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 ~
We can understand the origin of magnetic field in heavy-ion collisions by considering collision of two ions of radius
It has been known for a long time that classical electrodynamics breaks down at the critical (Schwinger) field strength
Throughout this paper, the heavy-ion collision axis is denoted by
Heavy-ion collision geometry as seen along the collision axis
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 [
Numerical integration in (
Magnetic field
Nuclear charge density
The mean absolute value of (a) magnetic field and (b) electric field at
Figure
Electromagnetic fields created in heavy-ion collisions were also examined in more elaborated approaches in [
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
In medium, magnetic field created by a charge
Equation (
It is instructive to compare time dependence of magnetic field created by moving charges in vacuum and in plasma. In vacuum, setting
In matter
Relaxation of magnetic field at
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.
So far I treated quark-gluon plasma as a static medium. Expanding medium temperature and hence conductivity are functions of time. In Bjorken scenario [
Magnetic field in expanding medium is still governed by (
The results of a numerical calculation of (
The dynamics of magnetic field relaxation in conducting plasma can be understood in a simple model [
Schwinger mechanism of pair production [
Since space dimensions of QGP are much less than
In view of smallness of
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
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,
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
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 [
Since the time derivative of
The viscous pressure generated by a deviation from equilibrium is given by the tensor
Let us expand
In strong magnetic field we can determine
In the relaxation-time approximation we can write the collision integral as
Equation for the second correction to the equilibrium distribution
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
The viscous pressure tensor in vanishing magnetic field is isotropic in the
Suppose that the pressure is isotropic; that is, it depends on the coordinates
We see that at later times after the heavy-ion collision, flow velocity is proportional to
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 [
The structure of the viscous stress tensor in very strong magnetic field (
General problem of charged fermion radiation in external magnetic field was solved in [
A typical diagram contributing to the synchrotron radiation, that is, radiation in external magnetic field, by a quark is shown in Figure
A typical diagram contributing to the synchrotron radiation by a quark.
Geometry of a heavy-ion collision.
Energy loss by a relativistic quark per unit length is given by [
To apply this result to heavy-ion collisions we need to write down the invariant
In Figure
Energy loss
Energy loss due to the synchrotron radiation has a very nontrivial azimuthal angle and rapidity dependence that comes from the corresponding dependence of the
In magnetic field gluon spectrum is azimuthally asymmetric. It is customary to describe this asymmetry by Fourier coefficients of intensity defined as
Synchrotron radiation leads to polarization of electrically charged fermions, this is known as the Sokolov-Ternov effect [
The nature of this spin-flip transition is transparent in the nonrelativistic case, where it is induced by the interaction Hamiltonian [
Let
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
In this section I consider pair production by photon in external magnetic field [
Characteristic frequency of a fermion of species
Photon decay rate was calculated in [
Plotted in Figure
Decay rate of photons moving in reaction plane in magnetic field as a function of transverse momentum
Azimuthal distribution of the decay rate of photons at LHC is azimuthally asymmetric as can be seen in Figure
Azimuthal distribution of the decay rate of photons at different rapidities at LHC. Only contribution of the
To quantify the azimuthal asymmetry it is customary to expand the decay rate in Fourier series with respect to the azimuthal angle. Noting that
What is measured experimentally is not the decay rate, but rather the photon spectrum. This spectrum is modified by the survival probability
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
Magnetic field has a three-fold effect on quarkonium.
Some of the notational definitions used in this section:
In this section I focus on Lorentz ionization, which is an important mechanism of
Effective potential
If we now go to the reference frame where
Consider a quarkonium traveling with velocity
I assume that the force binding
It is natural to study quarkonium ionization in the comoving frame [
The effective potential
Ionization probability of quarkonium equals its tunneling probability through the potential barrier. The later is given by the transmission coefficient
In order to compare with the results obtained in [
An important limiting case is crossed fields
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
Dimensionless function
Dissociation rate of
Motion of a particle can be treated non-relativistically if its momentum is much less than its mass. In such a case
In Figure
Of special interest is the limit of weak binding
In the limit
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
With quark spin taken into account, the non-relativistic version of (
So far I have entirely neglected possible existence of electric field in the lab frame. This field, which we will be denoted by
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 [
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
Given the electromagnetic field in the laboratory frame
It is useful to introduce dimensionless parameters
In the case that mechanism (i) is responsible for generation of electric field,
To obtain the experimentally observed
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
(1)
(2)
One of the most interesting applications of this formalism is calculation of the dissociation rate of
Results of numerical calculations are exhibited in Figures
Contour plot of the dissociation rate of
(a) Angular distribution of
As the plasma temperature varies, so is the binding energy of quarkonium, although the precise form of the function
In the absence of electric fied
Spectrum of quarkonia surviving in the electromagnetic field is proportional to the survival probability
Although magnetic fields in
The effect of
In Section
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
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
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
Angular distribution of radiation is obtained by integrating over the photon energies and remembering that
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:
Another consequence of the conservation laws (
Substituting of (
The natural variables to study the synchrotron radiation are the photon energy
Figure
Spectrum of synchrotron radiation by
Azimuthal distribution of synchrotron radiation by
In order to compare the photon spectrum produced by synchrotron radiation to the photon spectrum measured in heavy-ion collisions, the
Azimuthal average of the synchrotron radiation spectrum of
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
The low energy part of the photon spectrum satisfies the condition
For a qualitative discussion it is sufficient to consider the
The upper summation limit in (
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1 | 1 | 1 | 1 | 1 | 1 | 1 | 15 | 15 | 15 |
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0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.4 | 0.4 | 0.4 |
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0.1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 2 | 1 |
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0.096 | 9.6 | 38 | 86 | 29 | 35 | 19 | 0.64 | 2.6 | 1.3 |
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30 | 40 | 90 | 150 | 120 | 200 | 90 | 8 | 12 | 16 |
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 [
Variation of the synchrotron spectrum with plasma temperature. Lower line:
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.
The theory of one-photon pair annihilation was developed in [
For
The results of the numerical calculations are represented in Figure
Photon spectrum in one-photon annihilation of
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 [
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.
This work was supported in part by the U.S. Department of Energy under Grant no. DE-FG02-87ER40371.