Cross Section Calculations in Theories of Self-Interacting Dark Matter

The lowest-order (tree-level) amplitude for the $\chi+\chi\to\chi+\chi$ reaction resulting from the interaction (5) has the form

$${\cal M}={\cal M}^{(t)}-{\cal M}^{(u)}\ ,$$

(27)

where ${\cal M}^{(t)}$ and ${\cal M}^{(u)}$ are the $t$-channel and $u$-channel contributions, and the relative minus sign accounts for exchanging identical fermions in the final state. The Lorentz-invariant differential cross setion is

$$\frac{d\sigma}{dt}=\frac{1}{16\pi\lambda(s,m_{\chi}^{2},m_{\chi}^{2})}\overline{\sum}|{\cal M}|^{2}\ ,$$

(28)

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+zx)$, and $\overline{\sum}$ denotes an average over initial spins and a sum over final spins. Here, $\lambda(s,m_{\chi}^{2},m_{\chi}^{2})=(s\beta_{\chi})^{2}$. For our discussion, it will be useful to distinguish the terms in $d\sigma/dt$ arising from $\overline{\sum}|{\cal M}^{(t)}|^{2}$, $\overline{\sum}|{\cal M}^{(u)}|^{2}$, and $\overline{\sum}[{\cal M}^{(t)*}{\cal M}^{(u)}+{\cal M}^{(u)*}{\cal M}^{(t)}]=2\overline{\sum}{\rm Re}[{\cal M}^{(t)*}{\cal M}^{(u)}]$. We denote these as $d\sigma^{(t)}/dt$, $d\sigma^{(u)}/dt$, and $d\sigma^{(tu)}/dt$, respectively. We find

$$\frac{d\sigma}{dt}=\frac{\pi\alpha_{\chi}^{2}}{(\beta_{\chi}s)^{2}}\,\bigg{[}\frac{(t-4m_{\chi}^{2})^{2}}{(t-m_{\phi}^{2})^{2}}+\frac{(u-4m_{\chi}^{2})^{2}}{(u-m_{\phi}^{2})^{2}}-\frac{1}{(t-m_{\phi}^{2})(u-m_{\phi}^{2})}\,\bigg{\{}\frac{1}{2}(t^{2}+u^{2}-s^{2})+8m_{\chi}^{2}s-8m_{\chi}^{4}\bigg{\}}\bigg{]}\ .$$

(29)

The first and second terms on the right-hand side (RHS) of Eq. (28) are $d\sigma^{(t)}/dt$ and $d\sigma^{(u)}/dt$, while the third term with curly brackets is $d\sigma^{(tu)}/dt$. Since the amplitude (27) is antisymmetric under interchange of identical particles in the final state, and equivalently under interchange of the $t$-channel and $u$-channel terms, it follows that the square of the amplitude is symmetric under this interchange. This symmetry under the interchange $t\leftrightarrow u$ is evident in the RHS of Eq. (28). The center-of-mass cross section, $(d\sigma/d\Omega)_{\rm CM}$, is related to $d\sigma/dt$ as

$$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}=\frac{\lambda(s,m_{\chi}^{2},m_{\chi}^{2})}{4\pi s}\,\frac{d\sigma}{dt}=\bigg{(}\frac{\beta_{\chi}^{2}s}{4\pi}\bigg{)}\,\frac{d\sigma}{dt}\ .$$

(30)

In terms of the center-of-mass scattering angle $\theta$, the symmetry of the RHS of Eq. (28) under the interchange $t\leftrightarrow u$ is expressed as the symmetry

$$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}(\theta)=\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}(\pi-\theta)\ .$$

(31)

Because of the identical particles in the final state, a scattering event in which a scattered $\chi$ particle emerges at angle $\theta$ is indistingishable from one in which a scattered $\chi$ emerges at angle $\pi-\theta$. The total cross section for the reaction (8) thus involves a symmetry factor of $1/2$ to compensate for the double-counting involved in the integration over the range $\theta\in[0,\pi]$:

$$\sigma=\frac{1}{2}\int d\Omega\,\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}(\pi-\theta)\ .$$

(32)

Owing to the symmetry (31), this is equivalent to a polar angle integration from 0 to $\pi/2$:

$$\frac{1}{2}\int_{-1}^{1}\,d\cos\theta\,\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}=\int_{0}^{1}\,d\cos\theta\,\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}\ .$$

(33)

(Recall that if the final state consisted of $n$ identical particles, the factor $1/2$ in Eq. (32) would be replaced by $1/n!$.)

In addition to the differential cross section $(d\sigma/d\Omega)_{CM}$, other related (center-of-mass) differential cross sections have been used in the study of the effects of self-interacting dark matter, motivated by earlier analyses of transport properties in gases and plasmas (e.g., Krstić and Schultz (1999) and references therein). A major reason for this was the desire to define a differential cross section that yields a useful description of the thermalization effect of DM-DM scattering, particularly in the case where the mass of the mediator particle is much smaller than the mass of the DM particle. In this case, to the extent that the scattering angle $\theta$ is close to 0 for distinguishable particles or close to 0 or $\pi$ for indistinguishable particles, the DM particle trajectories are not significantly changed by the scattering. To give greater weighting to large-angle scattering that thermalizes particles in a gas or plasma, researchers Krstić and Schultz (1999) have used the transfer (T) differential cross section,

$$\frac{d\sigma_{\rm T}}{d\Omega}=(1-\cos\theta)\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}$$

(34)

and the viscosity (V) differential cross section,

$$\frac{d\sigma_{\rm V}}{d\Omega}=(1-\cos^{2}\theta)\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}\ ,$$

(35)

(Although the same symbol, V, is used for the vector mediator and viscosity, the context will always make clear which meaning is intended.) For the same reason, namely that these describe thermalization effects better than the ordinary cross section, the transfer and viscosity cross sections been used in studies of DM-DM scattering (e.g., Mohapatra et al. (2002); Tulin et al. (2013b) and subsequent work).

Given the invariance of $(d\sigma/d\Omega)_{CM}$ under the transformation $\theta\to\pi-\theta$ and the fact that $\cos\theta$ is odd under this transformation, it follows that the integral of the product of $\cos\theta$ times $(d\sigma/d\Omega)_{CM}$ vanishes. Hence, the total cross section is equal to the total transfer cross section:

	$\displaystyle\sigma$	$\displaystyle=$	$\displaystyle\frac{1}{2}\int d\Omega\ \bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}=\frac{2\pi}{2}\int_{-1}^{1}\ d\cos\theta\,\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}$		(36)
		$\displaystyle=$	$\displaystyle\frac{2\pi}{2}\int_{-1}^{1}\ d\cos\theta\,(1-\cos\theta)\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM}=\sigma_{\rm T}\ .$		(38)

As was noted in Krstić and Schultz (1999), the transfer differential cross section does not correctly describe the scattering in the case of identical particles, since it does not maintain the $\theta\leftrightarrow\pi-\theta$ symmetry in the reaction. However, given Eq. (LABEL:sig_sigtransfer_equality), the resultant integral over angles is equal to the integral of the ordinary (unweighted) cross section, i.e., $\sigma_{T}=\sigma$. The viscosity differential cross section, with its angle-weighting factor of $(1-\cos^{2}\theta)=\sin^{2}\theta$ does maintain the $\theta\leftrightarrow\pi-\theta$ symmetry in the scattering of identical particles. In passing, we note that another type of differential cross section has also been considered that weights large-angle scattering Kahlhoefer et al. (2014), namely $(1-|\cos\theta|)(d\sigma/d\Omega)_{\rm CM}$; this also maintains the $\theta\to\pi-\theta$ symmetry of reaction (8).)

In the nonrelativistic (NR) limit $\beta_{\chi}\ll 1$, the kinematic invariants have the property that $s\gg\{|t|,\ |u|\}$; $m_{\chi}^{2}\gg\{|t|,\ |u|\}$; and $s\to(2m_{\chi})^{2}$. Hence, in this limit, the CM differential cross section reduces to

	$\displaystyle\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{\rm CM,NR}$	$\displaystyle=$	$\displaystyle{\alpha_{\chi}^{2}m_{\chi}^{2}}\bigg{[}\frac{1}{(t-m_{\phi}^{2})^{2}}+\frac{1}{(u-m_{\phi}^{2})^{2}}-\frac{1}{(t-m_{\phi}^{2})(u-m_{\phi}^{2})}\bigg{]}$		(41)
		$\displaystyle=$	$\displaystyle{\sigma_{0}}\bigg{[}\frac{1}{(1+r\sin^{2}(\theta/2))^{2}}+\frac{1}{(1+r\cos^{2}(\theta/2))^{2}}-\frac{1}{(1+r\sin^{2}(\theta/2))(1+r\cos^{2}(\theta/2))}\bigg{]}\ ,$		(43)

where

$$\sigma_{0}=\frac{\alpha_{\chi}^{2}m_{\chi}^{2}}{m_{\phi}^{4}}$$

(46)

and $r$ is the ratio

$$r=\bigg{(}\frac{\beta_{\rm rel}m_{\chi}}{m_{\phi}}\bigg{)}^{2}\ .$$

(47)

The property that the transformation $\theta\to\pi-\theta$ (under which $\sin(\theta/2)\to\cos(\theta/2)$) interchanges the $t$ and $u$ channels is evident in Eq. (LABEL:dsigma_domega_nonrel), since it interchanges the first and second terms arising, respectively, from $|{\cal M}^{(t)}|^{2}$ and from $|{\cal M}^{(u)}|^{2}$, and leaves the third term arising from $-2{\rm Re}({\cal M}^{(t)*}{\cal M}^{(u)})$ invariant. Since all of the $\chi$-$\chi$ relative velocities $v_{\rm rel}$ in the relevant observational data are nonrelativistic, we will henceforth specialize to this case, taking the subscript NR to be implicit in the notation.

Since self-interacting dark matter has been studied extensively before, it is appropriate to discuss how our current results compare with and complement previous work. In (Eq. (25) of) the review Tulin and Yu (2018) on SIDM, the differential cross section in the center of mass for elastic DM self-scattering was given (in the same perturbative Born regime $\alpha_{\chi}m_{\chi}/m_{\phi}\ll 1$ as we use here) as

$$\frac{d\sigma}{d\Omega}=\frac{\alpha_{\chi}^{2}m_{\chi}^{2}}{[m_{\chi}^{2}v_{\rm rel}^{2}(1-\cos\theta)/2+m_{\phi}^{2}]^{2}}\equiv\frac{\sigma_{0}}{[r\sin^{2}(\theta/2)+1]^{2}}\ ,$$

(48)

where we transcribe the result from Tulin and Yu (2018) in our notation in the second term of Eq. (48). As is evident, this corresponds to the $t$-channel contribution in our full result (LABEL:dsigma_domega_nonrel). However, the true differential cross section for the DM self-scattering $\chi+\chi\to\chi+\chi$ must include not just the $t$-channel contribution but also the $u$-channel contribution, as we have done here. A subsequent study in Colquhoun et al. (2021) focused on a regime where nonperturbative effects are important and gave results for DM-DM scattering with both identical and non-identical particles. Our work is complementary to Colquhoun et al. (2021), since we choose parameters in Eq. (7) such that nonperturbative effects are not important.

Regarding the range of values of the ratio $r$ in Eq. (47), it is important to note that even in the nonrelativistic regime $\beta_{\rm rel}\ll 1$, it is not necessarily the case that the ratio $r$ is small. With the illustrative mass values in Eq. (7), and taking into account that for $v_{\rm rel}\sim 3\times 10^{3}$ km/s (i.e., $\beta_{\rm rel}\sim 10^{-2}$) for DM particles in galaxy clusters, it follows that $r\sim 10^{2}$ in this case. In contrast, for the analysis of DM self-interactions on length scales of order a few kpc within a galaxy, if $v_{\rm rel}\sim 30$ km/sec (i.e., $\beta_{\rm rel}\sim 10^{-4}$), then $r\sim O(10^{-2})$.

It is interesting to elucidate how the various contributions to the cross section from $|{\cal M}^{(t)}|^{2}$, $|{\cal M}^{(u)}|^{2}$, and $2{\rm Re}({\cal M}^{(t)*}{\cal M}^{(u)})$ behave as a function of $r$. We find that in the $r\ll 1$ regime relevant for the analysis of galactic data on the 1-10 kpc scale, the terms contributing to $(d\sigma/d\Omega)_{\rm CM}$ have the property that the $t$-channel term $|{\cal M}^{(t)}|^{2}$ and the $u$-channel term $|{\cal M}^{(u)}|^{2}$ give equal contributions, while the $t$-$u$ interference term $2{\rm Re}({\cal M}^{(t)*}{\cal M}^{(u)})$ gives a contribution equal in magnitude and opposite in sign to that from $|{\cal M}^{(u)}|^{2}$. As we denoted the three terms contributing to $d\sigma/dt$, we similarly label the three terms contributing to $(d\sigma/d\Omega)_{\rm CM}$ and the resultant total cross section with superscripts $(t)$, $(u)$, and $(tu)$, so that the respective contribution to the total cross section are

$$\sigma^{(i)}\equiv\frac{1}{2}\int\bigg{(}\frac{d\sigma^{(i)}}{d\Omega}\bigg{)}_{\rm CM}\,d\Omega\ ,\quad i=t,\ u,\ tu\ ,$$

(49)

and

$$\sigma=\sigma^{(t)}+\sigma^{(u)}+\sigma^{(tu)}\ .$$

(50)

We calculate

$$\sigma^{(t)}=\sigma^{(u)}=\frac{2\pi\sigma_{0}}{1+r}$$

(51)

and

$$\sigma^{(tu)}=-4\pi\sigma_{0}\,\frac{\ln(1+r)}{r(2+r)}\ ,$$

(52)

so that

$$\sigma=4\pi\sigma_{0}\bigg{[}\frac{1}{1+r}-\frac{\ln(1+r)}{r(2+r)}\bigg{]}\ .$$

(53)

For fixed $\sigma_{0}$, the total cross section $\sigma$ is a monotonically decreasing function of the ratio $r$. Concerning the individual contributions to $\sigma$, we observe that

$$\sigma^{(t)}=\sigma^{(u)}=-\sigma^{(tu)}=2\pi\sigma_{0}\quad{\rm at}\ r=0\ ,$$

(54)

so that for small $r$, there is a cancellation between the interference term $\sigma^{(tu)}$ and the $u$-channel term $\sigma^{(u)}$ (or equivalently, the $t$-channel term, since $\sigma^{(t)}=\sigma^{(u)}$). In contrast, for large $r$, $\sigma^{(t)}=\sigma^{(u)}$ decrease as $2\pi\sigma_{0}/r$, while $\sigma^{(tu)}$ decreases more rapidly, as $\sigma^{(tu)}\sim-4\pi\sigma_{0}\ln r/r^{2}$. The total cross section has the small-$r$ Taylor series expansion

$$\sigma=2\pi\sigma_{0}\Big{[}1-r+\frac{7}{6}r^{2}+O(r^{3})\Big{]}\quad{\rm for}\ r\ll 1\ .$$

(55)

For $r\gg 1$, $\sigma$ has the series expansion

	$\displaystyle\sigma$	$\displaystyle=$	$\displaystyle\frac{4\pi\sigma_{0}}{r}\bigg{[}1-\frac{(1+\ln r)}{r}+O\bigg{(}\frac{\ln r}{r^{2}}\bigg{)}\bigg{]}$		(58)
			$\displaystyle{\rm for\ NR\ regime\ and}\ r\gg 1\ .$

The prefactor in Eq. (58) is

$$\frac{4\pi\sigma_{0}}{r}=\frac{4\pi\alpha_{\chi}^{2}}{m_{\phi}^{2}\beta_{\rm rel}^{2}}\ .$$

(59)

To compare the full cross section with the result obtained by including only the contribution from the $t$-channel, we consider the ratio

$\displaystyle\frac{\sigma}{\sigma^{(t)}}=2\bigg{[}1-\frac{(1+r)\ln(1+r)}{r(2+r)}\bigg{]}\ .$

(60)

This ratio has the small-$r$ expansion

$$\frac{\sigma}{\sigma^{(t)}}=1+\frac{r^{2}}{6}-\frac{r^{3}}{6}+O(r^{4})\quad{\rm for}\ r\ll 1\ ,$$

(61)

so in the small-$r$ regime, $\sigma$ is approximately equal to $\sigma^{(t)}$. For the (nonrelativistic) large-$r$ regime, the ratio (60) has the expansion

	$\displaystyle\frac{\sigma}{\sigma^{(t)}}$	$\displaystyle=$	$\displaystyle 2\bigg{[}1-\frac{\ln r}{r}+\frac{\ln r-1}{r^{2}}+O\bigg{(}\frac{\ln r}{r^{3}}\bigg{)}\bigg{]}$		(64)
			$\displaystyle{\rm for\ NR\ regime\ and}\ r\gg 1\ .$

Thus, in this large-$r$ regime relevant for fits to observational data on galaxy clusters, the full $\chi$-$\chi$ scattering cross section is larger by approximately a factor of 2 then the result obtained by keeping only the contribution from the $t$-channel.

In order to compare the full calculation including contributions from both the $t$-channel and $u$-channel with a calculation that only includes the $t$-channel, we plot $\sigma$ versus $\sigma^{(t)}$ in Fig. 2 as a function of $v_{\rm rel}$. For this purpose, we use the illustrative values of masses and couplings in Eq. (7). In accordance with our result (113) below, we subsume the cases of a scalar and a vector mediator together and denote $m_{\xi}$ as the mass of $\phi$ or $V$. We note again that with these values, there is no significant Sommerfeld enhancement of the cross section, justifying our use of the lowest-order (tree-level) perturbatively computed amplitude in the scalar case. Separately, there is no Sommerfeld enhancement in the vector case since the scattering is repulsive. The dependence of the differential cross sections on the angle $\theta$ is shown in the comparative Fig. 2(d). As is evident from Fig. 2, for the range of relative velocities $v_{\rm rel}\mathrel{\raisebox{-2.58334pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}}10^{2}$ km/s relevant for dark matter scattering in the interior of galaxies and dwarf spheroidal satellites, $\sigma$ is close to $\sigma^{(t)}$, but as $v_{\rm rel}$ increases beyond about $10^{2}$ km/s, although both $\sigma$ and $\sigma^{(t)}$ decrease, the full cross section is larger than the result obtained by keeping only the $t$-channel contribution. This trend continues to values $v_{\rm rel}\sim O(10^{3})$ km/s relevant to dark matter effects in galaxy clusters. One should note that even for a fixed $v_{\rm rel}$, there is considerable diversity in the values of $\sigma/m_{DM}$ inferred from fits to galactic and cluster data (e.g., Kamada et al. (2017); Robertson et al. (2018); Zentner et al. (2022); Roper et al. (2022) and references therein). The curves marked QM_dist in Fig. 2 are the results that one would obtain in a quantum mechanical approach with a potential for the different situation with distinguishable particles (see Appendix). We show results for a specific set of $v_{\rm rel}$ values in Table 1.

Our result in Eq. (LABEL:dsigma_domega_nonrel) together with the definition (34) yields the differential transfer cross section in the relevant nonrelativistic limit. For the individual contributions from $|{\cal M}^{(t)}|^{2}$, $|{\cal M}^{(u)}|^{2}$, and $-2{\rm Re}({\cal M}^{(t)*}{\cal M}^{(u)})$, we calculate (in the nonrelativistic regime, as before),

$$\sigma^{(t)}_{\rm T}=\frac{4\pi\sigma_{0}}{r}\bigg{[}-\frac{1}{1+r}+\frac{\ln(1+r)}{r}\bigg{]}\ ,$$

(65)

$$\sigma^{(u)}_{\rm T}=\frac{4\pi\sigma_{0}}{r}\bigg{[}1-\frac{\ln(1+r)}{r}\bigg{]}\ ,$$

(66)

and

$$\sigma^{(tu)}_{\rm T}=-\frac{4\pi\sigma_{0}}{r}\bigg{[}\frac{\ln(1+r)}{2+r}\bigg{]}\ .$$

(67)

The prefactor in Eqs. (65)-(67) is given by Eq. (59). Note that, in contrast to the equality $\sigma^{(t)}=\sigma^{(u)}$ in Eq. (51), the individual contributions $\sigma^{(t)}_{\rm T}$ and $\sigma^{(u)}_{\rm T}$ to $\sigma_{\rm T}$ are not equal; i.e., $\sigma^{(t)}_{\rm T}\neq\sigma^{(u)}_{\rm T}$. This is a consequence of the fact that the definition of $d\sigma_{\rm T}/d\Omega$ fails to preserve the $\theta\to\pi-\theta$ symmetry of the actual differential cross section for the reaction (8). .

Summing these contributions, we find, in accordance with our general equality (LABEL:sig_sigtransfer_equality), the result

$\displaystyle\sigma_{\rm T}$

$\displaystyle=$

$\displaystyle\sigma=4\pi\sigma_{0}\bigg{[}\frac{1}{1+r}-\frac{\ln(1+r)}{r(2+r)}\bigg{]}\ .$

(68)

Since $\sigma_{\rm T}=\sigma$, the transfer cross section has the same small-$r$ and large-$r$ expansions as were displayed for $\sigma$ in Eqs. (55) and (58).

We may compare our result (LABEL:sigma_sigma_transfer_equality) for $\sigma_{T}$ with the result given, in the same Born regime, as Eq. (A1) in Ref. Tulin et al. (2013a) (denoted TYZ), which is the same as Eq. (5) in Ref. Feng et al. (2010) (denoted FKY) and reads (with their $R\equiv\sqrt{r}$ and $v=\beta_{\rm rel}$ in our notation)

	$\displaystyle\sigma_{T;FKY,TYZ}$	$\displaystyle=$	$\displaystyle\frac{8\pi\alpha_{\chi}^{2}}{m_{\chi}^{2}\beta_{\rm rel}^{4}}\bigg{[}\ln(1+r)-\frac{r}{(1+r)}\biggl{]}$		(71)
		$\displaystyle=$	$\displaystyle\frac{8\pi\alpha_{\chi}^{2}r}{m_{\chi}^{2}\beta_{\rm rel}^{4}}\bigg{[}\frac{\ln(1+r)}{r}-\frac{1}{1+r}\biggl{]}$		(73)
		$\displaystyle=$	$\displaystyle\frac{8\pi\alpha_{\chi}^{2}}{m_{\phi}^{2}\beta_{\rm rel}^{2}}\bigg{[}\frac{\ln(1+r)}{r}-\frac{1}{1+r}\biggl{]}\ .$		(75)

As is evident from a comparison of Eq. (LABEL:sigma_transfer_tyz) with our Eq. (65) (using the definition of our notation given in Eq. (59)), the result for the transfer cross section in Eq. (A1) of Ref. Tulin et al. (2013a) (or equivalently, Eq. (5) of Ref. Feng et al. (2010)) is what one would get for the DM self-scattering if one did the calculation for non-identical particles and hence only included the $t$-channel contribution and did not include the $1/2$ factor for identical particles in the final state in performing the integral over $d\Omega$. That is,

$$\sigma_{\rm T,TYZ,FKY}=2\sigma^{(t)}_{\rm T}\ .$$

(78)

To compare the full transfer cross section with the result obtained by just including the $t$-channel contribution, we examine the ratio

$$\frac{\sigma_{\rm T}}{\sigma^{(t)}_{\rm T}}=\frac{r\Big{[}\frac{1}{1+r}-\frac{\ln(1+r)}{r(2+r)}\Big{]}}{\Big{[}-\frac{1}{1+r}+\frac{\ln(1+r)}{r}\Big{]}}\ .$$

(79)

For small $r$, this ratio has the expansion

	$\displaystyle\frac{\sigma_{\rm T}}{\sigma^{(t)}_{\rm T}}=1+\frac{r}{3}+\frac{r^{2}}{9}+O(r^{3})\quad{\rm for}\ r\ll 1\ .$		(80)
			(81)
			(82)

For large $r$, we find

$$\frac{\sigma_{\rm T}}{\sigma^{(t)}_{\rm T}}\sim\frac{r}{\ln r-1}\quad{\rm for}\ r\gg 1\ .$$

(83)

Thus, although both our $\sigma_{\rm T}$ and the result $\sigma_{\rm T,FKY,TYZ}$ decrease with $v_{\rm rel}$ (and thus with $r$, for fixed $m_{\chi}$ and $m_{\xi}$), our result decreases substantially less rapidly for large $r$. With our parameters, this large-$r$ regime includes values of $v_{\rm rel}\sim O(10^{3})$ km/s typical of galaxy clusters. For example, at $v_{\rm rel}=3\times 10^{3}$ km/s (corresponding to $r=10^{2}$ with our choices for $m_{\chi}$ and $m_{\xi}$ in Eq. (7)), the ratio (79) has the value 26, or equivalently, $\sigma_{\rm T}/\sigma_{\rm T,FKY,TYZ}=13$, a substantial difference from unity. Therefore, in performing fits to observational data, if one uses the transfer cross section, we would advocate the use of Eq. (LABEL:sigma_sigma_transfer_equality) rather than the result in Eq. (A1) of Ref. Tulin et al. (2013a) for the large-$r$ regime, since they differ substantially.

In Fig. 2(b) we plot $\sigma_{\rm T}$ in comparison with $\sigma^{(t)}_{\rm T}$ over the same range of $v_{\rm vel}$ and thus also $\beta_{\rm rel}$ as for the regular CM cross section. The fact that the true $\sigma_{\rm T}$ decreases considerably less rapidly than the $t$-channel contribution used in Feng et al. (2010); Tulin et al. (2013a) is evident in this figure. This is also apparent in Table 1.

For the viscosity cross section we calculate the following contributions from the $t$-channel, $u$-channel, and $t$-$u$ interference:

	$\displaystyle\sigma^{(t)}_{\rm V}$	$\displaystyle=$	$\displaystyle\sigma^{(u)}_{\rm V}$		(84)
		$\displaystyle=$	$\displaystyle\frac{8\pi\sigma_{0}}{r^{2}}\bigg{[}-2+(2+r)\frac{\ln(1+r)}{r}\bigg{]}$		(86)

and

$$\sigma^{(tu)}_{\rm V}=\frac{8\pi\sigma_{0}}{r^{2}}\bigg{[}-1+\frac{2(1+r)\ln(1+r)}{(2+r)r}\bigg{]}\ ,$$

(87)

so that the total nonrelativistic viscosity cross section is

	$\displaystyle\sigma_{\rm V}$	$\displaystyle=$	$\displaystyle\sigma^{(t)}_{\rm V}+\sigma^{(u)}_{\rm V}+\sigma^{(tu)}_{\rm V}$		(88)
		$\displaystyle=$	$\displaystyle\frac{8\pi\sigma_{0}}{r^{2}}\bigg{[}-5+\frac{2(5+5r+r^{2})\ln(1+r)}{(2+r)r}\bigg{]}\ .$		(90)

As was the case with $\sigma$ and $\sigma_{\rm T}$, for fixed $m_{\chi}$ and $m_{\phi}$, the viscosity cross section $\sigma_{\rm V}$ is a monotonically decreasing function of $r$.

We remark on properties of the individual contributions $\sigma^{(t)}_{\rm V}$, $\sigma^{(u)}_{\rm V}$, and $\sigma^{(tu)}_{\rm V}$. The fact that $\sigma^{(t)}_{\rm V}=\sigma^{(u)}_{\rm V}$ is guaranteed by the property that $(d\sigma/d\Omega)_{\rm V}$ maintains the $\theta\to\pi-\theta$ symmetry of $(d\sigma/d\Omega)_{\rm CM}$, which interchanges the $t$- and $u$-channels. These contributions have the small-$r$ expansions

	$\displaystyle\sigma^{(t)}_{\rm V}$	$\displaystyle=$	$\displaystyle\sigma^{(u)}_{\rm V}=\frac{4\pi\sigma_{0}}{3}\bigg{[}1-r+\frac{9}{10}r^{2}+O(r^{3})\bigg{]}$		(95)
			$\displaystyle{\rm for\ }r\ll 1\$

and

	$\displaystyle\sigma^{(tu)}_{\rm V}$	$\displaystyle=$	$\displaystyle\sigma^{(u)}_{\rm V}=\frac{4\pi\sigma_{0}}{3}\bigg{[}-1+r-\frac{4}{5}r^{2}+O(r^{3})\bigg{]}$		(98)
			$\displaystyle{\rm for\ }r\ll 1\ .$

Hence,

	$\displaystyle\lim_{r\to 0}\sigma^{(t)}_{\rm V}$	$\displaystyle=$	$\displaystyle\lim_{r\to 0}\sigma^{(u)}_{\rm V}=-\lim_{r\to 0}\sigma^{(tu)}_{\rm V}$		(99)
		$\displaystyle=$	$\displaystyle\frac{4\pi\sigma_{0}}{3}\ .$		(101)

This is analogous to the relation that we found for the individual contributions to $\sigma$ in Eq. (54). Thus, the full viscosity cross section has the small-$r$ series expansion

$\displaystyle\sigma_{\rm V}$

$\displaystyle=$

$\displaystyle\frac{4\pi\sigma_{0}}{3}\bigg{[}1-r+r^{2}+O(r^{3})\bigg{]}\quad{\rm for}\ r\ll 1\ .$

(102)

At large $r$, $\sigma_{\rm V}$ has the series expansion

	$\displaystyle\sigma_{\rm V}$	$\displaystyle=$	$\displaystyle\frac{8\pi\sigma_{0}}{r^{2}}\bigg{[}2\ln r-5+\frac{2(3\ln r+1)}{r}+O\Big{(}\frac{\ln r}{r^{2}}\Big{)}\bigg{]}$		(107)
			$\displaystyle{\rm for}\ r\gg 1\ .$

The prefactor in Eq. (107) is $8\pi\sigma_{0}/r^{2}=8\pi\alpha_{\chi}^{2}/(\beta_{\rm rel}^{4}m_{\chi}^{2})$.