Prepared for submission to JHEP
arXiv:2203.01245v1 [hep-ph] 2 Mar 2022
QED and Lasers: A Tutorial
T. Heinzl,1
1 Centre
for Mathematical Sciences, School of Engineering, Computing and Mathematics,
University of Plymouth, PL4 8AA, UK
E-mail: theinzl@plymouth.ac.uk
Abstract: This is a write-up of a short tutorial talk on high-intensity QED, videopresented at the 2021 annual Christmas meeting of the Central Laser Facility at
Rutherford-Appleton Lab, UK. The first half consists of a largely historical introduction
to (quantum) electrodynamics focussing on a few key concepts. This well-established
theory is then compared to its strong-field generalisation when a high-intensity laser
is present. Some supplementary material and a fair amount of references have been
added.
In memoriam: Ernst Werner (1930-2021), ‘Doktorvater’ and mentor.
Contents
1 Introduction: A Story of Two Acronyms
1
2 Electrodynamics
2
3 Quantum Electrodynamics
3.1 Overview
3.2 Application: Compton Scattering
5
5
7
4 High-Intensity Quantum Electrodynamics
4.1 Introduction
4.2 Formalism
4.3 Application: Nonlinear Compton Scattering
4.4 Teaser: Radiation Reaction
10
10
12
14
17
5 Conclusion
18
1
Introduction: A Story of Two Acronyms
The title of this tutorial admittedly sounds a bit mundane – but then it is a fairly
precise description of what we will do: we will rather literally add a very strong laser
(field) to quantum electrodynamics (QED), the microscopic theory of light-matter interactions. With the preceding sentence we have defined the first acronym, QED.
Its invention has its own complicated history and arguably goes back all the way to
Planck’s analysis of black-body radiation and his introduction of energy quanta [1],
later to be identified with photons, the particles of light. The quantum field theoretical aspects (‘second quantisation’) were introduced in the Dreimännerarbeit by Born,
Heisenberg and Jordan [2], the latter being responsible for the section named ‘Coupled
harmonic oscillators. Statistics of Wave fields.’ In this section, Jordan quantised the
one-dimensional string which corresponds to a scalar, hence spin zero, field. Quantisation of the electromagnetic field, a spin-one vector field, was first achieved by Dirac a
year later in 1927 [3], which thus may be viewed as the proper year of birth for QED.
Useful collections of early papers on QED may be found in the source texts [4–6]. The
history of QED has been described in the magisterial tome by Schweber [7].
–1–
The laser has become a household name, so one tends to forget that it is another
acronym standing for ‘light amplification by the stimulated emission of radiation’. Stimulated emission is a quantum process that was first explained by Einstein in terms of
his eponymous coefficients [8]. Enhancing this effect employing population inversion
and a gain medium was achieved by 1960 when the first laser was built by Maiman [9].
A particularly important breakthrough in our context was achieved in 1985 with the
invention of chirped pulse amplification (CPA) by Strickland and Mourou [10], which
earned them the Nobel prize in 2018. For the purposes of this tutorial, we will take
such a CPA based high-intensity laser for granted and discuss how its presence affects
the fundamental processes of QED. More details on the fundamentals and applications
of lasers may be found in the comprehensive texts [11–13].
Incidentally, the births of QED and the laser are related theoretically: Dirac’s
inaugural 1927 paper [3] calculates the Einstein coefficients using QED. A modern
account of these calculations can be found at the very beginning of Schwartz’s recent
text on quantum field theory [14].
2
Electrodynamics
The modern theory of electrodynamics came into being with Maxwell’s equations as
summarised in Maxwell’s famous (but now little read) treatise of 1873 [15]. One of his
big achievements was the (theoretical) identification of light as electromagnetic radiation. This finally clarified the earlier observation by Weber [16] that electromagnetism
seemed to naturally involve a constant of nature with units of speed1 . Hertz’s discovery of electromagnetic radiation [18] was the experimental confirmation that finally
confirmed Maxwell’s prediction.
Maxwell’s treatise made for rather difficult reading as he did not have 3-vector
notation at hand. This was only introduced a decade later by Heaviside [19] who
(together with Hertz [20]) gave Maxwell’s equations their modern form2 . Thus, in
1894 Hertz could confidently state that “Maxwell’s theory is the system of Maxwell’s
equations”. Interestingly, Hertz, who died in 1894 aged 36, was not able to solve the
problem of the electrodynamics of moving bodies as he based his discussion in [21] on
Galilei symmetry. However, the symmetry transformations of Maxwell’s equations are
1
In Weber’s (wrong) electrodynamical
equation, based on action-at-a-distance rather than the field
√
concept, this speed corresponds to 2 times the speed of light, c. In an experiment with Kohlrausch
[17], they determined the speed implying a value of c = 3.1 × 108 m/s (if one anachronistically employs
modern terminology).
2
Nahin in his Heaviside biography [22] states that for a short while the reformulated Maxwell
equations were called the ‘Hertz-Heaviside equations’.
–2–
the Lorentz transformations discovered by Lorentz in 1895 [23]. The situation may be
analysed as follows: If Maxwell’s equations were Galilei covariant and thus had the
same form in all inertial frames related through Galilei transformations, it should be
possible to eliminate the speed of light, c, from the equations. Choosing HeavisideLorentz units and rescaling the magnetic field, B → cB, while leaving the electric
field, E, untouched, c can indeed be eliminated from all equations except for Faraday’s
induction law which becomes
1 ∂B
∇×E=− 2
.
(2.1)
c ∂t
In other words, sending c to infinity, we would end up with a Galilei covariant version
of electrodynamics losing the induction law (2.1) on the way [24, 25].
It took an Einstein to finally solve the problem of reconciling motion and electrodynamics in his aptly entitled paper On the Electrodynamics of Moving Bodies [26].
There, he formulated his two postulates of special relativity, namely that all laws of
physics take on the same form in inertial frames related by Lorentz transformations
(Lorentz covariance) and that the speed of light is universal, i.e. it has the same value
in all inertial frames (Lorentz invariance).
The final touch was provided by Minkowski [27] who introduced 4-vector notation
thus making Lorentz covariance manifest. All one has to do is to find an appropriate ‘zero component’ and ‘add’ it to Heaviside’s 3-vectors. 4-positions are space-time
vectors x = (ct, x) implying a 4-gradient ∂ = (∂ct , ∇) and so on. In doing so, one introduces what is now called ‘Minkowski space’, which Minkowski described as follows:
The consequences will be radical. Henceforth, ‘space’ and ‘time’, viewed as separate
entities, will turn into mere shadows entirely, and only a union of the two will retain
an independent meaning [28].
In the context of electrodynamics, one augments the electromagnetic current, j, by
0
j = cρ, where ρ is the charge density, implying a 4-current j = (cρ, j). Electric and
magnetic fields, E and B, do not transform as 3-vectors, but rather as the components
of an electromagnetic tensor, F = F (E, B) and its dual, F̃ = F (B, −E). The numbers
of degrees of freedom (six) match if F is anti-symmetric, F T = −F . With these ingredients, Maxwell’s equations can finally be written in a form that remains unchanged
upon changing frames. Thus, all inertial observers will agree on the following equations:
∂µ F µν = j ν ,
(2.2)
∂µ F̃ µν = 0 .
(2.3)
As important as covariant quantities (4-vectors, tensors, ...) are Lorentz invariant ones,
often referred to as scalars. In practical terms, these have fully contracted Lorentz indices, so all of these appear twice and are summed over. Regarding the electromagnetic
–3–
fields, the most important scalars are
1
1
S = − Fµν F µν = (E 2 − B 2 ) ,
4
2
1
P = − Fµν F̃ µν = E · B .
4
(2.4)
(2.5)
These may in turn be used to define Lorentz invariant field magnitudes, which are
basically given in terms of the real eigenvalues of F ,
√
E = ( S 2 + P 2 + S)1/2 ,
(2.6)
√
B = ( S 2 + P 2 − S)1/2 .
(2.7)
The scalar quantity S serves as the Maxwell Lagrangian in vacuum, i.e. in the absence of charges. To couple this to charged matter one needs another 4-vector, the
electromagnetic (or gauge) potential which combines the (rotational) scalar and 3vector potentials according to A = (φ, A). The field strength is then its covariant curl,
Fµν = ∂µ Aν − ∂ν Aµ . Note that this solves the homogeneous Maxwell equations by
construction as
∂µ F̃ µν = µνρσ ∂µ ∂ρ Aσ = 0
(2.8)
by the anti-symmetry of the Levi-Civita tensor (or, if you like, the Minkowski version
of div curl = 0). The coupling to matter is now given by the Lorentz invariant scalar
product of A and the electromagnetic current j which, for a relativistic point particle
of mass m and charge e on a space-time curve (world-line) x = x(τ ), takes on the form
Z
µ
j (x) = ec dτ ẋµ δ 4 (x − x(τ )) .
(2.9)
The coupling term in the action is thus
Z
Z
1
4
Sint =
d x j · A = e dτ ẋ · A .
c
(2.10)
The complete Lorentz invariant action for electrodynamics coupled to matter was first
written down by Schwarzschild in 1903 [29] and reads
Z
Z
2
S = dτ (−mc + eẋ · A) + d4 x S .
(2.11)
In the above, cdτ = ds = (dx · dx)1/2 denotes the invariant distance element with
proper time τ . We mention in passing that the action (2.11) is gauge invariant, that
is invariant under local transformations Aµ → Aµ + ∂µ Λ, which only add a total time
–4–
derivative to the Lagrangian, (ẋ·∂)Λ = dΛ/dτ , and hence leave the equations of motion
unchanged3 . The latter are obtained via Hamilton’s principle of least action and read
Aµ = jµ ,
e
mẍµ = F µν ẋν .
c
(2.12)
(2.13)
Equation (2.13) is the inhomogeneous wave equation which is solved by the LienardWiechert potential, formally Aµ = Aµ0 + −1 j µ , the inverse wave operator representing
the retarded Green function. The homogeneous solution, Aµ0 , obeys the vacuum wave
equation and corresponds to incoming radiation. The second equation of motion, (2.13),
is the covariant version of the Lorentz force law, the right-hand side being the Lorentz
4-force.
3
Quantum Electrodynamics
3.1
Overview
The following general rules apply when wants to quantise Maxwell theory:
Rule 1: Quantisation is required when the experimental resolution, hence the energy
of a given probe, is sufficiently large such that photons become the relevant degrees of
freedom. In this case, the classical wave picture has to be abandoned in favour of the
particle (photon) picture. Historically, this has been noticed in black-body radiation
(Planck’s law), the photo-electric effect as explained by Einstein and the Compton
effect, among many others.
Rule 2: Relativistic covariance has to be maintained throughout.
To implement these rules when quantising electrodynamics one might try to replace the particle equation of motion (2.13) by a relativistic version of the Schrödinger
equation. This was attempted by Dirac with his celebrated Dirac equation. It turned
out, however, that a formalism using single-particle wave functions obeying the Dirac
equation does not work: It does not allow for creation and annihilation processes where
particle number ceases to be conserved. In particular, particle number does not commute with the generators of Lorentz boosts: different observers will thus disagree in
their measurements of particle content! In more practical terms, once the localisation
energy exceeds a threshold of 2mc2 , the creation of an electron-positron pair of mass
2m will be possible. This has been seen in Klein’s paradox [34], where the relativistic
3
The notion of gauge invariance goes back to Weyl [30, 31] as nicely reviewed in [32, 33]
–5–
scattering at a potential step seems to violate unitarity in that reflection and transmission probabilities do not add up to unity. The resolution of the paradox requires a
‘leakage’ current due to the formation of pairs, a reaction channel that is absent in a
single-particle formalism.
Instead, one needs a relativistic many-body formalism based on ‘second quantisation’, where both light and matter have to be described by quantum fields with their
modes quantised according to their statistics: Bose-Einstein for photons and FermiDirac for electrons (and positrons). The result is quantum electrodynamics (QED),
the first realistic example of a relativistic quantum field theory which is the only consistent unification of quantum mechanics and special relativity. The physical content
of QED is encoded in its Lagrangian,
1
LQED = LMaxwell + LDirac + Lint = − F 2 + ψ̄(i~∂/ − mc)ψ + ej · A .
4
(3.1)
Each of these terms may be symbolically associated with a Feynman diagram,
LQED “
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+
+
(3.2)
the last term describing the fundamental QED vertex, the interaction between photons
and Dirac particles (electrons, positrons). The latter are encoded in the electromagnetic
or Dirac current, jµ = ψ̄γµ ψ with γµ denoting the four gamma matrices required for
a first-order relativistic wave equation. The form of the interaction is dictated by the
principle of gauge invariance which implies ‘minimal substitution’, i.e. the replacement
of mechanical by canonical momenta or derivatives by gauge covariant derivatives,
/ (The Feynman-‘slash’ notation represents contraction with the gamma
i∂/ → i∂/ − eA.
matrices, a
/ ≡ γ · a.)
Looking at the QED Lagrangian we can identify four parameters, ~, c, m and e
corresponding to Planck’s constant, the speed of light, the electron mass and charge,
respectively. The first two of these, ~ and c, reflect the union of quantum mechanics and
special relativity and can be set to unity without harm upon choosing natural units.
This renders the elementary charge, e, dimensionless, so that the only dimensionful
scale remaining is the mass m. Using dimensional analysis, one infers the important
basic quantities listed in Table 1.
The first entry, the electron rest energy, E0 = mc2 ' 0.5 MeV, provides the QED
energy scale. Adopting natural units henceforth, E0 = m, while its inverse defines
the QED length scale, λe = 1/m ' 400 fm. Energy per length, thus E0 /λe , is a
–6–
quantity
formula
name
energy scale
E0 ≡ mc2
electron rest energy
length scale
λe ≡ ~/mc
electron Compton wave length
field strength
coupling
eES ≡ E0 /λe = m2 c3 /~
α = EC /E0 = e2 /4π~c
Sauter-Schwinger field
fine structure constant
Table 1. Important quantities derived from basic QED parameters
force, which defines the QED field strength, ES = m2 /e = 1.3 × 1018 V/m. This field
magnitude is typical for QED across distances of the order of a Compton wavelength.
The challenge is to achieve such field strengths over macroscopic distances. Last but
not least, the fine structure constant is defined as the ratio of the Coulomb energy
between two electrons separated by a Compton wavelength and the electron rest energy,
α = e2 /4π = 1/137 1. The small value of this basic QED coupling guarantees that
perturbation theory in α works well and yields highly accurate results.
The reader may have noticed the subscript S associated with the QED field strength.
This is to flag the historical contributions of Sauter and Schwinger. The first introduced
this quantity as early as 1931 in his tunnelling interpretation of the Klein paradox [35].
Schwinger performed the first modern QED calculation of the pair creation rate, R, in
the presence of a uniform electric field, with the famous result [36]:
R ∼ E 2 exp(−πES /E) = E 2 exp(−πm2 /eE) .
(3.3)
This rate is nonperturbative in the coupling e and represents a huge exponential suppression for field strengths E ES . Schwinger’s rate (3.3) can be made Lorentz
invariant by employing the invariant fields (2.6) and (2.7) which leads to [37]
R ∼ EB coth(πB/E) exp(−πm2 /eE) .
3.2
(3.4)
Application: Compton Scattering
As an important application we will consider the elementary process of Compton scattering where an electron and a photon scatter off each other. This was first discovered
by Compton in 1923 [38] who shot X-rays at electrons at rest and observed a red-shift
of the X-ray photons corresponding to an energy-momentum transfer from the photons
to the electrons. The process is represented by the Feynman diagram of Fig. 1.
It shows an electron of 4-momentum p colliding with a photon of 4-momentum k
exchanging energy and momentum such that the particles end up with 4-momenta p0
–7–
𝑒 ! , 𝑝′
𝛾, 𝑘
𝛾, 𝑘′
𝑒 !, 𝑝
Figure 1. Feynman diagram for Compton scattering with 4-momentum assignments.
and k 0 , respectively. This description is relativistically covariant. Any choice of frame
is encoded in the parametrisation of the initial 4-momenta. Whatever this choice, one
always has energy momentum conservation,
p + k = p0 + k 0 .
(3.5)
One can actually do ‘better’ and introduce the Mandelstam invariants [39] which have
the same value in any frame,
s = (k + p)2 ≡ m2 (1 + 2η) ,
t = (k 0 − k)2 = −2k · k 0 ,
0 2
2
0
u = (p − k ) = m (1 − 2η ) .
(3.6)
(3.7)
(3.8)
Here we have also introduced the dimensionless energy variables
~k · p
η :=
,
(mc)2
~k · p0
η :=
,
(mc)2
0
(3.9)
which measure the electron momentum projection along the photon direction k in units
of (mc)2 . (We have temporarily reinstated ~ and c to flag that these are relativistic
quantum variables.)
The most important quantity is arguably s, defined in (3.6), which represents the
total energy (squared) of the particles in the centre-of-mass frame. In other words,
this is the available energy budget for the process expressed in a frame independent
manner. The Mandelstam variables are not independent but obey the constraint
s + t + u = 2m2 ,
(3.10)
which follows directly from the mass shell conditions,
p2 = p02 = m2 ,
k 2 = k 02 = 0 .
–8–
(3.11)
𝑠=0
𝑢=0
𝑡 = 4𝑚" : Pair threshold
(Breit-Wheeler)
𝑠 = 𝑠#$%
𝑢 = 𝑠 = 𝑚":
Thomson limit
𝑡=0
Kinematic
range
𝑠𝑢 = 𝑚! : Compton edge
Figure 2. Mandelstam plot for Compton scattering. (The pair threshold is not drawn to
scale to save space.)
Using (3.10) the invariant kinematics of the Compton process can be illustrated with
a Mandelstam plot as depicted in Fig. 2.
For any given point in the plane, the Mandelstam variables are given by the orthogonal distances to the axes s, t, u = 0, which form an equilateral triangle of height
2m2 . The allowed kinematic range for Compton scattering is given by the shaded area
enclosed by the (horizontal) axis t = 0 and the parabola su = m4 (in red) defining the
Compton edge. It corresponds to the maximal momentum transfer
4η 2 m2
− t = (s − m ) /s = 4(k · p) /s =
,
1 + 2η
2 2
2
(3.12)
or back-scattering kinematics, where the 3-momenta of incoming and outgoing photons
have opposite directions. The classical (Thomson) limit is located at the vertex of
the parabola where s = u = m2 and t = 0 (no momentum transfer or recoil). The
kinematic regions to the left and right of this point are referred to as the u and s
channel, respectively. The t channel, on the other hand, is given by the upper parabolic
region where t ≥ 4m2 , the pair creation threshold (not drawn to scale to save space).
This is the kinematic region for the crossed (pair creation) process, γ + γ → e+ + e− ,
–9–
first predicted by Breit and Wheeler in 1934 [40]. A version of this process with the
incoming photons still slightly virtual or off-shell (k 2 6= 0) has recently been observed
for the first time [41].
The total cross section for Compton scattering, first calculated by Klein and Nishina
in 1928, is depicted in Fig. 3 as a function of η in units of the classical Thomson cross
section,
8π α2
σTh =
,
(3.13)
3 m2
which is energy independent. Quantum effects thus lead to a significant reduction of
the cross section at high energy (η 1).
TOTAL CROSS SECTION (KLEIN-NISHINA)
Figure 3. Klein-Nishina cross section for QED Compton scattering in units of the classical
Thomson cross section.
4
4.1
High-Intensity Quantum Electrodynamics
Introduction
It is customary in physics to investigate the response of a system to an applied external
field. Well known examples include spin systems or superconductors in the presence of
– 10 –
an external magnetic field. In the context of QED, the study of external field problems
goes back at least to the work of Heisenberg and Euler on vacuum polarisation [42].
They assumed the external electric or magnetic fields to be constant in space and time.
Here we want to analyse how QED gets modified under the influence of a highintensity laser field. The simplest model for such a field is a plane wave characterised
by a wave vector k that is light-like or null, k 2 = 0. The associated 4-potential will only
depend on the invariant phase variable, k · x, and hence obeys the free wave equation,
Aµ = 0, while the field tensor is null as well. This means that the field invariants
(2.4) and (2.5) vanish,
S=P=0,
(4.1)
such that electric and magnetic fields, E and B, are orthogonal and of equal magnitude4 .
Null fields are hence rather elusive as has been known already to Heisenberg who noted
in 1934 that a single plane wave cannot polarise the vacuum. One needs at least two
of them brought into collision to ensure a non-vanishing energy density [43]. This has
been confirmed by a modern QED calculation due to Schwinger who showed that plane
waves lead to a vanishing effective action, hence the absence of vacuum polarisation
[44].
Thus, in order to define non-vanishing invariants, the (plane wave) fields alone are
not sufficient, and one has to employ a probe particle, say an electron, of momentum
p = mu. One can then define the laser frequency and the energy density ‘seen’ by this
probe (i.e. as measured in the instantaneous rest frame of the electron), namely
ωL := k · u ,
E02 := (u, F 2 , u)/c2 .
(4.2)
(4.3)
These can be made dimensionless upon introducing the parameters
~ωL
,
mc2
eE0 λe
χ :=
,
mc2
η :=
(4.4)
(4.5)
referred to as the quantum energy and the quantum nonlinearity parameters, respectively. (Note that η formally coincides with the definition (3.9), but with k now denoting the laser momentum.) Dividing the two one obtains the classical nonlinearity
parameter,
eE0 λL
a0 := χ/η =
,
(4.6)
mc2
4
Of course, this plane wave model is an over-simplification as it does not describe a focussed beam.
– 11 –
defined in terms of the field magnitude E0 ‘seen’ by the probe and the reduced laser
wavelength, λL = c/ωL . Thus, a0 is the dimensionless laser field amplitude (sometimes
denoted as ξ). The parameters above have to be added to those of standard QED which
obviously enlarges the parameter space to be studied. One is particularly interested in
the intensity, hence a0 , dependence at a given value of η, i.e. the centre-of-mass energy
of the combined laser-electron system, cf. (3.6) and the application further below.
4.2
Formalism
Before discussing an application let us have a brief look at the basic formalism of highintensity QED. In the QED Lagrangian, one splits the 4-potential into a plane wave
background field, Ā = Ā(k ·x), and a fluctuating photon, A. The background is null and
obeys the vacuum wave equation, Ā = 0. Its kinetic term, the invariant S̄, vanishes
while the mixed term, F F̄ , is a total derivative that does not contribute to the action.
Hence, the background field Ā enters the Lagrangian solely through its coupling to the
matter current, L̄int = eĀ · j. Setting eĀ =: a0 a, the Dirac equation thus becomes
/ ,
(i∂/ − m − a0 a
/)ψ = eAψ
(4.7)
while the equation for the QED photon, A, remains unchanged,
Aµ = eψ̄γµ ψ .
(4.8)
The important thing to note is that we are now dealing with two couplings, e and a0 .
While e < 1 (and in particular α = e2 /4π 1), the coupling a0 can be much larger
than unity. In other words, while perturbation theory in e (or, effectively, α) makes
sense, we need to treat a0 non-perturbatively, trying to sum up all orders in a0 . Thus,
the right-hand sides of (4.7) and (4.8) can be considered small but the zeroth-order
approximation to (4.7), obtained by setting e = 0 on the right, involves all orders in a0
on the left. Fortunately, there is an exact zeroth-order solution due to (and named after)
Volkov [45]. Perturbing on top of a background that is treated exactly corresponds to
working in the quantum mechanical Furry picture [46]. Rather than writing down a
formula, we will proceed graphically by employing the fact that the solutions of the
classical equations of motion (4.7) and (4.8) are found by summing all tree diagrams
formed from the leading-order solutions. Representing the Volkov solution by a double
line, the solutions to order e2 are depicted in Fig.s 4 and 5.
The first high-intensity QED calculation was done by Reiss who considered nonlinear Breit-Wheeler pair production in 1962 [47]. This was soon followed by extensive
work on photon emission and pair production due to Nikishov, Ritus, Narozhny and
others in the (then) Soviet Union [48–51] large parts of which are reviewed in [52].
– 12 –
<latexit sha1_base64="F08v8kTqzhMkzTLoiXy3xdvySWA=">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</latexit>
pxq “
Volkov
𝛾 emission
(NL Compton)
2𝛾 emission
(D NL Compton)
NL Trident PP
Figure 4. Solution of the electron equation of motion (4.7). The leading order on the right
represents the Volkov solutions while higher orders correspond to particle emission (a.k.a.
nonlinear or double nonlinear Compton scattering) or pair production (PP) processes.
Aµ pxq “
<latexit sha1_base64="NPDtDx0Y78v5n7GkghrZnn7NOws=">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</latexit>
QED
photon
NL BreitWheeler PP
Photo PP
Photo PP
Figure 5. Solution of the photon equation of motion (4.8). The leading order on the right
represents a free QED photon while higher orders correspond to particle emission and/or pair
production processes.
At the same time, important and lucid contributions analysing (nonlinear) Compton
scattering at high intensity were made by Brown and, in particular, Kibble [53–56].
The discussion of higher-order processes has included processes such as double
nonlinear Compton scattering [57–62] and the nonlinear trident process [63–71]. These
studies generalise their linear pendants which already are nontrivial due to the threeparticle final states. (See e.g. Ch. 11 of the classic QED text by Jauch and Rohrlich [72]
for an overview and early references.) For the nonlinear processes, the current state
of the art is presented in [73, 74] (and references therein). Photo-pair production (the
two-vertex processes in Fig. 5) was first considered by means of the optical theorem,
i.e. by cutting the relevant loop diagram [65]. Recently, it has been revisited under the
name of ‘photon trident’ using modern techniques [74].
– 13 –
4.3
Application: Nonlinear Compton Scattering
A natural question to ask is about the modifications to Compton scattering if one
of the incoming photons is replaced by a laser beam, i.e. a coherent superposition of
optical-frequency (hence low-energy) photons. An alternative point of view is to regard
this process as the emission of a photon by an electron dressed by, and exchanging 4momentum with, the laser field (the one-vertex process in Fig. 4).
The process in question replaces the Feynman diagram of Fig. 1 by a sum over the
diagrams depicted in Fig 6.
𝑛
𝑒′, 𝑞′
𝛾! , 𝑘
𝛾, 𝑘′
𝑒, 𝑞
Figure 6.
ments.
Feynman diagram for nonlinear Compton scattering with 4-momentum assign-
The Feynman graph corresponds to the n-photon process e + nγL → e0 + γ. As
any number n of laser photons may be involved, the total amplitude is obtained by
summing over all n. Formulae can be looked up in the original publications [48–50] or
in the textbook by Landau and Lifshitz [75]. We will not need the details with one
exception, the appearance of the quasi-momenta,
q := p +
a20
k,
2η
q 0 := p0 +
a20
k,
2η 0
(4.9)
in the energy-momentum balance
q + nk = q 0 = k 0 .
(4.10)
The quasi-momenta are thus intensity dependent, which is reflected in the altered
mass-shell condition,
q 2 = q 02 = m2 (1 + a20 ) =: m2∗ ,
(4.11)
The mass-shift, m → m∗ , was first discovered by Sengupta [76] and has been extensively
discussed by Kibble [54–56] so that m∗ also goes under the name of Kibble mass.
The effects caused by this modified mass can again be nicely illustrated in terms of a
– 14 –
Mandelstam plot [77]. To this end, one introduces the modified Mandelstam variables,
sn = (nk + q)2 = m2 (1 + a20 + 2nη) ,
(4.12)
tn = (k 0 − nk)2 = −2nk · k 0 ,
(4.13)
0 2
2
un = (nk − q ) = m (1 +
a20
0
− 2nη ) .
(4.14)
These obey the sum rule
sn + tn + un = 2m2∗ ,
(4.15)
which is independent of the photon number n, hence the same for all processes represented by Fig. 6. Crucially, this alters the height of the unilateral triangle in the
Mandelstam plot which thus serves as an invariant signature for the mass shift, see
Fig. 7. The n dependence of the Mandelstam variables corresponds to the appearance
of higher harmonics, labelled by sn , each with its own kinematic range (highlighted in
orange in Fig. 7).
𝑢=0
𝑠=0
𝑠$ 𝑠% 𝑠& Harmonics
QED
Mass shift:
2𝑚! 𝑎!"
𝑡=0
Harmonic
ranges
HI Compton edge:
𝑠! 𝑢! = 𝑚#∗
Figure 7.
Mandelstam plot for nonlinear Compton scattering. Compared to standard
QED, one should note the intensity dependent mass shift and the appearance of harmonics,
each with their own kinematic ranges.
Compared to linear Compton scattering (standard QED), the range in Mandelstamt gets enlarged corresponding to a larger momentum transfer. This implies a red-shift
– 15 –
(blue-shift) of the fundamental, n = 1, Compton edge in the scattered photon (electron)
spectra. Fig. 8 compares the photon spectra for linear Compton as well as nonlinear
Thomson and Compton scattering employing the parameters of the LUXE experiment
[78]. On top, we have listed the formulae for the different Compton edges. One can
clearly distinguish between classical intensity effects (parametrised by a0 ) and quantum
effects (parametrised by η).
2𝜂
1 + 𝑎"!
2𝜂
1 + 2𝜂 + 𝑎"!
2𝜂
1 + 2𝜂
𝑑𝑃
𝑑𝜂′
𝜂′
Figure 5. Photon energy spectrum due to the interaction of an electron with energy 16.5 GeV i
Figure 8. Scattered photon spectra for electron laser scattering (adapted from [78]). Vertical
di↵erent physical
models: linear QED (red dashed), nonlinear classical (blue dotted), nonlinea
lines demarcate the kinematic edges for QED Compton scattering (red-dashed), nonlinear
QED (black solid).
position
of theand
first
harmonic
peak,
often
called the Compton edge, i
ThomsonThe
scattering
(blue-dashed)
nonlinear
Compton
scattering
(black).
determined kinematically and plotted as a grey vertical line in each case.
As before, we can compare the total cross sections. The formulae are quite messy,
so we just have a look at the corrections to leading-order in a0 and η [79]. Fig. 9
provides
overview of the results
for thescattering,
different regions
parameterharmonic
space (in the
In linear QED,
i.e.anperturbative
Compton
theofposition
edges are foun
2n⌘
a
η
plane).
0
0
at, ulin.QED = 1+2n⌘0 . Nonlinear e↵ects appear through the introduction of ⇠ dependence in th
The corrections are in one-to-one correspondence with the three spectra of Fig. 8:
position of the Compton
edge. In the nonlinear classical theory, Thomson scattering, there is
The Klein-Nishina cross-section for (linear) QED Compton scattering was already
2n⌘0
clear red-shifting
due
to
the
field
dependence,
with
the
harmonic
edges
at,
u
=
nonlin.class
1+⇠ 2
shown in Fig. 3. For nonlinear (high-intensity) QED it gets replaced by the nonlinear
Recoil e↵ects are
included
in thewhich
transition
nonlinear
electrodynamics
Compton
cross section
has thefrom
nonlinear
Thomsonclassical
cross section
as its classical to nonlinea
2n⌘0
QED leading limit
to a(ηfurther
→ 0). red-shifting of the Compton edge to, unonlin.QED = 2n⌘ +1+⇠
2 . Th
0
red-shifting of the nonlinear QED Compton edge compared to linear QED is most readil
explained by the electron gaining an e↵ective mass [33] during the interaction, which is du
to the interaction with many background field photons.
The shift of the leading Compton edge (n– =
16 1)
– in the electron energy spectrum is shown i
fig. 6. The electron beam has an initial energy of 16.5 GeV, with a finite size of el = 5 µm. Th
laser pulse is taken to have a Gaussian profile in both the longitudinal and transverse directions
The red solid line shows the theoretical prediction, with the dashed lines representing a 5%
𝜂
𝜎 ≈ 𝜎"# (1 − 2𝜂)
Klein-Nishina
𝜎 ≈ 𝜎"# (1 − 2𝜂 + $% 𝑎&! )
NL Compton
𝜎 = 𝜎"#
Thomson
𝜎 ≈ 𝜎"# (1 + $% 𝑎&! )
NL Thomson
𝑎!
Figure 9. Leading-order corrections to the Thomson cross section induced by intensity and
quantum effects.
4.4
Teaser: Radiation Reaction
It turns out that there is a classical correction to the Thomson cross section that is
missing in Fig. 9, namely that induced by radiation reaction. It was first calculated by
Dirac in 1938 when he posed his relativistic equation of motion [80] that replaces (2.13).
Taking into account the effects of radiation through elimination of the Liénard-Wiechert
potentials he obtained what is now called the Lorentz-Abraham-Dirac equation,
mẍ = eF ẋ + Frad .
(4.16)
We have refrained from explicitly writing down the correction term, Frad . It suffices
to say that it is proportional to the third (proper) time derivative of position x (which
causes problems in its own right) and a time parameter,
τ0 := 32 αλe ' 2 fm/c .
(4.17)
Note that the powers of ~ cancel in this expression. From his equation (4.16), Dirac
finds that the Thomson cross section should be replaced by
σRR ≡
σTh
σTh
2
=
'
σ
1
−
αη
.
Th
3
1 + 2 k · u τ0
1 + 23 αη
(4.18)
Again, factors of ~ have all cancelled. It is an unsolved question how this cross section
arises within QED. Recent investigations suggest [81–83] that it may result from an
– 17 –
all-orders resummation of an appropriate class of Feynman diagrams including loops.
This resummation should yield the summed geometric series (in 23 αη) represented by
Dirac’s cross section, σRR .
5
Conclusion
In the course of the last decade or so, high-intensity QED has become a mature field
of research with new dedicated experiments close to realisation so that the findings
reported in [84–88] will soon be confirmed and/or improved upon. As a result, it is
not possible to do justice to the whole field in a half-hour tutorial session. The required focussing on just a few aspects has hence led to a number of omissions which
include: laser induced pair production and related processes, phenomena related to
vacuum polarisation and light-by-light scattering as well as higher-order processes and
fundamental questions related to them such as the breakdown of strong-field perturbation theory. One also needs to address the limitations of the approximations made,
in particular the plane wave and external field approximations. The plane wave model
cannot describe focussed beams, while an external field by definition is blind to backreaction phenomena caused by that very field. Space and time limitations have also led
to a focus on theory, for which the author apologises to his experimental colleagues, in
particular to those involved in the planned LUXE experiment.
There are a number of recent reviews which cover the omissions made here and
include references to the original works [89–97]. The reader is encouraged to find
further information and inspiration there.
Acknowledgements
The author thanks Alex Robinson for the invitation to give this presentation. He is
indebted to Ben King for a careful reading of the manuscript and a number of useful
comments.
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