The spin alignment of rho mesons in a pion gas

The orbital angular momentum and spin are intrinsically connected with each other, as demonstrated in the Barnett effect (RevModPhys.7.129, ) and Einstein-de-Haas effect (dehaas:1915, ) in materials. In peripheral collisions of heavy ions, a part of the orbital angular momentum (OAM) in the initial state can be distributed into the strong interaction matter via spin-orbit couplings in the form of the hadron’s spin polarization with respect to the direction of OAM (reaction plane), which is called the global polarization (Liang:2004ph, ; Liang:2004xn, ; Betz:2007kg, ; Gao:2007bc, ; Becattini:2007sr, ). The spin polarization of hyperons can be measured through their weak decays in which the parity symmetry is broken (PhysRevLett.36.1113, ). The global polarization of $\Lambda$ hyperons (including anti-paricles) has been measured by STAR collaboration in Au+Au collisions at 3-200 GeV (STAR:2017ckg, ; STAR:2018gyt, ), by HADES collaboration in Au+Au and Ag+Ag collisions at 2.42-2.55 GeV (HADES:2022enx, ) and by ALICE collaboration in Pb+Pb collisions at 5.02 TeV (ALICE:2021pzu, ). The global polarization of $\Xi$ and $\Omega$ hyperons (including anti-particles) has also been measured by STAR collaboration in Au+Au collisions at 200 GeV (STAR:2020xbm, ). These experimental measurements have been explained by various theoretical models (mainly hydrodynamics and transport models) (Karpenko:2016jyx, ; Li:2017slc, ; Xie:2017upb, ; Sun:2017xhx, ; Baznat:2017jfj, ; Shi:2017wpk, ; Xia:2018tes, ; Wei:2018zfb, ; Fu:2020oxj, ; Ryu:2021lnx, ; Fu:2021pok, ; Deng:2021miw, ; Becattini:2021iol, ; Wu:2022mkr, ). We refer the readers to some recent review articles in this field (Wang:2017jpl, ; Florkowski:2018fap, ; Gao:2020lxh, ; Huang:2020dtn, ; Gao:2020vbh, ; Becattini:2020ngo, ; Becattini:2022zvf, ).

Most vector mesons decay through strong interaction that preserves the parity symmetry, so the spin polarization of vector mesons cannot be measured in the same way as hyperons. The spin density matrix $\rho_{\lambda_{1}\lambda_{2}}$ for the spin-1 vector meson is a $3\times 3$ complex matrix with unit trace, $\mathrm{tr}\rho=1$, where $\lambda_{1}$ and $\lambda_{2}=0,\pm 1$ denote the spin states along the spin quantization direction. The 00-element $\rho_{00}$ for the vector meson can be measured by the angular distribution of its decay product or daugther particle (Schilling:1969um, ; Liang:2004xn, ; Yang:2017sdk, ; Tang:2018qtu, ), so $\rho_{00}-1/3$ is an observable that can describe the spin alignment of the vector meson. If $\rho_{00}=1/3$, the angular distribution of the daughter particle is isotropic and the vector meson has no spin alignment. If $\rho_{00}>1/3$, the polarization vector of the meson is aligned more in the spin quantization direction. If $\rho_{00}<1/3$, the polarization vector of the meson is aligned more in the transverse direction perpendicular to the spin quantization direction. The global spin alignment of $\phi$ and $K^{0*}$ mesons has recently been measured by STAR collaboration (STAR:2022fan, ). It is found that $\rho_{00}^{\phi}$ is significantly larger than 1/3 at lower energies, while $\rho_{00}^{K^{0*}}$ is consistent with 1/3.

There are many sources to the spin alignment of vector mesons (Yang:2017sdk, ; Xia:2020tyd, ; Gao:2021rom, ; Muller:2021hpe, ; Li:2022vmb, ; Wagner:2022gza, ; Kumar:2022ylt, ; Dong:2023cng, ; Kumar:2023ghs, ; Gao:2023wwo, ). In Ref. (Sheng:2019kmk, ), some of us proposed that a large deviation of $\rho_{00}$ from 1/3 for $\phi$ mesons may possibly come from the $\phi$ field, a strong force field with vacuum quantum number induced by the current of pseudo-Goldstone bosons. Such a proposal is based on a nonrelativistic quark coalescence model for the spin density matrix of vector mesons (Yang:2017sdk, ; Sheng:2019kmk, ), which is only valid for static vector mesons. In Ref. (Sheng:2022ffb, ), the relativistic version of the quark coalescence model has been constructed based on the spin Boltzmann equation with collisions. The model is successful in describing the experimental data for $\rho_{00}$ for $\phi$ mesons (Sheng:2022wsy, ). Recently some of us made a prediction for the rapidity dependence of the spin alignment with the same set of parameters (Sheng:2023urn, ), which was later confirmed by the preliminary data of STAR (Xi:2023quarkmatter, ). We refer the readers to some recent review articles about the spin alignment of vector mesons (Chen:2023hnb, ; Wang:2023fvy, ; Sheng:2023chinphyb, ).

In this paper, we try to study the spin alignment of the $\rho^{0}$ meson in a pion gas. As is well-known, the lifetime of the $\rho^{0}$ meson is very short and mainly decays inside the medium. As the result, the interaction between $\rho^{0}$ and $\pi^{\pm}$ mesons in the hadron phase of heavy-ion collisions has significant impact on the spin alignment of the rho meson. This is very different from the $\phi$ meson which is mainly formed by hadronization of quarks. This study is relevant to the search for the chiral magnetic effect (CME) (Kharzeev:2004ey, ; Kharzeev:2007jp, ; Fukushima:2008xe, ) since the decay of $\rho^{0}$ to $\pi^{\pm}$ provides a significant contribution to the background in the $\gamma$ correlator (STAR:2013ksd, ; STAR:2013zgu, ; Wang:2016iov, ) and the spin alignment of $\rho^{0}$ may have an effect on CME observables (Tang:2019pbl, ; Shen:2022gtl, ).

The paper is organized as follows. In Sec. II, an effective Lagrangian is given for the $\rho\pi\pi$ coupling (Fujiwara:1984pk, ). In Sec. III, from the Kadanoff-Baym (KB) equation for Green’s functions for pseudoscalar and vector mesons in the closed-time-path (CTP) formalism (Kadanoff2018QuantumSM, ), we derive the spin Boltzmann equations for vector mesons with collisions (Sheng:2022ffb, ). In Sec. IV, we derive the collision terms at the leading order (LO) and next-to-leading order (NLO) with the medium effect. The numerical results are given in Sec. V. In the final section, Sec. VI, are the conclusion and discussion.

The sign convention for the metric tensor is $g_{\mu\nu}=g^{\mu\nu}=\mathrm{diag}\left(1,-1,-1,-1\right)$, where we use Greek letters to denote four-dimension indices of vectors or tensors. The four-momentum is defined as $p=p^{\mu}=\left(p^{0},\mathbf{p}\right)$ and $p_{\mu}=\left(p^{0},-\mathbf{p}\right)$, where $p^{0}$ is the particle’s energy. For an on-shell particle, we have $p^{0}=E_{p}=\sqrt{\mathbf{p}^{2}+m^{2}}$.

We consider the chiral effective theory with SU(2) flavor symmetry. The $\rho$ meson is introduced via the hidden gauge field. The effective Lagrangian for a system of $\rho^{0}$, $\pi^{+}$ and $\pi^{-}$ mesons reads

$\displaystyle\mathcal{L}$

$\displaystyle=$

$\displaystyle\mathcal{L}_{\rho}+\mathcal{L}_{\pi}+\mathcal{L}_{\mathrm{int}},$

(1)

where $\mathcal{L}_{\rho}$, $\mathcal{L}_{\pi}$ and $\mathcal{L}_{\mathrm{int}}$ are the Lagrangians for free $\rho^{0}$, free $\pi^{\pm}$, and their interaction, respectively. They are given by

	$\displaystyle\mathcal{L}_{\rho}=$	$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m_{\rho}^{2}A_{\mu}A^{\mu},$
	$\displaystyle\mathcal{L}_{\pi}=$	$\displaystyle\partial_{\mu}\phi^{\dagger}\partial^{\mu}\phi-m_{\pi}^{2}\phi^{\dagger}\phi,$
	$\displaystyle\mathcal{L}_{\mathrm{int}}=$	$\displaystyle ig_{\rho\pi\pi}A^{\mu}\Big{(}\phi^{\dagger}\partial_{\mu}\phi-\phi\partial_{\mu}\phi^{\dagger}\Big{)},$		(2)

where $A_{\mu}$ is the real vector field for $\rho^{0}$, $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength tensor, $m_{\rho}=770$ MeV and $m_{\pi}=139$ MeV are masses of the rho meson and pion respectively, $\phi$ ($\phi^{\dagger}$) denotes the complex scalar field for $\pi^{+}$ ($\pi^{-}$), and $g_{\rho\pi\pi}\approx 5.9$ is the coupling constant for the $\rho\pi\pi$ vertex. The Lagrangian (1) is our starting point to derive the collision terms.

In this section we will introduce Wigner functions and spin kinetic or Boltzmann equations for vector mesons. The spin kinetic or Boltzmann equations can be derived from the KB equation in the CTP formalism (Martin:1959jp, ; Keldysh:1964ud, ; Kadanoff2018QuantumSM, ; Chou:1984es, ; Blaizot:2001nr, ; Berges:2004yj, ; Cassing:2008nn, ; Cassing:2021fkc, ). The spin kinetic or Boltzmann equations with collision terms are recent focus and have been derived for spin-1/2 massive fermions (Yang:2020hri, ; Sheng:2021kfc, ) and for vector mesons (Sheng:2022ffb, ; Sheng:2022wsy, ; Wagner:2023cct, ) in the CTP formalism. They can also be derived in other methods for spin-1/2 massive fermions (Weickgenannt:2019dks, ; Li:2019qkf, ; Sheng:2022ssd, ; Weickgenannt:2020aaf, ; Weickgenannt:2021cuo, ; Lin:2021mvw, ; Lin:2022tma, ; Wagner:2022amr, ) and for vector mesons (Wagner:2023cct, ). The building blocks of kinetic or Boltzmann equations are Wigner functions in phase space that are defined from two-point Green’s functions (Vasak:1987um, ; Heinz:1983nx, ; Blaizot:2001nr, ; Wang:2001dm, ; Gao:2012ix, ; Chen:2012ca, ; Becattini:2013fla, ; Gao:2019znl, ; Weickgenannt:2019dks, ; Hattori:2019ahi, ; Wang:2019moi, ; Weickgenannt:2020aaf, ; Yang:2020hri, ; Liu:2020flb, ; Weickgenannt:2021cuo, ; Sheng:2021kfc, ), see, e.g., Refs. (Gao:2020pfu, ; Hidaka:2022dmn, ) for recent reviews.

The real vector and complex scalar fields can be quantized as

	$\displaystyle A^{\mu}(x)$	$\displaystyle=$	$\displaystyle\sum_{\lambda=0,\pm 1}\int\frac{d^{3}p}{\left(2\pi\hbar\right)^{3}2E_{p}^{\rho}}$		(3)
			$\displaystyle\times\left[\epsilon^{\mu}(\lambda,{\bf p})a_{V}(\lambda,{\bf p})e^{-ip\cdot x/\hbar}+\epsilon^{\mu\ast}(\lambda,{\bf p})a_{V}^{\dagger}(\lambda,{\bf p})e^{ip\cdot x/\hbar}\right],$
	$\displaystyle\phi(x)$	$\displaystyle=$	$\displaystyle\int\frac{d^{3}k}{\left(2\pi\hbar\right)^{3}2E_{k}^{\pi}}\left[a({\bf k})e^{-ik\cdot x/\hbar}+b^{\dagger}({\bf k})e^{ik\cdot x/\hbar}\right],$		(4)

where $E_{p}^{\rho}=\sqrt{\mathbf{p}^{2}+m_{\rho}^{2}}$ and $E_{k}^{\pi}=\sqrt{\mathbf{k}^{2}+m_{\pi}^{2}}$ are the energies of $\rho$ and $\pi$ respectively, $\lambda$ denotes the spin state with respect to the spin quantization direction, and $\epsilon^{\mu}(\lambda,{\bf p})$ is the polarization vector

$\displaystyle\epsilon^{\mu}(\lambda,\mathbf{p})$

$\displaystyle=$

$\displaystyle\left(\frac{\mathbf{p}\cdot\boldsymbol{\epsilon}_{\lambda}}{m_{\rho}},\boldsymbol{\epsilon}_{\lambda}+\frac{\mathbf{p}\cdot\boldsymbol{\epsilon}_{\lambda}}{m_{\rho}(E_{p}+m_{\rho})}\mathbf{p}\right),$

(5)

with $\boldsymbol{\epsilon}_{\lambda}$ being the polarization three-vector of the vector meson in its rest frame and given by

	$\displaystyle\boldsymbol{\epsilon}_{0}=$	$\displaystyle(0,1,0),$
	$\displaystyle\boldsymbol{\epsilon}_{+1}=$	$\displaystyle-\frac{1}{\sqrt{2}}(i,0,1),$
	$\displaystyle\boldsymbol{\epsilon}_{-1}=$	$\displaystyle\frac{1}{\sqrt{2}}(-i,0,1).$		(6)

Here $\boldsymbol{\epsilon}_{0}$ is the spin quantization direction and is chosen to be $+y$ direction. The polarization vector $\epsilon^{\mu}(\lambda,{\bf p})$ has following properties

	$\displaystyle p_{\mu}\epsilon^{\mu}(\lambda,{\bf p})=0$
	$\displaystyle\epsilon(\lambda,{\bf p})\cdot\epsilon^{*}(\lambda^{\prime},{\bf p})=-\delta_{\lambda\lambda^{\prime}}$
	$\displaystyle\sum_{\lambda}\epsilon^{\mu}(\lambda,{\bf p})\epsilon^{\nu*}(\lambda,{\bf p})=-\left(g^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{m_{\rho}^{2}}\right).$		(7)

Then we can define the two-point Green’s functions on the CTP for the vector and pseudoscalar meson,

	$\displaystyle G_{CTP}^{\mu\nu}(x_{1},x_{2})$	$\displaystyle=$	$\displaystyle\left\langle T_{C}A^{\mu}(x_{1})A^{\nu}(x_{2})\right\rangle,$		(8)
	$\displaystyle S_{CTP}(x_{1},x_{2})$	$\displaystyle=$	$\displaystyle\left\langle T_{C}\phi(x_{1})\phi^{\dagger}(x_{2})\right\rangle.$		(9)

The two-point Green’s functions $G_{\mu\nu}^{\lessgtr}$ for the vector meson at the leading order are given as (Sheng:2022ffb, ),

	$\displaystyle G_{\mu\nu}^{<}(x,p)$	$\displaystyle=$	$\displaystyle 2\pi\hbar\sum_{\lambda_{1},\lambda_{2}}\delta\left(p^{2}-m_{\rho}^{2}\right)\left\{\theta(p^{0})\epsilon_{\mu}\left(\lambda_{1},{\bf p}\right)\epsilon_{\nu}^{\ast}\left(\lambda_{2},{\bf p}\right)f_{\lambda_{1}\lambda_{2}}(x,{\bf p})\right.$		(10)
			$\displaystyle\left.+\theta(-p^{0})\epsilon_{\mu}^{\ast}\left(\lambda_{1},-{\bf p}\right)\epsilon_{\nu}\left(\lambda_{2},-{\bf p}\right)\left[\delta_{\lambda_{2}\lambda_{1}}+f_{\lambda_{2}\lambda_{1}}(x,-{\bf p})\right]\right\},$
	$\displaystyle G_{\mu\nu}^{>}(x,p)$	$\displaystyle=$	$\displaystyle 2\pi\hbar\sum_{\lambda_{1},\lambda_{2}}\delta\left(p^{2}-m_{\rho}^{2}\right)\left\{\theta(p^{0})\epsilon_{\mu}\left(\lambda_{1},{\bf p}\right)\epsilon_{\nu}^{\ast}\left(\lambda_{2},{\bf p}\right)\left[\delta_{\lambda_{1}\lambda_{2}}+f_{\lambda_{1}\lambda_{2}}(x,{\bf p})\right]\right.$		(11)
			$\displaystyle\left.+\theta(-p^{0})\epsilon_{\mu}^{\ast}\left(\lambda_{1},-{\bf p}\right)\epsilon_{\nu}\left(\lambda_{2},-{\bf p}\right)f_{\lambda_{2}\lambda_{1}}(x,-{\bf p})\right\},$

where $f_{\lambda_{1}\lambda_{2}}(x,{\bf p})$ is the matrix valued spin dependent distribution (MVSD) for the rho meson,

$$f_{\lambda_{1}\lambda_{2}}(x,{\bf p})\equiv\int\frac{d^{4}u}{2(2\pi\hbar)^{3}}\delta(p\cdot u)e^{-iu\cdot x/\hbar}\left\langle a_{\rho}^{\dagger}\left(\lambda_{2},{\bf p}-\frac{{\bf u}}{2}\right)a_{\rho}\left(\lambda_{1},{\bf p}+\frac{{\bf u}}{2}\right)\right\rangle.$$

(12)

One can check that $f_{\lambda_{1}\lambda_{2}}(x,{\bf p})$ is an Hermitian matrix, $f_{\lambda_{1}\lambda_{2}}^{*}(x,{\bf p})=f_{\lambda_{2}\lambda_{1}}(x,{\bf p})$. The two-point Green’s function for $\pi^{\pm}$ at the leading order is

	$\displaystyle S^{<}(x,k)=$	$\displaystyle 2\pi\hbar\delta\left(k^{2}-m_{\pi}^{2}\right)$
		$\displaystyle\times\left\{\theta(k^{0})f_{\pi^{+}}(x,\mathbf{k})+\theta(-k^{0})\left[1+f_{\pi^{-}}(x,-\mathbf{k})\right]\right\},$		(13)
	$\displaystyle S^{>}(x,k)=$	$\displaystyle 2\pi\hbar\delta\left(k^{2}-m_{\pi}^{2}\right)$
		$\displaystyle\times\left\{\theta(k^{0})\left[1+f_{\pi^{+}}(x,\mathbf{k})\right]+\theta(-k^{0})f_{\pi^{-}}(x,-\mathbf{k})\right\},$		(14)

where $f_{\pi^{\pm}}(x,\mathbf{p})$ is the distribution for $\pi^{\pm}$. For notational convenience, we use $G$ and $p$ to denote the Green’s function and momentum for the rho meson respectively, while we use $S$ and $k$ to denote the Green’s function and momentum for $\pi^{\pm}$ respectively.

We start from the KB equation to derive the spin Boltzmann equation for the vector meson (Sheng:2022ffb, )

			$\displaystyle p\cdot\partial_{x}G^{<,\mu\nu}(x,p)-\frac{1}{4}\left[p^{\mu}\partial_{\eta}^{x}G^{<,\eta\nu}(x,p)+p^{\nu}\partial_{\eta}^{x}G^{<,\mu\eta}(x,p)\right]$		(15)
		$\displaystyle=$	$\displaystyle\frac{1}{4}\left[\Sigma_{\;\;\;\;\alpha}^{<,\mu}\left(x,p\right)G^{>,\alpha\nu}\left(x,p\right)-\Sigma_{\;\;\;\;\alpha}^{>,\mu}\left(x,p\right)G^{<,\alpha\nu}\left(x,p\right)\right]$
			$\displaystyle+\frac{1}{4}\left[G_{\ \ \ \ \alpha}^{>,\mu}\left(x,p\right)\Sigma^{<,\alpha\nu}\left(x,p\right)-G_{\ \ \ \ \alpha}^{<,\mu}\left(x,p\right)\Sigma^{>,\alpha\nu}\left(x,p\right)\right].$

In the above equation, the Poisson bracket terms are not considered. Multiplying $\epsilon_{\mu}^{*}\left(\lambda_{1},{\bf p}\right)\epsilon_{\nu}\left(\lambda_{2},{\bf p}\right)$ to both side of Eq. (15) and choose $p_{0}>0$ part, we obtain

	$\displaystyle p\cdot\partial_{x}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$	$\displaystyle=$	$\displaystyle-\frac{1}{4}\delta_{\lambda_{2}\lambda_{2}^{\prime}}\epsilon_{\mu}^{*}\left(\lambda_{1},{\bf p}\right)\epsilon^{\alpha}\left(\lambda_{1}^{\prime},{\bf p}\right)$		(16)
			$\displaystyle\times\left\{\left[\delta_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}+f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,{\bf p})\right]\Sigma_{\;\;\;\;\alpha}^{<,\mu}\left(x,p\right)-f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,{\bf p})\Sigma_{\;\;\;\;\alpha}^{>,\mu}\left(x,p\right)\right\}$
			$\displaystyle-\frac{1}{4}\delta_{\lambda_{1}\lambda_{1}^{\prime}}\epsilon_{\nu}\left(\lambda_{2},{\bf p}\right)\epsilon_{\alpha}^{\ast}\left(\lambda_{2}^{\prime},{\bf p}\right)$
			$\displaystyle\times\left\{\left[\delta_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}+f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,{\bf p})\right]\Sigma^{<,\alpha\nu}\left(x,p\right)-f_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}}(x,{\bf p})\Sigma^{>,\alpha\nu}\left(x,p\right)\right\}.$

The above equation is the spin Boltzmann equation for the vector meson in terms of MVSDs. The MVSDs of spin-1/2 fermions are defined in Refs. (Becattini:2013fla, ; Sheng:2021kfc, ) and those for vector mesons are defined in Refs. (Sheng:2022wsy, ; Sheng:2022ffb, ). The spin density matrix is just the normalized MVSD

$$\rho_{\lambda_{1}\lambda_{2}}=\frac{f_{\lambda_{1}\lambda_{2}}}{\sum_{\lambda}f_{\lambda\lambda}}=\frac{f_{\lambda_{1}\lambda_{2}}}{\mathrm{Tr}f}.$$

(17)

The spin alignment is given by the 00 element $\rho_{00}$.

We make a few remarks about the spin kinetic or Boltzmann equation (16). The collision terms in the right-hand side of Eq. (16) are the result of the on-shell approximation. In such an approximation, the retarded and advanced components of self-energies and two-point Green’s functions are neglected so that the collision terms only depend on the “<” and “>” components. Hence the contributions to the spin density matrix of vector mesons come from collisions of on-shell particles including the vector meson’s annihilation and production processes. The contribution from different retarded and advanced self-energies for transverse and longitudinal modes in equilbrium is called the off-shell contribution (Kim:2019ybi, ; Li:2022vmb, ; Dong:2023cng, ; Seck:2023oyt, ), which belongs to a different kind of the contribution from the one we consider in this paper.

In the next section we will derive the self-energy $\Sigma_{\mu\nu}$ and then collision terms incorporating the interaction part of the Lagrangian.

For clarification, we decompose the collision terms, the right-hand-side (r.h.s.) of Eq. (16), into $C_{\mathrm{coal}/\mathrm{diss}}$ and $C_{\mathrm{scat}}$ for the coalescence-dissociation and scattering processes respectively, where $C_{\mathrm{coal}/\mathrm{diss}}$ have contrbutions at LO and NLO, $C_{\mathrm{coal}/\mathrm{diss}}=C_{\mathrm{coal}/\mathrm{diss}}^{(0)}+C_{\mathrm{coal}/\mathrm{diss}}^{(1)}$, while $C_{\mathrm{scat}}$ is of NLO. Note that we only consider contributions up to NLO in this paper. Then Eq. (16) can be written as

$\displaystyle\frac{p}{E_{p}^{\rho}}\cdot\partial_{x}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$

$\displaystyle=$

$\displaystyle C_{\mathrm{coal}/\mathrm{diss}}+C_{\mathrm{scat}},$

(18)

where the spin indices $\lambda_{1}$, $\lambda_{2}$ and phase space variables $x,\mathbf{p}$ have been suppressed in collision terms. In this work, for simplicity, we adopt the gradient expansion in space and neglect spatial gradients of $f_{\lambda_{1}\lambda_{2}}$ at the leading order. This corresponds to the assumption that the system is homogeneous in space. So Eq. (18) becomes

$\displaystyle\partial_{t}f_{\lambda_{1}\lambda_{2}}(x,\mathbf{p})$

$\displaystyle=$

$\displaystyle C_{\mathrm{coal}/\mathrm{diss}}+C_{\mathrm{scat}}.$

(19)

We will evaluate $C_{\mathrm{coal}/\mathrm{diss}}$ and $C_{\mathrm{scat}}$ one by one.