Novel quantum phenomena induced by strong magnetic fields in heavy-ion collisions

Abstract

The relativistic heavy-ion collisions create both hot quark–gluon matter and strong magnetic fields, and provide an arena to study the interplay between quantum chromodynamics and quantum electrodynamics. In recent years, it has been shown that such an interplay can generate a number of interesting quantum phenomena in hadronic and quark–gluon matter. In this short review, we first discuss some properties of the magnetic fields in heavy-ion collisions and then give an overview of the magnetic field-induced novel quantum effects. In particular, we focus on the magnetic effect on the heavy flavor mesons, the heavy-quark transports, and the phenomena closely related to chiral anomaly.

Appendix: Mixing strengths from the Bethe–Salpeter amplitudes

It would be instructive to see the computation of the coupling strength at the three-point vertex among two quarkonia and an external magnetic field by using the Bethe–Salpeter amplitudes obtained in the ladder approximation and the heavy-quark limit [240, 241]. This computation involves a typical technique for the perturbation theory in the presence of a magnetic field.

For the static charmonium carrying a momentum $q = ( 2m - { \epsilon _0}, 0,0,0)$ with ${ \epsilon _0}$ being the binding energy, the Bethe–Salpeter amplitudes for $\eta _{\rm c}$ and $J/\psi$ are, respectively, given by

$$\begin{aligned} \Gamma ^5 (p,p-q)= & {} \left( \epsilon _0 + \frac{{\varvec{p}}^2}{m} \right) \! \sqrt{\frac{m_{ \scriptscriptstyle {c \bar{c}} }}{N_{\rm c}}} \, \psi _{ \scriptscriptstyle 1S }({\varvec{p}}) \, P_+ \gamma ^5 P_- , \end{aligned}$$

(62)

$$\begin{aligned} \Gamma ^\mu (p,p-q)= & {} \left( \epsilon _0 + \frac{{\varvec{p}}^2}{m} \right) \! \sqrt{\frac{m_{ \scriptscriptstyle {c \bar{c}} }}{N_{\rm c}}} \, \psi _{ \scriptscriptstyle 1S }({\varvec{p}}) \, P_+ \gamma ^\mu P_-, \end{aligned}$$

(63)

where we have the projection operators $P_\pm = \frac{1}{2} ( 1 \pm \gamma ^0 )$ and the ground-state wave function of the S-wave bound state $\psi _{ \scriptscriptstyle 1S }({\varvec{p}})$. The mass $m_{ \scriptscriptstyle {c \bar{c}} }$ is those of $\eta _{\rm c}$ and $J/\psi$, which are degenerated in the heavy-quark limit. The number of the color is $N_{\rm c}=3$.

Fig. 21

An effective coupling strength from triangle diagrams. Shaded vertices show form factors given by the Bethe–Salpeter amplitudes. (Reproduced from Ref. [30])

We show a calculation of a coupling strength in the mixing between $\eta _{\rm c}$ and the longitudinal $J/\psi$ from triangle diagrams (Fig. 21). Interactions between quarks and external magnetic fields are taken into account by employing the Fock–Schwinger gauge. In this gauge, the quark propagators with one and two insertions of constant external fields are expressed as [216]

$$\begin{aligned} S_1 (p)= & {} - \frac{i}{4} Q_\mathrm{em}F_{\alpha \beta } \frac{1}{(p^2-m^2+ i \varepsilon )^2}\nonumber \\&\times \left\{ \sigma ^{\alpha \beta } ({p\!\!/}+ m) + ( {p\!\!/}+ m) \sigma ^{\alpha \beta } \right\} \ , \end{aligned}$$

(64)

$$\begin{aligned} S_2 (p)= & {} - \frac{1}{4} Q_\mathrm{em}^2 F_{\alpha \beta } F_{\mu \nu } \frac{1}{(p^2-m^2+ i \varepsilon )^5}\nonumber \\& \times ({p\!\!/}+m) \left\{ f^{\alpha \beta \mu \nu } + f^{\alpha \mu \beta \nu } + f^{\alpha \mu \nu \beta } \right\} ({p\!\!/}+m) \ , \end{aligned}$$

(65)

where $Q_\mathrm{em}$ denotes an electromagnetic charge of a quark and the gamma matrix structures are given by

$$\begin{aligned} \sigma ^{\alpha \beta }= & {} \frac{i}{2} [\gamma ^\alpha , \gamma ^\beta ]\ , \end{aligned}$$

(66)

$$\begin{aligned} f^{{\alpha }{\beta } {\mu }{\nu }}= & {} {\gamma }^{\alpha } ( {p\!\!/}+ m) {\gamma }^{\beta } ( {p\!\!/}+ m) {\gamma }^{\mu } ( {p\!\!/}+ m) {\gamma }^{\nu } \ . \end{aligned}$$

(67)

The coupling strength can be read off from the sum of the two diagrams $i {\mathcal M} ^\mu = i {\mathcal M} _a^\mu + i {\mathcal M} _b^\mu$, where the each diagram is written down as

$$\begin{aligned} i {\mathcal M} _a^\mu= & {} - \!\! \int \! \frac{\hbox{d}^4p}{(2\pi )^4} \mathrm{Tr} \left[ \, \Gamma _5^\dagger (p-q,p) S_1(p) \right. \nonumber \\& \left. \times \Gamma ^\mu (p,p-q) S_0 (p-q) \, \right] \ , \end{aligned}$$

(68)

$$\begin{aligned} i {\mathcal M} _b^\mu= & {} - \!\! \int \! \frac{\hbox{d}^4p}{(2\pi )^4} \mathrm{Tr} \left[ \, \Gamma _5^\dagger (p+q,p) S_0 (p+q) \right. \nonumber \\& \left. \times \Gamma ^\mu (p,p+q) S_1(p) \, \right] \ . \end{aligned}$$

(69)

In the leading order of the heavy-quark expansion, the $p^0$-integral is easily performed, and we find

$$\begin{aligned} i {\mathcal M} _a^\mu= & {} i {\mathcal M} _b^\mu \nonumber \\= & {} 2 Q_\mathrm{em}\tilde{F}^{0\mu } \int \frac{ \hbox{d}^3{\varvec{p}}}{(2\pi )^3} \vert \psi _{ \scriptscriptstyle 1S }({\varvec{p}}) \vert ^2 \ . \end{aligned}$$

(70)

From the normalization of the wave function,

$$\begin{aligned} \int \frac{ \hbox{d}^3{\varvec{p}}}{(2\pi )^3} \vert \psi _{ \scriptscriptstyle 1S }({\varvec{p}}) \vert ^2 = 1, \end{aligned}$$

(71)

the amplitude is independent of the wave functions, and the sum of two triangle diagrams is obtained as

$$\begin{aligned} i {\mathcal M} ^\mu= & {} 4 Q_\mathrm{em}\tilde{F}^{0\mu } \ . \end{aligned}$$

(72)

By contracting with the polarization vector for the longitudinal (transverse) vector state $\epsilon ^\mu = (0,0,0,1)$ ($\tilde{\epsilon }^\mu = (0, n_\perp , 0)$), the amplitude vanishes for the transverse modes as

$$\begin{aligned} i {\mathcal M} _\mu \tilde{\epsilon }^\mu= & {} 0 , \end{aligned}$$

(73)

while the longitudinal mode has a nonvanishing amplitude

$$\begin{aligned} i {\mathcal M} _\mu \epsilon ^\mu= & {} 4 Q_\mathrm{em}\tilde{F}^{0\mu } \epsilon _\mu = 4 Q_\mathrm{em}B \, . \end{aligned}$$

(74)

Therefore, the coupling strength in Eq. (25) is found to be

$$ g_{{{\text{P}}V}} = 4Q_{{{\text{em}}}} . $$

(75)

The coupling strength depends only on the electric charge and is given by $g_{{_{\scriptscriptstyle {\mathrm {PV}}}}} = 8/3 \simeq 2.66$ ($g_{{_{\scriptscriptstyle {\mathrm {PV}}}}} = 4/3 \simeq 1.33$) for the transition between $\eta _{\rm c}$ and $J/\psi$ ($\eta _b$ and $\Upsilon$). This is consistent with the value obtained by fitting the measured radiative decay width [see Eq. (33)], but is slightly overestimated. The radiative decay widths in $J/\psi \rightarrow \eta _{\rm c} + \gamma$ and $\Upsilon \rightarrow \eta _b + \gamma$ computed with the coupling strength (75) agree with the leading-order results by the potential Non-Relativistic QCD (pNRQCD) [208, 209]. The overestimate can be improved with the inclusion of the subleading terms [208, 209]. Extension to the open heavy flavors was carried out in Ref. [31].