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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,
$$ \begin{array}{l} f_{\mathrm{EM}} = \dfrac{c}{8 \alpha_{\mathrm{EM}}} \dfrac{\sigma_{\mathrm{el}} n_{\mathrm{eq}}\left(1-n_{\mathrm{eq}}\right)}{T^3 p \cdot u} e Q_f F^{\mu \nu} p_\mu u_\nu\, , \end{array} $$ (1) 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
$$ \begin{split} {{\mathcal R}}^\gamma = &\dfrac{1}{2(2\pi)^3}\displaystyle\sum_i\int \dfrac{{\rm{d}}^3 {{\boldsymbol p}}_1}{2E_1(2\pi)^3} \dfrac{{\rm{d}}^3 {{\boldsymbol p}}_2}{2E_2(2\pi)^3}\dfrac{{\rm{d}}^3 {{\boldsymbol p}}_3}{2E_3(2\pi)^3}\times \\ & (2\pi)^4\delta^4(P_1+P_2-P_3-P)|{{\mathcal M}}_i|^2 \times \\ & f_1(P_1)f_2(P_2)[1\pm f_3(P_3)]\approx \dfrac{40\alpha \alpha_{\rm{s}}}{9\pi^2}{{\mathcal L}} f_q(P) I_c\, , \end{split} $$ (2) 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,
$$ \begin{split} {{\mathcal R}}^\gamma & \approx \dfrac{40\alpha \alpha_{\rm{s}}}{9\pi^2}{{\mathcal L}} [n_{\rm {eq}}(P) + f_{\rm {EM}}] [\bar{I}_c + {I}_c^{\rm {EM}}] \\ & =\dfrac{40\alpha \alpha_{\rm{s}}}{9\pi^2}{{\mathcal L}} \dfrac{T^2}{8}[n_{\rm {eq}}(P) + f_{\rm {EM}}] \\ & =\bar{{{\mathcal R}}}^\gamma+{{\mathcal R}}^\gamma_{\rm{EM}}\, , \end{split} $$ (3) 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,
$$ \begin{split} E_p\dfrac{{\rm{d}}^3 N}{{\rm{d}}^3 {{\boldsymbol p}}} = &\displaystyle\int\limits_{V} \bar{{{\mathcal R}}}^\gamma(P, X)+{{\mathcal R}}^\gamma_{\rm{EM}} \\ = & E_p\dfrac{{\rm{d}}^3 \bar{N}}{{\rm{d}}^3 {{\boldsymbol p}}}+E_p\dfrac{{\rm{d}}^3 N_{\rm{EM}}}{{\rm{d}}^3 {{\boldsymbol p}}}\, . \end{split} $$ (4) The elliptic flow of photon is defined as,
$$ \begin{split}{l} {\upsilon}_2^\gamma(p_{\rm{T}})= & \dfrac{\int {\rm{d}}y {\rm{d}}\phi_p \cos(2\phi_p) E_p {\rm{d}}^3N/{\rm{d}}^3{{\boldsymbol p}} }{\int {\rm{d}}y {\rm{d}}\phi_p E_p {\rm{d}}^3N/{\rm{d}}^3{{\boldsymbol p}}}\\ = & \dfrac{\bar {\upsilon}_2 + {{\mathcal A}} {\upsilon}_2^{\rm{EM}}}{1 + {{\mathcal A}}}, \end{split} $$ (5) where,
$$ \begin{split} {{\mathcal A}} = \dfrac{\int {\rm{d}}y {\rm{d}}\phi_p E_p {\rm{d}}^3N_{\rm{EM}}/{\rm{d}}^3{{\boldsymbol p}} }{\int dy {\rm{d}}\phi_p E_p {\rm{d}}^3\bar{N}/{\rm{d}}^3{{\boldsymbol p}}} \end{split}, $$ (6) $$ \begin{array}{l} {\upsilon}_2^{\rm{EM}} = \dfrac{\int {\rm{d}}y {\rm{d}}\phi_p \cos(2\phi_p) E_p {\rm{d}}^3N_{\rm{EM}}/{\rm{d}}^3{{\boldsymbol p}} }{\int {\rm{d}}y {\rm{d}}\phi_p E_p {\rm{d}}^3N_{\rm{EM}}/{\rm{d}}^3{{\boldsymbol p}}}. \end{array} $$ (7) $ \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.
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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
$$ \begin{array}{l} p^\mu = (p_{\rm{T}}\cosh (y-\eta_s), p_{\rm{T}}\cos\phi_p, p_{\rm{T}} \sin\phi_p, p_{\rm{T}} \sinh(y-\eta_s))\, , \\ u^\mu = (1, 0, 0, 0)\, ,\\[-12pt] \end{array} $$ (8) 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,
$$ \begin{array}{l} f_{{{{\rm{EM}}}}}^{\rm{Bjo}}\propto e Q_f B_y \dfrac{\sinh\eta_s }{ \cos(Y-\eta_s) } n_{\rm eq}\cos\phi_p\,, \end{array} $$ (9) 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],
$$ \begin{array}{l} n_{\rm eq} = A_0(\tau, \eta_s, p_{\rm{T}}, Y) + A_1(\tau, \eta_s, p_{\rm{T}}, Y) \cos\phi_p\, , \end{array} $$ (10) 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
$$ \begin{split} f_{{{{\rm{EM}}}}} \propto & eQ B_y \dfrac{\sigma_{\text{el}}}{T^3} \dfrac{\sinh\eta_s }{ \cosh(y-\eta_s) } (A_0 + A_1 \cos\phi_p )\cos\phi_p \\ = & eQ B_y \dfrac{\sigma_{\text{el}}}{T^3} \dfrac{\sinh\eta_s }{ \cosh(y-\eta_s) } \bigg[\dfrac{A_1}{2} +A_0 \cos\phi+ \\ & \dfrac{A_1}{2}\cos 2\phi \bigg] . \end{split} $$ (11) 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.
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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
$$ \begin{split} s(\tau_0, {\boldsymbol{x}}_\perp, \eta_s) \propto \; & f(\eta_s)[\chi N_{\rm coll} + (1-\chi)(N^+_{\rm part} f^+(\eta_s) + \\ & N^-_{\rm part} f^-(\eta_s))] , \end{split} $$ (12) 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,
$$ \begin{array}{l} f(\eta_s) = \exp\left(-\theta(|\eta_s|-\eta_M)\dfrac{(|\eta_s|-\eta_M)^2}{2\sigma_\eta^2}\right) \end{array} $$ (13) accounts for the longitudinal spectrum of charged hadrons, while
$$ \begin{array}{l} f^+(\eta_s) = \begin{cases} 0\, , \quad\quad\quad\quad\;\; \eta_s<-\eta_T\\ \dfrac{\eta_T+\eta_s}{2\eta_T}\, , \quad\quad -\eta_T \leqslant \eta_s \leqslant \eta_T\\ 1\, ,\quad\quad\quad\quad\;\; \eta_s>\eta_T \end{cases} \end{array} $$ (14) 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],
$$ \begin{array}{l} eB(\eta_s) = \overline{eB} \Gamma(\tau = 0.4 \text{ fm}/c, \eta_s)\, , \end{array} $$ (15) 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).
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摘要: 在相对论重离子对撞中,必然有电磁场产生。尽管伴随夸克胶子等离子体演化的电磁场可能很弱,但它们有可能对电磁探针有重要意义。本工作首次提出了弱磁场和背景介质的纵向动力学的耦合效应。阐明了当夸克-胶子等离子体存在弱外部磁场时,由磁场诱导的光子是高度各向异性的。另一方面,来自夸克-胶子等离子体的弱磁光子发射对光子产量只有很小的修正。在具有倾斜构型的火球经流体动力学演化之后,可以很好地重现实验测量的直接光子椭圆流。同时,在流体动力学阶段使用的时间平均磁场不大于介子质量平方的百分之几。Abstract: There must be electromagnetic fields created during high-energy heavy-ion collisions. Although the electromagnetic field may become weak with the evolution of the quark-gluon plasma (QGP), compared to the energy scales of the strong interaction, they are potentially important to some electromagnetic probes. In this work, we propose the coupled effect of the weak magnetic field and the longitudinal dynamics of the background medium for the first time. We demonstrate that the induced photon spectrum can be highly azimuthally anisotropic when the quark-gluon plasma is in the presence of a weak external magnetic field. On the other hand, the weak magnetic photon emission from quark-gluon plasma only leads to a small correction to the photon production rate. After hydrodynamic evolution with a tilted fireball configuration, the experimentally measured direct photon elliptic flow is well reproduced. Meanwhile, the used time-averaged magnetic field in the hydrodynamic stage is found no larger than a few percent of the pion mass square.
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Key words:
- heavy-ion collision /
- the direct photon /
- weak magnetic field /
- elliptic flow
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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]. (color online)
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]. (color online)
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