# Note on Klein-Nishina effect in strong-field QED:
the case of nonlinear Compton scattering

U. Hernandez Acosta1,2, B. Kämpfer1,3 1Helmholtz-Zentrum Dresden-Rossendorf,
01328 Dresden, Germany  2Center for Advanced Systems Understanding (CASUS),
Helmholtz-Zentrum Dresden-Rossendorf, Untermarkt 20, 02826 Görlitz, Germany
3Institut für Theoretische Physik, TU Dresden, 01062 Dresden, Germany

###### Abstract

Suitably normalized differential probabilities of one-photon emission in
external electromagnetic fields are compared to quantify the transit of
nonlinear Compton scattering to linear Compton scattering, described by the
Klein-Nishina formula, and to constant crossed field treatment. The known
Klein-Nishina suppression at large energies is further enforced by increasing
field intensity. In view of the Ritus-Narozhny conjecture, we demonstrate that
different paths in the field intensity vs. energy plane towards large values
of the quantum non-linearity parameter $\chi$ facilitate significantly
different asymptotic dependencies, both in the Klein-Nishina regime and the
constant crossed field regime and in between.

strong-field QED: nonlinear Compton scattering, linear Compton scattering,
Klein-Nishina effect, constant crossed field, Ritus-Narozhny conjecture

##  I Introduction

The Klein-Nishina effect means a dropping cross section of the linear Compton
scattering Klein:1929lcc  from Thomson’s value at small energies towards high
energies. In astrophysics, this dropping reduces the energy loss of ultra-
relativistic electrons and positrons Fang:2020dmi  and the inverse Compton
scattering rates with ultra-high energy photons, most notable for energies in
the order of PeV LHAASO:2023gne ; LHAASO:2021gok . Additionally, in certain
sites, ultra-intense electromagnetic background fields are present and enable
or influence elementary charged-particle–photon interactions. Pulsars in
general and magnetars in particular represent such sites. In earth-bound
laboratories similarly extreme conditions can be achieved, hence linking to or
exploring analog astrophysics scenarios. Beam-beam interactions in
accelerators Yakimenko:2018kih  or beam-laser interactions Abramowicz:2021zja
; Salgado:2021fgt ; Storey:2023wlu  or high-intensity laser-electron (plasma
Brodin:2022dkd ; Seipt:2023bcw ; Zhang:2020lxl ) experiments Poder:2017dpw ;
Cole:2017zca  or laser-laser interactions Blinne:2018nbd ; Gies:2017ezf ;
Gies:2017ygp  are to be mentioned in that context.

A special branch of phenomenology is tied to strong-field Quantum Electro
Dynamics (sfQED) (see Fedotov:2022ely  for a recent survey with an extensive
citation list and HernandezAcosta:2023msl  for a concise formulation in
momentum space) aiming at describing the fundamental electromagnetic
interactions in or at a background field. The latter one could be of
electromagnetic (or electro-weak) nature or being gravity or something -
hypothetically - beyond the Standard Model of Particle Physics. We confine
ourselves to electromagnetic backgrounds, i.e. sfQED.111 Many entries on the
topic are cited in Fedotov:2022ely . Further surveys are provided by
Gonoskov:2021hwf ; Titov:2015pre ; DiPiazza:2011tq ; Kaminski:2009wwd .

A straightforward question is then whether a noticeable background field has a
significant impact on elementary processes. We narrow here the problem to the
impact of an electromagnetic field on the photon emission off electrons and
consider specifically Compton scattering.222 A few entries for Compton as one-
photon emission are Ritus85 ; Lotstedt:2009zz ; Mackenroth:2010jr ;
Hartin:2011vr ; Heinzl:2009nd ; Seipt:2010ya ; Seipt:2011zz ; Seipt:2011dx ;
Seipt:2013taa ; Seipt:2016rtk ; Titov:2015tdz ; Dinu:2018efz ;
Ilderton:2018nws ; Ilderton:2020dhs ; Nielsen:2021nuo ; Gelfer:2022nqy ;
Kampfer:2020cbx ; Acosta:2021iyu ; Titov:2023yev ; Angioi:2016vir ;
Blackburn:2018sfn ; for two-photon Compton (two-vertex diagram) see, e.g.,
Herrmann:1971jc ; Guccione-Gush:1975juz ; Akhiezer:1985 ; Ehlotzky:1989 ;
Mackenroth:2012rb ; Seipt:2012tn ; Dinu:2018efz ; Seipt:2013hda , and higher
orders in Lotstedt:2012zz ; Lotstedt:2013uya ; Dinu:2019pau .

One aspect concerns the Klein-Nishina effect in a strong background field.
Another issue is related to the Ritus-Narozhny conjecture, arguing that the
account of the background field within the Furry picture is restricted to not
too intense fields Ritus:1972ky ; Fedotov:2016afw ; Heinzl:2021mji ;
Mironov:2020gbi . In doing so we resort to fairly known formalism as a vehicle
to elucidate, numerically, links between established descriptions of linear
Compton scattering, nonlinear Compton scattering, and photon emission in a
constant crossed field. This subject is also of interest for approximations
implemented in simulation codes - both for astrophysics applications and laser
physics in particular (cf. DiPiazza:2018bfu ; DiPiazza:2017raw ;
Gelfer:2022nqy ; Nielsen:2021nuo ; Blackburn:2021cuq ; Dinu:2019pau ;
Ilderton:2018nws ; Heinzl:2020ynb ).

Our note is organized as follows. In section II, we recall formulas describing
the normalized probabilities of linear and nonlinear Compton scattering as
well as photon emission in a constant crossed field. Subsequently, the
dependence on a Lorentz invariant quantity characterizing the emitted photon
(Section III), the energy dependence (Section IV) and external field intensity
dependence (Section V) are analyzed. Section VI demonstrates how various paths
approach a large value of the so-called quantum non-linearity parameter $\
chi$ leaves a strong impact on the asymptotic. After a brief discussion in
Section VII, we conclude in Section VIII. A few relevant numbers are listed in
Appendix A, and Appendix B recaps relations of plane wave and constant crossed
field quantities.

##  II Exclusive one-photon emission: recollection of linear Compton vs.
nonlinear Compton vs. constant crossed field

The Klein-Nishina (KN) effect means a dropping cross section of the linear
Compton scattering Klein:1929lcc  from Thomson’s value
$\sigma^{\mathrm{Thomson}}=\frac{8\pi}{3}r_{e}^{2}$ at small energies, $p\cdot
k/m^{2}=(s-m^{2})/m^{2}\to 0$, towards high energies:333 We employ units with
$\hbar=c=1$ and note the electron radius of $r_{e}=\alpha/m$ with Sommerfeld’s
fine-structure constant $\alpha\approx 1/137$ (ignoring for the sake of
clarity the running coupling) and electron mass $m$; the electron’s and
photon’s $in$ four-momenta are $p$ and $k$, respectively (for $out$-states,
the four-momenta receive primes). The Mandelstam variable $s=(p+k)^{2}$ refers
to the notion of “energy”.

$\displaystyle\sigma^{\mathrm{KN}}$ $\displaystyle=\frac{2\pi
r_{e}^{2}}{u_{KN}}\left[\left(1-\frac{4}{u_{\mathrm{KN}}}-\frac{8}{u_{\mathrm{KN}}^{2}}\right)\log(1+u_{\mathrm{KN}})+\frac{1}{2}+\frac{8}{u_{\mathrm{KN}}}-\frac{1}{2(1+u_{\mathrm{KN}})^{2}}\right]$
(1) $\displaystyle=\left\\{\begin{array}[]{ll}\frac{8\pi
r_{e}^{2}}{3}(1-u_{\mathrm{KN}})&\mathrm{for}\,u_{\mathrm{KN}}\ll 1\\\ \pi
r_{e}^{2}\frac{1}{u_{\mathrm{KN}}}\left(2\log
u_{\mathrm{KN}}+1\right)&\mathrm{for}\,u_{\mathrm{KN}}\gg
1\end{array}\right.,$ (4)

where $u_{\mathrm{KN}}:=2p\cdot k/m^{2}$, see empty squares in Fig. 1-left. At
the origin of such behavior is the differential cross section LLIV

$\displaystyle\frac{d\sigma^{\mathrm{KN}}}{du}$
$\displaystyle=\frac{2\pi\alpha^{2}}{p\cdot
k}\frac{1}{(1+u)^{2}}\left(1+\underline{\frac{u^{2}}{2(1+u)}}-2\frac{u}{u_{KN}}\left[1-\frac{u}{u_{\mathrm{KN}}}\right]\right)$
(5) $\displaystyle=\frac{2\pi\alpha^{2}}{p\cdot
k}\times\left\\{\begin{array}[]{ll}1\,-2u(1+u_{\mathrm{KN}}^{-1})+u^{2}\left(\frac{7}{2}\frac{4}{u_{\mathrm{KN}}}+\frac{2}{u_{\mathrm{KN}}^{2}}\right)+\mathcal{O}(u^{3})&\mathrm{for}\,u\to
0\\\
\frac{2+2u_{\mathrm{KN}}+u_{\mathrm{KN}}^{2}}{2(1+u_{\mathrm{KN}})^{3}}\stackrel{{\scriptstyle
u_{\mathrm{KN}}\gg 1}}{{\longrightarrow}}\frac{1}{4}\frac{m^{2}}{p\cdot
k}&\mathrm{for}\,u=u_{\mathrm{KN}}\\\ \end{array}\right.,$ (8)

where $u\in[0,u_{KN}]$ denotes the Ritus invariant $u=\frac{k\cdot
k^{\prime}}{k\cdot p^{\prime}}$ Ritus85  which is related, via energy-momentum
balance $p+k=p^{\prime}+k^{\prime}$, to the lightfront momentum fraction of
the $out$-photon $\mathfrak{s}=\frac{k\cdot k^{\prime}}{k\cdot p}$ via
$\mathfrak{s}=\frac{u}{1+u}$. The underlined term in Eq. (5) gives rise to the
logarithmic high-energy behavior $\sigma^{\mathrm{KN}}\propto(p\cdot
k/m^{2})^{-1}\log(p\cdot k/m^{2})$. With increasing energy, the differential
cross section drops at small values of $u$ as $\propto(p\cdot k/m^{2})^{-1}$
which turns, for $u>1$, into $\propto(p\cdot k/m^{2})^{-2}$, see Fig. 1-right.
Quite qualitatively, one can argue that, with increasing energy, more and more
exclusive and inclusive channels open with the consequence that one particular
exclusive channel receives less probability of keeping the unitarity of the S
matrix. In the case of linear Compton scattering, often the electron recoils
are said to reduce the $out$ photon energy.

Figure 1: Left panel: Total KN cross section Eq. (1) (squares) and the IPA
cross section
$\sigma^{\mathrm{IPA}}=\int_{0}^{\infty}du\,\frac{d\sigma^{\mathrm{IPA}}}{du}$
for various values of the field intensity parameter $a_{0}=1$, 10, $10^{2}$
and $10^{3}$ (solid curves, from top to bottom). Right panel: Differential KN
cross section Eq. (5) as a function of $u$ for various values of the energy.
The dashed curve connects the loci of kinematic limits $u=2p\cdot k/m^{2}$. In
view of applications in high-intensity laser physics and high-energy
astrophysics (cf. Appendix A), we have displayed the KN spectra for a fairly
large energy range.

The above (differential) cross section is for the leading-order tree-level
term of exclusive $2\to 2$ scattering
$e^{-}+\gamma\to{e^{-}}^{\prime}+\gamma^{\prime}$ within perturbative QED,
organized in powers of $\alpha$ (for higher orders, cf. Lee:2021iid ). It is
known LLIV ; Ritus85  to emerge from nonlinear Compton scattering,
$e_{A}^{-}\to{e_{A}^{-}}^{\prime}+\gamma^{\prime}$, as leading-order weak-
field term, where $e_{A}^{-}$ is for the electron wave function in an
external, classical background field $A$ which may refer to a monochromatic
laser field in a plane-wave model as the so-called infinite plane-wave
approximation (IPA). Having this relation in mind, one can define a suitably
normalized probability (henceforth dubbed ”cross section”) which makes this
relation explicit:444 See Bragin:2020akq  for the definition of “local cross
sections” for processes in an external field. Instead of using wave packages,
we employ a circularly polarized, monochromatic plane-wave field as a proxy of
a mono-energetic photon beam under special circumstances. The intensity
parameter is $a_{0}$.

$\displaystyle\frac{d\sigma^{\mathrm{IPA}}}{du}$
$\displaystyle=\frac{2\pi\alpha^{2}}{p\cdot
k}\frac{1}{(1+u)^{2}}\sum_{n=1}^{N\to\infty}\Theta(u_{n}-u)\mathcal{F}_{n}(z_{n}),$
(9)
$\displaystyle\mathcal{F}_{n}=-\frac{2}{a_{0}^{2}}J_{n}^{2}(z_{n})+A\left[J_{n+1}^{2}(z_{n})+J_{n-1}^{2}(z_{n})-2J_{n}^{2}(z_{n})\right]$
(10) $\displaystyle
z_{n}=\frac{2na_{0}}{\sqrt{1+a_{0}^{2}}}\sqrt{\frac{u}{u_{n}}\left(1-\frac{u}{u_{n}}\right)}$
(11) $\displaystyle u_{n}=\frac{2n}{1+a_{0}^{2}}\frac{p\cdot k}{m^{2}}\quad\to
n\geq n_{min}\equiv\frac{1}{2}u(1+a_{0}^{2})\frac{m^{2}}{p\cdot k}$ (12)
$\displaystyle A=1+\frac{u^{2}}{2(1+u)},$ (13)

see Acosta:2021iyu ; Kampfer:2020cbx  and Ritus85  for the basics. The
background field is characterized by the classical field strength parameter
$a_{0}$, and the meaning of the variable $u$ is as above. The differential
cross section depends separately on two variables: the energy variable $p\cdot
k/m^{2}$ and intensity parameter $a_{0}$, which can be combined with the so-
called “quantum non-linearity” parameter $\chi=a_{0}\,p\cdot k/m^{2}$ and a
second one. The total cross section $\sigma^{\mathrm{IPA}}$ is compared with
$\sigma^{\mathrm{KN}}$ in Fig. 1-left. In addition to the KN effect, the
external field causes a further suppression, increasing with $a_{0}$. In line
with the above-quoted argument, one can expect that, according to unitarity.

Sometimes, it is claimed in the literature that large values of $\chi$ would
refer to high intensities, and the differential probability behind Eq. (9)
turns in the expression for one-photon emission in a constant crossed field
(CCF). Transferred to “cross section”, linked in an ad hoc manner to Eq. (9),
with some seemingly abuse of relations of $\chi$ and $a_{0}$ and energy (see
however King:2014wfa  for a rationale and discussion in Appendix B), one has

$\displaystyle\frac{d\sigma^{\mathrm{CCF}}}{du}$
$\displaystyle=-\frac{4\pi\alpha^{2}}{m^{2}a_{0}}\frac{1}{\chi}\frac{1}{(1+u)^{2}}\mathcal{F}_{\mathrm{CCF}}(\hat{z}),$
(14)
$\displaystyle\mathcal{F}_{\mathrm{CCF}}(\hat{z})=\frac{2}{\hat{z}}\left[1+\frac{u^{2}}{2(1+u)}\right]\Phi^{\prime}(\hat{z})+\int_{\hat{z}}^{\infty}dy\,\Phi(y),\quad\Phi=\pi
Ai$ (15)
$\displaystyle\hat{z}=\left(\frac{u}{\chi}\right)^{2/3},\quad\chi=a_{0}\frac{p\cdot
k}{m^{2}}\mathrm{~{}is~{}independent~{}of~{}}\omega,$ (16)

which in turn obeys the asymptotic Ritus85

$\displaystyle\frac{d\sigma_{\mathrm{large}-u}^{\mathrm{CCF}}}{du}=\frac{2\sqrt{\pi}\alpha^{2}}{m^{2}a_{0}}\chi^{-1/2}u^{-3/2}\exp\left\\{-\frac{2u}{3\chi}\right\\},\quad\mathrm{for~{}}u\gg
1,\chi.$ (17)

Such a discussion is important w.r.t. the Ritus-Narozhny conjecture
Ritus:1972ky ; Narozhnyi:1980dc  (cf. Fedotov:2022ely  for an up-to-date
discussion with citations therein) which argues that the underlying Furry
picture555 Meaning here that the external field is exactly accounted for in
the quasi-classical electron wave function. For one-photon emission, the
Furry-picture Feynman diagram reads  where the double line is for the wave
function $\Psi$ solving the Dirac equation
$(i\not{\partial}_{x}-e\not{A}-m)\Psi(x)=0$ for an electron (charge $e$) in
the external field $A(x)$.  breaks down at $\alpha\chi^{2/3}\geq 1$ and loop
corrections become severe. In fact, Refs. Ilderton:2019kqp ; Podszus:2018hnz
point out that the large-$\chi$ regime may be achieved by large energy or high
intensity which both have very distinctive asymptotic, i.e. $\chi$ alone is
clearly not sufficient to characterize the process. Besides the principal
importance of such considerations, they are also tightly related to the design
of approximation schemes implemented in simulation codes.

Given that specific framework, we quantify the relations of the above-quoted
cross sections on a differential level and point out the detailed conditions
for $d\sigma^{\mathrm{IPA}}/du\to d\sigma^{\mathrm{CCF}}/du$ and
$d\sigma^{\mathrm{IPA}}/du\to d\sigma^{\mathrm{KN}}/du$. We also show that
approaching large values of $\chi$ along different paths in the
$a_{0}$-$p\cdot k/m^{2}$ plane facilitates different asymptotic.

##  III The $\mathbf{u}$ dependence

Figure 2 exhibits the differential cross section $d\sigma^{\mathrm{IPA}}/du$
as a function of $u$ for various values of $a_{0}=10^{0}\cdots 10^{3}$.
Squares display the KN cross section, while the CCF cross section is marked by
dots. The left panel is for $p\cdot k/m^{2}=10^{-1}$, while the right panel
has $p\cdot k/m^{2}=10^{1}$. With increasing values of $a_{0}$ the “spectra”
are shifted to the left, as also do the harmonic structures. The behavior in
the IR region, i.e. at small $u$, is determined by

$\displaystyle\frac{d\sigma^{\mathrm{IPA}}}{du}\bigg{|}_{u\to
0}\approx\frac{2\pi\alpha^{2}}{p\cdot
k}\left\\{1-u\left[2+\frac{1+2a_{0}^{2}}{p\cdot
k/m^{2}}\right]\right\\}\approx\frac{d\sigma^{\mathrm{KN}}}{du}\bigg{|}_{\mathrm{small~{}}u}-\frac{2\pi\alpha^{2}}{p\cdot
k}\,u\,\frac{2a_{0}^{2}}{p\cdot k/m^{2}},$ (18)

and obeys the $a_{0}$ independent limit $\lim_{u\to
0}\frac{d\sigma^{\mathrm{IPA}}}{du}=\frac{d\sigma^{\mathrm{KN}}}{du}=\frac{2\pi\alpha^{2}}{p\cdot
k}$. The last term in Eq. (18) quantifies the $a_{0}$ dependence. That is,
upon integration, the total IPA cross section contains inevitably a KN-like
contribution. To quantify that we exhibit by the thin dashed curve the
demarcation where 10% (on left/above) vs. 90% (on right/below) of the total
IPA cross section is located which clearly stems from the large-$u$ sections,
despite the small differential cross section.666 In doing so we inspect the
partially integrated cross section
$\sigma(U):=\int_{0}^{U}du\,\frac{d\sigma^{\mathrm{IPA}}}{du}$, normalized to
the total cross section, which resembles a $\tanh$ function.  At sufficiently
large values of $u$, the IPA differential cross section is identical to the
CCF expression (marked by dotted curves), remarkably also for
$a_{0}=\mathcal{O}(1)$. The overall patterns are the same, irrespective of
whether the energy is small (as in the left panel, where $\chi=10^{-1}\cdots
10^{2}$ at $p\cdot k/m^{2}=10^{-1}$) or large (as in right panel, where
$\chi=10^{1}\cdots 10^{4}$ at $p\cdot k/m^{2}=10^{1}$ ).

Figure 2: Differential cross section as a function of $u$ for $p\cdot
k/m^{2}=0.1$ (left panel), 1 (middle panel) and 10 (right panel). IPA, for
various values of $a_{0}=1,10^{1},10^{2},10^{3}$ and $10^{4}$ (from top to
bottom), is exhibited by solid curves. The squares are for KN. Circles depict
CCF beyond the harmonic structures. At smaller values of $u$, CCF fails badly
by a strong overshot, even diverging for $u\to 0$. Right/below the thin dashed
curves, 90% of the total cross sections is located.

As an interim summary, we mention:
$d\sigma^{\mathrm{IPA}}/du\stackrel{{\scriptstyle\mathrm{small}~{}u}}{{\xrightarrow{\hskip
25.6073pt}}}d\sigma^{\mathrm{KN}}/du$,
$d\sigma^{\mathrm{IPA}}/du\stackrel{{\scriptstyle\mathrm{large}~{}u}}{{\xrightarrow{\hskip
25.6073pt}}}d\sigma^{\mathrm{CCF}}/du$ when keeping fixed $a_{0}$ and $p\cdot
k/m^{2}$; the notions “small”/“large” depend separately on $a_{0}$ and $p\cdot
k/m^{2}$. The harmonic structure at $u=u_{1}$ may serve as demarcation via

$\displaystyle\frac{d\sigma^{\mathrm{IPA}}}{du}\bigg{|}_{u=u_{1}}=\frac{2\pi\alpha^{2}}{p\cdot
k}\frac{1}{(1+u_{1})^{2}}\left(1+\frac{u_{1}^{2}}{2(1+u_{1})}\right)\approx\frac{d\sigma^{\mathrm{KN}}}{du}\bigg{|}_{u=0}\times\left\\{\begin{array}[]{ll}1&\mathrm{for~{}}u_{1}\ll
1,\\\ \frac{1+a_{0}^{2}}{4}\frac{m^{2}}{p\cdot k}&\mathrm{for~{}}u_{1}\gg
1,\\\ \end{array}\right.$ (21)

where $u_{1}\stackrel{{\scriptstyle\gg}}{{\ll}}1$ corresponds to $\frac{p\cdot
k}{m^{2}}\stackrel{{\scriptstyle\ll}}{{\gg}}1+a_{0}^{2}$.

##  IV The energy dependence

Turning now to the energy dependence we exhibit in Fig. 3 the differential IPA
cross section for $u=0.1$ (left panel), $u=1$ (middle panel) and $u=10$ (right
panel). For a given value of $a_{0}$, the IPA cross section matches the CCF
expression at small energies, while the high-energy tails coincide with KN.
The demarcation is again provided by the first harmonic structures depending
on $a_{0}$ and $p\cdot k/m^{2}$ at a given value of $u$ via
$u=u_{1}=\frac{2}{1+a_{0}^{2}}p\cdot k/m^{2}$. As mentioned above, besides the
total cross section, the differential KN cross section also drops with
increasing energy $p\cdot k/m^{2}$. On top of that dropping is the suppression
of the differential IPA and CCF cross sections at small energies.

Figure 3: Differential IPA cross section (solid curves) for $u=0.1$ (left
panel), 1 (middle panel) and 10 (right panel) as a function of energy for
various values of $a_{0}=1,10^{1},10^{2},10^{3}$ and $10^{4}$ (from top to
bottom). The squares are for KN, and dots depict CCF beyond harmonic
structures.

As an interim summary, we mention:
$d\sigma^{\mathrm{IPA}}/du\stackrel{{\scriptstyle\mathrm{small}~{}p\cdot
k/m^{2}}}{{\xrightarrow{\hskip 42.67912pt}}}d\sigma^{\mathrm{CCF}}/du$,
$d\sigma^{\mathrm{IPA}}/du\stackrel{{\scriptstyle\mathrm{large}~{}p\cdot
k/m^{2}}}{{\xrightarrow{\hskip 42.67912pt}}}d\sigma^{\mathrm{KN}}/du$ offer a
different perspective when keeping fixed $u$ and $a_{0}$. Here, “small
energies” mean $p\cdot k/m^{2}<\frac{1}{2}u(1+a_{0}^{2})$, while “large
energies” mean $p\cdot k/m^{2}>\frac{1}{2}u(1+a_{0}^{2})$. Since this is the
edge of the first harmonic, $p\cdot k/m^{2}$ needs to be somewhat smaller for
CCF and somewhat bigger for KN.

##  V The $\mathbf{a_{0}}$ dependence

To identify further regions of dominating CCF behavior and discriminate
against the KN regime, we display in Fig. 4 the differential cross section at
$u=0.1$ (right panel), $u=1$ (middle panel) and $u=10$ (right panel) as
contour plots over the $a_{0}$-$p\cdot k/m^{2}$ plane. The differential cross
section clearly shows that even for such large values of $\chi$ as $10^{4}$
one encounters the KN regime at large energy $p\cdot k/m^{2}>10^{2}$. The KN
regime is that where the contour lines are vertical, i.e. independent of
$a_{0}$. In contrast, even for small values of $\chi$, e.g. $10^{0}$, the
inclined contour lines at small energies, e.g. $p\cdot k/m^{2}=10^{-1}$, point
to the CCF regime. The line $u_{1}=u$, at given $u$, is the demarcation, with
a structure related to the lowest harmonics. Left to the dashed vertical line
the KN regime terminates due to kinematic reasons. In the deep blue region,
the exponential suppression (17) applies.

Figure 4: Contour plots of $d\sigma^{\mathrm{IPA}}/du$ over the
$a_{0}$-$p\cdot k/m^{2}$ pane. Lines of $\chi=const$ are also displayed.
Left/middle/right panels are for $u=0.1/1/10$.

Going, at $p\cdot k/m^{2}=const$, from the KN region to larger values of
$a_{0}$ in the CCF region, one crosses the inclined contour lines, meaning a
dropping $d\sigma^{\mathrm{IPA}}/du$. This, together with the $u$ dependence,
is at the origin of the dropping total cross section with increasing $a_{0}$
exhibited in Fig. 1-left.

Coming back to Figs. 2 and 3, one infers that increasing values of $a_{0}$
suppress the differential cross section (see Fig. 2 for $p\cdot k/m^{2}=const$
and Fig. 3 for $u=const$) at medium-$u$, while enabling non-zero contributions
beyond the KN limit, i.e. at $u>u_{\mathrm{KN}}$. The net effect is
nevertheless the suppression of the total cross section, as exhibited in Fig.
1-left. That is, the Klein-Nishina effect is amplified by the external field.
Both, differential and total cross section are modified. Reference
Lotstedt:2008  reports on a different example, where an external field
modifies the differential cross section, while the total cross section remains
nearly unchanged. This is similar to the findings in Seipt:2013hda . The
difference is in field-enabled vs. field-assisted processes.

As an interim summary let us mention that a more differential perspective,
useful for defining “KN regime” and “CCF regime” at $u=const$, is Fig. 4: The
KN regime is below the line $u_{1}(a_{0},p\cdot k/m^{2})=u$ and contour lines
are vertical, i.e. independent of $a_{0}$. The CCF regime, where contour lines
are inclined due to the $a_{0}$ dependence, is above the line $u_{1}=u$.

This supplements the brief summary of the previous section: When keeping fixed
$u$ and $p\cdot k/m^{2}$, then
$d\sigma^{\mathrm{IPA}}/du\stackrel{{\scriptstyle\mathrm{large}~{}a_{0}}}{{\xrightarrow{\hskip
28.45274pt}}}d\sigma^{\mathrm{CCF}}/du$,
$d\sigma^{\mathrm{IPA}}/du\stackrel{{\scriptstyle\mathrm{small}~{}a_{0}}}{{\xrightarrow{\hskip
28.45274pt}}}d\sigma^{\mathrm{KN}}/du$.

##  VI Many roads lead to large $\chi$

The $a_{0}$-$p\cdot k/m^{2}$ plane can be mapped out by many other
coordinates, e.g. $\chi=a_{0}\,p\cdot k/m^{2}$ plus a second one, such as
$a_{0}$ or $p\cdot k/m^{2}$ or (functions of) combinations thereof. An example
of orthogonal coordinates is the pair $\chi,\eta^{2}$ with
$\eta^{2}(a_{0},p\cdot k/m^{2})=(p\cdot k/m^{2})^{2}-a_{0}^{2}$ for
$\eta^{2}\in[-\infty,\infty]$ and our $\chi(a_{0},p\cdot
k/m^{2})=a_{0}\,p\cdot k/m^{2}$ for $\chi\in[0,\infty]$.

Trading CCF as proxy of $\frac{d\sigma^{IPA}}{du}$, the differential cross
section can be fitted by
$\frac{d\sigma^{\mathrm{CCF}}}{du}|_{u=const}=\mathcal{A}(u)\,\chi^{-x}$ for
$u\in[0.1,20]$ in the asymptotic region, i.e. at large $\chi$. Various
trajectories of going to large $\chi$ in the $a_{0}$ vs. $p\cdot k/m^{2}$
plane are exhibited in Fig. 5-left, for definiteness starting at $a_{0}=1$ and
$p\cdot k/m^{2}=1$. We find for a path along $a_{0}=const$ (black line):
$x=1/3$, along $a_{0}=(p\cdot k/m^{2})^{\beta}$: $x=2/3$ for $\beta=1/2$ (red
line), $x=5/6$ for $\beta=1$ (blue line), $x=1$ for $\beta=2$ (green line),
and along $p\cdot k/m^{2}=const$ (magenta line): $x=4/3$. That is, $x$ depends
strongly on the path of going to large $\chi$. The large-$\chi$ limits
considered in Ilderton:2019kqp ; Podszus:2018hnz  are contained as special
cases.

Figure 5-right exhibits the same information in orthogonal coordinates
$\eta^{2}>0$ vs. $\chi$. For $p\cdot k/m^{2}>a_{0}$, $\eta^{2}\approx const>0$
corresponds to going to larger $\chi$ at $p\cdot k/m^{2}=const$, and vice
versa $\eta^{2}\approx const<0$ corresponds to going to larger $\chi$ at
$a_{0}=const$ as long as $a_{0}>p\cdot k/m^{2}$. (See remark in figure caption
on the relation of $\eta^{2}\lessgtr 0$ sectors.) As said above, many ways
lead to large $\chi$ (and result in fairly different asymptotic regimes).

Figure 5: Left panel: The $a_{0}$-$p\cdot k/m^{2}$ plane is mapped out by
curves $\chi=const$ (grey lines). A few paths of going to large values $\chi$
are depicted, labeled by the parameter $x$ determining the asymptotic
$\frac{d\sigma^{\mathrm{CCF}}}{du}|_{u=const}\propto\chi^{-x}$. For a path
$a_{0}=\mathfrak{z}\left(\frac{p\cdot k}{m^{2}}\right)^{\beta}$, we find the
$\chi$ dependence
$\frac{d\sigma^{\mathrm{CCF}}}{du}\propto\mathfrak{z}^{-1/(1+\beta)}\chi^{-1/3-\beta/(1+\beta)}$.
Right panel: The same information but in orthogonal coordinates $\eta^{2}>0$
vs. $\chi$. Solid and dashed curves are for $a_{0}=const$ and $p\cdot
k/m^{2}=const$, respectively. (The near-vertical sections are replaced by
arrows.) Heavy dots depict the trajectories with $\beta=1/2$ and 2; the
trajectory with $\beta=1$ is along the $\eta^{2}=0$ line (not exhibited).
Mapping out the full $a_{0}$ vs. $p\cdot k/m^{2}$ plane displayed in the left
panel requires the complement $\eta^{2}<0$ vs. $\chi$ (not shown since it
arises from re-labeling $a_{0}\leftrightarrow p\cdot k/m^{2}$ for the
coordinate lines $a_{0}=const$ and $p\cdot k/m^{2}=const$ as well as
$1/2\leftrightarrow 2$ for the trajectories).

In a nutshell: along trajectories in the $a_{0}$-$p\cdot k/m^{2}$ plane, we
find $d\sigma^{\mathrm{CCF}}\propto\chi^{-x}$ with $x$ strongly depending on
the actually chosen path in traversing the lines $\chi=const$.

##  VII Discussion of dependencies and regimes

The here employed formulas (1 \- 16) of one-photon emission in some external
univariate null-fields display often structures such as $1+u$, $1+a_{0}^{2}$,
$1-u/u_{n}$, or $1+u_{\mathrm{KN}}$ etc. suggesting various regimes depending
on whether $1<u$ or $1>u$, $1<a_{0}^{2}$ or $1>a_{0}^{2}$, $u_{n}<1$ or
$u_{n}>1$ etc. Our numerical study has the goal of identifying the
corresponding regimes, beyond asymptotic behaviors, such as $u\to 0$,
$a_{0}\to 0$ or $a_{0}\to\infty$, $p\cdot k/m^{2}\to\infty$ etc. That is, we
quantify the triple $u$ \- $a_{0}$ \- $p\cdot k/m^{2}$ for moving in upper
part of the triangle
$d\sigma^{\mathrm{KN}}$$d\sigma^{\mathrm{CCF}}$$d\sigma^{\mathrm{IPA}}$ with
baseline $d\sigma^{\mathrm{KN}}\leftrightarrow d\sigma^{\mathrm{CCF}}$ given
by classical emission (e.g. described by photon emission along an electron
trajectory in the external field via Lienard-Wiechert potential which ignores
the recoil).

It comes as some surprise that
$d\sigma^{\mathrm{IPA}}_{\mathrm{circ.\,pol.}}\to d\sigma^{\mathrm{CCF}}$,
since circular polarization (circ. pol.) means a rotating electric field, i.e.
one would expect a closer correspondence of
$d\sigma^{\mathrm{IPA}}_{\mathrm{lin.\,pol.}}$ and $d\sigma^{\mathrm{CCF}}$
since CCF means fixed (space-time independent) electric ($E$) and magnetic
($B$) fields, which can be aligned with the linearly polarized (lin. pol.)
wave fields. In Appendix B, this issue is considered, as also the meaning of
$a_{0}$ w.r.t. CCF.

We prefer to consider the differential probability in the invariant Ritus
variable $u$ instead of using the scattering angle or frequency
$\omega^{\prime}$ of the emitted photon. For the mapping,
$u\mapsto\omega^{\prime}$, cf. the discussion in Acosta:2021iyu .

When turning from the monochromatic external field to plane-wave pulses, the
differential spectra become smoothed out, as quantified in Kampfer:2020cbx ,
i.e. the harmonic structures are less significant. In momentum balance
equations, the electron IPA quasi-momenta are replaced by momenta, as
occurring in CCF and KN, of course.

##  VIII Summary

Inspired by the ongoing discussion of the Ritus-Narozhny conjecture concerning
the validity of the Furry picture for processes at and in external fields, we
analyze numerical probabilities of linear and nonlinear Compton scattering. To
establish a link to the Klein-Nishina cross section for linear Compton
scattering, we chose suitably normalized probabilities as “cross sections”,
mainly based on the long-known relation of the weak-field limit of the
nonlinear Compton scattering matching the Klein-Nishina expression. Choosing
furthermore the monochromatic plane wave model as a proxy of a mono-energetic
photon beam, we show with established expressions for the one-photon emission
in pane waves and constant crossed field how and where the so-defined cross
sections coincide. We do our analysis differentially with a Lorentz invariant
quantity - the Ritus variable $u$ \- instead of frequency or/and scattering
angle of the produced photon emitted by an electron moving in the external
field or scattering at a photon beam. We identify those regions in the three-
dimensional parameter space $u$ (for the emitted photon) – $a_{0}$ (external
field intensity) – $p\cdot k/m^{2}$ (energy), where the constant crossed field
approach is trustable. Since the differential cross section matches inevitably
the Klein-Nishina cross section at certain (small) values of $u$, the total
cross section contains also such a contribution. We argue that in discussing
the Ritus-Narozhny conjecture one should resort to such a differential
consideration. It could be that the tails of the Klein-Nishina behavior
require adequate treatment in simulation codes, far beyond the improved
constant cross field approximations.

For significant external field intensities, $a_{0}>1$, the Klein-Nishina
effect, i.e. the suppression of (differential) cross section at high energies,
becomes enforced.

##  Appendix A A few Numbers

To justify the use of the considered fairly wide spread of energies and field
strengths we quote a few numbers relevant to high-intensity laser physics and
astrophysics.

|  | $\lambda\ [\mathrm{nm}]$ | $\omega\ [\mathrm{eV}]$ | $I\ [\frac{\mathrm{W}}{\mathrm{cm}^{2}}]$ |  $a_{0}$777 Using $a_{0}\approx 60\frac{\lambda}{\mu\mathrm{m}}\sqrt{I/(10^{22}\mathrm{W/cm}{}^{2})}$ with laser intensity $I$ and wave length $\lambda$. | $\ E\ [\mathrm{GeV}]\ $ | $\ pk/m^{2}\ $ | $\ \ \chi\ \ $ |  ref.
---|---|---|---|---|---|---|---|---|---
LUXE |  | 800 | 1.55 | $2\times 10^{21}$ | 21.5 | 17.5 | 0.2077 | 4.5 |  Abramowicz:2021zja
FACET-II |  |  |  |  |  |  |  |  |
| E320 | 800 | 1.55 | $2\times 10^{20}$ | 6.8 | 13 | 0.1543 | 1.05 |  Meuren:2020nbw
HED-HIBEF888The quoted electron energy is a projection. |  |  |  |  |  |  |  |  |
| EuXFEL | 0.1 | $13\times 10^{3}$ | $10^{21}$ | 0.002 | $\sim 0.1$ | 9.5 | 0.02 |  EuXFEL
| RELAX | 795 | 1.56 | $8\times 10^{21}$ | 42.7 | $\sim 0.1$ | 0.001 | 0.05 |  RELAX
ELI-NP |  | 814 | 1.52 | $10^{22}$ | 49.8 |  |  |  |  Gales:2018
CoReLS999Current maximum for optical lasers. |  | 800 | 1.55 | $1.1\times 10^{23}$ | 159.2 |  |  |  |  Yoon:2021ony
cosmic rays |  |  | $10^{6}$ |  |  | $10^{4}$ | $7.7\times 10^{7}$ |  |  Boncioli:2023gbl
vela pulsar |  |  | $10^{13}$ |  |  | $2\times 10^{4}$ | $1.5\times 10^{15}$ |  |  HESS:2023sxo
CMB |  |  | $2.34\times 10^{-4}$ |  |  | $10^{4}$ | $0.0179$ |  |  ParticleDataGroup:2022pth

In general, magnetars produce magnetic fields101010 Critical field strengths:
$B_{c}=m^{2}c^{3}/\hbar e\approx 4.4\times 10^{13}$ G and
$E_{c}=m^{2}c^{3}/\hbar e\approx 1.323\times 10^{18}$ V/m.  with
$B\approx\mathcal{O}(10^{14}\cdots 10^{15}~{}\mathrm{G})$ Li:2024oda .

##  Appendix B Linear vs. circular polarization and CCF

One must not consider CCF as $\omega\to 0$ limit Ritus85 , but compare the
plane-wave field and CCF at the same field strengths
$E:=\sqrt{E_{x}^{2}+E_{y}^{2}+E_{z}^{2}}$ and
$B:=\sqrt{B_{x}^{2}+B_{y}^{2}+B_{z}^{2}}$. Following the reasoning and
notation in Heinzl:2008rh  we recap the definition of a pane wave by the field
strength tensor $F^{\mu\nu}(x)=F_{1}(\omega n\cdot
x)f_{1}^{\mu\nu}+F_{2}(\omega n\cdot x)f_{2}^{\mu\nu}$ with
$f_{i}^{\mu\nu}:=n^{\mu}\epsilon_{i}^{\nu}-n^{\nu}\epsilon_{i}^{\mu}$ and
$n^{2}=0$, $n\cdot\epsilon_{i}=0$, $\epsilon_{1}\cdot\epsilon_{2}=0$, yielding
transversality $n^{\mu}F_{\mu\nu}=0$ and energy-momentum tensor
$T_{\mu\nu}=(\sum_{i=1,2}F_{i}^{2})n_{\mu}n_{\nu}$. The normalized gauge and
Lorentz invariant field intensity parameter reads
$a_{0}^{2}=\frac{e^{2}}{m^{2}}\frac{\langle\langle
p_{\mu}T^{\mu\nu}p_{\nu}\rangle\rangle}{(p\cdot k)^{2}}$ with
$\langle\langle\cdots\rangle\rangle$ meaning Lorentz invariant proper-time
($\tau)$ average. This facilitates the following relations:

| lin. pol. | circ. pol. | CCF
---|---|---|---
$F_{1}=\mathcal{C}\omega~{}\times$ | $\cos k\cdot x$ | $\cos k\cdot x$ | 1
$F_{2}=\mathcal{C}\omega~{}\times$ | $0$ | $-\sin k\cdot x$ | 0
$E_{x}=B_{y}=\mathcal{C}\omega~{}\times$ | $\cos\Omega\tau$ | $\cos\Omega\tau$ | 1
$E_{y}=B_{x}=\mathcal{C}\omega~{}\times$ | $0$ | $\sin\Omega\tau$ | 0
$E=B=\mathcal{C}\omega~{}\times$ | $\cos\Omega\tau$ | 1 | 1
$T_{\mu\nu}=\mathcal{C}^{2}k_{\mu}k_{\nu}~{}\times$ | $\cos^{2}k\cdot x$ | 1 | 1
$a_{0}^{2}=\frac{e^{2}}{m^{2}}\mathcal{C}^{2}~{}\times$ | $\frac{1}{2}$ | 1 | 1
$b_{1}=\mathcal{C}~{}\times$ | $1-\cos\Omega\tau$ | $1-\cos\Omega\tau$ | $\frac{1}{2}\Omega^{2}\tau^{2}$
| $\approx\frac{1}{2}\Omega^{2}\tau^{2}(1+\frac{1}{12}\Omega^{2}\tau^{2})$ | $\approx\frac{1}{2}\Omega^{2}\tau^{2}(1+\frac{1}{12}\Omega^{2}\tau^{2})$ |
$b_{2}=\mathcal{C}~{}\times$ | $0$ | $-(\Omega\tau-\sin\Omega\tau)$ | 0
|  | $\approx-\frac{1}{6}\Omega^{3}\tau^{3}(1-\frac{1}{20}\Omega^{2}\tau^{2})$ |
$b_{3}=\mathcal{C}^{2}~{}\times$ | $\frac{1}{4}(2\Omega\tau-\sin 2\Omega\tau)$ | $2(\Omega\tau-\sin\Omega\tau)$ | $\frac{1}{3}\Omega^{3}\tau^{3}$
| $\approx\frac{1}{3}\Omega^{3}\tau^{3}(1-\frac{1}{5}\Omega^{2}\tau^{2})$ | $\approx\frac{1}{3}\Omega^{3}\tau^{3}(1-\frac{1}{20}\Omega^{2}\tau^{2})$ |

where $\Omega\tau:=k\cdot x$ and $k=\omega n$. (We use coordinates with
$n=(1,0,0,1)$, $\epsilon_{1}=(0,1,0,0)$, $\epsilon_{2}=(0,0,1,0)$, hence
$E_{z}=B_{z}=0$.) The equality of $T_{\mu\nu}$, $E$ and $B$, and $a_{0}^{2}$
for circular polarization and CCF supports their close correspondence. The
electron trajectories determine the classical emission. They are described by
$x^{\mu}(\tau)=x^{\mu}(0)+p^{\mu}(0)\,\frac{\tau}{m}-\epsilon^{\mu\,i}b_{i}(\tau)\frac{e}{p\cdot
k}+k^{\mu}b_{3}(\tau)\frac{e^{2}}{2(p\cdot k)^{2}}$ with coefficients
$b_{1,2,3}$ listed above. In leading order in powers of $\Omega\tau$, all
$b_{1,3}$ agree and $b_{2}^{\mathrm{circ.~{}pol.}}$ is nonzero but suppressed,
$b_{2}^{\mathrm{circ.~{}pol.}}\approx-\frac{1}{3}b_{1}^{\mathrm{circ.~{}pol.}}\Omega\tau(1+\frac{1}{30}\Omega^{2}\tau^{2})$.
The periodicity of the IPA trajectories (figure-8/ellipse in $x$-$z$ plane and
a frame, where drift terms vanish, for lin./circ. polarizations) is at the
origin of the IPA quasi-momenta, see Appendix in Seipt:2012tn  for instance.
Note the cancellation of $p\cdot k$ in the definition of $a_{0}^{2}$ and
$a_{0}=\frac{e}{m}\mathcal{C}=\frac{m}{\omega}\frac{E}{E_{c}}$ due to
$\mathcal{C}=E/\omega$ for both circular polarization and CCF. As emphasized
above, $E$ and $\omega$ are primarily given by the considered plane wave and
used afterward to define the related $a_{0}$ for CCF.

###### Acknowledgements.

The authors gratefully acknowledge the former collaboration with D. Seipt, T.
Nousch, T. Heinzl, A. Otto and useful discussions with A. Ilderton, K.
Krajewska, M. Marklund, C. Müller, S. Rykovanov, and G. Torgrimsson. A.
Ringwald and B. King are thanked for explanations w.r.t. LUXE. The work is
supported by R. Sauerbrey and T. E. Cowan w.r.t. the study of fundamental QED
processes for HIBEF. Useful discussions with R. Schützhold, C. Kohlfürst, and
N. Ahmadiniaz are thanked for. The work of UHA was partly funded by the Center
for Advanced Systems Understanding (CASUS) that is financed by Germany’s
Federal Ministry of Education and Research (BMBF) and by the Saxon Ministry
for Science, Culture and Tourism (SMWK) with tax funds on the basis of the
budget approved by the Saxon State Parliament.

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