In high-energy heavy-ion experiments carried out at the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC), a novel state of matter, known as quark-gluon plasma (QGP), has been created, which with quark and gluon degrees of freedom^[1]. Studying the dynamic properties of QGP is an important means to enhance our understanding of quantum chromodynamics (QCD)^[2]. It has been established that QGP behaves as a perfect fluid theoretically, and viscous hydrodynamics has been proven to be a remarkably successful model that can account for many experimental observables related to charged hadrons^[3-4].

As penetrating probes, photons have a large mean-free path in the quark-gluon plasma. They are emitted immediately after their production and provide information about the system, rendering them a more direct means of studying the properties of QGP^[5-7]. Unfortunately, the hydrodynamic model fails to reproduce the photon spectrum. At the top RHIC energies, in the low $ p_{\rm{T}} $ region, experimentally measured direct photon yields ($ i.e. $, photon yields excluding those from hadron decays) exceed the current theoretical predictions. In particular, it has been found that direct photons exhibit a large elliptic flow $ {\upsilon}_2 $, comparable to that of pions^[8-11]. However, according to theoretical expectations, the direct photons should be isotropic because they are dominantly from the early stages, where the momentum anisotropy has not been fully developed. The observation of the yield excess and the large elliptic flow of direct photons, which contradicts theoretical expectations, is often referred to as "the direct photon puzzle"^[12-15].

Incorporating a significant emission anisotropy for the direct photons in hydrodynamic models is challenging. In the most updated multi-messenger calculation focusing on direct photons^[13], many ingredients have been taken into account: (i) the photon emission from the pre-equilibrium stage based on KoMPost is considered; (ii) the quark chemical equilibration time is also studied; (ⅲ) the updated next-next leading order perturbative QCD is used for prompt photons; and (ⅳ) the distribution functions with shear and bulk viscous corrections are used to calculate the photon emission rate. However, the yield excess at the low $ p_{\rm{T}} $ region is still notable. More importantly, the $ {\upsilon}_2^\gamma $ is still under-predicted.

The presence of a magnetic field provides a possible solution to the direct photon puzzle. In high-energy heavy-ion collisions, as a consequence of the relativistic motion of ions, magnetic fields are generated with extremely strong field strength, with $ |eB|\approx 10 m_\pi^2 $ at the top energies of RHIC and $ |eB|\approx 10^2 m_\pi^2 $ at the LHC, where $ m_\pi^2 $ is the pion mass^[16-19]. But the magnetic field decays drastically and is expected weak as the system starts to evolve hydrodynamically. For instance, at around 0.4 fm/c and in the center of the fireball, the residual strength of the magnetic field can drop to $ |eB|\approx 0.01 m_\pi^2 $ in a non-central Au+Au collision at the top RHIC energy. Nonetheless, after the pre-equilibrium stage, the detailed evolution of the magnetic fields in QGP remains undetermined, owing to the lack of knowledge of the electrical properties of the QGP medium^[20-26].

In the scenario of a strong magnetic field, the momentum anisotropy of photons can be actually generated, relying on mechanisms such as conformal anomaly or quark transitions between Landau levels, $ etc $^[27-32]. Meanwhile, the synchrotron radiation induced by a strong magnetic field presents naturally an elliptic mode^[33]. However, the required magnetic field is too strong and violates the weak magnetic field fact during the hydrodynamic stage. In this paper, we obey the weak magnetic field assumption where $ |eB|\ll m_\pi^2 $. In this particular scenario, while the photon emissions within the Quark-Gluon Plasma (QGP) experience a small correction owing to the magnetic field, a substantial anisotropy in the direct photon spectrum is observed.

The different scales of the magnetic field can modify the theoretical formalism dramatically. Especially, when $ |eB|\gg T\cdot \nabla $, where $ \nabla $ is the space gradient, magnetohydrodynamics should be taken into account, while when $ |eB|\gg g^2T^2 $, the quark Landau level excitations can not be neglected. In the hydrodynamic stage, the magnetic field is about $ |eB|\ll m_\pi^2 $. With such a weak magnetic field, the effect of the magnetic field should be a small correction to quark distribution. We solve the Boltzmann equation under an external magnetic field to obtain the magnetic correction,

where $ eQ_f $ indicates the corresponding electrical charge a quark, $ \sigma_{\rm el} $ is the electrical conductivity and $ u_\nu $ is flow four-velocity. Eq. (1) is consistent with the kinetic theory definition of charge current, $ j_{\mathrm{EM}}^i = \sigma_{\rm el} E^i = \sum_f Q_f\int \frac{\mathrm{d}^3 p }{(2\pi)^3 p^0} p^i f_{\mathrm{EM}} $. The constant $ c $ is used to match this equation and the number of quark flavors. To be consistent with the perturbative photon calculations, we take the electrical conductivity as $ \sigma_{\rm el}/T = 6 $ which is also perturbative^[34].

On the other hand, photons radiated from a thermalized QGP are mainly produced by $ 2\to2 $ scattering processes among quarks and gluons ($ 1+2\to 3 + \gamma $)^[35]. In the kinetic theory description, the production rate is

where the Compton and the quark-antiquark annihilation channels with respect to the scattering amplitudes $ |{{\mathcal M}}_i|^2 $ have been summed, and $ f_1 $, $ f_2 $ and $ f_3 $ are distribution functions of quarks and gluons, correspondingly. At the last line in Eq. (2), the small angle approximation is performed^[36-37], with $ {{\mathcal L}} $ a Coulomb logarithm, and $I_c = \int {\rm{d}}^3 {{\boldsymbol p}}/(2\pi)^3 [f_g+f_q]/p$ effectively characterizing the conversion between a quark-antiquark and a gluon in the thermalized QGP^[38].

After weak magnetic correction, the photon rate reads,

where $ n_{\rm {eq}} $is equilibrium distribution function, $ \bar{I}_c $ is $ T^2/8 $^[36] and $I_c^{\rm{EM}} = \int {\rm{d}}^3 {{\boldsymbol p}}/(2\pi)^3 f_{\rm {EM}}/p = 0$. The $ \bar{{{\mathcal R}}}^\gamma $ represents the photon rate without magnetic contributions and the $ {{\mathcal R}}^\gamma_{\rm{EM}} $ is photon radiation induced by the weak magnetic field. After a space-time integral with respect to the medium evolution, it leads to the photon invariant spectrum,

The elliptic flow of photon is defined as,

where,

$ \bar {\upsilon}_2 $ is elliptic flow without magnetic correction, $ {\upsilon}_2^{\rm{EM}} $ is photon anisotropy induced by the magnetic field and $ {{\mathcal A}} $ is magnetic photon emission yield over the background photon yield, namely without magnetic correction.

To explain the effect of weak magnetic photon emission from QGP, we consider the background medium in terms of the Bjorken flow, namely, a flow pattern with longitudinal expansion which is boost-invariant, and expansion in transverse directions is neglected. Written in the Milne coordinates $ (\tau, \eta_s) $, four-momentum and the flow four-velocity are

where $ y $ is the rapidity and $ p_{\rm{T}} $ the transverse momentum. Accordingly, in the presence of an external magnetic field orientated along the out-of-plane direction, the correction in the quark distribution function owing to a weak magnetic field becomes,

where for simplicity only factors of relevance are kept.

Owing to the rapidity-odd $ v_{1 \rm Ch}^{\rm odd} $ of the charged hadrons has been observed experimentally^[39-40] and it has been captured successfully with a tilted fireball configuration^[41], one can expand the background medium as^[42],

where $ A_1 $ is a rapidity-odd function that represents the tilted components which are responsible for $ v_{1 \rm Ch}^{\rm odd} $, $ A_0 $ should be an even function of rapidity as it is related to particle yields. Substitute back to Eq. (9), one has

The term outside the brackets is an odd function of $ y $ and $ \eta_s $ and the direct photons in experiments are measured in a symmetric rapidity window, $ Y\in[-Y_M, Y_M] $. Thus, when space-time integration is performed, only terms with rapidity-odd $ A_1 $ survive and $ A_0\cos\phi_p $ should vanish. Furthermore, Eq. (11) already implies that $ {\upsilon}_2^{{{\rm{EM}}}} = 0.5 $. The above analysis shows that when the rapidity-odd $ v_1 $ of the charged hadrons is coupled with a weak magnetic field, the elliptic flow of photons can be generated.

According to the Bjorken analysis, to get the momentum anisotropy emission of photons induced by a weak magnetic field, a tilted fireball is required to generate the rapidity-odd $ v_1 $ of hadrons. Following Ref. [41], we take the initial entropy density distribution

where $ N_{\rm coll} $, $ N_{\rm part}^+ $ and $ N_{\rm part}^- $ are the densities of binary collisions and participants of the forward and backward going nuclei, respectively. As in the standard Glauber model, entropy production receives contributions from binary collisions and participants, relatively determined by the constant $ \chi $. Longitudinal description in Eq. (12) is introduced via the functions $ f(\eta_s) $ and $ f^\pm(\eta_s) $. The symmetric longitudinal profile,

accounts for the longitudinal spectrum of charged hadrons, while

and $ f^-(\eta_s) = f^+(-\eta_s) $ give rise to rapidity-odd component. For a given collision centrality, the spatial geometry of the distribution relies entirely on these parameters, $ \eta_T $, $ \eta_M $, and $ \sigma_\eta $, which we choose as in Ref. [41]. Note, in particular, $ \eta_T $ determines the extent to which the fireball is tilted.

With this initial entropy density distribution, we solve 3+1 dimensional viscous hydrodynamics using the state-of-the-art MUSIC program^[43-44]. We calculate the weak magnetic photon emission between the initial time 0.4 fm/c and an effective temperature cut $ T_c = 145 $ MeV^[13].

The magnetic field is viewed as a constant in time but with $ \eta_s $ dependence as Lienard-Wiechert potential^[45],

where $ \overline{eB} $ is time-averaged magnetic field strength at $ \eta_s = 0 $.

In this paper, we didn't calculate the background photon yield and elliptic flow $ \bar{\upsilon}_2 $ but used the data extracted from the most updated hydrodynamical modeling in Ref. [13]. To get the elliptic flow of photon after magnetic correction in Eq. (5), we calculate the $ {{\mathcal A}} $ and $ {\upsilon}_2^{\rm EM} $ as shown in Eqs. (6) and (7).

Figure 1 shows the photon elliptic flow at RHIC 200 GeV AuAu collisions for 0~20%, 20%~40%, and 40%~60% centrality classes. The experiment data are well reproduced for all centrality classes after weak magnetic correction. The green dashed line is the data from Ref. [13] that is $ \bar{\upsilon}_2 $ without magnetic correction. The blue line is the $ {\upsilon}_2 $ in the presence of a weak magnetic field when $ \eta_T $ is 40% of beam rapidity $ y_{\rm beam} $. The brown line represents $ \eta_T = y_{\rm beam}-2.5 $. We find that the fireball with smaller $ \eta_T $ gives a larger enhancement on the $ {\upsilon}_2 $ under the same magnetic field. It is expected because the smaller $ \eta_T $ means the fireball is more tilted, and under the more tilted configuration the required B field is smaller to obtain the same $ {\upsilon}_2 $ increment. As centrality grows, the time-averaged $ |\overline{eB}^0_y| $ systematically increases, from $ |\overline{eB}^0_y| = 0.02 m_\pi^2 $ at the 0~20% centrality class, $ |\overline{eB}^0_y| = 0.04 m_\pi^2 $ at the 20%~40% centrality class, to $ |\overline{eB}^0_y| = 0.06 m_\pi^2 $ at the 40~60% centrality class. All these values satisfy the weak magnetic field condition, $ |eB|/m_\pi^2\ll 1 $. Weak magnetic photon emission leads to a minor increase in the direct photon yields. In the Fig. 2, the averaged yield increment is about 10% in all three centralities.

Figure 1.  The direct photon elliptic flow transverse momentum dependence at RHIC 200 GeV AuAu collisions. The green dashed line is theoretical calculations without magnetic field contribution. The blue solid line and brown dashed line are the results with weak magnetic correction but under different tilted configurations. The experiment data is from the PHENIX collaboration^[9].

Figure 2.  The direct photon yield ratio between that with and without weak magnetic contributions transverse momentum dependence at RHIC 200 GeV AuAu collisions. The blue solid line and brown dashed line represent the different tilted configurations.

The direct photon elliptic flow is shown similarly for the PbPb collisions at $ \sqrt{s_{{\rm{NN}}}} = 2.76 $ TeV in Fig. 3. Comparing the RHIC data, there exist large experimental uncertainties from the LHC measurements. Nevertheless, with the effect of weak magnetic photon emissions, the resulting elliptic flow is improved significantly. The time-averaged $ |\overline{eB}^0_y| $ in the centrality classes 0~20% and 20%~40%, is $ 0.03 m_\pi^2 $ and $ 0.05m_\pi^2 $, respectively. Similarly, the weak magnetic field has limited contributions to the photon yield for the PbPb collisions at $ \sqrt{s_{{\rm{NN}}}} = 2.76 $ TeV in Fig. 4.

Figure 3.  The direct photon elliptic flow transverse momentum dependence at LHC 2.76 TeV PbPb collisions. The green dashed line is theoretical calculations without magnetic field contribution. The blue solid line and brown dashed line are the results with weak magnetic correction but under different tilted configurations. The experiment data is from ALICE collaboration^[11].

Figure 4.  The direct photon yield ratio between that with and without weak magnetic contributions transverse momentum dependence at LHC 2 760 GeV PbPb collisions. The blue solid line and brown dashed line represent the different tilted configurations.

In this paper, instead of a strong magnetic field assumption which has been considered previously, we propose the effect of weak magnetic photon emission, originating from the interplay of a weak external magnetic field and the longitudinal dynamical evolution of the quark-gluon plasma. The weak magnetic photon emission results in an extra source of photon production from the quark-gluon plasma, with a large elliptic flow. In cases of Bjorken flow and more realistic 3+1D hydrodynamical evolution simulated via MUSIC, the effects of weak magnetic photon emission are justified. The experimentally measured direct photon elliptic flow at RHIC and LHC can be well reproduced. Meanwhile, the time-averaged magnetic field is still under the weak magnetic field assumption $ |eB|/m_\pi^2\ll 1 $.

The non-trivial coupling effect between the weak magnetic field and the longitudinal dynamics of the fireball can be generalized to high-order harmonic flow. For example, the longitudinally dependent elliptic moment in QGP would generate direct photon $ v_3^\gamma $, while the longitudinal dynamics of a triangular moment can contribute to $ v_4^\gamma $, $ etc $. In the future, to explore these correlations, one needs to perform event-by-event simulations^[46]. On the other hand, we are interested in the effect on other electromagnetic signals, such as local spin polarization^[47] and di-lepton polarization.

The current calculation is still crude: The space-time profile of the magnetic field is simplified. Dissipations from the magnetic forces correct only the quark distribution function, while their influences on the hydrodynamic equations of motion have been neglected. Effects from the electrical field have not been taken into account. We hope we can develop a systematic hydrodynamic model incorporating the electromagnetic effect.

Acknowledgments We are grateful for very helpful discussions with Charles Gale, Xu-Guang Huang, Lipei Du, Kaijia Sun, and Yifeng Sun. This work is supported in part by the NSFC Grants through No. 11975079.