11institutetext:  University of Crete, Department of Physics & Institute of
Theoretical & Computational Physics, 70013 Herakleio, Greece
22institutetext: Institute of Astrophysics, Foundation for Research and
Technology-Hellas, 71110 Heraklion, Crete, Greece
33institutetext: Max-Planck-Institut für extraterrestrische Physik, Postfach
1312, 85741 Garching, Germany

#  Relativistic quantum-mechanical versus classical magnetic resonant
scattering cross sections

N. A. Loudas N. D. Kylafis and J. E. Trümper 1122112233

(Received ; accepted )

###### Abstract

Context.  Radiative transfer calculations in strong (few $\times 10^{12}$ G)
magnetic fields, observed in X-ray pulsars, require accurate resonant
differential scattering cross sections. Such cross sections exist, but they
are quite cumbersome.

Aims.  Here we compare the classical (non-relativistic) with the quantum-
mechanical (relativistic) resonant differential scattering cross sections and
offer a prescription for the use of the much simpler classical expressions
with impressively accurate results.

Methods.  We have expanded the quantum-mechanical differential cross sections
and kept terms up to first order in $\epsilon\equiv E/m_{e}c^{2}$ and
$B\equiv{\cal B}/{\cal B}_{cr}$, where $E$ is the photon energy and ${\cal
B}_{cr}$ is the critical magnetic field, and recovered the classical
differential cross sections plus terms that are due to spin flip, which is a
pure quantum-mechanical phenomenon.

Results.  Adding by hand the spin-flip terms to the polarization-dependent
classical differential cross sections, we find that they are in excellent
agreement with the quantum mechanical ones for all energies near resonance and
all angles. We have plotted both of them and the agreement is impressive.

Conclusions.  We give a prescription for the use of the classical differential
cross sections that guarantees very accurate results.

###### Key Words.:

accretion – pulsars: general – stars: magnetic field – stars: neutron –
X-rays: stars

##  1 Introduction

Cyclotron lines are prominent features in the spectra of X-ray pulsars. The
first cyclotron line was discovered in Hercules X-1 (Trümper et al. 1977,
1978) and since then over 35 accreting magnetic neutron stars exhibit electron
cyclotron lines, sometimes with harmonics (Staubert et al. 2019). They are
also called cyclotron resonance scattering features (CRSFs).

Despite the many years that have passed since the first observation of a CRSF,
it is still unclear where these features form and with what mechanism. Early
on, it was suggested (Basko & Sunyaev 1976) that the CRSFs are produced in the
radiative shock in the accretion column. However, no calculation has been done
so far for the simultaneous production of the power-law spectrum and the
cyclotron line in a radiative shock. Instead, several calculations have been
performed using a slab, illuminated from one side (Ventura et al. 1979; Nagel
1981; Nishimura 2008; Araya & Harding 1999, Araya-Gochez & Harding 2000;
Schoenherr et al. 2007).

Another interesting idea was given by Poutanen et al. (2013), who proposed
that the CRSF is produced by reflection of the continuum emitted at the
radiative shock on the surface of the neutron star. Again, no radiative
transfer calculation has been reported yet on this mechanism.

A possible reason for the limited detailed calculations is the resonant
differential cross sections. Not their unavailability, but their complexity.
The expressions derived by Herold (1979), Daugherty & Harding (1986), Bussard
et al. (1986), Harding & Daugherty (1991), Sina (1996), Gonthier et al.
(2014), Mushtukov et al. (2016), and Schwarm (2017) for the complete, Quantum
Electrodynamic, differential Compton resonant cross sections are quite general
and, because of this, cumbersome and rather impractical. However, this
generality is not needed in most cases.

Most of the cyclotron lines that have been observed so far (for a review see
Staubert et al. 2019) are at cyclotron energies $E_{c}\ll m_{e}c^{2}$,
implying magnetic fields ${\cal B}\ll{\cal B}_{cr}$, where ${\cal
B}_{cr}=m_{e}^{2}c^{3}/e\hbar$ is the critical magnetic field. For such cases,
an expansion of the full quantum mechanical cross sections up to first order
in $\epsilon\equiv E/m_{e}c^{2}$ and $B\equiv{\cal B}/{\cal B}_{cr}$ gives
simpler, but nevertheless accurate, expressions for the necessary
calculations.

Known in the literature are the much simpler classical (Thomson) cross
sections (Canuto, Lodenquai, & Ruderman 1971; Blandford & Scharlemann 1976;
Nobili, Turolla, & Zane 2008a). The question then arises: can a prescription
be found, such that the use of the simple classical cross sections in
radiative transfer calculations gives very accurate results? The present work
answers this question positively.

In § 2, we discuss the polarization-dependent cross sections. First, we give
the classical ones. Then, we discuss the quantum-mechanical cross sections,
that have been derived either with the Johnson & Lippmann (1949) wave
functions or with the physically more meaningful Sokolov & Ternov (1968)
formalism, and compare them numerically for $E\ll m_{e}c^{2}$ and ${\cal
B}\ll{\cal B}_{cr}$. We expand to first order in the small parameters
$\epsilon$ and $B$ the cross sections given by Harding & Daugherty (1991) and
Sina (1996). Finally, we offer a prescription with which the classical cross
sections can be used for extremely accurate results. In § 3, we summarize our
work.

##  2 The cross sections

###  2.1 Classical

The classical (Thomson) differential cross sections for polarized resonant
scattering are given by Nobili, Turolla, & Zane (2008a; see also Canuto,
Lodenquai, & Ruderman 1971; Blandford & Scharlemann 1976). After integrating
over the azimuthal angle $\phi$ we have

${{d\sigma_{11}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi r_{0}c}\over
8}L(\omega,\omega_{r})\cos^{2}\theta\cos^{2}\theta^{\prime},$ $None$
${{d\sigma_{12}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi r_{0}c}\over
8}L(\omega,\omega_{r})\cos^{2}\theta,$ $None$
${{d\sigma_{21}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi r_{0}c}\over
8}L(\omega,\omega_{r})\cos^{2}\theta^{\prime},$ $None$
${{d\sigma_{22}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi r_{0}c}\over
8}L(\omega,\omega_{r}),$ $None$

where the index 1 (2) stands for the ordinary (extraordinary) mode, $r_{0}$ is
the classical electron radius,

$L(\omega,\omega_{r})={{\Gamma/2\pi}\over{(\omega-\omega_{r})^{2}+(\Gamma/2)^{2}}}$
$None$

is the normalized Lorentz profile with $\omega=E/\hbar$ the photon frequency,
$\omega_{r}=E_{r}/\hbar$ the resonant frequency, which in the non-relativistic
regime is the cyclotron frequency $\omega_{c}=e{\cal B}/m_{e}c$,
$\Gamma=4e^{2}\omega_{c}^{2}/(3m_{e}c^{3})$ accounts for the finite transition
life-time of the excited state (e.g. Daugherty & Ventura 1978; Ventura 1979),
and $\theta$ and $\theta^{\prime}$ are the incident and scattered angles,
respectively, with respect to the direction of the magnetic field $\@vec{\cal
B}$.

The polarization averaged differential cross section is given by

${{d\sigma}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi r_{0}c}\over
16}L(\omega,\omega_{r})(1+\cos^{2}\theta)(1+\cos^{2}\theta^{\prime}).$ $None$

###  2.2 Relativistic quantum mechanical

Expressions for the differential Compton scattering cross sections as
functions of polarization, energy $E$, and magnetic field $\cal B$ have been
derived by Harding & Daugherty (1991), using the Johnson & Lippmann (1949)
wavefunctions. Similar, but more accurate, expressions have been derived by
Sina (1996) using the Sokolov & Ternov formalism (for a detailed discussion
see Gonthier et al. 2014). At large values of $\cal B$, there are differences
between the two (Schwarm 2017).

Since we are interested at non-relativistic energies and sub-critical magnetic
fields, we compare numerically the expressions of Harding & Daugherty (1991)
with those of Sina (1996) for $B=0.03$. In Fig. 1, we compare
$d\sigma_{11}/d\cos\theta^{\prime}$ near resonance for four values of the
incident angle $\theta$ with respect to the magnetic field, and four values of
the scattered angle $\theta^{\prime}$. The black lines correspond to Harding &
Daugherty (1991), while the red ones to Sina (1996). The curves essentially
overlap, but for a detailed comparison at resonance we show in Fig. 2 the
ratio of the two curves for the cases displayed in Fig. 1. The differences at
resonance are less than 3%. We remark that similar agreement exists for the
other three differential cross sections.

Figure 1:  Polarization-dependent differential cross section
$(1/\sigma_{T})\leavevmode\nobreak\ d\sigma_{11}/d\cos\theta^{\prime}$ as a
function of photon energy $E=\hbar\omega$ for a set of incident angles
$\theta$ and scattered angles $\theta^{\prime}$. The set of angles is
$0,30,60$, and $90$ degrees, with $\theta$ changing vertically and
$\theta^{\prime}$ horizontally. The red solid lines are produced from the
expression of Sina (1996), while the black dashed ones are produced from the
expression of Harding & Daugherty (1991). In all cases, the resonant frequency
is given by expression (7). The vertical line indicates the resonant frequency
$\omega_{r}$, which for $\theta\neq 0$ is smaller than $\omega_{c}$. Here
$E_{c}=\hbar\omega_{c}=15.33$ keV.  Figure 2:  Ratio of the Harding &
Daugherty (1991) polarization-dependent differential cross section
$(1/\sigma_{T})\leavevmode\nobreak\ d\sigma_{11}/d\cos\theta^{\prime}$ to
those of Sina (1996) for the cases shown in Fig. 1.

In Fig. 3, we show in the form of heat maps the ratios at resonance of the
differential cross sections calculated from the expressions of Harding &
Daugherty (1991) to those of Sina (1996) for $B=0.03$. It is evident that, for
all incident angles $\theta$ and all scattered angles $\theta^{\prime}$, the
ratio is between 1.000 and 1.027. Thus, for $B\ll 1$, it is irrelevant which
of the two formalisms is used.

Figure 3:  Ratio of the Harding & Daugherty (1991) polarization-dependent
differential cross sections at resonance to those of Sina (1996) for different
incident and scattered angles $\theta$ and $\theta^{\prime}$, respectively. In
all cases, the resonant frequency is given by expression (7) and $B=0.03$.

In order to find simpler expressions, that are nevertheless quite accurate for
$E\ll m_{e}c^{2}$ and ${\cal B}\ll{\cal B}_{cr}$, we have expanded the
expressions of Harding & Daugherty (1991) to first order in the small
parameters $\epsilon=E/m_{e}c^{2}$ and $B={\cal B}/{\cal B}_{cr}$. This is
done in Appendix A. Here we give the results.

${{d\sigma_{11}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{1\to 1}\cdot L_{-}+h^{1\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$
${{d\sigma_{12}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{1\to 2}\cdot L_{-}+h^{1\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$
${{d\sigma_{21}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{2\to 1}\cdot L_{-}+h^{2\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$
${{d\sigma_{22}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{2\to 2}\cdot L_{-}+h^{2\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where $L_{-}(\omega,\omega_{r}),\,L_{+}(\omega,\omega_{r})$ and
$L_{mix}(\omega,\omega_{r})$ are the normalized Lorentz profiles, which are
characterized by the decay widths $\Gamma^{\prime}_{-},\,\Gamma^{\prime}_{+}$,
and $\Gamma^{\prime}_{mix}$, respectively, and are given in Appendix A. The
resonant frequency $\omega_{r}$ is given in eq. (7) below. In addition,
$g^{s\to s^{\prime}}(\theta,\theta^{\prime},B)$ and $h^{s\to
s^{\prime}}(\theta,\theta^{\prime},B)$ are first degree polynomials in $B$,
and are given in Appendix A. Note that, $s$ represents the polarization mode
of the incident photon, whereas $s^{\prime}$ that of the scattered photon.

In Appendix B, we have expanded the expressions of Sina (1996) to first order
in $\omega=E/(\hbar m_{e}c^{2})$ and $B={\cal B}/{\cal B}_{cr}$. The results
are:

${{d\sigma_{11}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal{G}}^{1\to 1}\cdot L_{-}+h^{1\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$
${{d\sigma_{12}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal{G}}^{1\to 2}\cdot L_{-}+h^{1\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$
${{d\sigma_{21}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal{G}}^{2\to 1}\cdot L_{-}+h^{2\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$
${{d\sigma_{22}}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal{G}}^{2\to 2}\cdot L_{-}+h^{2\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where
$L_{-}(\omega,\omega_{r}),\,L_{+}(\omega,\omega_{r}),\,L_{mix}(\omega,\omega_{r})$
and $h^{s\to s^{\prime}}$ are the same as in (4a)-(4d), while the correction
functions ${\cal G}^{s\to s^{\prime}}$ are slightly different from the
$g^{s\to s^{\prime}}$ ones and are given in Appendix B. Note that, in order to
avoid confusion between the different notations, we have employed the notation
of Harding & Daugherty (1991) in eqs. (5) instead of the notation of Sina
(1996) that we systematically use in Appendix B.

It is important to notice two things: a) The similarity of eqs. (5) with those
of (4). They are identical, except for the small differences between $g$ and
$\cal G$. This, of course, is not surprising given the ratio plots in Fig. 2
and the heat plots in Fig. 3. b) The expressions in eqs. (4) and (5) are much
simpler than the full expressions given by Harding & Daugherty (1991) and Sina
(1996), respectively.

###  2.3 Even simpler expressions

Looking at expressions (4) and (5), we have wondered whether all the terms in
them are crucial. Thus, we have written the extremely simple expressions that
follow.

${{d\sigma_{11}}\over{d\cos\theta^{\prime}}}\approx 2\pi{{3\pi r_{0}c}\over
8}\left(\cos^{2}\theta\cos^{2}\theta^{\prime}\cdot L_{-}+{B\over 2}\cdot
L_{+}\right),$ $None$ ${{d\sigma_{12}}\over{d\cos\theta^{\prime}}}\approx
2\pi{{3\pi r_{0}c}\over 8}\left(\cos^{2}\theta\cdot L_{-}+{B\over
2}\cos^{2}\theta^{\prime}\cdot L_{+}\right),$ $None$
${{d\sigma_{21}}\over{d\cos\theta^{\prime}}}\approx 2\pi{{3\pi r_{0}c}\over
8}\left(\cos^{2}\theta^{\prime}\cdot L_{-}+{B\over 2}\cos^{2}\theta\cdot
L_{+}\right),$ $None$ ${{d\sigma_{22}}\over{d\cos\theta^{\prime}}}\approx
2\pi{{3\pi r_{0}c}\over 8}\left(L_{-}+{B\over
2}\cos^{2}\theta\cos^{2}\theta^{\prime}\cdot L_{+}\right).$ $None$

The terms proportional to $L_{-}$ are identical to the classical cross
sections (1), because $L_{-}$ is equal to $L$ given by eq. (2). The terms
proportional to $L_{+}$ are non-classical, because they contain the decay
width $\Gamma_{+}$, which is associated with the spin flip.

In Fig. 4, we compare expression (6b) with expression (5b) for $B=0.03$ and in
Fig. 5, we present the ratio of these two formulae. The differences are
impressively negligible. The same is true for the heat maps at resonance (Fig.
6).

Figure 4:  Polarization-dependent resonant differential cross section
$(1/\sigma_{T})\leavevmode\nobreak\ d\sigma_{12}/d\cos\theta^{\prime}$ as a
function of photon energy $E=\hbar\omega$ for a set of incident angles
$\theta$ and scattered angles $\theta^{\prime}$. The set of angles is
$0,30,60$, and $90$ degrees, with $\theta$ changing vertically and
$\theta^{\prime}$ horizontally. The red solid lines are produced from the
first order expansion (5b) of Sina (1996), while the black dashed ones are
produced from the simplified expression (6b). In all cases, the resonant
frequency is given by expression (7). The vertical line indicates the resonant
frequency $\omega_{r}$, which for $\theta\neq 0$ is smaller than $\omega_{c}$.
Here $E_{c}=\hbar\omega_{c}=15.33$ keV.  Figure 5:  Ratio of the polarization-
dependent differential cross sections $d\sigma_{12}/d\cos\theta^{\prime}$
given by (5b) to the simplified expression (6b) for the cases shown in Fig. 4.
Figure 6:  Ratio at resonance of the exact polarization-dependent differential
cross sections (5) to the approximate expressions (6) for different incident
and scattered angles $\theta$ and $\theta^{\prime}$, respectively. In all
cases, the resonant frequency is given by expression (7) and $B=0.03$.

For quick calculations with $B\ll 1$, where high accuracy is not demanded, one
may safely use the very simple expressions (6), while for more accurate ones,
expressions (5) are recommended. Of course, the highest accuracy is provided
by the expressions of Sina (1996). It is our hope that these very simple
expressions, together with the prescription that we give in the next
subsection, will make resonant Compton scattering calculations easily doable.

###  2.4 The prescription

If, in a calculation, one wants to use the much simpler differential cross
sections (1a) - (1d), then for accurate results the following prescription
should be followed.

1) The spin-flip terms must be added by hand. Thus, eqs. (6a) - (6d) should be
used. Of course, the exact results (5a) - (5d) are not that complicated.

2) For the resonant frequency $\omega_{r}$, the relativistically correct
expression

$\omega_{r}={{m_{e}c^{2}}\over{\hbar}}{{2B}\over{1+\sqrt{1+2B\sin^{2}\theta}}}$
$None$

should be used in the Lorentz profile (2) and not the cyclotron frequency
$\omega_{c}=E_{c}/\hbar$.

3) In the classical (Thomson) limit, there is no change in the photon energy
after scattering, i.e. $\epsilon^{\prime}=\epsilon$. However, naturally there
is an energy change (see § Appendix A) and the prescription dictates that the
energy $\epsilon^{\prime}$ of the photon after scattering should be taken
equal to

$\epsilon^{\prime}={{\epsilon^{2}\sin^{2}\theta+2\epsilon}\over{1+\epsilon(1-\cos\theta\cos\theta^{\prime})+\sqrt{f_{1}(\epsilon,\theta,\theta^{\prime})+f_{2}(\epsilon,\theta,\theta^{\prime})}}},$
$None$

where

$f_{1}(\epsilon,\theta,\theta^{\prime})=1+2\epsilon\cos\theta^{\prime}(\cos\theta^{\prime}-\cos\theta),$
$None$

and

$f_{2}(\epsilon,\theta,\theta^{\prime})=\epsilon^{2}(\cos\theta-\cos\theta^{\prime})^{2}.$
$None$

If the polarization of the CRSF is not of interest, then the polarization-
averaged cross section can be used and it is

${{d\sigma}\over{d\cos\theta^{\prime}}}=2\pi{{3\pi r_{0}c}\over
16}(1+\cos^{2}\theta)(1+\cos^{2}\theta^{\prime})\left[L_{-}+{B\over 2}\cdot
L_{+}\right].$ $None$

Note that for $B\ll 1$, the polarization-averaged cross section (3) is very
accurate. No additional terms due to spin flip are necessary.

In this paper, we have restricted ourselves to cyclotron energies
$E_{c}\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}50$ keV, because most of the observed
CRSFs are in this energy range (Staubert et al. 2019). However, a few CRSFs
with $E_{c}>50$ keV have been observed (Staubert et al. 2019), plus the
recently reported (Ge et al. 2020) champion with $E_{c}=90.32$ keV. For this
reason, in Fig. 7 we compare the relativistically correct
$d\sigma_{12}/d\cos\theta^{\prime}(\theta=0,\theta^{\prime}=\pi/2)$ of Sina
(1996) with the modified classical one (eq. 6b) for $E_{c}=25$ keV (left
panel), 50 keV (middle panel), and 100 keV (right panel). Clearly, 100 keV is
not much less than $m_{e}c^{2}$. Nevertheless, we show it for the readers to
see the magnitude of the discrepancy between classical and quantum-mechanical
cross sections.

Figure 7:  Top panels: Comparison of the relativistically correct
$d\sigma_{12}/d\cos\theta^{\prime}(\theta=0,\theta^{\prime}=\pi/2)$ from Sina
(1996) (blue solid line) with the modified classical one (eq. 6b) (red dashed
line) for $E_{c}=25$ keV (left panel), 50 keV (middle panel), and 100 keV
(right panel). Bottom panels: Corresponding ratios of the curves shown in the
top panels

##  3 Summary

We have shown numerically that, for photon energies $E\ll mc^{2}$ and magnetic
field strengths ${\cal B}\ll{\cal B}_{cr}$, the relativistic polarization-
dependent resonant differential cross sections of Harding & Daugherty (1991),
derived with the Johnson & Lippmann (1949) wave functions, are in excellent
agreement with the ones of Sina (1996), derived with the Sokolov & Ternov
(1968) formalism.

We have shown analytically that the expansion up to first order in
$\epsilon=E/m_{e}c^{2}$ and $B={\cal B}/{\cal B}_{cr}$ of the expressions of
Harding & Daugherty (1991) and of Sina (1996) lead to nearly identical
results.

We have provided simple, heuristic, but very accurate, expressions for the
polarization-dependent resonant differential cross sections, together with a
prescription that should be followed, for easy to perform detailed
calculations involving resonant scattering at $E_{c}\mathbin{\lower
3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char 60\relax$}\hss}\mathchar
29208\relax$}}50$ keV.

###### Acknowledgements.

We thank an anonymous referee for pointing out to us the Sokolov & Ternov
(1968) formalism. We also thank Roberto Turolla and Alexander Mushtukov for
useful discussions concerning the Sokolov & Ternov (1968) formalism. We are
indebted to Alice Harding for sending us the unpublished PhD Thesis of Ramin
Sina and to Peter Gonthier for e-mail exchanges. N.D.K. would like to thank
Sterl Phinney for his notes on the classical magnetic scattering cross
sections.

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##  Appendix A Approximations resulting from the Harding & Daugherty (1991)
expressions

We start from eq. (11) of Harding & Daugherty (1991) that has been derived in
the electron’s rest frame, considering that the electron initially occupies
the ground state. In a unit system where $\hbar=c=m_{e}=1$, this equation can
be written as follows

$\dfrac{d\sigma_{ss^{\prime}}^{0,l}}{d\Omega^{\prime}}=\dfrac{3\sigma_{T}}{4}\dfrac{\omega^{\prime}}{\omega}\dfrac{\exp\left(-(\omega^{2}\sin^{2}\theta+\omega^{\prime
2}\sin^{2}\theta^{\prime})/2B\right)}{1+\omega-\omega^{\prime}-(\omega\cos\theta-\omega^{\prime}\cos\theta^{\prime})\cos\theta^{\prime}}\\\
\times\left|\sum_{n=0}^{\infty}\sum_{i=1}^{2}\left(F^{(1)}_{n,i}e^{i\Phi}+F^{(2)}_{n,i}e^{-i\Phi}\right)\right|^{2},$

where $\sigma_{T}$ is the Thomson cross section, $\omega$, $\omega^{\prime}$
are the incident and scattered photon frequencies in units of
$m_{e}c^{2}/\hbar$, $\theta$ and $\theta^{\prime}$ are the incident and
scattered photon angles with respect to the magnetic field direction, $B$ is
the magnetic field strength $\cal B$ (in units of ${\cal B}_{cr}$), $\Phi$ is
proportional to $\sin(\phi-\phi^{\prime})$, $\phi$ and $\phi^{\prime}$ are the
azimuthal angles of the incident and the scattered photon, respectively. The
terms $F^{(1)}_{n,i}$, $F^{(2)}_{n,i}$ are complex functions of $\theta$,
$\theta^{\prime}$, $\omega$, $\omega^{\prime}$, $B$, $\phi$, $\phi^{\prime}$,
$s$, $s^{\prime}$ and can be found in the Appendix of Harding & Daugherty
(1991), where the upper indices 1, 2 are referred to the 1st and 2nd Feynman
diagram. Furthermore, $s$ and $s^{\prime}$ stand for the incident and
scattered photon’s polarization mode and $l$ is the final electron’s Landau
state. Note that, the infinite sum in eq. (A1) is carried out over
intermediate Landau states with principal quantum number $n$, whereas the sum
over $i={1,2}$ is for the possible electron spin orientations (spin-up, spin-
down) in the intermediate state.

Equation (A1) is quite general, but this generality is not needed in most
cases. Specifically, for resonant scattering of photons with energy
$E\mathbin{\lower 3.0pt\hbox{$\hbox to0.0pt{\raise 5.0pt\hbox{$\char
60\relax$}\hss}\mathchar 29208\relax$}}m_{e}c^{2}$ in a magnetic field smaller
than or comparable to the critical one ${\cal B}_{cr}=m_{e}^{2}c^{3}/e\hbar$,
Nobili, Turolla, & Zane (2008b, hereafter NTZ08b) pointed out that, for such
magnetic fields, the probability of exciting Landau levels above the second
($n=2$) is very small, thus the infinite sum over $n$ in eq. (A1) becomes
finite ($n\leq 2$).

Moreover, for photon energies $E\ll m_{e}c^{2}$, and magnetic fields ${\cal
B}\ll{\cal B}_{cr}$, that we are interested in, the cross section expressions
can be simplified even more, keeping only the term $n=1$ in the sum over all
the possible intermediate states and setting $l=0$ for the final electron
Landau state. Besides that, as NTZ08b stated, the $F^{(1)}_{n=1,i}$ terms
exhibit a divergent behavior near resonant frequency (see their eqs. 7 and 8),
while the terms $F^{(2)}_{n=1,i}$, remain instead finite. Given that, and as
long as we are interested in resonant scattering (i.e. photon frequency near
the resonant one), the above arguments lead us to a major simplification.

Neglecting all the non-resonant terms and the contributions of ($n\neq 1$) in
eq. (A1), we deduce the following expression

$\dfrac{d\sigma_{ss^{\prime}}^{0,l=0}}{d\Omega^{\prime}}=\dfrac{3\sigma_{T}}{4}\dfrac{\omega^{\prime}}{\omega}\dfrac{\exp\left(-(\omega^{2}\sin^{2}\theta+\omega^{\prime
2}\sin^{2}\theta^{\prime})/2B\right)}{1+\omega-\omega^{\prime}-(\omega\cos\theta-\omega^{\prime}\cos\theta^{\prime})\cos\theta^{\prime}}\\\
\times\left|F^{(1)}_{1,1}+F^{(1)}_{1,2}\right|^{2}.$

NTZ08b have managed to write the above equation in a simpler way, obtaining
compact expressions for the $F^{(1)}_{1,i}$ terms. Specifically, using eq.
(14) of NTZ08b in eq. (10) of NTZ08b one gets

${{d\sigma_{ss^{\prime}}}\over{d\Omega^{\prime}}}={{3\sigma_{T}}\over{16\pi}}{{\epsilon^{\prime}}\over{\epsilon}}{{(2+\epsilon-\epsilon^{\prime})\exp\left(-{{\epsilon^{2}\sin^{2}\theta+{\epsilon^{\prime}}^{2}\sin^{2}\theta^{\prime}}\over{2B}}\right)}\over{1+\epsilon-\epsilon^{\prime}-(\epsilon\cos\theta-\epsilon^{\prime}\cos\theta^{\prime})\cos\theta^{\prime}}}\\\
\times\left({{1+E_{1}}\over{2E_{1}}}\right)^{2}\left|{{T_{+}^{s\to
s^{\prime}}}\over{1+\epsilon-E_{1}+i\Gamma_{+}/2}}+{{T_{-}^{s\to
s^{\prime}}}\over{1+\epsilon-E_{1}+i\Gamma_{-}/2}}\right|^{2},$

where, for convenience, we retain the notation of NTZ08b, but for simplicity
we drop the indices referring to $n=1$ and $l=0$. So, $\epsilon$,
$\epsilon^{\prime}$ are the incident and scattered photon energies in units of
$m_{e}c^{2}$, $\theta$ and $\theta^{\prime}$ are the incident and scattered
photon angles with respect to the magnetic field direction, $B$ is the
magnetic field strength $\cal B$ (in units of ${\cal B}_{cr}$) and
$\Gamma_{+}$, $\Gamma_{-}$ are the relativistic decay rates corresponding to
the n=1 intermediate state and are given by Herold, Ruder, & Wunner (1982, see
also Harding & Daugherty 1991; Pavlov et al. 1991; Baring et al. 2005;
NTZ08b), where the index ”+” stands for the electron in the intermediate state
with spin-up whereas the index ”-” stands for spin-down. In addition, $s$ and
$s^{\prime}$ refer to the incident and scattered photon polarization modes,
and the explicit expressions for the $T_{+}^{s\to s^{\prime}}$, $T_{-}^{s\to
s^{\prime}}$ terms are given in the Appendix of NTZ08b for $n=1$ and $l=0$.

We remark that we do not use the relativistic polarization-dependent resonant
Compton differential cross section expressions of NTZ08b, because, for their
purposes, they substituted all the Lorentz profiles with a $\delta-$function,
and as a result the information about the exact shape of the different line
profiles is lost.

The energy $E_{1}$ is the electron intermediate state and, in units of
$m_{e}c^{2}$, it is given by eq. (12) of NTZ08b (see also eq. 4 of Harding &
Daugherty 1991).

$E_{1}=\sqrt{1+\epsilon^{2}\cos^{2}\theta+2B}.$ $None$

The energy $\epsilon^{\prime}$ of the photon after scattering, in units of
$m_{e}c^{2}$, is given by eq. (6) of NTZ08b (see also eq. 12 of Harding &
Daugherty 1991)

$\epsilon^{\prime}={{\epsilon^{2}\sin^{2}\theta+2\epsilon}\over{1+\epsilon(1-\cos\theta\cos\theta^{\prime})+\sqrt{f_{1}(\epsilon,\theta,\theta^{\prime})+f_{2}(\epsilon,\theta,\theta^{\prime})}}},$
$None$

where

$f_{1}(\epsilon,\theta,\theta^{\prime})=1+2\epsilon\cos\theta^{\prime}(\cos\theta^{\prime}-\cos\theta),$
$None$

and

$f_{2}(\epsilon,\theta,\theta^{\prime})=\epsilon^{2}(\cos\theta-\cos\theta^{\prime})^{2}.$
$None$

After a lengthy but straightforward calculation, and a trivial integration
over the angle $\phi^{\prime}$, one can write eq. (A3) in the following way

${{d\sigma_{ss^{\prime}}}\over{d\cos\theta^{\prime}}}={{3\pi\sigma_{T}}\over{16}}{{(1+E_{1})^{2}}\over{E_{1}\sqrt{1+2B\sin^{2}\theta}}}{{\epsilon^{\prime}}\over{\epsilon}}A\\\
\times\left[{{(T_{+}^{s\to s^{\prime}})^{2}}\over{\Gamma_{+}}}{\cal
L}_{+}+{{(T_{-}^{s\to s^{\prime}})^{2}}\over{\Gamma_{-}}}{\cal
L}_{-}+2{{T_{+}^{s\to s^{\prime}}T_{-}^{s\to
s^{\prime}}}\over{\Gamma_{mix}}}{{{\cal L}_{+}{\cal L}_{-}}\over{{\cal
L}_{mix}}}\right],$

where the effective decay rates $\Gamma^{e}_{\pm}$, which result from the
change in the Lorentz profiles argument (see NTZ08b), must be used in the
Lorentz profiles (see eqs. A8-A10)

$\Gamma^{e}_{\pm}=\dfrac{E_{1}(\epsilon_{r})}{\sqrt{1+2B\sin^{2}\theta}}\Gamma_{\pm}$
$None$

and we have defined the following function

$A={{(2+\epsilon-\epsilon^{\prime})\exp\left(-{{\epsilon^{2}\sin^{2}\theta+{\epsilon^{\prime}}^{2}\sin^{2}\theta^{\prime}}\over{2B}}\right)}\over{1+\epsilon-\epsilon^{\prime}-(\epsilon\cos\theta-\epsilon^{\prime}\cos\theta^{\prime})\cos\theta^{\prime}}}.$
$None$

The quantities ${\cal L}_{+}(\epsilon,\epsilon_{r})$, ${\cal
L}_{-}(\epsilon,\epsilon_{r})$ & ${\cal L}_{mix}(\epsilon,\epsilon_{r})$ are
the dimensionless Lorentz profiles and are given by

${\cal
L}_{+}(\epsilon,\epsilon_{r})={{\Gamma^{e}_{+}/2\pi}\over{(\epsilon-\epsilon_{r})^{2}+(\Gamma^{e}_{+}/2)^{2}}},$
$None$ ${\cal
L}_{-}(\epsilon,\epsilon_{r})={{\Gamma^{e}_{-}/2\pi}\over{(\epsilon-\epsilon_{r})^{2}+(\Gamma^{e}_{-}/2)^{2}}},$
$None$ ${\cal
L}_{mix}(\epsilon,\epsilon_{r})={{\Gamma^{e}_{mix}/2\pi}\over{(\epsilon-\epsilon_{r})^{2}+(\Gamma^{e}_{mix}/2)^{2}}},$
$None$

where $\Gamma^{e}_{mix}$ is calculated by

$\Gamma^{e}_{mix}=\dfrac{E_{1}(\epsilon_{r})}{\sqrt{1+2B\sin^{2}\theta}}\Gamma_{mix},$
$None$

with

$\Gamma_{mix}=\sqrt{\Gamma_{+}\Gamma_{-}},$ $None$

and $\epsilon_{r}$ is the resonant energy in units of $m_{e}c^{2}$. It is
given by eq. (8) of NTZ08b (see also eq. 5 of Harding & Daugherty 1991)

$\epsilon_{r}=\epsilon_{1}={{2B}\over{1+\sqrt{1+2B\sin^{2}\theta}}}.$ $None$

Having obtained the fully relativistic cross sections we proceed to the
derivation of eqs. (4a)-(4d).

Expansion up to first order in the small parameters $\epsilon$,
$\epsilon^{\prime}$, and $B$ yields

${\epsilon^{\prime}\over\epsilon}\approx
1-\dfrac{\epsilon}{2}(\cos\theta-\cos\theta^{\prime})^{2},$ $None$
${{(1+E_{1})^{2}}\over{E_{1}\sqrt{1+2B\sin^{2}\theta}}}{{\epsilon^{\prime}}\over{\epsilon}}A\approx
8\left[1-B\left(2\sin^{2}\theta+(\cos\theta-\cos\theta^{\prime})^{2}\right)\right].$
$None$

To lowest order, $\Gamma_{+}$, $\Gamma_{-}$ are given in Herold, Ruder, &
Wunner (1982) (see also eqs. 15, 16 of Harding & Daugherty 1991 and eq. 31 of
NTZ08b)

$\Gamma_{+}\approx 2\alpha B^{3}/3,$ $None$ $\Gamma_{-}\approx 4\alpha
B^{2}/3,$ $None$

and substituting the above expressions into eq. (A11b) we get

$\Gamma_{mix}\approx\dfrac{4}{3}\alpha B^{2}\sqrt{B/2},$ $None$

where $\alpha$ is the fine structure constant. For a discussion regarding the
cyclotron line widths see our Appendix C.

Note that the classical cross sections (1) and the expressions (4) are
proportional to $r_{0}c$, while the quantum-mechanical ones are proportional
to $\sigma_{T}$. This means that the Lorentz profiles with variable the photon
frequency (i.e. $L_{+}(\omega,\omega_{r})$, $L_{-}(\omega,\omega_{r})$ &
$L_{mix}(\omega,\omega_{r})$) have dimensions of time and their relation with
the dimensionless ones (i.e. ${\cal L}_{+}(\epsilon,\epsilon_{r})$, ${\cal
L}_{-}(\epsilon,\epsilon_{r})$ & ${\cal L}_{mix}(\epsilon,\epsilon_{r})$) are
(see Appendix C)

${\cal
L}_{+}(\epsilon,\epsilon_{r})\approx{{m_{e}c^{2}}\over\hbar}L_{+}(\omega,\omega_{r}),$
$None$ ${\cal
L}_{-}(\epsilon,\epsilon_{r})\approx{{m_{e}c^{2}}\over\hbar}L_{-}(\omega,\omega_{r}),$
$None$ ${\cal
L}_{mix}(\epsilon,\epsilon_{r})\approx{{m_{e}c^{2}}\over\hbar}L_{mix}(\omega,\omega_{r}),$
$None$

where the Lorentz profiles with argument the photon frequency are given by

$L_{+}(\omega,\omega_{r})={{\Gamma^{\prime}_{+}/2\pi}\over{(\omega-\omega_{r})^{2}+(\Gamma^{\prime}_{+}/2)^{2}}},$
$None$
$L_{-}(\omega,\omega_{r})={{\Gamma^{\prime}_{-}/2\pi}\over{(\omega-\omega_{r})^{2}+(\Gamma^{\prime}_{-}/2)^{2}}},$
$None$
$L_{mix}(\omega,\omega_{r})={{\Gamma^{\prime}_{mix}/2\pi}\over{(\omega-\omega_{r})^{2}+(\Gamma^{\prime}_{mix}/2)^{2}}},$
$None$

and

$\Gamma^{\prime}_{+}=\dfrac{m_{e}c^{2}}{\hbar}\Gamma_{+},$ $None$
$\Gamma^{\prime}_{-}=\dfrac{m_{e}c^{2}}{\hbar}\Gamma_{-}\approx\Gamma=4e^{2}\omega^{2}_{c}/(3m_{e}c^{3}),$
$None$ $\Gamma^{\prime}_{mix}=\dfrac{m_{e}c^{2}}{\hbar}\Gamma_{mix}.$ $None$

Using the above, one finds that eq. (A6) is approximated by the following one

$\dfrac{d\sigma_{ss^{\prime}}}{d\cos\theta^{\prime}}\approx\dfrac{3\pi\sigma_{T}}{2}\dfrac{m_{e}c^{2}}{\hbar}W(B,\theta,\theta^{\prime})\\\
\times\left[{{(T_{+}^{s\to
s^{\prime}})^{2}}\over{\Gamma_{+}}}L_{+}+{{(T_{-}^{s\to
s^{\prime}})^{2}}\over{\Gamma_{-}}}L_{-}+2{{T_{+}^{s\to s^{\prime}}T_{-}^{s\to
s^{\prime}}}\over{\Gamma_{mix}}}{{L_{+}L_{-}}\over{L_{mix}}}\right],\\\
=2\pi\dfrac{3\pi
r_{o}c}{8}\left(\dfrac{16\alpha}{3}\right)W(B,\theta,\theta^{\prime})\\\
\times\left[{{(T_{+}^{s\to
s^{\prime}})^{2}}\over{\Gamma_{+}}}L_{+}+{{(T_{-}^{s\to
s^{\prime}})^{2}}\over{\Gamma_{-}}}L_{-}+2{{T_{+}^{s\to s^{\prime}}T_{-}^{s\to
s^{\prime}}}\over{\Gamma_{mix}}}{{L_{+}L_{-}}\over{L_{mix}}}\right],$

where

$W(B,\theta,\theta^{\prime})=1-B\left(2\sin^{2}\theta+(\cos\theta-\cos\theta^{\prime})^{2}\right),$
$None$

and we have used that

$\epsilon\approx\epsilon_{r}\approx B.$ $None$

The expressions for $T_{+}^{s\to s^{\prime}}$ and $T_{-}^{s\to s^{\prime}}$
have strong polarization dependence and, as we mentioned earlier, are given in
the Appendix of NTZ08b. In order to obtain equations (4), we will work
separately for each pair $s,s^{\prime}$.

###  A.1 Transition $1\to 1$

By expanding the terms $T_{+}^{1\to 1}$, $T_{-}^{1\to 1}$ in the small
parameters $\epsilon,\epsilon^{\prime}$, and $B$, and employing eqs. (A15) and
(A21), one obtains the following approximations

$\dfrac{(T^{1\to
1}_{+})^{2}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\right),$
$None$ $\dfrac{(T^{1\to
1}_{-})^{2}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\cos^{2}\theta\cos^{2}\theta^{\prime}\\\
\times\bigg{[}1-B\left(2+\sin^{2}\theta+\dfrac{2\sin^{2}\theta^{\prime}\cos\theta}{\cos\theta^{\prime}}-\sin^{2}\theta^{\prime}\right)\bigg{]},$
$\dfrac{2T^{1\to 1}_{+}T^{1\to
1}_{-}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta\cos\theta^{\prime}.$
$None$

Equations (A22), along with (A19), lead us to derive the following
approximation for $d\sigma_{11}/d\cos\theta^{\prime}$

${{d\sigma_{11}}\over{d\cos\theta^{\prime}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{1\to 1}\cdot L_{-}+h^{1\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

$g^{1\to
1}(\theta,\theta^{\prime},B)=\cos^{2}\theta\cos^{2}\theta^{\prime}\bigg{[}1-B\bigg{(}3\sin^{2}\theta-\sin^{2}\theta^{\prime}\\\
+(\cos\theta-\cos\theta^{\prime})^{2}+2+2\sin^{2}\theta^{\prime}\dfrac{\cos\theta}{\cos\theta^{\prime}}\bigg{)}\bigg{]},$

and

$h^{1\to 1}(B)=\dfrac{B}{2}.$ $None$

###  A.2 Transition $1\to 2$

We apply the same methodology to all the other cases. Hence, by expanding the
terms $T_{+}^{1\to 2}$, $T_{-}^{1\to 2}$ in the small parameters
$\epsilon,\epsilon^{\prime}$, and $B$, and using eqs. (A15) and (A21), one
finds the following approximations

$\dfrac{(T^{1\to
2}_{+})^{2}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\cos^{2}\theta^{\prime}\right),$
$None$ $\dfrac{(T^{1\to
2}_{-})^{2}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\cos^{2}\theta\left[1-B\left(2+\sin^{2}\theta\right)\right],$
$\dfrac{2T^{1\to 2}_{+}T^{1\to
2}_{-}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta\cos\theta^{\prime}.$
$None$

Then, by substituting eqs. (A24) into eq. (A19) one gets the following
approximation

${{d\sigma_{12}}\over{d\cos\theta^{\prime}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{1\to 2}\cdot L_{-}+h^{1\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

$g^{1\to
2}(\theta,\theta^{\prime},B)=\cos^{2}\theta\bigg{[}1-B\bigg{(}3\sin^{2}\theta+2\\\
+(\cos\theta-\cos\theta^{\prime})^{2}\bigg{)}\bigg{]},$

and

$h^{1\to 2}(\theta^{\prime},B)=\dfrac{B}{2}\cos^{2}\theta^{\prime}.$ $None$

###  A.3 Transition $2\to 1$

Similarly, by expanding the terms $T_{+}^{2\to 1}$, $T_{-}^{2\to 1}$ in the
small parameters $\epsilon,\epsilon^{\prime}$, and $B$, and taking into
account eqs. (A15) and (A21), one deduces the following approximations

$\dfrac{(T^{2\to
1}_{+})^{2}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\cos^{2}\theta\right),$
$None$ $\dfrac{(T^{2\to
1}_{-})^{2}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\cos^{2}\theta^{\prime}\\\
\times\bigg{[}1-B\left(2-\sin^{2}\theta^{\prime}+2\sin^{2}\theta^{\prime}\dfrac{\cos\theta}{\cos\theta^{\prime}}\right)\bigg{]},$
$\dfrac{2T^{2\to 1}_{+}T^{2\to
1}_{-}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta\cos\theta^{\prime}.$
$None$

Using eqs. (A26) and (A19), one finds

${{d\sigma_{21}}\over{d\cos\theta^{\prime}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{2\to 1}\cdot L_{-}+h^{2\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

$g^{2\to
1}(\theta,\theta^{\prime},B)=\cos^{2}\theta^{\prime}\bigg{[}1-B\bigg{(}2\sin^{2}\theta+(\cos\theta-\cos\theta^{\prime})^{2}\\\
+2-\sin^{2}\theta^{\prime}+2\sin^{2}\theta^{\prime}\dfrac{\cos\theta}{\cos\theta^{\prime}}\bigg{)}\bigg{]},$

and

$h^{2\to 1}(\theta,B)=\dfrac{B}{2}\cos^{2}\theta.$ $None$

###  A.4 Transition $2\to 2$

Following the same procedure as above, for the terms $T_{+}^{2\to 2}$,
$T_{-}^{2\to 2}$ and employing eqs. (A15) and (A21), one derives that

$\dfrac{(T^{2\to
2}_{+})^{2}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\cos^{2}\theta\cos^{2}\theta^{\prime}\right),$
$None$ $\dfrac{(T^{2\to
2}_{-})^{2}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\bigg{(}1-2B\bigg{)},$
$None$ $\dfrac{2T^{2\to 2}_{+}T^{2\to
2}_{-}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta\cos\theta^{\prime}.$
$None$

Thus, the substitution of eqs. (A28) into (A19), yields

${{d\sigma_{22}}\over{d\cos\theta^{\prime}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}g^{2\to 2}\cdot L_{-}+h^{2\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta\cos\theta^{\prime}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

$g^{2\to
2}(\theta,\theta^{\prime},B)=1-B\left[2\sin^{2}\theta+(\cos\theta-\cos\theta^{\prime})^{2}+2\right],$
$None$

and

$h^{2\to
2}(\theta,\theta^{\prime},B)=\dfrac{B}{2}\cos^{2}\theta\cos^{2}\theta^{\prime}.$
$None$

##  Appendix B Approximations resulting from the Sina (1996) expressions

We start from eq. (3.25) of Sina (1996), written in the electron’s rest frame
(ground state), considering a unit system where $\hbar=m_{e}=c=1$, (i.e.
energy is measured in $m_{e}c^{2}$, frequency is measured in
$m_{e}c^{2}/\hbar$ and the magnetic field strength ${\cal B}$ is measured in
${\cal B}_{cr}=e^{-1}$) and neglect terms that are related to the 2nd Feynman
diagram, as well as the terms that the intermediate electron Landau state has
a principal quantum number $n\neq 1$ (i.e. keeping only the terms that exhibit
a divergence near resonant frequency with $n=1$, as we did in Appendix A). So,
the infinite sum over $n$ for all possible intermediate states collapses to
the following

$\dfrac{d\sigma_{ss^{\prime}}}{d\Omega_{f}}=\dfrac{\alpha^{2}}{1-\beta_{f}\cos\theta_{f}}\dfrac{\omega_{f}}{\omega_{i}}\left|Z_{c1}(n=1)\right|^{2},$
$None$

where $Z_{c1}(n=1)$ is given by eq. (3.15) of Sina (1996) and is a sum over
the possible electron spin orientations $s_{n}$ in the intermediate state (for
spin-up $s_{n}=+1$ whereas for spin-down $s_{n}=-1$)

$Z_{c1}(n=1)=\sum_{s_{n}}\dfrac{D^{f,n=1,s_{f},s_{n}}(k_{f})H^{n=1,i,s_{n},s_{i}}(k_{i})}{\omega_{i}+E_{i}-E_{n=1,1}+i\Gamma^{n=1}_{s_{n}}/2}.$
$None$

We retain the notation of Sina (1996), so $\omega_{i}$ is the incident photon
frequency, $E_{i}$ is the electron’s energy before scattering, $\theta_{i}$,
$\theta_{f}$ are the incident and scattered photon angles with respect to the
magnetic field direction, the indices $i$ and $f$ stand for the initial and
final electron Landau states (in our case they are equal to 0, i.e. ground
state), the $s_{i}$ and $s_{f}$ are the initial and final electron’s spin
orientations and are equal to -1 (since only spin-down is allowed in the
ground state), and $k_{i}$, $k_{f}$ are the incident and scattered electron
wavenumbers, respectively. Furthermore, $E_{n=1,1}$ ($E_{1}$, hereafter) is
the electron’s energy in the intermediate state with $n=1$ and is given by eq.
(3.3) of Sina (1996) (see also eq. 3.1 of Schwarm 2017)

$E_{1}\equiv E_{n=1,1}=\sqrt{1+\omega^{2}_{i}\cos^{2}\theta_{i}+2B},$ $None$

where a misprint has been corrected and we have substituted eq. (3.4) of Sina
(1996) into eq. (3.3) of Sina (1996) and took into account that the initial
electron momentum $p_{i}$ is equal to zero, since we work in the electron’s
rest frame. Moreover, $\beta_{f}$ can be written as $p_{f}/E_{f}$ (see eq. B18
of Gonthier et al. 2014), where $E_{f}$ and $p_{f}$ are the final electron
energy and parallel component of momentum (with respect to the magnetic field
direction) and are calculated by imposing the conservation of energy and by
eq. (3.29) of Sina (1996). Thus

$E_{f}=1+\omega_{i}-\omega_{f},$ $None$

and

$p_{f}=\omega_{i}\cos\theta_{i}-\omega_{f}\cos\theta_{f}.$ $None$

Note that $\omega_{f}$ is the scattered photon frequency and is given by eq.
(3.28) of Sina (1996), although it is actually our eq. (A5) written in a
different form, so we do not present it here and $\Gamma^{n=1}_{s_{n}=+1}$,
$\Gamma^{n=1}_{s_{n}=-1}$ ($\Gamma_{+}$, $\Gamma_{-}$, hereafter) are the
relativistic decay widths for the electron in the intermediate state with
spin-up and spin-down, respectively, and are given by Herold, Ruder, & Wunner
(1982). Fig. 3.4 of Schwarm (2017) verifies that the decay widths of Sina
(1996) are in full agreement with the ones of Herold, Ruder, & Wunner (1982)
that we have used in Appendix A. This is reasonable and absolutely
predictable, since both works have employed electron wavefunctions of Sokolov
& Ternov (1968).

Substituting the above into eqs. (B1) and using that the initial electron
energy $E_{1}$ is equal to 1 and $\alpha^{2}=3\sigma_{T}/8\pi$ in this system
of units, we obtain

$\dfrac{d\sigma_{ss^{\prime}}}{d\Omega_{f}}=\dfrac{3\sigma_{T}}{8\pi}\dfrac{1+\omega_{i}-\omega_{f}}{1+\omega_{i}-\omega_{f}-(\omega_{i}\cos\theta_{i}-\omega_{f}\cos\theta_{f})\cos\theta_{f}}\\\
\times\dfrac{\omega_{f}}{\omega_{i}}\bigg{|}\dfrac{D^{f=0,n=1,s_{f}=-1,s_{n}=+1}(k_{f})H^{n=1,i=0,s_{n}=+1,s_{i}=-1}(k_{i})}{1+\omega_{i}-E_{1}+i\Gamma_{+}/2}\\\
+\dfrac{D^{f=0,n=1,s_{f}=-1,s_{n}=-1}(k_{f})H^{n=1,i=0,s_{n}=-1,s_{i}=-1}(k_{i})}{1+\omega_{i}-E_{1}+i\Gamma_{-}/2}\bigg{|}^{2},$

where the complex functions $D^{f,n,s_{f},s_{n}}(k_{f})$ and
$H^{n,i,s_{n},s_{i}}(k_{i})$ have strong polarization dependence, since they
depend on the polarization modes of the incident and scattered photons, and
are given in Appendix D of Sina (1996). Specifically, the
$D^{f,n,s_{f},s_{n}}(k_{f})$ terms exclusively depend on the final photon
polarization mode and are calculated by eqs. (D.61), (D.66) of Sina (1996)

$D_{\perp}^{f=0,n=1,s_{f}=-1,s_{n}}(k_{f})=\\\
i\bigg{[}\big{(}C_{1,f=0}C_{4,n=1}+C_{3,f=0}C_{2,n=1}\big{)}\Lambda_{-1,1}(k_{f})-\\\
\big{(}C_{2,f=0}C_{3,n=1}+C_{4,f=0}C_{1,n=1}\big{)}\Lambda_{0,0}(k_{f})\bigg{]},$
$D_{\parallel}^{f=0,n=1,s_{f}=-1,s_{n}}(k_{f})=\\\
\cos\theta_{f}\bigg{[}\big{(}C_{1,f=0}C_{4,n=1}+C_{3,f=0}C_{2,n=1}\big{)}\Lambda_{-1,1}(k_{f})+\\\
\big{(}C_{2,f=0}C_{3,n=1}+C_{4,f=0}C_{1,n=1}\big{)}\Lambda_{0,0}(k_{f})\bigg{]}-\\\
\sin\theta_{f}\bigg{[}\big{(}C_{1,f=0}C_{3,n=1}+C_{3,f=0}C_{1,n=1}\big{)}\Lambda_{-1,0}(k_{f})-\\\
\big{(}C_{2,f=0}C_{4,n=1}+C_{4,f=0}C_{2,n=1}\big{)}\Lambda_{0,1}(k_{f})\bigg{]},$

while the terms $H^{n,i,s_{n},s_{i}}(k_{i})$ exclusively depend on the initial
photon polarization mode and are given by eqs. (D.60), (D.65) of Sina (1996)

$H_{\perp}^{n=1,i=0,s_{n},s_{i}=-1}(k_{i})=\\\
i\bigg{[}\big{(}C_{1,n=1}C_{4,i=0}+C_{3,n=1}C_{2,i=0}\big{)}\Lambda_{0,0}(k_{i})-\\\
\big{(}C_{2,n=1}C_{3,i=0}+C_{4,n=1}C_{1,i=0}\big{)}\Lambda_{-1,1}(k_{i})\bigg{]},$
$H_{\parallel}^{n=1,i=0,s_{n},s_{i}=-1}(k_{i})=\\\
\cos\theta_{i}\bigg{[}\big{(}C_{1,n=1}C_{4,i=0}+C_{3,n=1}C_{2,i=0}\big{)}\Lambda_{0,0}(k_{i})+\\\
\big{(}C_{2,n=1}C_{3,i=0}+C_{4,n=1}C_{1,i=0}\big{)}\Lambda_{-1,1}(k_{i})\bigg{]}-\\\
\sin\theta_{i}\bigg{[}\big{(}C_{1,n=1}C_{3,i=0}+C_{3,n=1}C_{1,i=0}\big{)}\Lambda_{-1,0}(k_{i})-\\\
\big{(}C_{2,n=1}C_{4,i=0}+C_{4,n=1}C_{2,i=0}\big{)}\Lambda_{0,1}(k_{i})\bigg{]},$

where the lower index “$\parallel$” stands for the ordinary polarization mode
whereas the lower index “$\perp$” stands for the extraordinary polarization
mode. Moreover the quantities $C_{k}$ with k=1,2,3,4, are the wavefunction
coefficients of Sokolov & Ternov (1968) and are given by eqs. (B61)-(B65) of
Sina (1996), whereas the $\Lambda_{i,j}$ functions are proportional to
Laguerre polynomials and are given in Appendix D of Sina (1996). Note that,
one can also obtain all of the above using Appendix B of Gonthier et al.
(2014).

After a lengthy, but straightforward calculation, we can write eq. (B4) in the
following form

$\dfrac{d\sigma_{ss^{\prime}}}{d\cos\theta_{f}}=\dfrac{3\pi\sigma_{T}}{2}\dfrac{E_{1}}{\sqrt{1+2B\sin^{2}\theta_{i}}}\dfrac{\omega_{f}}{\omega_{i}}{\cal
A}\\\ \times\left[{{{\cal T}_{+}^{s\to s^{\prime}}}\over{\Gamma_{+}}}{\cal
L}_{+}+{{{\cal T}_{-}^{s\to s^{\prime}}}\over{\Gamma_{-}}}{\cal
L}_{-}+2{{{\cal T}_{mix}^{s\to s^{\prime}}}\over{\Gamma_{mix}}}{{{\cal
L}_{+}{\cal L}_{-}}\over{{\cal L}_{mix}}}\right],$

where we have done a trivial integration over the scattered photon azimuthal
angle $\phi_{f}$. Also, for simplicity, we have defined the following
quantities

${\cal
A}=\dfrac{(1+\omega_{i}-\omega_{f})\exp(-(\omega^{2}_{i}\sin^{2}\theta_{i}+\omega^{2}_{f}\sin^{2}\theta_{f})/2B)}{1+\omega_{i}-\omega_{f}-(\omega_{i}\cos\theta_{i}-\omega_{f}\cos\theta_{f})\cos\theta_{f}},$
$None$ ${\cal T}_{+}^{s\to
s^{\prime}}=\exp((\omega^{2}_{i}\sin^{2}\theta_{i}+\omega^{2}_{f}\sin^{2}\theta_{f})/2B)\\\
\times\left|D_{s^{\prime}}^{f=0,n=1,s_{f}=-1,s_{n}=+1}H_{s}^{n=1,i=0,s_{n}=+1,s_{i}=-1}\right|^{2},$
${\cal T}_{-}^{s\to
s^{\prime}}=\exp((\omega^{2}_{i}\sin^{2}\theta_{i}+\omega^{2}_{f}\sin^{2}\theta_{f})/2B)\\\
\times\left|D_{s^{\prime}}^{f=0,n=1,s_{f}=-1,s_{n}=-1}H_{s}^{n=1,i=0,s_{n}=-1,s_{i}=-1}\right|^{2},$
$2{\cal T}_{mix}^{s\to
s^{\prime}}=\exp((\omega^{2}_{i}\sin^{2}\theta_{i}+\omega^{2}_{f}\sin^{2}\theta_{f})/2B)\\\
\times\bigg{[}\bigg{(}(D_{s^{\prime}}^{f=0,n=1,s_{f}=-1,s_{n}=+1}H_{s}^{n=1,i=0,s_{n}=+1,s_{i}=-1}\big{)}^{*}\\\
\times
D_{s^{\prime}}^{f=0,n=1,s_{f}=-1,s_{n}=-1}H_{s}^{n=1,i=0,s_{n}=-1,s_{i}=-1}\bigg{)}+\\\
\bigg{(}D_{s^{\prime}}^{f=0,n=1,s_{f}=-1,s_{n}=+1}H_{s}^{n=1,i=0,s_{n}=+1,s_{i}=-1}\\\
\times\big{(}D_{s^{\prime}}^{f=0,n=1,s_{f}=-1,s_{n}=-1}H_{s}^{n=1,i=0,s_{n}=-1,s_{i}=-1}\big{)}^{*}\bigg{)}\bigg{]}.$

Note that the dimensionless Lorentz profiles ${\cal
L}_{+}(\omega_{i},\omega_{r})$, ${\cal L}_{-}(\omega_{i},\omega_{r})$, and
${\cal L}_{mix}(\omega_{i},\omega_{r})$ that are shown in eq. (B7a) are the
same as the ones in Appendix A, since in this unit system the dimensionless
photon frequencies $\omega_{i},\omega_{f}$ and the dimensionless resonant
frequency, have the same values with the corresponding dimensionless energies
and, as we mentioned earlier, the decay widths are identical to those shown in
Appendix A (and so are the effective decay widths given by eqs. A7a, A11b).
Furthermore, the dimensionless resonant frequency $\omega_{r}$ is given by eq.
(5) of Harding & Daugherty (1991) and is actually the same as the
dimensionless resonant energy (eq. A12) in Appendix A

$\omega_{r}=\dfrac{2B}{1+\sqrt{1+2B\sin^{2}\theta_{i}}}.$ $None$

Having obtained eqs. (B7), we are able to proceed to the derivation of eqs.
(5a)-(5d). Expansion up to first order in the small parameters $\omega_{i}$,
$\omega_{f}$, and $B$ yields

${\omega_{f}\over\omega_{i}}\approx
1-\dfrac{\omega_{i}}{2}(\cos\theta_{i}-\cos\theta_{f})^{2},$ $None$
$\dfrac{\omega_{f}E_{1}\cdot{\cal
A}}{\omega_{i}\sqrt{1+2B\sin^{2}\theta_{i}}}\approx
1-B\left(\sin^{2}\theta_{i}+\cos^{2}\theta_{f}-2\cos\theta_{i}\cos\theta_{f}\right)$
$None$ $\omega_{i}\approx\omega_{r}\approx B.$ $None$

Using the above, one finds that eq. (B7a) can be approximated by

$\dfrac{d\sigma_{ss^{\prime}}}{d\cos\theta_{f}}\approx\dfrac{3\pi\sigma_{T}}{2}{\cal
W}(\theta_{i},\theta_{f},B)\\\ \times\left[{{{\cal T}_{+}^{s\to
s^{\prime}}}\over{\Gamma_{+}}}{\cal L}_{+}+{{{\cal T}_{-}^{s\to
s^{\prime}}}\over{\Gamma_{-}}}{\cal L}_{-}+2{{{\cal T}_{mix}^{s\to
s^{\prime}}}\over{\Gamma_{mix}}}{{{\cal L}_{+}{\cal L}_{-}}\over{{\cal
L}_{mix}}}\right],$

and recalling the dimensions of each quantity by taking into account the
procedure as well as the results of Appendix A, we obtain the following
approximation

$\dfrac{d\sigma_{ss^{\prime}}}{d\cos\theta_{f}}\approx 2\pi\dfrac{3\pi
r_{o}c}{8}\left(\dfrac{16\alpha}{3}\right){\cal W}(\theta_{i},\theta_{f},B)\\\
\times\left[{{{\cal T}_{+}^{s\to s^{\prime}}}\over{\Gamma_{+}}}L_{+}+{{{\cal
T}_{-}^{s\to s^{\prime}}}\over{\Gamma_{-}}}L_{-}+2{{{\cal T}_{mix}^{s\to
s^{\prime}}}\over{\Gamma_{mix}}}{{L_{+}L_{-}}\over{L_{mix}}}\right],$

where

${\cal
W}(\theta_{i},\theta_{f},B)=1-B\left(\sin^{2}\theta_{i}+\cos^{2}\theta_{f}-2\cos\theta_{i}\cos\theta_{f}\right).$
$None$

From now on, all the quantities have their physical dimensions and as a result
the Lorentz profiles that are shown in eq. (B13a) have dimensions of time and
are calculated by eqs. (A17), where, in the notation that we use in this
Appendix, the arguments of $L_{+}(\omega_{i},\omega_{r})$,
$L_{i}(\omega_{i},\omega_{r})$, and $L_{mix}(\omega_{i},\omega_{r})$ are
$\omega_{i}$ and $\omega_{r}$. As we said earlier, the terms ${\cal
T}_{+}^{s\to s^{\prime}}$, ${\cal T}_{-}^{s\to s^{\prime}}$, and ${\cal
T}_{mix}^{s\to s^{\prime}}$ have strong polarization dependence and for this
reason we are going to derive the approximations separately for each possible
combination of the incident and scattered photon polarization modes.

###  B.1 Transition $1\to 1$

By expanding the terms ${\cal T}_{+}^{1\to 1}$, ${\cal T}_{-}^{1\to 1}$, and
${\cal T}_{mix}^{1\to 1}$ in the small parameters $\omega_{i},\omega_{f}$, and
$B$, and employing eqs. (A15) and (B11), one obtains the following
approximations

$\dfrac{{\cal T}^{1\to
1}_{+}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\right),$
$None$ $\dfrac{{\cal T}_{-}^{1\to
1}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\cos^{2}\theta_{i}\cos^{2}\theta_{f}\\\
\times\bigg{[}1-B\left(\sin^{2}\theta_{i}+\dfrac{2\sin^{2}\theta_{f}\cos\theta_{i}}{\cos\theta_{f}}-\sin^{2}\theta_{f}\right)\bigg{]},$
$\dfrac{2{\cal T}_{mix}^{1\to
1}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta_{i}\cos\theta_{f}.$
$None$

Equations (B14) along with (B13) lead us to derive the following approximation
for $d\sigma_{11}/d\cos\theta_{f}$

${{d\sigma_{11}}\over{d\cos\theta_{f}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal G}^{1\to 1}\cdot L_{-}+h^{1\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta_{i}\cos\theta_{f}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

${\cal G}^{1\to
1}(\theta_{i},\theta_{f},B)=\cos^{2}\theta_{i}\cos^{2}\theta_{f}\bigg{[}1-B\bigg{(}3\sin^{2}\theta_{i}-\sin^{2}\theta_{f}\\\
+(\cos\theta_{i}-\cos\theta_{f})^{2}-1+2\sin^{2}\theta_{f}\dfrac{\cos\theta_{i}}{\cos\theta_{f}}\bigg{)}\bigg{]},$

and

$h^{1\to 1}(B)=\dfrac{B}{2}.$ $None$

###  B.2 Transition $1\to 2$

We apply the same methodology to all the other cases. Hence, by expanding the
terms ${\cal T}_{+}^{1\to 2}$, ${\cal T}_{-}^{1\to 2}$, and ${\cal
T}_{mix}^{1\to 2}$ in the small parameters $\omega_{i},\omega_{f}$, and $B$,
and employing eqs. (A15) and (B11), one finds the following approximations

$\dfrac{{\cal T}^{1\to
2}_{+}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\cos^{2}\theta_{f}\right),$
$None$ $\dfrac{{\cal T}^{1\to
2}_{-}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\cos^{2}\theta_{i}\left(1-B\sin^{2}\theta_{i}\right),$
$None$ $\dfrac{2{\cal T}_{mix}^{1\to
2}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta_{i}\cos\theta_{f}.$
$None$

Then, by substituting eqs. (B16) into eq. (B13a) one gets the following
approximation

${{d\sigma_{12}}\over{d\cos\theta_{f}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal G}^{1\to 2}\cdot L_{-}+h^{1\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta_{i}\cos\theta_{f}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

${\cal G}^{1\to
2}(\theta_{i},\theta_{f},B)=\cos^{2}\theta_{i}\bigg{[}1-B\bigg{(}3\sin^{2}\theta_{i}-1\\\
+(\cos\theta_{i}-\cos\theta_{f})^{2}\bigg{)}\bigg{]},$

and

$h^{1\to 2}(\theta_{f},B)=\dfrac{B}{2}\cos^{2}\theta_{f}.$ $None$

###  B.3 Transition $2\to 1$

Similarly, by expanding the terms ${\cal T}_{+}^{2\to 1}$, ${\cal T}_{-}^{2\to
1}$, and ${\cal T}_{mix}^{2\to 1}$ in the small parameters
$\omega_{i},\omega_{f}$, and $B$, and taking into account eqs. (A15) and
(B11), one deduces the following approximations

$\dfrac{{\cal T}_{+}^{2\to
1}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\cos^{2}\theta_{i}\right),$
$None$ $\dfrac{{\cal T}_{-}^{2\to
1}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha}\cos^{2}\theta_{f}\\\
\times\bigg{[}1-B\left(2\sin^{2}\theta_{f}\dfrac{\cos\theta_{i}}{\cos\theta_{f}}-\sin^{2}\theta_{f}\right)\bigg{]},$
$\dfrac{2{\cal T}_{mix}^{2\to
1}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta_{i}\cos\theta_{f}.$
$None$

Using eqs. (B18) and (B13a), one finds

${{d\sigma_{21}}\over{d\cos\theta_{f}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal G}^{2\to 1}\cdot L_{-}+h^{2\to 1}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta_{i}\cos\theta_{f}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

${\cal G}^{2\to
1}(\theta_{i},\theta_{f},B)=\cos^{2}\theta_{f}\bigg{[}1-B\bigg{(}2\sin^{2}\theta_{i}+(\cos\theta_{i}-\cos\theta_{f})^{2}\\\
-1-\sin^{2}\theta_{f}+2\sin^{2}\theta_{f}\dfrac{\cos\theta_{i}}{\cos\theta_{f}}\bigg{)}\bigg{]},$

and

$h^{2\to 1}(\theta_{i},B)=\dfrac{B}{2}\cos^{2}\theta_{i}.$ $None$

###  B.4 Transition $2\to 2$

Following the same procedure as above for the terms ${\cal T}_{+}^{2\to 2}$,
${\cal T}_{-}^{2\to 2}$, and ${\cal T}_{mix}^{2\to 2}$ and employing eqs.
(A15) and (B11), one derives that

$\dfrac{{\cal T}_{+}^{2\to
2}}{\Gamma_{+}}\approx\dfrac{3}{16\alpha}\left(\dfrac{B}{2}\cos^{2}\theta_{i}\cos^{2}\theta_{f}\right),$
$None$ $\dfrac{{\cal T}_{-}^{2\to 2}}{\Gamma_{-}}\approx\dfrac{3}{16\alpha},$
$None$ $\dfrac{2{\cal T}_{mix}^{2\to
2}}{\Gamma_{mix}}\approx\dfrac{3}{16\alpha}\sqrt{2B}\cos\theta_{i}\cos\theta_{f}.$
$None$

Thus, the substitution of eqs. (B20) into (B13a), yields

${{d\sigma_{22}}\over{d\cos\theta_{f}}}\approx 2\pi{{3\pi
r_{o}c}\over{8}}\Big{[}{\cal G}^{2\to 2}\cdot L_{-}+h^{2\to 2}\cdot L_{+}\\\
+\sqrt{2B}\cos\theta_{i}\cos\theta_{f}\dfrac{L_{-}\cdot
L_{+}}{L_{mix}}\Big{]},$

where

${\cal G}^{2\to
2}(\theta_{i},\theta_{f},B)=1-B\left[2\sin^{2}\theta_{i}+(\cos\theta_{i}-\cos\theta_{f})^{2}-1\right],$
$None$

and

$h^{2\to
2}(\theta_{i},\theta_{f},B)=\dfrac{B}{2}\cos^{2}\theta_{i}\cos^{2}\theta_{f}.$
$None$

##  Appendix C Cyclotron line widths

The spin dependent relativistic cyclotron widths $\Gamma_{+},\,\Gamma_{-}$ for
the transitions from the 1st excited intermediate electron state to the
fundamental $n=0$ Landau state are given by eq. (17) of Herold et al. (1982)
or/and by eq. (3) of Pavlov et al. (1991). So, we do not present the complete
formulae here.

Known in the literature are the non-relativistic cyclotron line widths, which
consist of the dominant terms and are valid for small magnetic fields (${\cal
B}\ll{\cal B}_{cr}$) (see Herold et al. 1982; NTZ08b).

$\Gamma_{+}\approx 2\alpha B^{3}/3,$ $None$ $\Gamma_{-}\approx 4\alpha
B^{2}/3.$ $None$

For users who would like to have more accuracy, we provide below next order
corrections in $B$. These corrections have been derived from numerical fits to
the relativistic transition rates of Herold et al. (1982).

$\Gamma_{+}\approx\dfrac{2}{3}\alpha B^{3}(1-2.9B),$ $None$
$\Gamma_{-}\approx\dfrac{4}{3}\alpha B^{2}(1-2.7B).$ $None$

In Fig. 8 we display the relativistic decay rates of Herold (1982), along with
the non-relativistic ones and the expressions C2, which have a first order
correction in $B$.

Figure 8:  Comparison of the relativistic transition rates of Herold et al.
(1982) (blue solid lines) with the non-relativistic ones (eq. A15) (black
dashed lines) and with the expressions given by eq. (C2) (red dash-dotted
lines). Spin up refers to spin parallel to the magnetic field, whereas spin
down refers to anti-parallel spin. The $\Gamma$’s are divided by the cyclotron
frequency $\omega_{c}$ for presentation reasons.

Clearly the non-relativistic expressions given by eq. (C1) can be used in the
Lorentz profiles (defined by eq. A17) in the case where $B\ll 1$


DAC

    Dynamic Algorithm Configuration
RL

    Reinforcement Learning
MDP

    Markov Decision Process
PPO

    Proximal Policy Optimization

# Instance Selection for Dynamic Algorithm Configuration with Reinforcement
Learning: Improving Generalization

Carolin Benjamins  Leibniz University HannoverGermany ,  Gjorgjina Cenikj
Jožef Stefan International Postgraduate School
Jožef Stefan InstituteSlovenia ,  Ana Nikolikj  Jožef Stefan International
Postgraduate School
Jožef Stefan InstituteSlovenia ,  Aditya Mohan  Leibniz University
HannoverGermany ,  Tome Eftimov  Jožef Stefan InstituteSlovenia and  Marius
Lindauer  Leibniz University Hannover
L3S Research CenterGermany

(2024)

###### Abstract.

Dynamic Algorithm Configuration (DAC) addresses the challenge of dynamically
setting hyperparameters of an algorithm for a diverse set of instances rather
than focusing solely on individual tasks. Agents trained with Deep
Reinforcement Learning (RL) offer a pathway to solve such settings. However,
the limited generalization performance of these agents has significantly
hindered the application in DAC. Our hypothesis is that a potential bias in
the training instances limits generalization capabilities. We take a step
towards mitigating this by selecting a representative subset of training
instances to overcome overrepresentation and then retraining the agent on this
subset to improve its generalization performance. For constructing the meta-
features for the subset selection, we particularly account for the dynamic
nature of the RL agent by computing time series features on trajectories of
actions and rewards generated by the agent’s interaction with the environment.
Through empirical evaluations on the Sigmoid and CMA-ES benchmarks from the
standard benchmark library for DAC, called DACBench, we discuss the potentials
of our selection technique compared to training on the entire instance set.
Our results highlight the efficacy of instance selection in refining DAC
policies for diverse instance spaces.

dynamic algorithm configuration, reinforcement learning, instance selection,
generalization

††journalyear: 2024††copyright: rightsretained††conference: Genetic and
Evolutionary Computation Conference; July 14–18, 2024; Melbourne, VIC,
Australia††booktitle: Genetic and Evolutionary Computation Conference (GECCO
’24 Companion), July 14–18, 2024, Melbourne, VIC, Australia††doi:
10.1145/3638530.3654291††isbn: 979-8-4007-0495-6/24/07††ccs: Computing
methodologies Reinforcement learning††ccs: Theory of computation Design and
analysis of algorithms

##  1\. Introduction

Dynamic Algorithm Configuration (DAC) offers an automated solution to the task
of setting algorithm hyperparameters dynamically, by determining well-
performing hyperparameter schedules or policies. One way to learn such
policies is through Reinforcement Learning (RL) (Biedenkapp et al., 2020;
Adriaensen et al., 2022). While conceptually appealing, RL algorithms have the
notorious tendency to significantly overfit their training environments (Zhang
et al., 2018; Justesen et al., 2022; Kirk et al., 2023). As a consequence, RL
methods for DAC suffer from a lack of generalization to instances not seen
during training, thereby limiting their applicability.

We take a step towards improving the generalization performance of RL policies
on new test instances by subselecting representative training instances using
SELECTOR (Cenikj et al., 2022). To capture the dynamic nature of RL, we use
trajectory-based representations generated by the RL algorithm after training
on the full instance set.

Figure 1. Our proposed flow of subselecting representative instances with
SELECTOR for DAC with RL

Concretely, we make the following contributions: i) For DAC with RL, we
present a principled framework to select representative instances to train on
to improve generalization to the instance space; ii) we propose a new domain-
agnostic approach for generating instance meta-features that encode the
dynamics of the DAC problem; iii) we demonstrate superior performance training
on the subselected instance set; iv) we analyze the selected instances; and v)
we provide an insight on how to use the framework SELECTOR.

Reproducibility: Code and data is available here:
https://github.com/automl/instance-dac.

##  2\. Method

The goal of our study is to improve the generalization of an RL agent in
Dynamic Algorithm Configuration (DAC), as measured by the performance of a
policy on a test set of target instances. Figure 1 shows the outline of our
method. Overall, we use SELECTOR to sample a subset of representative training
instances, to which we then allocate more training resources. We fix the total
number of times the RL agent interacts with the environment before the
subselection and after the subselection to be the same. This means for the
same training budget, we train on fewer but more representative instances
after subselection.

To enable this workflow, we start by training an RL agent on the train
instance set $\mathcal{I}_{train}$. A key element is using meta-features based
on the data from the trajectory generated by the RL agent as it interacts with
the algorithm. We do this by evaluating the trained agent on the train
instance set and producing rollout trajectories, specifically the actions
taken by the agent and the reward received for each action. This data encodes
the agent’s behavior for each training instance. We feed these meta-feature
data for all training instances to SELECTOR, which subselects instances from
the train instance set to form the reduced, subselected instance set
$\mathcal{I}_{selected}\subseteq\mathcal{I}_{train}$. Intuitively, these
instances capture the essential aspects of the dynamics observed by the agent
during training and should, therefore, enable better generalization. We
finally train the RL agent again on the subselected instance set to obtain the
final policy, which we can subsequently evaluate on the held-out test set of
instances.

#### Meta-Feature Representations

SELECTOR requires the representation of data instances (in our case, episodes
from training the RL agent) to be numerical features. In prior work, instance
meta-features were obtained via a manual approach (Bischl et al., 2016),
possibly not always reflecting the agent’s interaction with the environment.
Our approach, however, uses features from the data generated by the agent as
it interacts, thus capturing the agent’s dynamic behavior. We explore the
following representations:

Raw Representations are the raw actions and rewards observed during training.
These representations are constructed by simply concatenating the sequence of
actions taken by the agent and the corresponding rewards obtained in each
iteration.

Catch22 Representation are time-series features extracted from the raw actions
and rewards observed during training. These features capture a broad spectrum
of time-series characteristics, including the distribution of values in the
time series, linear and nonlinear temporal autocorrelation properties, scaling
of fluctuations, and other relevant properties. Another advantage to using
time-series features is the ability to characterize and compare variable-
length episodes. We use the catch22 (Lubba et al., 2019) library to extract 22
time-series features from the observed sequences of actions and rewards
together with mean and standard deviation, resulting in 24 features. Note that
we could use any other time-series features.

Both representations (raw and catch22) can also be combined with instance
features describing the problem instance and are not directly related to the
behavior of the RL agent. An example of such features can be the slope and
shift of a sigmoid problem instance.

#### SELECTOR

We execute the SELECTOR methodology using the different aforementioned
representations to represent the instances from the training set. We use the
Dominating Sets (DS; (Esfahanian, 2013)) and Maximal Independent Set (MIS;
(Byskov, 2003)) algorithms with different similarity thresholds, specifically,
$0.7,0.8,0.9$, and $0.95$.

##  3\. Experiments

For evaluating our method, we rely on the benchmark library DACBench (Eimer et
al., 2021), which features DAC benchmarks from different AI domains. We first
cover the evaluation protocol, then the DAC benchmarks used, Sigmoid and CMA-
ES, and finally, detail the training of the RL agent.

#### Evaluation Protocol

Our overall objective is to assess the generalization performance on the test
instance set $\mathcal{I}_{test}$. Therefore, we evaluate the agent trained on
the full, original train instance set $\mathcal{I}_{train}$ and the agent
trained on the subselected set
$\mathcal{I}_{selected}\subseteq\mathcal{I}_{train}$ once again on the test
instance set $\mathcal{I}_{test}$. For an empirical upper limit to performance
on the test instance set, we additionally train _Instance-Specific Agents
(ISAs)_. Each ISA is an RL agent trained on one instance of the test instance
set and evaluated on that specific instance, serving as a reference. This
construction of ISA exploits the notorious property of the RL agents to
overfit their training instance: _Each ISA demonstrates the possible reward
that an RL agent can accumulate when trained solely on this instance_. In
other words, they serve as an empirical performance upper bound that should be
hard to achieve for a DAC agent being trained across a variety of training
instances. In addition, we also compare to RL agents trained on $5$ random
subsets of $10\%$ of the train instance set $\mathcal{I}_{train}$, which is a
similar fraction of instances selected by SELECTOR.

We perform experiments on the Sigmoid benchmark, where a Sigmoid curve with
varying slope and shift should be approximated, and on CMA-ES, where the step-
size $\sigma$ is adapted. In the following paragraphs, we further explain
these benchmarks.

#### Sigmoid

This benchmark challenges DAC agents to approximate a Sigmoid function in
different dimensions. It is an artificial white-box benchmark that was
proposed to study DAC with full control over the application (Biedenkapp et
al., 2020). A Sigmoid function is characterized by its shift and slope and has
function values between $0$ and $1$. Actions are discrete and evenly space the
interval $[0,1]$. For example, for an action space of $5$ actions, the actions
would be $a\in\\{0,0.25.,0.5,0.75,1\\}$. We approximate Sigmoids in two
dimensions, with $5$ and $10$ actions, respectively. The state features
consists of the remaining budget, the shift and slope for each dimension, and
the action for each dimension. The difficulty of the problem can be increased
by increasing the dimensionality. The training and test instance sets comprise
$300$ instances of two-dimensional Sigmoids.

#### CMA-ES

CMA-ES (Covariance Matrix Adaption Evolution Strategy) (Hansen, 2006) is an
evolutionary algorithm for continuous black-box problems which can be non-
linear and non-convex. In DACBench (Eimer et al., 2021), the step-size
$\sigma\in[0,10]$ of CMA-ES can be adapted, which is a continuous action
space. Others adapt the step-size via a heuristic (Igel et al., 2007; Hansen,
2008) or guided policy search (Shala et al., 2020). As a state, the RL agent
receives the generation size, the current step-size $\sigma$, the remaining
optimization budget, as well as the function and instance ID. The reward is
the negative minimum function value observed so far since CMA-ES is a
minimizer and the RL agent is a maximization algorithm. The train and test
instance set comprises ten synthetic blackbox optimization benchmarking (BBOB)
functions (Hansen et al., 2020) – Sphere, Ellipsoidal, Rastrigin, Büche-
Rastrigin, Linear Slope, Attractive Sector, Step Ellipsoidal, original and
rotated Rosenbrock and Ellipsoidal. All of these functions are either
separable or have low or moderate conditioning, except for the last one with
high conditioning, $\in\mathbb{R}^{10}$. The train set features four instances
of each function, and the test set one instance.

#### Training Details

We repeat our training and evaluation pipeline for $10$ random seeds. Our
training details are as follows: We train a PPO (Schulman et al., 2017) agent
for $10\,000$ environment steps in Sigmoid, equaling $1\,000$ episodes, with
each episode having a length of $10$. For CMA-ES, we train the agent for
$1\,000\,000$ steps. However, here, we have variable episode lengths. We
evaluate each trained agent with $10$ evaluation episodes per instance. Based
on the evaluation rollout data, we run SELECTOR $5$ times and normalize the
agent’s performance per instance. We then compute bootstrapped mean, median,
and IQM with $5\,000$ samples using the library rliable (Agarwal et al., 2021)
for the evaluation performance. We additionally use fANOVA (Hutter et al.,
2014) with standard settings to analyze the sensitivity of SELECTOR to its own
hyperparameters, namely feature types, the method of selection, the source of
features, and the threshold.

#### Instance representation and selection

The chosen benchmark suites encompass training RL in distinct environments:
one involving discrete actions (Sigmoid) and the other involving continuous
actions (CMA-ES). We employ different representations to depict the behavior
of the RL agent. Based on the actions (A) and rewards (R) recorded on
evaluation rollouts, we either use the raw (flattened) vectors for fixed-
length episodes or the catch22 time-series features for variable-length
episodes. We can also concatenate action and reward vectors (RA) and add
instance features (I) if applicable.

In both benchmark suites and their respective instance representations, we
employ the SELECTOR method (both MIS and DS with similarity thresholds
$\in\\{0.7,0.8,0.9,0.95\\}$ for creating the graph) to choose subsets of
instances for retraining the RL agent.

###  3.1. Results and Discussion

Figure 2. Sigmoid Performance Figure 3. CMA-ES Performance

On both benchmarks, Sigmoid and CMA-ES, training on subselected instances from
SELECTOR generalizes better to the test instance set than training on the full
instance set, see Figure 2 for Sigmoid and Figure 3 for CMA-ES. First of all,
this supports our hypothesis that training an DAC agent with RL on a well-
constructed subset of instances can be better than simply training on an
arbitrary instance set. Secondly, the extraction of trajectory information is
sufficiently informative to construct this set.

Interestingly, the performance of the ISA for CMA-ES is worse than the
performance of SELECTOR. Initially, our aim was to construct ISA so that we
get an empirical approximation of a theoretical upper limit; thus our DAC
agent on SELECTOR should not be able to outperform ISA. We hypothesize that
the diversity in trajectories from multiple instances instead of only one
instance allows the optimization process of the RL agent to escape potential
local minima in policy space that the ISA agents get stuck in. This
corroborates the successful methodology of learning the step size with guided
policy search (Shala et al., 2020), where they guide the optimization and
start from a suitable point in the policy space.

Depending on the benchmark we observe different best performing variants of
SELECTOR. According to the IQM, SELECTOR with (MIS, Catch22, R, $0.7$) for
Sigmoid and SELECTOR with (DS, Catch22, R or RA, $0.8$) for CMA-ES performed
best. So, it is important to study the hyperparameter (HP) sensitivity of our
approach. For Sigmoid the type of representation is important, using only
actions or combinations with reward yields best results. The other HPs do not
have a major impact on Sigmoid. For CMA-ES, the subselection method on the
similarity graph (DS or MIS) is the most important HP. Again, representations
using actions and rewards together works best. A reasonable robust and general
choice would be to use rewards and actions as features sources combined with
DS.

The size of the instance set shows strong variation for the threshold of
SELECTOR for Sigmoid, but not so much for CMA-ES, as shown in Figure 4.
Peaking closer into Sigmoid, Figure 4 (right) indicates that instance features
and trajectory features are not very correlated. A small instance set with a
high threshold induces a dense graph, i.e. instances are pretty similar in
terms of instance features which does not necessarily mean trajectory features
are similar.

In addition, the instances selected by SELECTOR evenly cover the full instance
set, capturing the diversity that is most apparent for the second dimension
(Figure 5). For CMA-ES, often only one instance of the BBOB functions 7, 8, 9
is selected. These functions have a more complex local structure compared to
the first functions but still are similar in global structure, rendering them
suitable to represent the instance set.

Figure 4. (Left) Size of subselected instance set for different SELECTOR
thresholds per benchmark. (Right) Size of subselected instance sets for
Sigmoid for different representations.  Figure 5. Selected instances by
SELECTOR for Sigmoid. A small but diverse set of instances is selected.

#### Limitations and Future Work

One limitation of our method is that it requires training the RL agent twice
as well as training SELECTOR. We plan to investigate the benefits that can be
potentially gained from _early-stopping_ , such as only training the agent for
half of the training budget. Potentially, the benefits of SELECTOR could be
attained in the same training budget as a standard baseline agent. We
additionally plan to meta-learn well-performing presets for SELECTOR to create
a truly end-to-end training and selection pipeline. Lastly, we would like to
approach handling instances also at the level of the RL algorithm: For
benchmarks like CMA-ES, we have problems with different reward scales,
potentially hindering learning, which we could normalize per instance.

##  4\. Conclusion

In this work, we demonstrate the potential of instance selection in enhancing
the generalization capabilities of Reinforcement Learning (RL) for Dynamic
Algorithm Configuration (DAC). We first train an RL agent on a train set of
instances and then generate rollout trajectories by evaluating the trained
agent on the same set of instances. Since these trajectories capture the
agent’s behavior on the training instances, we use this data to create time-
series features that capture the _dynamic_ behavior of the RL policy. We then
subselect a representative set of training instances and retrain the RL agent
on these instances to obtain better generalization performance on unseen new
instances. By meticulously selecting representative instances for training, we
not only address the challenge of overrepresentation in training instances but
also demonstrate superior performance to agents trained on specific instances
on CMA-ES. Our approach marks a step forward in the application of RL to DAC,
offering a scalable solution that can adapt to the ever-changing complexities
of hyperparameter control using RL.

##  5\. Acknowledgements

Funding in direct support of this work: Slovenian Research Agency: research
core funding No. P2-0098, young researcher grants No. PR-12393 to GC and No.
PR-12897 to AN, project No. J2-4460, and a bilateral project between Slovenia
and Germany grant No. BI-DE/23-24-003. DAAD: 57654659.

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  *[DAC]: Dynamic Algorithm Configuration
  *[RL]: Reinforcement Learning