Real-time dynamics of the Chiral Magnetic Effect
In quantum chromodynamics, a gauge field configuration with nonzero topological charge generates a difference between the number of left- and right-handed quarks. When a (electromagnetic) magnetic field is added to this configuration, an electromagnetic current is induced along the magnetic field; this is called the chiral magnetic effect. We compute this current in the presence of a color flux tube possessing topological charge, with a magnetic field applied perpendicular to it. We argue that this situation is realized at the early stage of relativistic heavy-ion collisions.
Introduction. The theory of the strong interactions, quantum chromodynamics (QCD), is an Yang-Mills theory coupled to fermions (quarks). An intriguing aspect of Yang-Mills theories is their relation to topology. This reveals itself in the existence of gauge field configurations carrying topological charge BPST (). This charge is quantized as an integer if these configurations interpolate between two of the infinite number of degenerate vacua of the Yang-Mills theory CDG (). Expressed in terms of the field strength tensor the topological charge reads ; here denotes the coupling constant and the dual field strength tensor equals .
By interacting with fermions the fields induce parity and charge-parity () odd effects H (). This can be seen by the following exact equation (valid for each quark flavor separately) which is a result of the axial anomaly S51 (); ABJ (): where is the quark mass, and denotes the axial current density in the background of a gauge field configuration . Let us define the chirality density and the chirality . Integrating the anomaly equation over space and time gives for massless quarks , where denotes the change in chirality over time. For massless quarks, the chirality is equal to the difference between the number of particles plus antiparticles with right-handed and left-handed helicity. Again for right-handed helicity implies that spin and momentum are parallel whereas they are antiparallel for left-handed helicity.
The gauge fields are included in the path integral, and as a result they contribute to the amplitudes of physical processes. Experimental evidence for these configurations is however indirect. The clearest confirmation follows from the large mass of pseudoscalar meson compared to the , , and mesons H (). In this Letter we will discuss an alternative way in which topological configurations of gauge fields in QCD, i.e. gluon fields could be studied in heavy-ion collisions.
Using high-energy heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) one can investigate the behavior of QCD at high energy densities. Very strong color electric and color magnetic fields are produced during these collisions, whose strength is characterized by the gluon saturation momentum . In addition, extremely strong (electromagnetic) magnetic fields are present in non-central collisions, albeit for a very short time. In gold-gold collisions at RHIC energies the magnitude of this magnetic field at the typical time scale after the collision is of the order of which corresponds to KMW (); SIT (). Such extremely strong magnetic fields are able to polarize to some degree the bulk of the produced quarks which have typical momenta of a few hundred MeV. More specifically, quarks with positive (negative) charge have a tendency to align their spins parallel (anti-parallel) to the magnetic field. As a result, assuming the produced quarks can be treated as massless, a positively (negatively) charged quark with right-handed helicity will have its momentum parallel (antiparallel) to the magnetic field. For quarks with left-handed helicity this is exactly opposite. Hence a quark and anti-quark both having the same helicity will move in opposite directions with respect to the magnetic field. This implies that an electromagnetic current is generated along the magnetic field if there is an imbalance in the helicity, i.e. a nonzero chirality. Because gauge fields with generate chirality, they will therefore induce an electromagnetic current along a magnetic field. This mechanism which signals - and -odd interactions has been named the chiral magnetic effect KMW (); KKZ (). In an extremely strong magnetic field , so strong that all quarks are fully polarized, it follows from the arguments presented above that for each quark flavor separately the induced current equals , where is the charge of the quark.
The magnetic field in heavy ion collisions is pointing in a direction perpendicular to the reaction plane; this is the plane in which the impact parameter and the beam axis lie. As a result of the chiral magnetic effect the charge asymmetry between the two sides of the reaction plane will be generated. The sign of this asymmetry will fluctuate from collision to collision since (assuming the so-called angle vanishes and there is no global violation of parity) the probability of generating either positive or negative is equal. Using the observable proposed in V04 () the STAR collaboration has analyzed charge correlations star (). The results are qualitatively in agreement with the predictions of the chiral magnetic effect; the search for alternative explanations and additional manifestations of local parity violation is underway WBKL ().
Several quantitative theoretical studies of the chiral magnetic effect have appeared in the literature FKW (); adscft (); lattice (); inst (). Most of the analytic studies are based on introducing a chiral asymmetry by hand, after which the equilibrium response to a magnetic field is studied FKW (); adscft () (see also add ()). In this Letter we will for the first time investigate a situation in which the chirality is generated dynamically in real-time in the presence of a magnetic field. For this we will take the simplest Yang-Mills gauge field configuration carrying topological charge, that is one which describes a color flux tube having constant Abelian field strength, i.e. with and constant and homogeneous. Furthermore, we will take only the -components of the color electric () and color magnetic () field nonzero. Perpendicular to this field configuration we will apply an electromagnetic field pointing in the direction (see Fig. 1). Note that hereafter we write to denote a color magnetic field and for an electromagnetic one. Such color flux tubes, which carry topological charge and are homogeneous over a spatial scale , naturally arise in the glasma glasma (), the dense gluonic state just after the collision, where . The induced current itself can generate electromagnetic and color fields, which can alter the dynamics. We will ignore this back-reaction, which can be justified as long as the induced current is small compared to the currents that create the external color and magnetic fields. Furthermore we will also ignore the production of gluons in the color flux-tube.
Calculation. Using a color rotation we can choose only the third component of nonvanishing. Since the generator of the Lie algebra is diagonal, the different color components decouple. As a result for each quark flavor separately the problem is equivalent to a quantum electrodynamics (QED) calculation, in which the magnetic field with and the electric field with . Here labels the different color components, and denotes the electric charge of a particular quark. We will define to be the coordinate frame in which the electromagnetic field has this form.
We hence need to compute the induced electromagnetic current density in . To do this we will start in a different coordinate system in which and . In this frame it is rather straightforward to do calculations. Then by applying a Lorentz transformation we can obtain the results in as is illustrated in Fig. 2. We will switch on the electric field in uniformly at a time in the distant past, i.e. . In this way the situation in is completely homogeneous.
The production of pairs in gives rise to an homogeneous electromagnetic current density . Because of symmetry reasons the only nonvanishing component of this current lies in the -direction. Furthermore, each time a pair is created the current will grow. Eventually when both components of the pair are accelerated by the electric field to (nearly) the speed of light, the net effect of the creation of one single pair will be that the total current has increased by two units of . Therefore, sufficiently long after the switch-on, the change in current density in the -direction becomes times the rate per unit volume of pair-production, to be precise . This equation has been verified explicitly numerically in T08 (). We have also found it to be correct analytically, even for F10 ().
Before we compute the induced currents in let us point out that the rate is consistent with the anomaly equation. In the limit of a very large magnetic field () all produced pairs will reside in the lowest Landau level causing maximal chiral asymmetry. Since each pair then produces two units of , the pair production rate should then be equal to half the chirality rate. Taking the limit in Eq. (1) gives
which is indeed in agreement with the anomaly equation (see Introduction) in the limit of , since the chiral current vanishes because of homogeneity. It turns out that Eq. (2) also exactly gives the chirality rate for nonzero and any and F10 ().
As is indicated in Fig. 2 we can go from frame to by applying a boost with rapidity in the -direction. In the new coordinate system obtained by this boost, the electric and magnetic field respectively read and . Since points in the -direction, the direction of will not change after the boost in the -direction. However because the boost implies that , the current density rate is modified to . The current density has now also obtained a gradient in the -direction (). This and other inhomogeneities in arise because the uniform switch-on of at implies an inhomogeneous switch-on of part of and at .
To arrive in frame we have to apply a rotation with angle around the -axis such that the electric field points in the -direction. The angle follows from Fig. 2 and satisfies and . The current density rate becomes after the rotation
We can eliminate by expressing the above in terms of the fields in . The magnetic field is , implying that . Because both and are Lorentz invariant, one finds , and .
Now we can put all our results together. After summing over colors the -component of the current vanishes (), implying that the only remaining component lies in the -direction. Using that we obtain after summing over colors,
where and have dependence on and . The rate of chirality production in becomes . Inserting Eq. (2) yields for the rate of current over chirality density generation
Discussion. Equation (4) clearly shows that an external magnetic field induces a current perpendicular to the color flux tube. To summarize our findings we display in Fig. 3 for three different values of the rate of generation of this current normalized to Eq. (5), the rate of chirality production. We will now analyze our results and show that indeed behaves as the chiral magnetic effect predicts.
First of all let us take either or , which implies that no chirality is generated. If then , for either or . In all these cases indeed vanishes as follows from Eq. (4). This is obvious when since in that case no particles are produced as follows from Eq. (1). Also as expected vanishes if there is no perpendicular magnetic field which can be seen from Fig. 3 as well.
Secondly, in the limit of , we have so that from Eq. (5) it follows that . This indicates that for large magnetic fields the current rate is indeed exactly given by the chirality rate in agreement with the prediction outlined in the introduction. Therefore the curves in Fig. 3 approach unity for when both and are large.
A finite mass reduces the chirality and indeed also as can be seen from Eq. (4). In fact Eq. (5) shows for any value of the mass the current is proportional to the chirality. Hence the curves displayed in Fig. 3 are independent of mass. Moreover let us point out that the direction of the current is independent of the sign of the quark charge, but does depend on the direction of the magnetic field and the sign of the chirality, i.e. ). For small compared to both and , we have and so that
The linear dependence on for small is clearly visible in Fig. 3. The small kink at and is due to the fact that and vary rapidly around when is small compared to , which is equivalent to .
The generation of a current by the transformation from frame to is a very general result of Lorentz invariance, and is equivalent to the Lorentz force in frame . Therefore any charged colored particle that is present in the color flux-tube plus magnetic field background will experience a force in the -direction if . To illustrate this we can consider the whole calculation for fictional colored and electrically charged scalar particles. In that case there is no anomaly so that no chirality is generated. The results for scalar fermions can be obtained by replacing by in Eq. (1) Dunne () and subsequently in all other equations. The ratio between the scalar and fermion current density rate becomes simply , which is approximately for and for . Clearly scalar particles behave completely different from the predictions of the chiral magnetic effect, moreover the scalar contribution to is always smaller than that of fermions and even exponentially suppressed for .
Let us finally stress that our quantitative results are strictly speaking only valid for the rather special inhomogeneous switch-on of the fields in the color-flux tube. Nevertheless, as is the case at later times in heavy-ion collisions, if is small compared to the color fields, the effects of the inhomogeneous switch-on are marginal. Therefore it is very likely that the result for small , Eq. (6), is also correct for a homogeneous switch-on. To further address this issue one can start from an inhomogeneous switch-on in that becomes homogeneous in . However, this situation is more complicated, at present we are unfortunately unable to solve it exactly.
To conclude, we have shown by a dynamical calculation that if topological charge is present in a magnetic field, an electromagnetic current will be generated along this magnetic field. This very natural mechanism is called the chiral magnetic effect and signals - and -odd interactions. As such it could be an explanation for the charge correlations in heavy-ion collisions observed by the STAR collaboration star (). For a broader range of physics applications the intermediate QED results might be of use for testing the axial anomaly with strong field lasers.
Acknowledgments. We are grateful to Antti Gynther, Larry McLerran, Anton Rebhan and Andreas Schmitt for discussions. The work of H.J.W. was supported by the Alexander von Humboldt Foundation. K.F. was supported by the Japanese MEXT grant No. 20740134 and also in part by the Yukawa International Program for Quark Hadron Sciences. This manuscript has been authored under Contract No. #DE-AC02-98CH10886 with the U.S. Department of Energy.
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