# Discovery of higher order quantum electrodynamics effect for the vacuum pair
production

Wangmei Zha<EMAIL_ADDRESS>University of Science and Technology of China,
Hefei, China  Zebo Tang<EMAIL_ADDRESS>University of Science and
Technology of China, Hefei, China

###### Abstract

The quantum electrodynamics (QED) vacuum behaves like a polarizable medium
which leads to novel quantum effect, such as lepton pair production from
vacuum in a extreme strong electromagnetic field. This phenomena became known
as Schwinger mechanism and have not yet been observed, which probes the
unknown nonperturbative and nonlinear regime of QED. With the developments in
laser physics, it seems that the field strength could be approached in the
foreseeable future, and the interest has grown enormously both in theory and
experiment. Currently, on the earth-based experiments, the critical
electromagnetic field can only be realized in relativistic heavy ion
collisions. However, the duration time for the strong field in heavy-ion
collisions is very short, which makes the perturbative theory still effective
in contrary to the nonperturbative situation of the Schwinger mechanism.
Surely, there should be sizeable and interesting higher order effect due to
the large charge carried by ion (coupling constant $Z\alpha\sim$ 0.6), which
links the crossover from perturbative to nonperturbative regime of QED. In
1954, Bethe and Maximon studied the higher order QED effect for a similar
process — Bethe-Heitler process and made a prediction of negative correction.
However, the higher order effect has been searched without success since then
together with the efforts both from experimentalist and theorist, which
greatly challenges the existence of higher order correction for vacuum pair
production. In this paper, we show that the lowest order QED calculations for
lepton pair vacuum production in heavy-ion collisions are about 20$\%$ higher
than the combined world-wide data with a seven sigma-level of significance and
the corresponding higher order QED results are consistent with data. We claim
the discovery of higher order effect for the QED pair production, which
settles the dust of previous debates for several decades. The verification of
higher order QED effect is a fundamental scientific problem, which is an
important milestone towards the nonperturbative and nonlinear regime of QED
vacuum.

Quantum electrodynamics (QED) is a quantum field theory of the electromagnetic
force [1, 2, 3], which describes the gauge invariant interaction of charged
particles with photons. Due to the quantum fluctuations, the QED vacuum
behaves like a polarizable medium, leading to novel quantum effects, such as
Casimir effect [4], elastic photon-photon scattering [5, 6] and pair
production from vacuum in extreme strong electromagnetic field [7]. The vacuum
pair production is a process in which virtual dipole pairs in the vacuum can
be accelerated apart by the external field, emerging as real pair. The
paradigm of vacuum pair production is the Schwinger mechanism: the production
of electron-positron pairs in a constant spatially uniform electric field. The
pair production rate per volume for Schwinger mechanism is given by [8]:

$\frac{d^{4}n_{e^{+}e^{-}}}{d^{3}xdt}\sim\frac{c}{4\pi^{3}\lambda^{4}}exp(-\pi\frac{E_{c}}{E}),$
(1)

where $E_{c}$ the critical field strength, $m$ is the electron mass, and
$\lambda$ is the Compton wavelength of electron. In the equation, there is no
power series expansion in terms of the external field strength $E$, which
clearly demonstrates the nonperturbative and nonlinear nature of the
production. Currently, our understanding of perturbative regime of QED is
extreme good, however, very little is known about the nonperturvative regime.
To reach a sizeable production rate from Schwinger mechanism, as shown in Eq.
1, the external field strength should be stronger than the critical value
$E_{c}$($\simeq 1.6\times 10^{16}$ V/cm). In the realm of astrophysics, it is
generally believed that the critical field is realized during the process of
gravitational collapse leading to a Black Hole [9], creating a dense plasma of
electrons, positrons and photons, which results in the phenomenon of Gamma Ray
Bursts (GBRs) [10, 11, 12]. Back to the Earth-based laboratories, the extreme
critical value prevents the experimentalist to directly discovery this
fundamental effect for more than 65 years.

With the technological advances in laser science [13, 14, 15], it seems that
the critical field strength could be realized in the near future. However, the
maximum field strength created by the existing PetaWatt class lasers is still
orders of magnitude smaller than the critical value. Currently, in the
laboratory, the critical field strength can only be reached in the strong
Coulumb field of relativistic heavy ion collisions [12]. The electromagnetic
fields accompanied in these collisions are of order of [16]

$E_{max}\simeq\frac{Ze\gamma}{b^{2}},$ (2)

where $Ze$ is the electric charge carried by the colliding ion, $b$ is the
impact parameter, and $\gamma$ is the Lorentz factor of nuclei in center of
mass frame. For gold-gold collisions at highest energies from the Relativistic
Heavy Ion Collider (RHIC), with the values of $Z=79$, $b=15fm$ and
$\gamma=108$, the maximum field strength is of $E_{max}=5.3\times 10^{16}$
V/cm. For the lead-lead collisions at the Large Hadron Collider (LHC), we get
$E_{max}=1.4\times 10^{18}$ V/cm with the values of $Z=82$, $b=15fm$ and
$\gamma=2706$. The field strengths generated at RHIC and LHC are comparable to
the critical field $E_{c}$ for Schwinger mechanism. However, the duration time
of the strong field is very short, which makes the perturbative theory still
appropriate in contrast to the nonperturbative situation of the Schwinger
production. The theoretical investigation of vacuum pair production in heavy-
ion collisions goes back to the early days of QED. The lowest order (Born
approximation) results were given by Landau and Lifshitz [17] and Racah [18]
in the thirties of the past century. In the calculations, as illustrated in
Fig. 1, the pair production is treated via the collision of two quasi-real
light quanta from the strong Coulumb field surrounding the heavy ions.

Figure 1: (Color online) An illustration of vacuum pair production process
from the strong Coulomb field of fast moving nuclei in heavy ion collisions.

Due to the large charge carried by the heavy ion, the effect coupling
$Z\alpha$ ($\sim$ 0.6 for gold and lead) for the production process is close
to 1. This suggests significant higher order effect, which links the crossover
from perturbative to nonperturbative regime of QED. This is analogous to the
case of Quantum Chromodynamics (QCD) [19, 20] from high energy to low energy
area. In 1954, the pioneer studies [21, 22] for higher order effect of QED
were made by Bethe, Maximon and Davies in a similar process — Bethe-Heitler
process [21] (the photoproduction of electron-positron pairs in the nuclear
Coulomb field). Higher order effects were treated using the Sommerfeld-Maue
wave functions, which are appropriate solutions of the Dirac equation at high
energy. This approach takes higher order effects into account to all orders
and can be related to the usual Feynman graph technique [23]. A sizeable
negative correction was found to the Bethe-Heilter formula. Come back to the
case of vacuum pair production in relativistic heavy ion collisions, the
correction should be stronger, since the projectile quasi-real photon is also
attached to the heavy ion in contrast to the Bethe-Heilter process. The
theoretical progresses of higher order QED effects in heavy-ion collisions are
reviewed in Ref. [16].

The experimental investigations of vacuum pair production in heavy ion
collisions were spurred in the late 20th century by the relativistic heavy ion
facilities Alternating Gradient Synchrotron (AGS) at Brookhaven, the Super
Proton Synchrotron (SPS) at the European Organization for Nuclear Research
(CERN) and more recently RHIC at Brookhaven and LHC at CERN. Over the decades,
various measurements of lepton pair production [24, 25, 26, 27, 28, 29, 30]
has been made in ultra-peripheral collisions (UPC) based on these facilities.
Surprisingly, all the measurements are found to be in good agreement with the
lowest order calculation. The searching for higher order effect seems to be
hopeless, just as the statement in Ref. [16]: “In April 1990 a workshop took
place in Brookhaven with the title ’Can RHIC be used to test QED?’. We think
that after about 17 years the answer to this question is ’no’.”. This is very
puzzling: if the higher order effects are not there, how could we expect the
nonperturbative vacuum pair production from Schwinger mechanism? And exploring
the nonperturbative feature of QED is one of the primary goal for the planing
ultrastrong laser facilities in the not-so-far future.

In this paper, we have performed the latest lowest order QED calculations for
lepton pair production in heavy-ion collisions and found that the results are
about seven standard deviations larger than the combined world-wide data.
Furthermore, the higher order QED corrections to all orders have been carried
out, which reduce the lowest order results sizeably ($~{}\sim 20\%$). Taken
the higher order effect into account, the corresponding results are consistent
with data, which claims the discovery of higher order effect for the QED pair
production, and settles the dust of previous debates for several decades. We
also discuss the missing part in the old lowest order calculations, which
prevents us from the observation of higher order QED effect.

The total cross section of vacuum pair production in lowest order due to the
Coulomb fields of two colliding nuclei has been carried out for the first time
by Landau and Lifshitz [17] in 1934. However, the experimental measurements
are usually performed in limited kinematics phase due to the limited
acceptance covered by the detector system, which calls for differential
theoretical calculations. We treat the electromagnetic fields of the heavy
ions as external fields and employ ”straight line approximation” since the
deflection angle due to the Coulomb scattering is very small. Following the
derivation of Refs. [31, 32], with the direct and cross Feynman diagrams, one
can get the differential probability
$\frac{d^{6}P(\vec{b})}{d^{3}p_{+}d^{3}p_{-}}$ in lowest order QED for
symmetric heavy-ion collisions ($Z=Z_{A}=Z_{B}$) as

$\frac{d^{6}P(\vec{b})}{d^{3}p_{+}d^{3}p_{-}}=\int
d^{2}b\frac{d^{6}\widetilde{P}(\vec{q})}{d^{3}p_{+}d^{3}p_{-}}e^{i{\vec{q}}\cdot{\vec{b}}},$
(3)

where $p_{+}$ and $p_{-}$ are the momenta of the created leptons. The
introduced Fourier transformation of differential probability
$\frac{d^{6}\widetilde{P}(\vec{q})}{d^{3}p_{+}d^{3}p_{-}}$ is given by

$\begin{split}\frac{d^{6}P(\vec{q})}{d^{3}p_{+}d^{3}p_{-}}&=(Z\alpha)^{4}\frac{4}{\beta^{2}}\frac{1}{(2\pi)^{6}2\epsilon_{+}2\epsilon_{-}}\int
d^{2}q_{1}\\\ &F(N_{0})F(N_{1})F(N_{3})F(N_{4})[N_{0}N_{1}N_{3}N_{4}]^{-1}\\\
&\times{\rm{Tr}}\\{(\not{p}_{-}+m)[N_{2D}^{-1}\not{u}_{1}(\not{p}_{-}-\not{q}_{1}+m)\not{u}_{2}+\\\
&N_{2X}^{-1}\not{u}_{2}(\not{q}_{1}-\not{p}_{+}+m)\not{u}_{1}](\not{p}_{+}-m)[N_{5D}^{-1}\not{u}_{2}\\\
&(\not{p}_{-}-\not{q}_{1}-\not{q}+m)\not{u}_{1}+N_{5X}^{-1}\not{u}_{1}(\not{q}_{1}+\not{q}-\not{p}_{+}\\\
&+m)\not{u}_{2}]\\},\end{split}$ (4)

with

$\begin{split}&N_{0}=-q_{1}^{2},N_{1}=-[q_{1}-(p_{+}+p_{-})]^{2},\\\
&N_{3}=-(q_{1}+q)^{2},N_{4}=-[q+(q_{1}-p_{+}-p_{-})]^{2},\\\
&N_{2D}=-(q_{1}-p_{-})^{2}+m^{2},\\\ &N_{2X}=-(q_{1}-p_{+})^{2}+m^{2},\\\
&N_{5D}=-(q_{1}+q-p_{-})^{2}+m^{2},\\\
&N_{5X}=-(q_{1}+q-p_{+})^{2}+m^{2},\end{split}$ (5)

where the longitudinal components of $q_{1}$ are given by
$q_{10}=\frac{1}{2}[(\epsilon_{+}+\epsilon_{-})+\beta(p_{+z}+p_{-z})]$,
$q_{1z}=q_{10}/\beta$, $\epsilon_{+}$ and $\epsilon_{-}$ are the energies of
the produced leptons, $\beta$ is the velocity of heavy ion, $m$ is the mass of
lepton, and the form factor $f(N_{k})$ is Fourier transform of the charge
distribution in nucleus. In the calculations, we employ the Woods-Saxon form
[33] to model the charge distribution of nucleus in spherical coordinates:

$\rho_{A}(r)=\frac{\rho^{0}}{1+\exp[(r-R_{\rm{WS}})/d]},$ (6)

where the radius $R_{WS}$ and skin depth $d$ are based on fits to electron-
scattering data [34] and $\rho^{0}$ is the normalization factor. The photon
propagators $F(N_{k})$ attached to the Coulomb field of heavy ions can be
written as

$F(N_{k})=\frac{Z\alpha}{N_{k}}=\frac{Z\alpha}{\frac{w_{k}^{2}}{\gamma^{2}}+p_{Tk}^{2}},\
k=0,1,3,4,$ (7)

where $Z$ is the charge of nucleus, $\alpha$ ($\sim\frac{1}{137}$) is the fine
structure constant, $w_{k}$ and $p_{Tk}$ are the energy and transverse
momentum of the photons and $\gamma$ is the Lorentz contraction factor,

It was pointed out by Ivanov, Schiller, and Serbo [35] that the higher order
effect of lepton pair production in heavy-ion collisions was analogous to the
well know Bethe-Heilter process on a heavy target [21], which shows a negative
higher order (Coulomb) correction proportional to $Z^{2}$. These authors went
on to calculate the higher order correction to the total cross section by
considering higher-order Feynman diagrams in a leading logarithm approximation
and also found a significant negative contribution. Lee and Milstein [36]
later came to essentially the same result for higher order correction by
constructing an appropriate regularized transverse integral in the low
transverse momentum approximation which analytically solves the higher order
correction to all orders. They replaced the photon propagator shown in Eq. 7
by a properly regularized expression

$F(N_{k})=\frac{1}{2}\int d\rho\rho J_{0}(p_{Tk}\rho)\\{exp[2iZ\alpha
K_{0}(\rho\omega_{k}/\gamma)]-1\\},$ (8)

where $J_{0}$ and $K_{0}$ are modified Bessel functions. This expression goes
over into Eq. 7 in the perturbative limit ($Z\alpha\rightarrow 0$). In the
calculations, we employ Eq. 8 to estimate the higher order corrections.

As revealed in Eq. 4, there are apparent numerical difficulties in evaluating
the multiple dimension integration due to the oscillating factor
$e^{i{\vec{q}}\cdot{\vec{b}}}$. A more simple model, EPA approach, is widely
used, which is identical to the lowest order QED calculation for the cross
section estimation. Since most the experimental results are compared to EPA
calculations to test the validity of lowest order QED, we briefly introduce it
here. In the approach, the vacuum pair production in heavy-ion collisions can
be factorized into a semiclassical and quantum part. The semiclassical part
deals with the distribution of quasi-real photons induced by the colliding
ions, while the quantum part handles the interactions of photon-photon. It
gives:

$\displaystyle\sigma(A+A\rightarrow A+A+l^{+}l^{-})$ (9) $\displaystyle=\int
d\omega_{1}d\omega_{2}n(\omega_{1})n(\omega_{2})\sigma(\gamma\gamma\rightarrow
l^{+}l^{-}),$

where $\omega_{1}$ and $\omega_{2}$ are the photon energies from the two
colliding beams, and $\sigma(\gamma\gamma\rightarrow l^{+}l^{-})$ is the
photon-photon reaction cross-section for lepton pair. The photon flux induced
by the heavy ions can be modelled using the Weizsäcker-Williams method [37]:

$\displaystyle
n(\omega,r)=\frac{4Z^{2}\alpha}{\omega}\bigg{|}\int\frac{\vec{q}_{\bot}}{(2\pi)^{2}}\vec{q}_{\bot}\frac{f(\vec{q})}{q^{2}}e^{i\vec{q}_{\bot}\cdot\vec{r}}\bigg{|}^{2}$
(10) $\displaystyle\vec{q}=(\vec{q}_{\bot},\frac{\omega}{\gamma})$

where $n(\omega,r)$ is the flux of photons with energy $\omega$ at distant $r$
from the center of nucleus. For the point-like charge distribution, the photon
flux is given by the simple formula

$n(\omega,r)=\frac{Z^{2}\alpha}{\pi^{2}\omega r^{2}}x^{2}K_{1}^{2}(x),x=\omega
r/\gamma.$ (11)

The elementary cross-section to produce a lepton pair with lepton mass $m$ and
pair invariant mass $W$ can be determined by the Breit-Wheeler formula [38].

$\displaystyle\sigma(\gamma\gamma\rightarrow l^{+}l^{-})=$ (12)
$\displaystyle\frac{4\pi\alpha^{2}}{W^{2}}[(2+\frac{8m^{2}}{W^{2}}-\frac{16m^{4}}{W^{4}})\text{ln}(\frac{W+\sqrt{W^{2}-4m^{2}}}{2m})$
$\displaystyle-\sqrt{1-\frac{4m^{2}}{W^{2}}}(1+\frac{4m^{2}}{W^{2}})].$

Figure 2: (Color online) Comparison of the ratios of the measured cross
sections from world-wide experiments [39, 40, 41, 42] to the lowest order QED
calculations for lepton pair production in $p+p(\bar{p})$ collisions. The
error bar represents the total uncertainty for each measurement, which
includes the statistical and systematic errors. The yellow band denotes the
uncertainty for lowest order QED calculations.

Various exclusive experimental measurements [39, 40, 41, 42] for lepton pair
production in $p+p(\bar{p})$ have been made, despite of the tiny cross section
due to $Z=1$. The coupling constant in these collisions is in perturbative
limit ($Z\alpha\rightarrow 0$), which provides excellent baseline to test the
validity of lowest order QED. Fig. 2 shows comparison of the ratios of the
measured cross sections from world-wide experiments [39, 40, 41, 42] to the
lowest order QED calculations for lepton pair production in $p+p(\bar{p})$
collisions. The error bar represents the total uncertainty for each
measurement, in which the statistical and systematic errors are added in
quadrature. The exclusive measurements are performed in UPC events, which
exclude the inelastic collisions to reject hadronic background. In the
calculations, the UPC trigger probability in impact parameter space can be
modelled by

$P_{\rm{non-inelastic}}=|1-\rm{exp}(-b^{2}/(2B))|^{2},$ (13)

where $B$ is determined by the experimental measurements [43, 44]. The
standard dipole form factor of proton (anti-proton) is utilized for the lowest
order QED calculation. The theoretical uncertainties are estimated by varying
the parameters in the proton form factor and the UPC trigger probability,
which are found to be less than $2\%$. The yellow band in the figure
represents the uncertainty for lowest order QED calculations. As shown in
figure, the lowest order QED calculations describe the world-wide measurements
very well. The world-wide results are combined with unequal weights determined
by the errors to improve the precision of measurement. The combined result is
consistent with the lowest order QED calculation within one standard
deviation.

Figure 3: (Color online) Comparison of the ratios of the measured cross
sections from world-wide experiments [26, 27, 28, 29, 30] and the predicted
higher order QED results to the lowest order QED calculations for lepton pair
production in $A+A$ collisions. The error bar represents the total uncertainty
for each measurement, which includes the statistical and systematic errors.
The yellow bands denote the uncertainties for QED calculations.

The comparison of the ratios of the measured cross sections from world-wide
experiments [26, 27, 28, 29, 30] and the predicted higher order QED results to
the lowest order QED calculations for lepton pair production in heavy ion
collisions is shown in Fig. 3. In the calculation, optical Glauber model [33]
is employed to determine the UPC trigger probability of heavy ion collision in
impact parameter space. The theoretical uncertainties are estimated by varying
the parameters in the Woods-Saxon distributions of heavy nuclei, which
simultaneously changes the form factor of heavy nuclei and the UPC trigger
probability in impact parameter space. The uncertainties are found to be less
than $2.5\%$, which are represented as yellow bands in the figure. As
demonstrated in the figure, the measurements are systematically smaller than
the lowest order QED predictions and the QED results with higher order
correction describe the data very well. The combined data is seven standard
deviations smaller than the lowest order calculation and consistent with the
higher order result within one standard deviation, which claims the discovery
of higher order effect for the QED pair production in heavy ion collisions.

As shown in Fig. 3, the higher order corrections are significant ($~{}\sim
20\%$). This raises a puzzle: why the higher order effect is not observed
previously? Most the experimental measurements are compared to a industry
standard model — STARLight [45], which incorporates the equivalent photon
approximation (EPA) approach to calculate the lepton pair production. The EPA
approach is identical to the lowest order QED calculation for cross section
estimation. In STARLight model, it treats the nucleus as a point-like charge
for evaluating the photon flux. To avoid the singularities in the spatial
distribution of photon flux, the pair production within the geometrical radius
of the nucleus is ignored. However, according to our study, the production
within the geometrical radius of the nucleus is not negligible. As an
illustration, for the dimuon production in $Pb+Pb$ collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV with the fiducial acceptance of ATLAS ($p_{T\mu}>5$
GeV/c, $|\eta_{\mu}|<2.4$), the fraction of production within the nuclei is
about $28\%$. Coincidently, the higher order correction to the lowest QED
result is about $20\%$, which is comparable to the fraction of production
ignored in STARLight model. The two missing parts (the production within the
nuclei and the higher order correction) compensate with each other, which
makes the STARLight model effective to describe world-wide data. This prevents
us from the observation of higher order effect.

We report the lowest-order QED calculations for lepton pair production both in
proton-proton (anti-proton) and heavy ion collisions. The lowest-order
predictions describe the world-wide measurements in $p+p(\bar{p})$ collisions
($Z\alpha\rightarrow 0$) very well, however, overestimate the production in
heavy ion collision ($Z\alpha\sim 0.6$) by about $20\%$ with a seven sigma-
level of significance. The corresponding higher order QED results can
reproduce the world-wide measurements within one standard deviation. These
findings lend credence to claim the discovery of higher order effect for the
QED pair production under the strongest electromagnetic field in laboratory,
which have waited more than half a century for verification and pave the way
for future tests of QED in the unexplored nonperturbative regimes.
Furthermore, by colliding different species of nuclei, the coupling constant
for the QED pair production can be varied to investigate the higher order
effect towards the nonperturbative regime, which provides a nice reference for
the study of QCD from perturbative to nonperturbative area. The investigations
towards nonperturbative QED cover complementary aspects, which shed new light
on the understanding of the nuclear and laser physics processes, of heavy ion
collisions as well as neutron stars formation and gravitational collapse,
supernovae and GRBs phenomena.

We thank Prof. Zhangbu Xu, Prof. Lijuan Ruan, Prof. Spencer Klein and Dr.
Daniel Brandenburg for useful discussions. This work was funded by the
National Natural Science Foundation of China under Grant Nos. 11775213 and
11720101001, and MOST under Grant No. 2018YFE0104900.

## References

  * Schwinger [1949] J. Schwinger, Quantum electrodynamics. iii. the electromagnetic properties of the electron—radiative corrections to scattering, Phys. Rev. 76, 790 (1949).
  * Feynman [1949a] R. P. Feynman, The theory of positrons, Phys. Rev. 76, 749 (1949a).
  * Feynman [1949b] R. P. Feynman, Space-time approach to quantum electrodynamics, Phys. Rev. 76, 769 (1949b).
  * Casimir [1948] H. Casimir, On the Attraction Between Two Perfectly Conducting Plates, Indag. Math. 10, 261 (1948).
  * Aaboud  _et al._ [2017] M. Aaboud _et al._ (ATLAS), Evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC, Nature Phys. 13, 852 (2017).
  * Aad  _et al._ [2019] G. Aad _et al._ (ATLAS), Observation of light-by-light scattering in ultraperipheral Pb+Pb collisions with the ATLAS detector, Phys. Rev. Lett. 123, 052001 (2019).
  * Schwinger [1951] J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  * Dunne [2004] G. V. Dunne, Heisenberg-Euler effective Lagrangians: Basics and extensions, in  _From fields to strings: Circumnavigating theoretical physics. Ian Kogan memorial collection (3 volume set)_, edited by M. Shifman, A. Vainshtein, and J. Wheater (2004) pp. 445–522.
  * Damour and Ruffini [1975] T. Damour and R. Ruffini, Quantum electrodynamical effects in kerr-newmann geometries, Phys. Rev. Lett. 35, 463 (1975).
  * Fazio and Stecker [1970] G. G. Fazio and F. W. Stecker, Predicted High Energy Break in the Isotropic Gamma Ray Spectrum: a Test of Cosmological Origin, Nature 226, 135 (1970).
  * Piran [2004] T. Piran, The physics of gamma-ray bursts, Rev. Mod. Phys. 76, 1143 (2004).
  * Ruffini  _et al._ [2010] R. Ruffini, G. Vereshchagin, and S.-S. Xue, Electron–positron pairs in physics and astrophysics: From heavy nuclei to black holes, Phys. Rep. 487, 1 (2010).
  * Maiman [1960] T. H. Maiman, Stimulated Optical Radiation in Ruby, Nature 187, 493 (1960).
  * Strickland and Mourou [1985] D. Strickland and G. Mourou, Compression of amplified chirped optical pulses, Opt. Commun. 55, 447 (1985), [Erratum: Opt.Commun. 56, 219–221 (1985)].
  * Danson  _et al._ [2019] C. N. Danson _et al._ , Petawatt and exawatt class lasers worldwide, High Power Laser Sci. Eng. 7, e54 (2019).
  * Baur  _et al._ [2007] G. Baur, K. Hencken, and D. Trautmann, Electron–positron pair production in ultrarelativistic heavy ion collisions, Phys. Rep. 453, 1 (2007).
  * Landau and Lifshitz [1934] L. D. Landau and E. M. Lifshitz, On the production of electrons and positrons by a collision of two particles, Phys. Z. Sowjet. 6, 244 (1934).
  * Racah [1937] G. Racah, Sulla Nascita di Coppie per Urti di Particelle Elet-trizzate, Nuovo Cim. 14, 93 (1937).
  * Politzer [1973] H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
  * Gross and Wilczek [1973] D. J. Gross and F. Wilczek, Ultraviolet behavior of non-abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
  * Bethe and Maximon [1954] H. A. Bethe and L. C. Maximon, Theory of bremsstrahlung and pair production. i. differential cross section, Phys. Rev. 93, 768 (1954).
  * Davies  _et al._ [1954] H. Davies, H. A. Bethe, and L. C. Maximon, Theory of bremsstrahlung and pair production. ii. integral cross section for pair production, Phys. Rev. 93, 788 (1954).
  * Ivanov and Melnikov [1998] D. Ivanov and K. Melnikov, Lepton pair production by a high energy photon in a strong electromagnetic field, Phys. Rev. D 57, 4025 (1998).
  * Vane  _et al._ [1992] C. R. Vane _et al._ , Electron-positron pair production in coulomb collisions of ultrarelativistic sulfur ions with fixed targets, Phys. Rev. Lett. 69, 1911 (1992).
  * Bauer  _et al._ [1994] R. Bauer _et al._ , Measurement of electromagnetically produced $e^{+}e^{-}$ pairs in distant s-pt collisions, Phys. Lett. B 332, 471 (1994).
  * Adams  _et al._ [2004] J. Adams _et al._ (STAR), Production of $e^{+}e^{-}$ pairs accompanied by nuclear dissociation in ultraperipheral heavy-ion collisions, Phys. Rev. C 70, 031902 (2004).
  * Afanasiev  _et al._ [2009] S. Afanasiev _et al._ (PHENIX), Photoproduction of $J/\psi$ and of high mass $e^{+}e^{-}$ in ultra-peripheral Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV, Phys. Lett. B 679, 321 (2009).
  * Abbas  _et al._ [2013] E. Abbas _et al._ (ALICE), Charmonium and $e^{+}e^{-}$ pair photoproduction at mid-rapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV, Eur. Phys. J. C 73, 2617 (2013).
  * Sirunyan  _et al._ [2019] A. M. Sirunyan _et al._ (CMS), Evidence for light-by-light scattering and searches for axion-like particles in ultraperipheral PbPb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV, Phys. Lett. B 797, 134826 (2019).
  * Aad  _et al._ [2020] G. Aad _et al._ (ATLAS), Exclusive dimuon production in ultraperipheral Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV with ATLAS, arXiv (2020), 2011.12211 .
  * Hencken  _et al._ [1995] K. Hencken, D. Trautmann, and G. Baur, Impact parameter dependence of the total probability for the electromagnetic electron - positron pair production in relativistic heavy ion collisions, Phys. Rev. A 51, 1874 (1995).
  * Alscher  _et al._ [1997] A. Alscher _et al._ , Multiple electromagnetic electron positron pair production in relativistic heavy ion collisions, Phys. Rev. A 55, 396 (1997).
  * Miller  _et al._ [2007] M. L. Miller _et al._ , Glauber modeling in high energy nuclear collisions, Ann. Rev. Nucl. Part. Sci. 57, 205 (2007).
  * Barrett and Jackson [1977] R. C. Barrett and D. F. Jackson,  _Nuclear Sizes and Structure_ (Oxford University Press, 1977).
  * Ivanov  _et al._ [1999] D. Ivanov, A. Schiller, and V. Serbo, Large Coulomb corrections to the $e^{+}e^{-}$ pair production at relativistic heavy ion colliders, Phys. Lett. B 454, 155 (1999).
  * Lee  _et al._ [2004] R. Lee, A. Milstein, and V. Strakhovenko, High-energy expansion of Coulomb corrections to the $e^{+}e^{-}$ photoproduction cross-section, Phys. Rev. A 69, 022708 (2004).
  * Krauss  _et al._ [1997] F. Krauss _et al._ , Photon and gluon induced processes in relativistic heavy-ion collisions, Prog. Part. Nucl. Phys. 39, 503 (1997).
  * Breit and Wheeler [1934] G. Breit and J. A. Wheeler, Collision of two light quanta, Phys. Rev. 46, 1087 (1934).
  * Aaltonen  _et al._ [2009] T. Aaltonen _et al._ (CDF), Observation of exclusive charmonium production and $\gamma+\gamma$ to $\mu^{+}\mu^{-}$ in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV, Phys. Rev. Lett. 102, 242001 (2009).
  * Abulencia  _et al._ [2007] A. Abulencia _et al._ (CDF), Observation of Exclusive Electron-Positron Production in Hadron-Hadron Collisions, Phys. Rev. Lett. 98, 112001 (2007).
  * Aad  _et al._ [2015] G. Aad _et al._ (ATLAS), Measurement of exclusive $\gamma\gamma\rightarrow\ell^{+}\ell^{-}$ production in proton-proton collisions at $\sqrt{s}=7$ TeV with the ATLAS detector, Phys. Lett. B 749, 242 (2015).
  * Chatrchyan  _et al._ [2012] S. Chatrchyan _et al._ (CMS), Search for Exclusive or Semi-Exclusive Photon Pair Production and Observation of Exclusive and Semi-Exclusive Electron Pair Production in $pp$ Collisions at $\sqrt{s}=7$ TeV, JHEP 11, 080.
  * Zyla  _et al._ [2020] P. Zyla _et al._ (Particle Data Group), Review of Particle Physics, PTEP 2020, 083C01 (2020).
  * Aad  _et al._ [2011] G. Aad _et al._ (ATLAS), Measurement of the Inelastic Proton-Proton Cross-Section at $\sqrt{s}=7$ TeV with the ATLAS Detector, Nature Commun. 2, 463 (2011).
  * Klein  _et al._ [2017] S. R. Klein _et al._ , STARlight: A Monte Carlo simulation program for ultra-peripheral collisions of relativistic ions, Comput. Phys. Commun. 212, 258 (2017).
  * Bethe and Heitler [1934] H. Bethe and W. Heitler, On the Stopping of fast particles and on the creation of positive electrons, Proc. Roy. Soc. Lond. A 146, 83 (1934).