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Advances in High Energy Physics
Volume 2013 (2013), Article ID 490495, 34 pages
http://dx.doi.org/10.1155/2013/490495
Review Article

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

Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA

Received 31 December 2012; Accepted 11 April 2013

Academic Editor: Jan E. Alam

Copyright © 2013 Kirill Tuchin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ~ at RHIC and ~ 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 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 with electric charge ( is the magnitude of electron charge) at impact parameter . According to the Biot and Savart law they create magnetic field that in the center-of-mass frame has magnitude and points in the direction perpendicular to the reaction plane (span by the momenta of ions). Here is the Lorentz factor. At RHIC heavy ions are collided at 200 GeV per nucleon, hence . Using for gold and  fm we estimate  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  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 [3]. In cgs units the corresponding magnetic field is  G. Because , 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 27 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 . Average magnetic field then points in the -direction; see Figures 1 and 7. Plane is the reaction plane, and is the impact parameter.

490495.fig.001
Figure 1: Heavy-ion collision geometry as seen along the collision axis . 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 collisions, magnetic field was first estimated in [9] who also pointed out a possibility of formation of -condensate [9, 10]). Magnetic field at point created by two heavy ions moving in the positive or negative -direction can be calculated using the Liénard-Wiechert potentials as follows: with , where sums run over all protons in each nucleus, their positions and velocities being and . The magnitude of velocity is determined by the collision energy and the proton mass . 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 . 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 . 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.

fig2
Figure 2: Magnetic field (multiplied by ) at the origin produced in collision of two gold ions at beam energies (a)  GeV and (b)  GeV. Adapted from [5]. Note that is the same in Gauss and Lorentz-Heaviside units in contrast to .
1.2.2. Event-by-Event Fluctuations in Proton Positions

Nuclear charge density provides only event-averaged distribution of protons. The actual distribution in a given event is different form implying that in a single event there is not only magnetic field along the -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 at the origin (denoted by a black dot in Figure 1) in collisions at  GeV.

fig3
Figure 3: The mean absolute value of (a) magnetic field and (b) electric field at and as a function of impact parameter for collision at  GeV.

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

Electromagnetic fields created in heavy-ion collisions were also examined in more elaborated approaches in [1113]. 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] where is plasma temperature and 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 is constant as a function of time . 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 moving in -direction with velocity is a solution of the following equations: where we used the Ohm’s law to describe currents induced in the medium. Position of the observation point is specified by the longitudinal and transverse coordinates and . Taking curl of the second equation in (7) and substituting (6) we get The particular solution reads where Green’s function satisfies the following equation: which is solved by where . Plugging this into (9) and substituting for the expression in the square brackets in (9) its Fourier image, we obtain We are interested in the -component of the field. Noting that , where is the azimuthal angle in the transverse plane, and integrating over we derive where , and we introduced notation where is the dielectric constant of the plasma with the following frequency dependence:

Equation (14) is actually valid for any functional form of [19], which can be easily verified by using electric displacement instead of in (7). In this case (16) can be viewed as a low frequency expansion of . 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 we would have derived (14) with . After integration over 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 in (13) and integrating first over and then over give where we used . This coincides with (3) for a single proton when we take . Consider field strength (17) at the origin . At times the field is constant, while at it decreases as . At the time the ratio between these two is which is a very small number (~10−6 at RHIC).

In matter . Let me write the modified Bessel function appearing in (14) as follows: Substituting (19) into (14) and using (16), we have () Closing the contour in the lower half-plane of complex picks a pole at . We have At this function vanishes at and and has maximum at the time instant which is much larger than . The value of the magnetic field at this time is (Here is the base of natural logarithm.) This is smaller than the maximum field in vacuum 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.

490495.fig.004
Figure 4: Relaxation of magnetic field at in vacuum (blue), in static conducting medium at  MeV (red) and at  MeV (brown), and in the expanding medium (green). Units of is .  fm, (gold nucleus), (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, , where is the particle number density and is plasma volume. Since and at early times expansion is one-dimensional it follows that . (Eventually, we will consider the midrapidity region , therefore distinction between the proper time and is not essential.) Equation (5) implies that . I will parameterize conductivity as follows: where I took  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 . Then employing (5) we estimate MeV. Let me define another parameter that I will need in the forthcoming calculation:

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 as follows: Field satisfies equation where Its solution can be written as in terms of Green’s function satisfying To solve this equation we represent as three-dimensional Fourier integral with respect to the space coordinates and Laplace transform with respect to the “time” coordinate: with the contour 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 Therefore, Substituting (31) and (34) into (29) we obtain upon integration over the volume and time as follows: where is the step function. Taking consequent integrals over and gives Consider now . Integrating over azimuthal angle and then over as in (13), (14) yields where .

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 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 magnetic field permits the plasma. The problem is to find the time dependence of the field at . In this model, the field sources turn off at and do not at all contribute to the field at . Electromagnetic field is governed by the following equations: that lead to the diffusion equation for , after we neglect the second-time derivative as discussed previously as follows: For simplicity we treat electrical conductivity as constant. Initial condition at reads where the Gaussian profile is chosen for illustration purposes, and is the nuclear radius. Solution to the problem (39), (40) is where Green’s function is Integrating over the entire volume we derive It follows from (43) that as long as , where is a characteristic time and magnetic field 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 right after the collision to at  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 , where is mass of lightest electrically charged particle. Indeed, in order to excite a fermion out of the Dirac sea, electric force must do work along the path satisfying If , then . The maximal value of is the fermion Compton’s wavelength implying that the minimum (or critical) value of electric field is Notice that in stronger fields . 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 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  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 . Work done by electric field is where 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 . Employing the estimate that we obtained after (45) implies . Following [22] I define the inhomogeneity parameter that describes the effect of spatial variation of electric field on the pair production rate. For electrons  MeV in QGP  fm at we have . 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 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.

fig5
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 . For electrons  fm. Plasma undergoes phase transition at about . 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 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, -plane in Figure 1. Charged particles moving along the magnetic field direction are not influenced by the magnetic Lorentz force, while those moving the -plane (i.e., the reaction plane) are affected the most. The result is azimuthally anisotropic flow of expanding plasma in the -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 . 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 and bulk viscosity . 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 [3033]. To apply this approach to QGP in strong magnetic field we start with kinetic equation for the distribution function of a quark flavor of charge as follows: where is the collision integral and is the electromagnetic tensor, which contains only magnetic field components in the laboratory frame. Ellipsis in the argument of indicates the distribution functions of other quark flavors and gluons (I will omit them in the following). The equilibrium distribution reads where is the macroscopic velocity of the fluid, is particle momentum, , and is the mass density. Since , 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 : where is a deviation from equilibrium. Differentiating (49) we find Since and it follows that Thus, in the comoving frame Substituting (53) in (50) yields where I defined and used .

Since the time derivative of 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 , we cast (54) in the form

The viscous pressure generated by a deviation from equilibrium is given by the tensor Effectively it can be parameterized in terms of the viscosity coefficients as follows (we neglect the bulk viscosities): where the linearly independent tensors are given byHere is a unit vector in the direction of magnetic field. For the calculation of the shear viscosities , we can set and .

Let us expand to the second order in velocities in terms of the tensors as follows: Then, substituting (60) into (58) and requiring consistency of (57) and (58) yield 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 [30].

2.2.2. Collisionless Plasma

In strong magnetic field we can determine by the method of consecutive approximations. Writing and substituting into (56), we find 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 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 Using (60), we get Substituting (64) into (63) and using (59a), (59b), (59c), (59d), and (59e) yield where I used the following identities . Clearly, (65) is satisfied only if . Concerning the other two coefficients, we use the identitiesthat we substitute into (65) to derive Since , we obtain Using (49), (68) in (61) in the comoving frame (of course s do not depend on the frame choice) and integrating using 3.547.9 of [34], we derive [35] The nonrelativistic limit corresponds to in which case we get In the opposite ultrarelativistic case (high-temperature plasma) where is the number density.

2.2.3. Contribution of Collisions

In the relaxation-time approximation we can write the collision integral as where is an effective collision rate. Strong field limit means that where is the synchrotron frequency. Whether itself is function of the field depends on the relation between the Larmor radius , where is the particle velocity in the plane orthogonal to and the Debye radius . If then the effect of the field on the collision rate can be neglected [30]. Assuming that (74) is satisfied, the collision rate reads where is the transport cross-section, which is a function of the saturation momentum [36, 37]. We estimate , with  GeV and with pressure  GeV/; we get MeV. Inequality (73) is well satisfied since [5, 11], and 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 , where is the mean free path and is the plasma size is assumed to hold. Altogether we have .

Equation for the second correction to the equilibrium distribution follows from (62) after substitution (72) Now, plugginginto (76) yields where I used . It follows that With the help of (80), (49), and (65) we obtain [35]

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 where is the viscous pressure tensor, is mass density, and 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 Because of the approximate boost invariance of the heavy-ion collisions, we can restrict our attention to the two dimensional flow in the -plane corresponding to the central rapidity region.

The viscous pressure tensor in vanishing magnetic field is isotropic in the -plane and is given by where the superscript 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 with , we can write where we also used (82). Notice that 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]: Additionally, we need to set the initial conditions The solution to the the problem (85), (86) isHere Green’s function is given by and the diffusion coefficient by

Suppose that the pressure is isotropic; that is, it depends on the coordinates , only via the radial coordinate ; accordingly we pass from the integration variables and to in (87a) and (87b) correspondingly. At later times we can expand Green’s function (88) in inverse powers of . The first terms in the r.h.s. of (87a) and (87b) are subleading, and we obtain and by the same token where denotes the boundary beyond which the density of the plasma is below the critical value. We observe that . Consequently, the azimuthal anisotropy of the hydrodynamic flow is [35] 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 , where 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 [3840]. 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 [3840] 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 [4547]. 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 [5254]).

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 , where is the number of external field lines. In strong field, powers of 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 . 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 defined as where the initial quark 4-momentum is and is the quark charge in units of the absolute value of the electron charge . At high energies, The regime of weak fields corresponds to , while in strong fields . In our case, (at RHIC), and therefore . In terms of , spectrum of radiated gluons of frequency can be written as [45] where is the intensity and is quark’s energy in the final state. 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 should be small compared to the quark energy itself .

490495.fig.006
Figure 6: A typical diagram contributing to the synchrotron radiation by a quark.
490495.fig.007
Figure 7: Geometry of a heavy-ion collision. 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 , which is labeled as .

Energy loss by a relativistic quark per unit length is given by [48] In two interesting limits, energy loss behaves quite differently. At we have [48] In the strong field limit energy loss is independent of the quark mass, whereas in the weak field case it decreases as . Since , limit of corresponds to the classical energy loss.

To apply this result to heavy-ion collisions we need to write down the invariant in a suitable kinematic variables. The geometry of a heavy-ion collision is depicted in Figure 7. Magnetic field is orthogonal to the reaction plane span by the impact parameter vector and the collision axis (-axis). For a quark of momentum we define the polar angle with respect to the -axis and azimuthal angle with respect to the reaction plane. In this notation, and , where . Thus, . Conventionally, one expresses the longitudinal momentum and energy using the rapidity as and , where . We have

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 quark with  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.

fig8
Figure 8: Energy loss per unit length by quarks moving in constant external magnetic field as a function of (a) at RHIC at , (b) at LHC at , (c) azimuthal angle with respect to the reaction plane for  GeV and , and (d) fractional energy loss versus rapidity at  GeV and .

Energy loss due to the synchrotron radiation has a very nontrivial azimuthal angle and rapidity dependence that comes from the corresponding dependence of the -parameter (98). As can be seen in Figure 8(c), energy loss has a minimum at that corresponds to quark’s transverse momentum being parallel (or antiparallel) to the field direction. It has a maximum at when is perpendicular to the field direction and thus lying in the reaction plane. It is obvious from (98) that at midrapidity the azimuthal angle dependence is much stronger pronounced than in the forward/backward direction. Let me emphasize that the energy loss (96) divided by , that is, scales with . In turn, 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 Azimuthal averaged intensity is . In strong fields , and we can write We have At the Fourier coefficients can be calculated analytically using formula 3.631.9 of [34] as follows: where is Euler’s beta function. The corresponding numerically values of the lowest harmonics are .

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] where is a unit axial vector that coincides with the direction of quark spin in its rest frame and 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: It is seen that negatively charged quarks and antiquarks (e.g., and ) tend to align against the field, while the positively charged ones (e.g., and ) align parallel to the field.

Let 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 the spinasymmetry defined as vanishes. Equation (103) implies that at later times, a beam of nonpolarized fermions develops a finite asymmetry given by [46] where is the characteristic time over which the maximal possible asymmetry is achieved. This time is extremely small on the scale of . For example, it takes only  fm for a quark of momentum  GeV at at RHIC to achieve the maximal asymmetry of %. 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 by replacing . For muons we get  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  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 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 in the process , where stands for a charged fermion, as a function of photon’s transverse momentum , rapidity , and azimuthal angle . 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 of mass and charge ( is the absolute value of electron charge) moving in external magnetic field (in a plane perpendicular to the field direction) is where is the fermion energy. Here—in the spirit of the adiabatic approximation— 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 . This implies that For RHIC it is equivalent to , for LHC .

Photon decay rate was calculated in [45] and, using the quasi-classical method, in [56]. It reads where summation is over fermion species and the invariant parameter is defined as with the initial photon 4-momentum . With notations of Figure 7, , where . Thus, . Employing and , we write

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

fig9
Figure 9: Decay rate of photons moving in reaction plane in magnetic field as a function of transverse momentum : (a) at RHIC, (b) at LHC. Broken lines from bottom to top give contributions of , and 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 field direction, that is, in the direction orthogonal to the reaction plane. At the dependence of is very weak which is reflected in nearly symmetric azimuthal shape of the dashed line in Figure 10.

490495.fig.0010
Figure 10: Azimuthal distribution of the decay rate of photons at different rapidities at LHC. Only contribution of the channel is shown. Note that the rate vanishes at and , that is, in the direction of . However, it happens at so small values of 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 is an even function of , we have In strong fields . For example, for at RHIC at and  GeV we get . Therefore, we can expand the rate (110) at large as [45] At the Fourier coefficients can be calculated analytically using formula 3.631.9 of [34] where is the Euler’s beta function and is defined in (114), Substituting these expressions into (113) we find The first few terms in this expansion read

What is measured experimentally is not the decay rate, but rather the photon spectrum. This spectrum is modified by the survival probability which is obviously azimuthally asymmetric. To quantify this asymmetry, we write using (113) where is the survival probability averaged over the azimuthal angle. Since , as can be seen in Figure 9, we can estimate using (114) and (115) In particular, the “elliptic flow” coefficient is [55] For example, at  GeV and  fm/c one expects % at RHIC and % 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 over which it significantly varies is much larger than the quarkonium radius. If is magnetic field life-time, then . For a quarkonium with binding energy and radius , the quasi-static approximation applies when . Estimating conservatively  fm we get for : , which is comfortably large to justify the quasi-static approximation, where I assumed that is given by its vacuum value. As temperature increases drops. Temperature dependence of is model dependent; however, it is certain that eventually it vanishes at some finite temperature . Therefore, only in the close vicinity of , that is, at very small binding energies, the quasi-static approximation is not applicable. I thus rely on the quasi-static approximation to calculate 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 in the comoving frame satisfies (for weakly bound states), where is quark mass; see (144). This effect causes a significant increase in quarkonium dissociation rate; numerical calculation for is shown in Figure 13.(2)Zeeman effect. Energy of a quarkonium state depends on spin , orbital angular momentum , and total angular momentum . In a magnetic field these states split; the splitting energy in a weak field is , where is projection of the total angular momentum on the direction of magnetic field, is quark mass, and is Landé factor depending on , , and in a well-known way; see, for example, [59]. For example, with , and () splits into three states with and with mass difference  GeV, where we used . 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 [60]. This is 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: and are velocity and momentum of quarkonium in the lab frame; is its mass; is the momentum of quark or anti-quark in the comoving frame; is its mass; is the magnetic field in the lab frame, and are electric and magnetic fields in the comoving frame; is the quarkonium Lorentz factor; and is a parameter defined in (139). I use Gauss units throughout the section; note that expressions , , and 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 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 ) is a sum of its potential energy in the binding potential and its energy in the electric field , where is the electric field direction; see Figure 11. Since becomes large and negative at large and negative (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 as it goes away. By picking up a light quark out of vacuum it can hadronize into a -meson.

490495.fig.0011
Figure 11: Effective potential for (except the blue line where ). The width of the potential barrier decreases with and increases with . corresponds to the binding energy in units of .

If we now go to the reference frame where and there is only magnetic field (we can always do so since ), 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 -mesons. This happens in spite of the fact that mass is smaller than masses of two -mesons due to additional momentum 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 in constant magnetic field . Let and be magnetic and electric fields in the comoving frame, and let subscripts and denote field components parallel and perpendicular to correspondingly. Then,