=𝑚²𝜏₀𝜌𝑒𝐸1−𝜌²1−𝜖₀𝜖₀−𝑞_𝑚𝜌, (132) where we denoted 𝑓_𝑚=𝑓(𝑞_𝑚). This agrees with results of [62].

5.3.3. Special Case: Crossed Fields

An important limiting case is crossed fields 𝐸=𝐵. Since also 𝐄⟂𝐁 (see Section 5.3.1), both field invariants vanish. Nevertheless, quarkonium ionization probability is finite [62]. This limit is obtained by taking 𝜌→1 in the equations from the previous section. Employing (131a) and (131b) we get the following condition for extremum: 𝜖²₀−1+2𝑞²_𝑚−3𝜖₀𝑞_𝑚=0, (133) with the solution 𝑞_𝑚=143𝜖₀−𝜖²₀+8. (134) Substituting into (132) produces 𝑓_𝑚=23𝑚²𝑒𝐸−𝜖²₀+1+𝑞²_𝑚^3/2𝜖₀−𝑞_𝑚. (135)

5.4. Nonrelativistic Approximation

A very useful approximation of the relativistic formulas derived in the previous section is the nonrelativistic limit because (i) it provides a very good numerical estimate (see Figure 12), and (ii) it allows us to eliminate the parametric dependence in (128), (130) and write 𝑓(𝑞_𝑚) explicitly in terms of 𝜌 and 𝜖₀, and (iii) spin effects can be accounted for [57, 58].

Figure 12 

Dimensionless function 𝑓𝑚𝑒𝐸/𝑚2 versus 𝜖𝑏 for different values of 𝜌. The solid line is the full relativistic calculation, the dashed line is the nonrelativistic approximation. 𝐽/𝜓 binding energy in vacuum corresponds to 𝜖𝑏=0.68.

(a)

(b)

Figure 13 

Dissociation rate of 𝐽/𝜓 at 𝑒𝐵0=15𝑚2𝜋, 𝜙=𝜋/2 (in the reaction plane), and 𝜂=0 (midrapidity) as a function of (a) 𝑃⟂ at 𝜀𝑏=0.16 GeV and (b) 𝜀𝑏 at 𝑃⟂=1 GeV.

5.4.1. Arbitrary Binding

Motion of a particle can be treated non-relativistically if its momentum is much less than its mass. In such a case 𝜀₀≈𝑚 or 𝜀_𝑏=𝑚−𝜀₀≪𝑚. Additionally, motion of a charged particle in electromagnetic field is non-relativistic if 𝐸≪𝐵. Indeed, the average velocity of a nonrelativistic particle is of order 𝑣∼𝐸/𝐵=𝜌. Thus, the non-relativistic limit is obtained by taking the limits 𝜖_𝑏=𝜀_𝑏/𝑚≪1 and 𝜌≪1. In these limits the extremum conditions (131a), (131b) reduce to𝜏₀=2𝜖_𝑏+𝑞²_𝑚𝜌,tanh𝜏₀𝜏₀=𝜌𝜌−𝑞_𝑚. (136a)(136b)Out of two solution to (136a) we pick the following one: 𝑞_𝑚=−𝜏²₀𝜌²−2𝜖_𝑏. (137) The sign of 𝑞_𝑚 is fixed using (136b) by noticing that tanh𝜏₀/𝜏₀<1. Eliminating 𝑞_𝑚 gives 𝜏²₀−𝜏₀coth𝜏₀−1²=𝛾², (138) where 𝛾=2𝜖_𝑏𝜌, (139) where 𝛾 is analogous to the adiabaticity parameter of Keldysh [64]. Taking the non-relativistic limit of (132) and using (137) yields 𝑓_𝑚=2𝑚²2𝜖_𝑏^3/2𝑔𝛾,3𝑒𝐸 (140) where 𝑔(𝛾) is the Keldysh function [64] 𝑔𝛾=3𝜏₀12𝛾1−𝛾𝜏²₀𝛾²−1^1/2. (141)

In Figure 12 we show the dimensionless ratio 𝑓_𝑚𝑒𝐸/𝑚² as a function of the binding energy 𝜖_𝑏 (in units of 𝑚) for several values of 𝜌. The vacuum binding energy of 𝐽/𝜓 corresponds to 𝜖_𝑏=0.68. We observe an excellent agreement between the full relativistic calculation and the non-relativistic approximation. At 𝜌=0.9 and 𝜖_𝑏=0.68 the difference between the two lines is 10% and can be further improved by considering higher order corrections to 𝑓_𝑚 [63].

5.4.2. Weak Binding

Of special interest is the limit of weak binding 𝛾≪1; that is, 2𝜖_𝑏≪𝜌. Expanding (138) at small 𝛾 and 𝜏₀ yields 𝜏₀1=𝛾1+𝛾18², (142) and substituting into (141) and subsequently into (140) yields 𝑓_𝑚=23𝑚²𝑒𝐸2𝜖_𝑏^3/2. (143) Hence, the quarkonium dissociation probability reads [65] −2𝑤=exp32𝜖_𝑏𝑚^3/2.𝑚𝑒𝐸 (144) Since the quasi-classical approximation employed in this paper is valid if 𝑓(𝑞_𝑚)≫1, it follows that the binding energy must satisfy (𝑒𝐸)^2/3𝑚^1/3≪𝜀_𝑏≪𝜌²𝑚. (145) Note also that we work in the approximation of the short-range binding potential meaning that 2𝜖_𝑏≪1/(𝑚𝑅); see Section 5.1.

5.4.3. Strong Binding

In the limit 𝛾≫1, (138) and (141) imply that 𝜏₀=𝛾²2,𝑔𝛾=3𝛾8. (146) Substituting (146) into (140), we derive 𝑓_𝑚=𝜀²_𝑏𝐵𝑒𝐸𝐸. (147) Thus, quarkonium dissociation probability in the case of strong binding is −𝜀𝑤=exp^2𝑏𝐵𝑒𝐸𝐸. (148) This formula is valid when 𝜌²𝑚,𝑒𝐸𝜌≪𝜀_𝑏≪1𝑅. (149)

5.4.4. Contribution of Quark Spin

So far I have neglected the contribution of quark spin. In order to take into account the effect of spin interaction with the external field, we can use squared Dirac equation for a bispinor 𝜓 as follows: 𝜀−𝑒𝜑²−(𝐩−𝑒𝐀)²−𝑚²+𝑒𝚺⋅𝐁−𝑖𝑒𝜶⋅𝐄𝜓=0, (150) where 𝚺=𝝈00𝝈,𝜶=𝝈00−𝝈. (151) Operators Σ_𝑦 and 𝛼_𝑥 do not commute. Therefore, in order to apply the WKB method for calculation of the ionization probability one actually needs to square (150), which leads to a differential equation of the fourth order in derivatives. The problem becomes more tractable in the non-relativistic case and for crossed fields. Spin effects in crossed fields were discussed in [66].

With quark spin taken into account, the non-relativistic version of (124) becomes 1𝑝2𝑚^𝑧+𝑒𝐵𝑥²+𝑝²_𝑥𝜇−𝑒𝐸𝑥−𝑠𝐬⋅𝐁=−𝜀_𝑏, (152) and hence 𝑝²_𝑥=2𝑚−𝜀_𝑏+𝜇𝑠𝐬⋅𝐁+𝑒𝐸𝑥−𝑝_𝑧+𝑒𝐵𝑥², (153) where 𝜇 is the quark magnetic moment and 𝑠 is the projection of spin in the direction of the magnetic field. For a point quark, 𝜇=𝜇_𝐵=𝑒ℏ/2𝑚𝑐. The effect of quark spin on quarkonium dissociation probability can be taken into account by replacing 𝜀_𝑏→𝜀^′_𝑏=𝜀_𝑏−(𝜇/𝑠)𝐬⋅𝐁 in formulas for 𝑓_𝑚. With this replacement, all results of this section apply to a particle with spin. Note that effective binding energy 𝜀^′_𝑏 decreases if spin is parallel to the magnetic field and increases if it is antiparallel. In particular, in the case of weak binding 𝑤=_𝑠=±1/2−2exp32𝜀_𝑏𝑚+2𝑠𝑒𝐵^3/2.𝑚𝑒𝐸 (154) Since the non-relativistic limit provides a good approximation of the full relativistic formulas, we will implement the quark spin dependence using the non-relativistic prescription [57, 58].

5.5. Effect of Electric Field Produced in the Lab Frame

5.5.1. Origin of Electric Field in the Lab Frame

So far I have entirely neglected possible existence of electric field in the lab frame. This field, which we will be denoted by 𝐄₀, can have two origins: (i) asymmetry of nucleon distributions in the colliding heavy ions (see Figure 3(b)) and (ii) chiral magnetic effect (CME) [5, 67–72], which has recently attracted a lot of attention. In a nutshell, if a metastable 𝑃- and 𝐶𝑃-odd bubble is induced by axial anomaly in the hot nuclear matter, then in the presence of external magnetic field 𝐁₀ the bubble generates an electric field which is parallel to the magnetic one. According to [68] the value of the electric field 𝐄₀ in the bubble is 𝐄₀=−𝑁_𝑐𝑓𝑒²_𝑓4𝜋²Θ𝑁_𝑓𝐁₀2=−3𝛼Θ𝜋𝐁₀, (155) where the sum runs over quark flavors 𝑓 and it is assumed that only three lightest flavors contribute. The value of the Θ-angle fluctuates from event to event. CME refers to the macroscopic manifestation of this effect—separation of electric charges with respect to the reaction plane. This effect is a possible explanation of experimentally observed charge asymmetry fluctuations [73–75].

No matter what is the origin of electric field in the lab frame, it averages to zero over an ensemble of events. We are interested to know the effect of this field on quarkonium dissociation—this is the problem we are turning to now [58].

5.5.2. Quarkonium Dissociation Rate

Ionization probability of quarkonium equals its tunneling probability through the potential barrier. In the WKB approximation the later is given by the transmission coefficient and was calculated in Section 5.3. In this method contribution of the quark spin can be easily taken into account. Another method of calculating the ionization probability, the imaginary time method [76–78], was employed in [61–63]. It also yielded in the non-relativistic approximation the preexponential factor that appears due to the deviation of the quark wave function from the quasi-classical approximation. Such a calculation requires matching quark wave function inside and outside the potential barrier [65]. Extension of this approach to the relativistic case is challenging due to analytical difficulties of the relativistic two-body problem. Fortunately, it was argued in Section 5.3, that the non-relativistic approximation provides a very good accuracy in the 𝜀_𝑏≪𝑚 region, which is relevant in the quarkiononium dissociation problem [57, 62].

Given the electromagnetic field in the laboratory frame 𝐁₀, 𝐄₀, the electromagnetic field 𝐁, 𝐄 in the comoving frame moving with velocity 𝐕 is given by𝐄=𝐸₀𝛾_𝐿𝐛₀+𝜌₀⁻¹𝐕×𝐛₀−𝛾_𝐿−1𝐕𝐕⋅𝐛₀𝑉²,𝐁=𝐵₀𝛾_𝐿𝐛₀−𝜌₀𝐕×𝐛₀−𝛾_𝐿−1𝐕𝐕⋅𝐛₀𝑉², (156a)(156b) where 𝐛₀=𝐁₀/𝐵₀ is a unit vector in the magnetic field direction, 𝜌₀=𝐸₀/𝐵₀=2𝛼|Θ|/3𝜋 (see (155)) and 𝛾_𝐿=1/1−𝑉². It follows from (156a) and (156b) that𝐸=𝐸₀1+𝛾²_𝐿𝐛₀×𝐕²1+𝜌₀⁻²,𝐵=𝐵₀1+𝛾²_𝐿𝐛₀×𝐕²1+𝜌²₀. (157a)(157b) Using (157a) and (157b), we find that the angle 𝜃 between the electric and magnetic field in the comoving frame is cos𝜃=𝐄⋅𝐁=1𝐸𝐵1+𝛾²_𝐿𝐛₀×𝐕²1+𝜌₀⁻²1+𝛾²_𝐿𝐛₀×𝐕²1+𝜌²₀, (158) where we used the relativistic invariance of 𝐄⋅𝐁.

It is useful to introduce dimensionless parameters 𝛾, 𝜖 and 𝜌 as [62] 1𝛾=𝜌2𝜀_𝑏𝑚𝐸,𝜌=𝐵,𝜖=𝑒𝐸𝑚²𝑚2𝜀_𝑏^3/2, (159) where 𝑚 is quark mass and 𝜀_𝑏 is quarkonium binding energy. I will treat the quarkonium binding potential in the non-relativistic approximation, which provides a very good accuracy to the dissociation rate [57, 62]. The quarkonium dissociation rate in the comoving frame in the non-relativistic approximation is given by [61] 𝑤=8𝜀_𝑏𝜖𝑃𝛾,𝜃𝐶²𝛾,𝜃𝑒^{−(2/3𝜖)𝑔(𝛾,𝜃)}, (160) where function 𝑔 reads 𝑔=3𝜏₀12𝛾1−𝛾𝜏²₀𝛾²−1^1/2𝜏sin𝜃−²⁰3𝛾²cos²𝜃. (161) and functions 𝑃 and 𝐶 are given in the following formulas: 𝛾𝑃=²𝜏₀𝜏₀coth𝜏₀+sinh𝜏₀cosh𝜏₀𝜏₀−2sin²𝜃+sinh²𝜏₀cos²𝜃^−1/2,𝜏𝐶=expln⁰+2𝛾^𝜏0₀𝛾𝑑𝜏𝜉(𝜏)−1𝜏₀,𝜉(𝜏)=1−𝜏4𝜏²₀−𝜏²²cos²𝜃+𝜏²₀cosh𝜏₀−cos𝜏sinh𝜏₀²−sinh𝜏sinh𝜏₀−𝜏𝜏₀²sin²𝜃^1/2. (162) The contribution of quark spin is taken into account by replacing 𝜀_𝑏→𝜀^′_𝑏=𝜀_𝑏−(𝑒/𝑚)𝐬⋅𝐁 [57]. Function 𝑔 represents the leading quasi-classical exponent, 𝑃 is the pre-factor for the 𝑆-wave state of quarkonium, and 𝐶 accounts for the Coulomb interaction between the valence quarks. Parameter 𝜏₀ satisfies the following equation: 𝜏²₀−sin²𝜃𝜏₀coth𝜏₀−1²=𝛾² (163) which establishes its dependence on 𝜃 and 𝛾. Note that in the limit 𝐸→0 the dissociation rate (160) exponentially vanishes. This is because pure magnetic field cannot force a charge to tunnel through a potential barrier.

In the case that mechanism (i) is responsible for generation of electric field, 𝐄₀ is the field permitting the entire plasma in a single event. Event average is then obtained by averaging (160) over an ensemble of events. In the case that mechanism (ii) is operative, averaging is more complicated. Equation (160) gives the quarkonium dissociation rate in a bubble with a given value of Θ. Its derivation assumes that the dissociation process happens entirely inside a bubble and that Θ is constant inside the bubble. Since in a relativistic heavy-ion collision many bubbles can be produced with a certain distribution of Θs (with average ⟨Θ⟩=0) more than one bubble can affect the dissociation process. This will result in a distractive interference leading to reduction of the 𝐶𝑃-odd effect on quarkonium dissociation. However, if a typical bubble size 𝑅₀ is much larger than the size of quarkonium 𝑅_𝐽, then the dissociation is affected by one bubble at a time independently of others, and hence the interference effect can be neglected. In this case (74) provides, upon a proper average, a reasonable estimate of quarkonium dissociation in a heavyion collision. We can estimate the bubble size as the size of the sphaleron, which is of the order of the chromo-magnetic screening length ~1/𝑔²𝑇, whereas the quarkonium size is of the order 𝛼_𝑠/𝜀_𝑏. Consequently, at small coupling and below the zero-field dissociation temperature (i.e., when 𝜀_𝑏 is not too small) 𝑅₀ is parametrically much larger than 𝑅_𝐽. A more quantitative estimate of the sphaleron size is 𝑅₀≃1.2/𝛼_𝑠𝑁_𝑐𝑇≃0.4 fm [79]; whereas 𝑅_𝐽≃𝛼_𝑠/𝜀_𝑏≃0.1--0.2 fm. Thus, based on this estimate bubble interference can be neglected in the first approximation. However, since the ratio 𝑅_𝐽/𝑅₀ is actually not so small this effect nevertheless warrants further investigation.

To obtain the experimentally observed 𝐽/𝜓 dissociation rate we need to average (74) over the bubbles produced in a given event and then over all events. To this end it is important to note that because the dissociation rate depends only on 𝜌²₀ it is insensitive to the sign of the 𝐄₀ field, or, in other words, it depends only on absolute value of Θ but not on its sign. Therefore, it stands to reason that although the precise distribution of Θs is not known, (74) gives an approximate event average with parameter Θ representing a characteristic absolute value of the theta angle.

5.5.3. Limiting Cases

Before I proceed with the numerical calculations, let us consider for illustration several limiting cases. If quarkonium moves with non-relativistic velocity, then in the comoving frame electric and magnetic fields are approximately parallel 𝜃≈0, whereas in the ultrarelativistic case they are orthogonal 𝜃≈𝜋/2; see (158). In the later case the electromagnetic field in the comoving frame does not depend on 𝐸₀ as seen in (157a) and (157b) and therefore the dissociation rate becomes insensitive to the CME. In our estimates I will assume that 𝜌₀<1 which is the relevant phenomenological situation. Indeed, it was proposed in [68] that 𝜌₀∼𝛼≪1 produces charge fluctuations with respect to the reaction plane of the magnitude consistent with experimental data.

(1) 𝜃≳0; that is, electric and magnetic fields are approximately parallel. This situation is realized in the following two cases. (i) Non-relativistic quarkonium velocities: 𝑉≪𝜌₀ or (ii) motion of quarkonium at small angle 𝜙 to the direction of the magnetic field 𝐛₀: 𝜙≪𝜌₀/𝛾_𝐿𝑉. In both cases 𝐸≈𝐸₀ and 𝐵≈𝐵₀. This is precisely the case where the dissociation rate exhibits its strongest sensitivity to the strength of the electric field 𝐄₀ generated by the local parity violating QCD effects. Depending on the value of the 𝛾 parameter defined in (159) we can distinguish the case of strong electric field 𝛾≫1 and weak electric field 𝛾≪1 [63]. In the former case, 𝑔=(3/8)𝛾, 𝑃=(8/𝑒)^1/2𝛾𝑒^−𝛾2/2, and 𝐶=𝑒^𝜋𝛾/2/𝛾. Substituting into (74) the dissociation rate reads 𝑤=8𝜀_𝑏𝜖𝛾8𝑒𝑒^−𝛾2/2𝑒^{−𝛾/4𝜖}=16𝜀²_𝑏𝑚𝑒𝐵₀8𝑒𝑒^{−𝜀𝑏/𝜌20𝑚}𝑒^{−𝜀2𝑏/𝜌0𝑒𝐸0},𝛾≫1. (164) In the later case, 𝑔=𝑃=𝐶=1 and 𝑤=8𝜀_𝑏𝜖𝑒^−2/3𝜖=8𝜀_𝑏𝑚²𝑒𝐸₀2𝜀_𝑏𝑚^3/2𝑒^{−(2𝑚2/3𝑒𝐸0)(2𝜀𝑏/𝑚)3/2},𝛾≪1, (165) where the electromagnetic field in the comoving frame equals one in the laboratory frame as was mentioned previously.

(2) 𝜃∼𝜋/2; that is, electric and magnetic fields are approximately orthogonal (note that the limit 𝛾≫1 is different in 𝜃=𝜋/2 and 𝜃<(𝜋/2) cases [61]). This occurs for an ultra-relativistic motion of quarkonium 𝑉→1. In this case 𝐵=𝐸=𝐵₀𝛾_𝐿𝐛₀×𝐕1+𝜌²₀. (166) This case was discussed in detail in our previous paper [57]. In particular for 𝛾≪1, we get 𝑤=8𝜀_𝑏𝑚²𝑒𝐸2𝜀_𝑏𝑚^3/2𝑒^{−(2𝑚2/3𝑒𝐸)(2𝜀𝑏/𝑚)3/2}. (167) Due to (164) and (167) dependence of 𝑤 on 𝐸₀ is weak unless 𝜌₀≫1.

5.6. Dissociation Rate of 𝐽/𝜓

One of the most interesting applications of this formalism is calculation of the dissociation rate of 𝐽/𝜓 which is considered a litmus test of the quark-gluon plasma [80]. Let 𝑧 be the heavy ions collision axis; heavy-ion collision geometry implies that 𝐛₀⋅𝐳̂=0. The plane containing 𝑧-axis and perpendicular to the magnetic field direction is the reaction plane. We have 𝐛₀×𝐕²=𝑉²_𝑧+𝑉²_⟂sin²𝜙, (168) where 𝜙 is the angle between the directions of 𝐁₀ and 𝐕_⟂, and I denoted vector components in the 𝑥𝑦-plane by the subscript ⟂. We can express the components of the quarkonium velocity 𝐕 in terms of the rapidity 𝜂 as 𝑉_𝑧=tanh𝜂, 𝑉_⟂=𝑃_⟂/(𝑀_⟂cosh𝜂), where 𝐏 and 𝑀 are the quarkonium momentum and mass and 𝑀²_⟂=𝑀²+𝑃²_⟂.

Results of numerical calculations are exhibited in Figures 13–15 [58]. In Figure 13 I show the dissociation rate of 𝐽/𝜓 for several values of the electric field 𝐄₀ induced by the Chiral Magnetic Effect. Note that the typical size of the medium traversed by a quarkonium in magnetic field can be estimated very conservatively as a few fm. Therefore, 𝑤∼0.3--0.5 fm⁻¹ corresponds to complete destruction of 𝐽/𝜓s. This means that in the magnetic field of strength 𝑒𝐵₀∼15𝑚²_𝜋 all 𝐽/𝜓s with 𝑃_⟂≳0.5 GeV are destroyed independently of the strength of 𝐸₀. Since magnetic field strength decreases towards the QGP periphery, most of 𝐽/𝜓 surviving at later times originate from that region. Effect of electric field 𝐄₀ is strongest at low 𝑃_⟂, which is consistent with our discussion in the previous section. The dissociation rate at low 𝑃_⟂ exponentially decreases with decrease of 𝐸₀. Probability of quarkonium ionization by the fields below 𝐸₀≲0.1𝐵₀(i.e., 𝜌₀≲0.1) is exponentially small. This is an order of magnitude higher than the estimate 𝜌₀∼𝛼 of electric field due to CME effect as proposed in [68].

Figure 14 

Contour plot of the dissociation rate of 𝐽/𝜓 as a function of 𝜀𝑏 and 𝑒𝐸0 at 𝑒𝐵0=15𝑚2𝜋, 𝜙=𝜋/2 (in the reaction plane), 𝜂=0 (midrapidity), and 𝑃⟂=0.1 GeV. Numbers inside boxes indicate the values of 𝑤 in 1/fm.

(a)

(b)

Figure 15 

(a) Angular distribution of 𝐽/𝜓 dissociation rate at 𝑒𝐵0=15𝑚2𝜋, 𝜂=0 at different 𝐸0 and 𝑃⟂ (in GeV’s). Magnetic field 𝐁0 points in the positive vertical direction. Reaction plane coincides with the horizontal plane. (b) Rescaled second Fourier-harmonic 𝑣2 of the azimuthal distribution as a function of 𝑃⟂. ⟨𝑃⟩ is the azimuthal average of the survival probability, and 𝑡 is the time spent by 𝐽/𝜓 in the 𝑃-odd bubble.

As the plasma temperature varies, so is the binding energy of quarkonium, although the precise form of the function 𝜀_𝑏(𝑇) is model dependent. The dissociation rate picks at some 𝜀⁰_𝑏<𝜀^vac_𝑏 (see Figure 13(b)), where 𝜀^vac_𝑏 is the binding energy in vacuum, indicating that 𝐽/𝜓 breaks down even before 𝜀_𝑏 drops to zero, which is the case at 𝐁₀=0. This 𝜀⁰_𝑏 is a strong function of 𝐸₀ as can be seen in Figure 14. It satisfies the equation 𝜕𝑤/𝜕𝜀_𝑏=0. In the case 𝛾≪1 (165) and (167) imply that 𝜀⁰_𝑏=𝑚25𝑒𝐸2𝑚^22/3,𝛾≪1. (169) At 𝛾≫1 and 𝜃=𝜋/2 we employ (164) to derive the condition (𝜀⁰_𝑏)²+𝑒𝐵𝜀⁰_𝑏/2𝑚−𝑒𝐸²/𝐵=0. In view of (166) 𝐸≈𝐵, and we obtain 𝜀⁰_𝑏=𝑒𝐵4𝑚16𝑚²≈√𝑒𝐵+1−1𝑒𝐵,𝛾≫1, (170) where in the last step I used that 𝑒𝐵≪𝑚². For a given function 𝜀_𝑏(𝑇) one can convert 𝜀⁰_𝑏 into the dissociation temperature, which is an important phenomenological parameter.

In the absence of electric fied 𝐄₀, the dissociation probability peaks in the direction perpendicular to the direction of magnetic field 𝐛₀, that is, in the reaction plane. Dissociation rate vanishes in the 𝐛₀ direction. Indeed, for 𝐕⋅𝐛₀=0 (157a) and (157b) imply that 𝐸=0. This feature is seen in Figure 15(a). At finite 𝐄₀ the dissociation probability is finite in the 𝐛₀ direction making the azimuthal distribution more symmetric. The shape of the azimuthal distribution strongly depends on quarkonium velocity, while at low 𝑉 the strongest dissociation is in the direction of the reaction plane, at higher 𝑉 the maximum shifts towards small angles around the 𝐛₀ direction. Extrema of the azimuthal distribution are roots of the equation 𝜕𝑤/𝜕𝜙=0. At 𝛾≪1 it yields minimum at 𝜙₀=0, maximum at 𝜙₀=𝜋/2 and another maximum that satisfies the condition (neglecting the spin-dependence of 𝜀_𝑏) 𝑒𝐸₀1+𝛾²_𝐿𝑉²_𝑧+𝑉²_⟂sin²𝜙₀1+𝜌₀⁻²=2𝑚²32𝜀_𝑏𝑚^3/2. (171) In order to satisfy (171) 𝜙₀ must decrease when 𝑉 increases and vise versa. This features are seen in Figure 15(a).

Spectrum of quarkonia surviving in the electromagnetic field is proportional to the survival probability 𝑃=1−𝑤𝑡, where 𝑡 is the time spent by the quarkonium in the field. Consider 𝑃 as a function of the angle 𝜒 between the quarkonium velocity and the reaction plane 𝜒=𝜋/2−𝜙. Fourier expansion of 𝑃 in 𝜒 reads 𝑃𝜒=12𝑃₀+^∞_𝑛=1𝑃_𝑛,𝑃cos𝑛𝜒_𝑛=1𝜋^𝜋_−𝜋𝑃𝜒cos𝑛𝜒𝑑𝜒. (172) Ellipticity of the distribution is characterized by the “elliptic flow" coefficient 𝑣₂ defined as 𝑣₂=𝑃₂(1/2)𝑃₀=∫^𝜋_−𝜋(1−𝑤𝑡)cos2𝜒𝑑𝜒𝜋⟨𝑃⟩𝑡=−𝜋⟨𝑃⟩^𝜋_−𝜋𝑤cos2𝜒𝑑𝜒, (173) where ⟨𝑃⟩ denotes average of 𝑃 over the azimuthal angle. These formulas are applicable only as long as 𝑤𝑡<1 because otherwise there are no surviving quarkonia. In Figure 15(b) [58] I show 𝑣₂⟨𝑃⟩/𝑡, which is independent of 𝑡, as a function of 𝑃_⟂. As expected, in the absence of the CME, 𝑣₂ is negative at low 𝑃_⟂ and positive at high 𝑃_⟂. 𝑣₂ changes sign at 𝑃_⟂ that depends on the strength of the electric field. It decreases as 𝐸₀ increases until at 𝐸₀≃𝐵₀ it becomes positive at all 𝑃_⟂. Figure 15(b) provides the low bound for 𝑣₂ because ⟨𝑃⟩<1 and 𝑡≳1 fm. We thus expect that magnetic field strongly modifies the azimuthal distribution of the produced 𝐽/𝜓s. Role of the magnetic field in generation of azimuthal anisotropies in heavy-ion collisions has been pointed out before in [29, 55].

In summary, we observed that 𝐽/𝜓 dissociation energy increases with magnetic field strength and quarkonium momentum. As a consequence, quarkonia dissociate at lower temperature than one would have expected based on calculations neglecting magnetic field [57, 58]. Figure 13 indicates that in heavy-ion collisions at the LHC, all 𝐽/𝜓’s moving with 𝑃_⟂>0.5 GeV in the reaction plane would dissociate with probability of order unity even if the QGP effect was completely negligible. If electric field fluctuations shown in Figure 3 are taken into account, then even low 𝑃_⟂𝐽/𝜓’s are destroyed. However, chiral magnetic effect has negligible effect on 𝐽/𝜓 dissociation.

Although magnetic fields in 𝑝𝑝 and 𝑝𝐴 collisions are much weaker than in 𝐴𝐴 collisions, they are still strong enough to cause 𝐽/𝜓 dissociation at sufficiently high momenta 𝑃_⟂. A truly spectacular feature of such process would be 𝐽/𝜓 decay into two heavier 𝐷-mesons.

The effect of 𝐽/𝜓 dissociation in a magnetic field vanishes in the direction parallel to the magnetic field, that is, perpendicular to the reaction plane. Therefore, 𝐽/𝜓 dissociation gives negative contribution to the total azimuthal asymmetry coefficient 𝑣₂. It remarkable that presence of electric field reverses this effect making 𝑣₂ positive.

6. Electromagnetic Radiation by Quark-Gluon Plasma in Magnetic Field

6.1. Necessity to Quantize Fermion Motion

In Section 3 we discussed synchrotron radiation of gluons by fast quarks. Our main interest was the energy loss problem. In this section we turn to the problem of electromagnetic radiation by QGP, namely, radiation of photons by thermal fermions [7]. In this case quasi-classical approximation that we employed in Sections 3 and 4 is no longer applicable, and one has to take into account quantization of fermion motion in magnetic field.

Electromagnetic radiation by quarks and antiquarks of QGP moving in external magnetic field originates from two sources: (i) synchrotron radiation and (ii) quark and antiquark annihilation. QGP is transparent to the emitted electromagnetic radiation because its absorption coefficient is suppressed by 𝛼². Thus, QGP is shinning in magnetic field. The main goal of this paper is to calculate the spectrum and angular distribution of this radiation. In strong magnetic field it is essential to account for quantization of fermion spectra. Indeed, spacing between the Landau levels is of the order 𝑒𝐵/𝜀 (𝜀 being quark energy), while their thermal width is of the order 𝑇. Spectrum quantization is negligible only if 𝑒𝐵/𝜀≪𝑇 which is barely the case at RHIC and certainly not the case at LHC (at least during the first few fms of the evolution). Fermion spectrum quantization is important not only for hard and electromagnetic probes but also for the bulk properties of QGP.

6.2. Synchrotron Radiation

Motion of charged fermions in external magnetic field, which I will approximately treat as spatially homogeneous, is quasi-classical in the field direction and quantized in the reaction plane, which is perpendicular to the magnetic field and span by the impact parameter and the heavy-ion collision axis. In high energy physics one usually distinguishes the transverse plane, which is perpendicular to the collision axis and span by the magnetic field and the impact parameter. In this section I use notation in which three vectors are discriminated by the bold face and their component along the field direction by the plain face. Momentum projections onto the transverse plane are denoted by subscript ⟂.

In the configuration space, charged fermions move along spiral trajectories with the symmetry axis aligned with the field direction. Synchrotron radiation is a process of photon 𝛾 radiation by a fermion 𝑓 with electric charge 𝑒_𝑓=𝑧_𝑓𝑒 in external magnetic field 𝐵 as follows: 𝑓𝑒_𝑓,𝑗,𝑝⟶𝑓𝑒_𝑓,𝑘,𝑞+𝛾(𝐤), (174) where 𝐤 is the photon momentum, 𝑝,𝑞 are the momentum components along the magnetic field direction, and indicies 𝑗,𝑘=0,1,2,… label the discrete Landau levels in the reaction plane. The Landau levels are given by 𝜀_𝑗=𝑚²+𝑝²+2𝑗𝑒_𝑓𝜀𝐵,_𝑘=𝑚²+𝑞²+2𝑘𝑒_𝑓𝐵. (175) In the constant magnetic field only momentum component along the field direction is conserved. Thus, the conservation laws for synchrotron radiation read 𝜀_𝑗=𝜔+𝜀_𝑘,𝑝=𝑞+𝜔cos𝜃, (176) where 𝜔 is the photon energy and 𝜃 is the photon emission angle with respect to the magnetic field. Intensity of the synchrotron radiation was derived in [81]. In [82–85] it was thoroughly investigated as a possible mechanism for 𝛾-ray bursts. In particular, synchrotron radiation in electromagnetic plasmas was calculated. Spectral intensity of angular distribution of synchrotron radiation by a fermion in the 𝑗th Landau state is given by 𝑑𝐼^𝑗𝑑𝜔𝑑Ω=_𝑓𝑧²_𝑓𝛼𝜋𝜔^2𝑗_𝑘=0Γ_𝑗𝑘ℳ_⟂²+ℳ_‖²𝛿𝜔−𝜀_𝑗+𝜀_𝑘, (177) where Γ_𝑗𝑘=(1+𝛿_𝑗0)(1+𝛿_𝑘0) accounts for the double degeneration of all Landau levels except the ground one. The squares of matrix elements ℳ, which appear in (177), corresponding to photon polarization perpendicular and parallel to the magnetic field are given by, respectively, 4𝜀_𝑗𝜀_𝑘ℳ_⟂²=𝜀_𝑗𝜀_𝑘−𝑝𝑞−𝑚²𝐼²_{𝑗,𝑘−1}+𝐼²_{𝑗−1,𝑘}+22𝑗𝑒_𝑓𝐵2𝑘𝑒_𝑓𝐵𝐼_{𝑗,𝑘−1}𝐼_{𝑗−1,𝑘},4𝜀_𝑗𝜀_𝑘ℳ_‖²=cos²𝜃𝜀_𝑗𝜀_𝑘−𝑝𝑞−𝑚²𝐼²_{𝑗,𝑘−1}+𝐼²_{𝑗−1,𝑘}−22𝑗𝑒_𝑓𝐵2𝑘𝑒_𝑓𝐵𝐼_{𝑗,𝑘−1}𝐼_{𝑗−1,𝑘}−2cos𝜃sin𝜃𝑝2𝑘𝑒_𝑓𝐵𝐼_{𝑗−1,𝑘}𝐼_{𝑗−1,𝑘−1}+𝐼_j,𝑘−1𝐼_𝑗,𝑘+𝑞2𝑗𝑒_𝑓𝐵𝐼_𝑗,𝑘𝐼_{𝑗−1,𝑘}+𝐼_{𝑗−1,𝑘−1}𝐼_{𝑗,𝑘−1}+sin²𝜃𝜀_𝑗𝜀_𝑘+𝑝𝑞−𝑚²𝐼²_{𝑗−1,𝑘−1}+𝐼²_𝑗,𝑘+22𝑗𝑒_𝑓𝐵2𝑘𝑒_𝑓𝐵𝐼_{𝑗−1,𝑘−1}𝐼_𝑗,𝑘, (178) where for 𝑗≥𝑘, 𝐼_𝑗,𝑘≡𝐼_𝑗,𝑘(𝑥)=(−1)^𝑗−𝑘𝑘!𝑒𝑗!^−𝑥/2𝑥^{(𝑗−𝑘)/2}𝐿_𝑘^𝑗−𝑘(𝑥), (179) and 𝐼_𝑗,𝑘(𝑥)=𝐼_𝑘,𝑗(𝑥) when 𝑘>𝑗. (𝐼_𝑗,−1 are identically zero.) The functions 𝐿_𝑘^𝑗−𝑘(𝑥) are the generalized Laguerre polynomials. Their argument is 𝜔𝑥=²2𝑒_𝑓𝐵sin²𝜃. (180)

Angular distribution of radiation is obtained by integrating over the photon energies and remembering that 𝜀_𝑘 also depends on 𝜔 by virtue of (175) and (176) as follows: 𝑑𝐼^𝑗𝑑Ω=_𝑓𝑧²_𝑓𝛼𝜋^𝑗_𝑘=0𝜔^∗𝜀_𝑗−𝜔^∗𝜀_𝑗−𝑝cos𝜃−𝜔^∗sin²𝜃×Γ_𝑗𝑘ℳ_⟂²+ℳ_‖², (181) where photon energy 𝜔 is fixed to be 𝜔^∗=1sin²𝜃𝜀_𝑗−𝜀−𝑝cos𝜃_𝑗−𝑝cos𝜃²−2𝑒_𝑓𝐵𝑗−𝑘sin²𝜃^1/2. (182)

In the context of heavy-ion collisions the relevant observable is the differential photon spectrum. For ideal plasma in equilibrium each quark flavor gives the following contribution to the photon spectrum: 𝑑𝑁^synch𝑑𝑡𝑑Ω𝑑𝜔=_𝑓^∞_−∞𝑒𝑑𝑝^𝑓𝐵2𝑁_𝑐𝑉2𝜋²×^∞𝑗_𝑗=0𝑘=0𝑑𝐼^𝑗𝜔𝑑𝜔𝑑Ω2−𝛿_𝑗,0𝑓𝜀_𝑗1−𝑓𝜀_𝑘, (183) where 2𝑁_𝑐 accounts for quarks and antiquarks each of 𝑁_𝑐 possible colors, and (2−𝛿_𝑗,0) sums over the initial quark spin. Index 𝑓 indicates different quark flavors. 𝑉 stands for the plasma volume. The statistical factor 𝑓(𝜀) is 𝑓(𝜀)=1𝑒^𝜀/𝑇.+1 (184) The 𝛿-function appearing in (177) can be re-written using (175) and (176) as 𝛿𝜔−𝜀_𝑗+𝜀_𝑘=_±𝛿𝑝−𝑝^∗_±𝑝/𝜀_𝑗−𝑞/𝜀_𝑘, (185) where𝑝^∗_±=cos𝜃𝑚²_𝑗−𝑚²_𝑘+𝜔²sin²𝜃±𝑚_𝑗+𝑚_𝑘²−𝜔²sin²𝜃𝑚_𝑗−𝑚_𝑘²−𝜔²sin²𝜃2𝜔sin²𝜃. (186) The following convenient notation was introduced: 𝑚²_𝑗=𝑚²+2𝑗𝑒_𝑓𝐵,𝑚²_𝑘=𝑚²+2𝑘𝑒_𝑓𝐵. (187) The physical meaning of (186) is that synchrotron radiation of a photon with energy 𝜔 at angle 𝜃 by a fermion undergoing transition from 𝑗th to 𝑘th Landau level is possible only if the initial quark momentum along the field direction equals 𝑝^∗_±.

Another consequence of the conservation laws (176) is that for a given 𝑗 and 𝑘 the photon energy cannot exceed a certain maximal value that will be denoted by 𝜔_{𝑠,𝑗𝑘}. Indeed, inspection of (186) reveals that this equation has a real solution only in two cases: (i)𝑚_𝑗−𝑚_𝑘≥𝜔sin𝜃,or(ii)𝑚_𝑗+𝑚_𝑘≤𝜔sin𝜃. (188) The first case is relevant to the synchrotron radiation, while the second one is revelant to the one-photon pair annihilation as discussed in the next section. Accordingly, allowed photon energies in the 𝑗→𝑘 transition satisfy 𝜔≤𝜔_{𝑠,𝑗𝑘}≡𝑚_𝑗−𝑚_𝑘=sin𝜃𝑚²+2𝑗𝑒_𝑓𝐵−𝑚²+2𝑘𝑒_𝑓𝐵.sin𝜃 (189) No synchrotron radiation is possible for 𝜔>𝜔_{𝑠,𝑗𝑘}. In particular, when 𝑗=𝑘,𝜔_{𝑠,𝑗𝑘}=0, that is, no photon is emitted, which is also evident in (182). The reason is clearly seen in the frame where 𝑝=0: since 𝜀_𝑗≥𝜀_𝑘, constraints (175) and (176) hold only if 𝜔=0.

Substituting of (177) into (183) yields the spectral distribution of the synchrotron radiation rate per unit volume 𝑑𝑁^synch=𝑉𝑑𝑡𝑑Ω𝑑𝜔_𝑓2𝑁_𝑐𝑧²_𝑓𝛼𝜋³𝑒_𝑓𝐵×^∞𝑗_𝑗=0𝑘=0𝜔1+𝛿_𝑘0𝜗𝜔_{𝑠,𝑖𝑗}×−𝜔𝑑𝑝_±𝛿𝑝−𝑝^∗_±𝑝/𝜀_𝑗−𝑞/𝜀_𝑘×ℳ_⟂²+ℳ_‖²𝑓𝜀_𝑗1−𝑓𝜀_𝑘, (190) where 𝜗 is the step function.

The natural variables to study the synchrotron radiation are the photon energy 𝜔 and its emission angle 𝜃 with respect to the magnetic field. However, in high energy physics particle spectra are traditionally presented in terms of rapidity 𝑦 (which for photons is equivalent to pseudo-rapidity) and transverse momentum 𝑘_⟂. 𝑘_⟂ is a projection of three-momentum 𝐤 onto the transverse plane. These variables are not convenient to study electromagnetic processes in external magnetic field. In particular, they conceal the azimuthal symmetry with respect to the magnetic field direction. To change variables, let 𝑧 be the collision axis, and let 𝐲 be the direction of the magnetic field. In spherical coordinates photon momentum is given by 𝐤=𝜔(sin𝛼cos𝜙𝐱+sin𝛼sin𝜙𝐲+cos𝛼𝐳̂), where 𝛼 and 𝜙 are the polar and azimuthal angles with respect to 𝑧-axis. The plane 𝑥𝑧 is the reaction plane. By definition, 𝐤⋅𝐲=cos𝜃 implying that cos𝜃=sin𝛼sin𝜙. Thus, 𝑘_⟂=𝑘²_𝑥+𝑘²_𝑦=𝜔cos𝜃𝛼sin𝜙,𝑦=−lntan2. (191) The second of these equations is the definition of (pseudo-) rapidity. Inverting (191) yields 𝜔=𝑘_⟂cosh𝑦,cos𝜃=sin𝜙.cosh𝑦 (192) Because 𝑑𝑦=𝑑𝑘_𝑧/𝜔 the photon multiplicity in a unit volume per unit time reads 𝑑𝑁^synch𝑑𝑉𝑑𝑡𝑑²𝑘_⟂𝑑𝑦=𝜔𝑑𝑁^synch𝑑𝑉𝑑𝑡𝑑³𝑘=𝑑𝑁^synch.𝑑𝑉𝑑𝑡𝜔𝑑𝜔𝑑Ω (193)

Figure 16 displays the spectrum of synchrotron radiation by 𝑢 quarks as a function of 𝑘_⟂ at fixed 𝜙 [7]. At midrapidity 𝑦=0 (192) implies that 𝑘_⟂=𝜔. Contribution of 𝑑 and 𝑠 quarks is qualitatively similar. At 𝑒𝐵≫𝑚², quark masses do not affect the spectrum much. The main difference stems from the difference in electric charge. In panel (a) only the contributions of the first ten Landau levels are displayed. The cutoff frequencies 𝜔_{𝑠,𝑗𝑘} can be clearly seen, and some of them are indicated on the plot for convenience. The azimuthal distribution is shown in Figure 17. Note that at midrapidity 𝜙=𝜋/2−𝜃. Therefore, the figure indicates that photon production in the direction of magnetic field (at 𝜙=𝜋/2) is suppressed. More photons are produced in the direction of the reaction plane 𝜙=0. This results in the ellipticity of the photon spectrum that translates into the positive “elliptic flow" coefficient 𝑣₂. It should be noted that the classical synchrotron radiation has a similar angular distribution.

(a)

(b)

Figure 16 

Spectrum of synchrotron radiation by 𝑢 quarks at 𝑒𝐵=𝑚2𝜋, 𝑦=0, and 𝜙=𝜋/3: (a) contribution of 10 lowest Landau levels 𝑗≤10; several cutoff frequencies are indicated; (b) summed over all Landau levels. 𝑚𝑢=3MeV, 𝑇=200MeV. Adopted from [7].

Figure 17 

Azimuthal distribution of synchrotron radiation by 𝑢-quarks at 𝑘⟂=0.2 GeV, 𝑒𝐵=𝑚2𝜋,and𝑦=0.𝑚𝑢=3 MeV. Adopted from [7].

In order to compare the photon spectrum produced by synchrotron radiation to the photon spectrum measured in heavy-ion collisions, the 𝑢, 𝑑, and 𝑠 quarks contributions were summed up. Furthermore, the experimental data from [8] was divided by 𝑉𝑡, where 𝑡 is the magnetic field relaxation time. The volume of the plasma can be estimated as 𝑉=𝜋𝑅²𝑡 with 𝑅≈5 fm being the nuclear radius. Therefore, 𝑑𝑁^𝛾_exp𝑑𝑉𝑑𝑡𝑑²𝑘_⟂=𝑑𝑦𝑑𝑁^𝛾_exp𝑑²𝑘_⟂1𝑑𝑦𝜋𝑅²𝑡²=𝑑𝑁^𝛾_exp𝑑²𝑘_⟂𝑑𝑦GeV14.9⁴1fm𝑡². (194) The results are plotted in Figure 18. In panel (a) it is seen that synchrotron radiation gives a significant contribution to the photon production in heavy-ion collisions at RHIC energy. This contribution is larger at small transverse momenta. This may explain enhancement of photon production observed in [8]. Panel (b) indicates the increase of the photon spectrum produced by the synchrotron radiation mechanism at the LHC energy. This increase is due to enhancement of the magnetic field strength, but mostly because of increase of plasma temperature. This qualitative features can be better understood by considering the limiting cases of low and high photon energies.

(a)

(b)

Figure 18 

Azimuthal average of the synchrotron radiation spectrum of 𝑢, 𝑑, 𝑠 quarks and their corresponding antiquarks. (a) 𝑒𝐵=𝑚2𝜋,𝑦=0 compared to the experimental data from [8] divided by 𝑉𝑡=25𝜋fm4 (dots) and 𝑉𝑡=9×25𝜋fm4 (stars); (b) 𝑒𝐵=𝑚2𝜋,𝑇=200 MeV, and 𝑦=0 (solid line) compared to 𝑒𝐵=15𝑚2𝜋,𝑇=400MeV, and 𝑦=0 (dashed line). 𝑚𝑢=3 MeV,   𝑚𝑑=5MeV, and  𝑚𝑠=92MeV. Adopted from [7].

One possible way to ascertain the contribution of electromagnetic radiation in external magnetic field is to isolate the azimuthally symmetric component with respect to the direction of the magnetic field. It seems that synchrotron radiation dominates the photon spectrum at low 𝑘_⟂. Thus, azimuthal symmetry can be easily checked by simply plotting the multiplicity versus 𝜔, 𝜃, and 𝜑, where 𝜔 is photon energy, 𝜃 is emission angle with respect to the magnetic field, and 𝜑 is azimuthal angle around the magnetic field direction (which is perpendicular both to the collision axis and to the impact parameter). In Figure 16(a) it is also seen that in these variables it may be possible to discern the cutoff frequencies 𝜔_{𝑠,𝑗𝑘} that appear as resonances (in Figure 16  𝑦=0 so 𝑘_⟂=𝜔). Note that averaging over the azimuthal angle 𝛼 around the collision axis direction destroys these features; see Figure 18.

6.2.1. Low Photon Energy

The low energy part of the photon spectrum satisfies the condition 𝜔≪𝑒_𝑓𝐵. The corresponding initial quark momentum components along the field 𝑝 and energy 𝜀_𝑗 follow from (186) and (175) and are given by 𝑝^∗_±≈𝑗−𝑘𝑒_𝑓𝐵(cos𝜃±1)𝜔sin²𝜃(𝜔),𝜀+𝒪_𝑗≈𝑝^∗_±+𝒪(𝜔). (195) Evidently, 𝜀_𝑗≫𝑒𝐵. In practice, magnetic field strength satisfies √𝑒𝐵≳𝑇, so that 𝜀_𝑗≫𝑇. Therefore, synchrotron radiation is dominated by fermion transitions from low Landau levels due to the statistical factors appearing in (183).

For a qualitative discussion it is sufficient to consider the 1→0 transition. In this case the matrix elements (178) read ℳ^1,02=12𝜀₁𝜀₀𝐼²_1,0𝜀₁𝜀₀−𝑝𝑞cos²𝜃−𝑚²+cos𝜃sin𝜃𝑞2e_𝑓𝐵𝐼_1,0𝐼_0,0. (196) Assuming that the field strength is supercritical; that is, 𝑒_𝑓𝐵≫𝑚², but keeping all powers of 𝜔 (for future reference) (186) reduces to 𝑝^∗_±≈12𝜔sin²𝜃2𝑒_𝑓𝐵(cos𝜃±1)+𝜔²sin²𝜃(cos𝜃∓1). (197) Furthermore, using the conservation laws (176) we obtain in this approximation 𝜀_1±=12𝜔sin²𝜃2𝑒_𝑓𝐵(cos𝜃±1)−𝜔²sin²𝜃(,𝑞cos𝜃∓1)_±=12𝜔sin²𝜃2𝑒_𝑓𝐵−𝜔²sin²𝜃(),𝜀cos𝜃±1_0±=𝑞. (198) The values of the nonvanishing matrix elements 𝐼_𝑗,𝑘 defined by (179) are 𝐼_1,0(𝑥)=−𝑥^1/2𝑒^−𝑥/2,𝐼_0,0(𝑥)=𝑒^−𝑥/2. (199) For 𝑗=1, 𝑘=0 we write using (189) 𝜔_𝑠,10=2𝑒_𝑓𝐵/sin𝜃. Then (180) implies 𝑥=𝜔²/𝜔²_𝑠,10. Substituting (197)–(199) into (196) gives ℳ_±^1,02=12𝑥𝑒^−𝑥()±()1−cos𝜃1+𝑥1−𝑥cos𝜃(1−𝑥)±(1+𝑥)cos²𝜃−2(1−𝑥)cos𝜃sin²𝜃()±().cos𝜃1−𝑥1+𝑥 (200) According to (190) the contribution of the 1→0 transition to the synchrotron radiation reads [7] 𝑑𝑁^synch,10=𝑉𝑑𝑡𝑑Ω𝑑𝜔_𝑓2𝑁_𝑐𝑧²_𝑓𝛼𝜋𝑒𝜔Γ^𝑓𝐵2𝜋²_±𝑓𝜀₁1−𝑓𝜀₀ℳ_±^1,02×(1−𝑥)cos𝜃±(1+𝑥)()𝜗𝜔−2𝑥cos𝜃∓1^𝑠,10−𝜔. (201) Consider radiation spectrum at 𝜃=𝜋/2, that is, perpendicular to the magnetic field. The spectrum increases with 𝑥 and reaches maximum at 𝑥=1. Since 𝑥=𝜔²/(2𝑒_𝑓𝐵), spectrum decreases with increase of 𝐵 at fixed 𝜔. This feature holds at low 𝑥 part of the spectrum for other emission angles and even for transitions form higher excited states. However, at high energies, it is no longer possible to approximate the spectrum by the contribution of a few low Landau levels. In that case the typical values of quantum numbers are 𝑗,𝑘≫1. For example, to achieve the numerical accuracy of 5%, sum over 𝑗 must run up to a certain 𝑗_max. Some values of 𝑗_max are listed in Table 1 [7].

Table 1 

The upper summation limit in (190) that yields the 5% accuracy. 𝑗_max is the highest Landau level of the initial quark that is taken into account at this accuracy. Throughout the table 𝑦=0.

6.2.2. High Photon Energy

The high energy tail of the photon spectrum is quasi-classical and approximately continuous. In this case the Laguerre polynomials can be approximated by the Airy functions or the corresponding modified Bessel functions. The angular distribution of the spectrum can be found in [82] 𝑑𝑁^synch𝑉𝑑𝑡𝑑Ω𝑑𝜔=_𝑓𝑧²_𝑓𝛼𝜋𝑛_𝑓𝜔𝑚²4𝑇³𝑒_𝑓𝐵𝑇sin𝜃𝑚³𝑒^−𝜔/𝑇, (202) provided that 𝜔≫𝑚𝑚𝑇/𝑒_𝑓𝐵sin𝜃. Here 𝑛_𝑓 is number density of flavor 𝑓, which is independent of 𝐵 as follows: 𝑛_𝑓=2⋅2𝑁_𝑐𝑒_𝑓𝐵4𝜋^2∞_𝑗=0^∞_−∞𝑑𝑝𝑒^{−𝜀𝑗/𝑇}≈4𝑁_𝑐𝜋²𝑇³. (203) Here summation over 𝑗 was replaced by integration. It follows that this part of the spectrum increases with magnetic field strength as √𝐵 and with temperature as √𝑇𝑒^−𝜔/𝑇. Therefore, variation of the spectrum with 𝑇 is much stronger than with 𝐵. The 𝑇 dependence is shown in Figure 19.

Figure 19 

Variation of the synchrotron spectrum with plasma temperature. Lower line: 𝑇=200 MeV, upper line: 𝑇=250 MeV; Other parameters are the same as in Figure 18(a). Adopted from [7].

Unlike time-dependence of magnetic field, time-dependence of temperature is non-negligible even during the first few fm/c. Final synchrotron spectrum, which is an average over all temperatures, is dominated by high temperatures/early times. However, the precise form of time-dependence of temperature is model-dependent. Therefore, the spectrum is presented at fixed temperatures, so that a reader can appreciate its qualitative features in a model-independent way.

6.3. Pair Annihilation

The theory of one-photon pair annihilation was developed in [86, 87]. It was shown in [88] that in the super-critical regime 𝑒𝐵≫𝑚² one-photon annihilation is much larger than the two-photon annihilation. In this section the one-photon annihilation of 𝑞 and 𝑞 pairs in the QGP is calculated.

For 𝑞𝑞 pair annihilation the conservation of energy and momentum is given by 𝜀_𝑗+𝜀_𝑘=𝜔,𝑝+𝑞=𝜔cos𝜃. (204) The spectral density of the annihilation rate per unit volume reads 𝑑𝑁^annih=𝑉𝑑𝑡𝑑𝜔𝑑Ω_𝑓𝛼𝑧²_𝑓𝜔𝑁_𝑐4𝜋𝑒_𝑓𝐵^∞∞_𝑗=0𝑘=0𝑑𝑝2𝑒_𝑓𝐵2𝜋²𝑓𝜀_𝑗𝑑𝑞2𝑒_𝑓𝐵2𝜋²𝑓𝜀_𝑘×𝛿𝑝+𝑞−𝜔cos𝜃𝛿𝜀_𝑗+𝜀_𝑘×𝒯−𝜔_⟂2+𝒯_‖², (205) where the matrix elements 𝒯 can be obtained from (178) by making substitutions 𝜀_𝑘→−𝜀_𝑘, 𝑞→−𝑞 and are given by 4𝜀_𝑗𝜀_𝑘𝒯_⟂²=𝜀_𝑗𝜀_𝑘−𝑝𝑞+𝑚²𝐼²_{𝑗,𝑘−1}+𝐼²_{𝑗−1,𝑘}−22𝑗𝑒_𝑓𝐵2𝑘𝑒_𝑓𝐵𝐼_{𝑗,𝑘−1}𝐼_{𝑗−1,𝑘},4𝜀_𝑗𝜀_𝑘𝒯_‖²=cos²𝜃𝜀_𝑗𝜀_𝑘−𝑝𝑞+𝑚²𝐼²_{𝑗,𝑘−1}+𝐼²_{𝑗−1,𝑘}+22𝑗𝑒_𝑓𝐵2𝑘𝑒_𝑓𝐵𝐼_{𝑗,𝑘−1}𝐼_{𝑗−1,𝑘}−2cos𝜃sin𝜃−𝑝2𝑘𝑒_𝑓𝐵𝐼_{𝑗−1,𝑘}𝐼_{𝑗−1,𝑘−1}+𝐼_{𝑗,𝑘−1}𝐼_𝑗,𝑘+𝑞2𝑗𝑒_𝑓𝐵𝐼_𝑗,𝑘𝐼_{𝑗−1,𝑘}+𝐼_{𝑗−1,𝑘−1}𝐼_{𝑗,𝑘−1}+sin²𝜃𝜀_𝑗𝜀_𝑘+𝑝𝑞+𝑚²𝐼²_{𝑗−1,𝑘−1}+𝐼²_𝑗,𝑘−22𝑗𝑒_𝑓𝐵2𝑘𝑒_𝑓𝐵𝐼_{𝑗−1,𝑘−1}𝐼_𝑗,𝑘, (206) with the same functions 𝐼_𝑖,j as in (179). Integration over 𝑞 removes the delta function responsible for the conservation of momentum along the field direction. The remaining delta function is responsible for energy conservation and can be written in exactly the same form as in (185) with particle energies and momenta now obeying the conservation laws (204). It is straightforward to see that momentum 𝑝^∗_± is still given by (190), (187). The photon spectrum produced by annihilation of quark in state 𝑗 with antiquark in state 𝑘 has a threshold 𝜔_{𝑎,𝑖𝑗} that is given by the case (ii) in (188) as follows: 𝜔≥𝜔_{𝑎,𝑖𝑗}=𝑚_𝑗+𝑚_𝑘=sin𝜃𝑚²+2𝑗𝑒_𝑓𝐵+𝑚²+2𝑘𝑒_𝑓𝐵.sin𝜃 (207) Thus, the spectral density of the annihilation rate per unit volume is 𝑑𝑁^annih=𝑉𝑑𝑡𝑑𝜔𝑑Ω_𝑓𝛼𝑧²_𝑓𝜔𝑁_𝑐4𝜋⁵𝑒_𝑓𝐵^∞∞_𝑗=0𝑘=0𝜗𝜔−𝜔_{𝑎,𝑖𝑗}×𝑑𝑝_±𝛿𝑝−𝑝^∗_±𝑝/𝜀_𝑗−𝑞/𝜀_𝑘×𝒯_⟂²+𝒯_‖²𝑓𝜀_𝑗𝑓𝜀_𝑘. (208) Passing to 𝑦 and 𝑝_⟂ variables in place of 𝜔 and 𝜃 is similar to (193).

The results of the numerical calculations are represented in Figure 20. Panel (a) shows the spectrum of photons radiated in annihilation of 𝑢 and 𝑢. We conclude that contribution of the annihilation channel is negligible as compared to the synchrotron radiation.

(a)

(b)

Figure 20 

Photon spectrum in one-photon annihilation of 𝑢 and 𝑢 quarks. 𝑒𝐵=𝑚2𝜋,𝑦=0. (a) 𝑘⟂-spectrum at 𝜙=𝜋/3, (b) azimuthal angule distribution at 𝑘⟂=1 GeV. Adopted from [7].

In summary, results of the calculations presented in this section indicate that photon production by QGP due to its interaction with external magnetic field gives a considerable contribution to the total photon multiplicity in heavy-ion collisions. This is seen in Figure 18 were the model calculation is compared with the experimental data [8]. The largest contribution to the photon multiplicity arises from photon momenta of the order of √𝑒𝐵. This may provide an explanation of the photon excess observed by the PHENIX experiment [8]. Similar mechanism may also be responsible for enhancement of low mass di-lepton production that proceeds via emission of virtual photon which subsequently decays into dilepton pair.

7. Summary

Analytical and numerical calculations indicate existence of extremely powerful electromagnetic fields in relativistic heavy-ion collisions. They are the strongest electromagnetic fields that exist in nature. They evolve slowly on characteristic QGP time scale and therefore have a profound effect on dynamics of QGP. In this review I described the recent progress in understanding of particle production in presence of these fields. Treating the fields as quasi-static and spatially homogeneous allowed us to use analytical results derived over the past half century. This is, however, the main source of uncertainty that can be clarified only in comprehensive numerical approach based on relativistic magnetohydrodynamics.

I discussed many spectacular effects caused by magnetic field. All of them have direct phenomenological relevance. Breaking of spherical symmetry by magnetic field in the direction perpendicular to the collision axis results in azimuthal asymmetry of particle production in the reaction plane. Fast quarks moving in magnetic field radiate a significant fraction of their energy. All electromagnetic probes are also naturally affected by magnetic field. Therefore, all experimental processes that are being used to study the properties of QGP have strong magnetic field dependence. In addition, the QCD phase diagram is modified by magnetic field as has been extensively studied using model calculations [89–110] and lattice simulations [111–123]. Entanglement of effects produced by magnetic field with conventional QGP ones makes it difficult to quantify the role of magnetic field in QGP dynamics. A unique observable is polarization of leptons escaping from QGP, which can be induced only by magnetic field see Section 3.3.

Profound influence of magnetic field on properties of QGP is truly remarkable. Hopefully, progress in theory will soon be matched by experimental investigations that will eventually discover properties of QCD at high temperatures and strong electromagnetic fields.

Acknowledgment

This work was supported in part by the U.S. Department of Energy under Grant no. DE-FG02-87ER40371.