Also at]International Centre for Advanced Training and Research in Physics
(CIFRA), P.O. Box MG12, 077125 Bucharest-Măgurele, Romania

# Constraints for electron-capture decays mimicking production of axion-like
particles in nuclei

Aagrah Agnihotri<EMAIL_ADDRESS>University of Jyväskylä,
Department of Physics, P.O. Box 35, FI-40014 Jyväskylä, Finland  Jouni Suhonen
<EMAIL_ADDRESS>[  University of Jyväskylä, Department of Physics, P.O.
Box 35, FI-40014 Jyväskylä, Finland  Hong Joo Kim<EMAIL_ADDRESS>Department of Physics, Kyungpook National University, Daegu 41566, Republic of
Korea

###### Abstract

We give for the first time, theoretical estimates of ground-state-to-ground-
state (GS-to-GS) electron-capture (EC) branch decay rates of 44Ti, 57Co, and
139Ce. The nuclear-structure calculations have been done exploiting the
nuclear shell model (NSM) with well-established Hamiltonians and an advanced
theory of $\beta$ decay. In the absence of experimental measurements of these
GS-to-GS branches, these estimates are of utmost importance for terrestrial
searches of axion-like particles (ALPs). Predictions are made for EC-decay
rates of 2nd-forbidden unique (FU) and 2nd-forbidden non-unique (FNU) EC
transitions that can potentially mimic nuclear axion production in experiments
designed to detect ALPs in nuclear environments.

The study of nuclear $\beta$ decays has important implications for Beyond the
Standard Model Physics (BSMP) [1, 2, 3, 4, 5]. Among different avenues of
searches for BSMP, where the study of nuclear $\beta$ decays plays an
important role, experimental searches for solar axions are among such
endeavors [6, 7]. Axions being potential dark-matter candidates [8], were
originally hypothesized to exist as a solution to the strong CP problem [9,
10, 11] and later generalized to classes of axion-like particles (ALPs), see
the review [12].

Cosmological and astrophysical constraints for ALPs are stringent, but it is
also necessary to perform laboratory experiments in searches for axions or
ALPs since these experiments are well-controlled, reproducible, and less
model-dependent. There are several approaches of laboratory experiments based
on axion-photon coupling, such as helioscopes, light-shining through walls,
interferometry, and solar-axion experiments as shown in the review [8]. To
cover higher mass regions with terrestrial sources, collider, and beam-dump
facilities [13], nuclear reactor facilities [14] and radio-active-source based
experiments are ongoing or proposed.

Among different proposed axion-production mechanisms, the nuclear-based
production mechanism is expected to generate axions in magnetic-
dipole(M1)-decaying excited states of nuclei. These axions, referred to as
solar axions, can also be investigated experimentally using suitable nuclear
M1 $\gamma$ transitions [6, 7]. In general, these M1 transitions can be used
for ALP search through ALP de-excitation with missing energy of the $\gamma$.
These experiments have the advantage of laboratory scale and direct coupling
to nuclear de-excitation transitions. The first experiment was performed by
the group of M. Minowa [6] using the 139La M1 transition fed by the 139Ce
electron-capture (EC) decay. This experiment could set the upper limit
$\Gamma_{\rm invisible}/\Gamma_{\rm total}<9.70\times 10^{-7}$ (95% C.L.) to
the invisible transition rate relative to the total transition rate. This
particular experiment had a limitation in sensitivity caused by the background
stemming from $\gamma$s and rare nuclear transitions. This external background
level can be significantly lowered by deep underground experiments with higher
radiation shielding. However, there is a severe internal background in the
form of a rare EC-decay branch to the ground state (GS) of 139La and it needs
either to be measured or to be calculated in a reliable way.

EC-decaying 44Ti, 57Co and 139Ce nuclei are among sources that feed nuclear
states with suitable measurable M1 properties for ALP detection. The known EC
branches of these sources include the allowed Gamow-Teller (GT) transitions
57Co$(7/2^{-}_{1})\rightarrow\,^{57}$Fe*$(5/2^{-}_{1})$,
57Co$(7/2^{-}_{1})\rightarrow\,^{57}$Fe*$(5/2^{-}_{2})$, and
139Ce$(3/2^{+}_{1})\rightarrow\,^{139}$La*$(5/2^{+}_{1})$, along with the 1st-
forbidden non-unique $\beta$ transitions
44Ti$(0^{+}_{1})\rightarrow\,^{44}$Sc*$(1^{-}_{1})$ and
44Ti$(0^{+}_{1})\rightarrow\,^{44}$Sc*$(0^{-}_{1})$. These EC-decays feed
directly or indirectly the excited states 44Sc*$(0^{-}_{1})$,
57Fe*$(3/2^{-}_{1})$ and 139La*$(5/2^{+}_{1})$ decaying by an M1 transition to
the GSs 57Fe$(1/2^{-}_{1})$ and 139La$(7/2^{+}_{1})$, and the first excited
state 44Sc$(1^{-}_{1})$. These M1 decays are expected to compete with the
production of low-mass solar axions or ALPs.

The experimental signature of the axionic deexcitation of M1-decaying excited
states, namely non-observation of de-excitation $\gamma$ rays or conversion
electrons, is also expected to be identical to the ground-state-to-ground-
state (GS-to-GS) branches of the EC-decaying sources under consideration.
These GS-to-GS branches include 2nd-forbidden non-unique (FNU) decays:
44Ti$(0^{+}_{1})\rightarrow\,^{44}$Sc$(2^{+}_{1})$ and
139Ce$(3/2^{+}_{1})\rightarrow\,^{139}$La$(7/2^{+}_{1})$, along with the 2nd-
forbidden unique (FU) decay
57Co$(7/2^{-}_{1})\rightarrow\,^{57}$Fe$(1/2^{-}_{1})$ [6]. These particular
transitions are much more hindered than the known EC branches (GT and 1st FNU
decays) and beyond detection thus far. This also means that in the search for
rare events of axionic de-excitation, estimation of the rates of these rare EC
decays is key for experimental confirmation of the detection of solar axions
or ALPs using these terrestrial sources.

Due to the rarity of these GS-to-GS EC decays, experimental branching ratios
(BRs) for these decays are not yet known, a fact that motivates us to
theoretically study these decays. The only known attempt to measure such a GS-
to-GS EC decay was made for 139Ce in [6], establishing the upper limit of BR
to be $5(5)\times 10^{-7}\%$. Our focus in this article is the estimation of
BRs for these unknown decays by using the nuclear shell model (NSM) and a
reasonable range $g^{\rm eff}_{\rm A}=0.5-1.27$ of effective values of the
weak axial coupling. In addition to the GS-to-GS transitions, we also evaluate
the other relevant EC branches present in the decay of the source isotopes in
order to validate our methodology for obtaining reasonable BR estimates.

Non-relativistic limit of the theory of $\beta$ decays is well established and
a detailed account is presented in [15]. A more concise and practical
exposition of the treatment of forbidden $\beta^{+}$/EC decays is given in
[16]. The half-life value for pure EC decays, $t^{\rm EC}_{1/2}$, is given as
$t^{\rm EC}_{1/2}=\kappa/\tilde{C}^{\mathrm{EC}}$, where $\kappa$ takes the
value of 6289 s and $\tilde{C}^{\mathrm{EC}}$ is the integrated shape function
which can be written as [16]

${\tilde{C}}^{\rm
EC}=\frac{\pi}{2}\sum_{x=1s,2s}n_{x}\beta_{x}^{2}(p_{\nu_{x}}/m_{e}c)^{2}C(p_{\nu_{x}})\,,$
(1)

when considering the leading channels, namely K capture (1s) and L1 capture
(2s). Here $n_{x}$ are relative occupancies for the respective orbitals,
$\beta_{x}$ are the electron Coulomb amplitudes, and $p_{\nu_{x}}$ is the
momentum of the neutrino when the electron is captured from the atomic orbital
$x$ [16]. The shape factor $C(p_{\nu_{x}})$ used in Eq. (1), containing
complicated combinations of phase-space factors and nuclear matrix elements
(NMEs), is given for forbidden EC transitions in detail in [16].

Table 1:  Experimental and computed values of energies for the states involved in EC decays in the three axion sources. Energies are given in MeV. Experimental values are taken from [17]. Nucleus | $J^{\pi}$ | Eexp. | Etheo.
---|---|---|---
${}^{44}_{22}$Ti22 | $0^{+}_{1}$ | 0 | 0
${}^{44}_{21}$Sc23 | $2^{+}_{1}$ | 0 | 0
| $1^{-}_{1}$ | 0.068 | 7.410
| $0^{-}_{1}$ | 0.146 | 9.093
${}^{57}_{27}$Co30 | $7/2^{-}_{1}$ | 0 | 0
${}^{57}_{26}$Fe31 | $1/2^{-}_{1}$ | 0 | 0
| $3/2^{-}_{1}$ | 0.0144 | 0.022
| $5/2^{-}_{1}$ | 0.136 | 0.063
| $5/2^{-}_{2}$ | 0.706 | 0.820
${}^{139}_{58}$Ce81 | $3/2^{+}_{1}$ | 0 | 0
${}^{139}_{57}$La82 | $7/2^{+}_{1}$ | 0 | 0
| $5/2^{+}_{1}$ | 0.166 | 0.082

Nuclear wave functions for the NMEs and electromagnetic (EM) observables were
computed using the NSM computer program NuShellX$@$MSU [18]. The nuclei 44Ti
and 44Sc were computed in the SDPFPN model space with the NOWPN interaction
and 36Ar core. The nucleus 57Co was computed using the FPPN model space and
GX1APN interaction. For 57Fe, KB3GPN interaction was used in the FPPN model
space with the truncation scheme of [19]. For 139La and 139Ce, the JJ55PN
model space was used with the SN100PN interaction. Four protons are fixed in
the 1g7/2 orbital for 139Ce. Experimental and computed energies of states
involved in all the discussed EC decays are given in Table 1. The values of
magnetic dipole ($\mu$) and electric quadrupole ($Q$) moments of the involved
states are given in Table 2. The moments $\mu$ and $Q$ are given since they
are indicative of the quality of the wave functions [20, 21]. In addition to
this, transition strengths and mixing ratios M1/E2 for the M1-decaying states
are given in Table 3.

Table 2:  Experimental and theoretical values of EM moments of the states involved in the EC decays to the three axion sources. The magnetic dipole ($\mu$) and electric quadrupole ($Q$) moments are given in units of nuclear magnetons ($\mu_{N}/c$) and $e$barns, respectively. In the calculations, an effective charge 1.5$e$ (0.5$e$) for proton (neutron) and bare $g$ factors were used [22]. Experimental values are taken from [17].  Nucleus | $J^{\pi}$ | $\mu_{\rm exp}$ | $\mu_{\rm theo}$ | $Q_{\rm exp}$ | $Q_{\rm theo}$
---|---|---|---|---|---
${}^{44}_{22}$Ti22 | $0^{+}_{1}$ | - | - | - | -
${}^{44}_{21}$Sc23 | $2^{+}_{1}$ | +2.498(5) | +2.605 | +0.10(5) | +0.0806
| $1^{-}_{1}$ | +0.342(6) | -1.287 | +0.21(2) | +0.0750
| $0^{-}_{1}$ | - | - | - | -
${}^{57}_{27}$Co30 | $7/2^{-}_{1}$ | +4.720(10) | +4.568 | +0.54(10) | +0.3876
${}^{57}_{26}$Fe31 | $1/2^{-}_{1}$ | +0.09064(7) | +0.130 | - | -
| $3/2^{-}_{1}$ | -0.15531(18) | -0.387 | +0.160(8) | +0.1726
| $5/2^{-}_{1}$ | +0.935(10) | +0.980 | - | -0.3173
| $5/2^{-}_{2}$ | - | +0.533 | - | -0.0533
${}^{139}_{58}$Ce81 | $3/2^{+}_{1}$ | +1.06(4) | +1.316 | - | +0.2523
${}^{139}_{57}$La82 | $7/2^{+}_{1}$ | +2.7791(2) | +1.756 | +0.206(4) | +0.1267
| $5/2^{+}_{1}$ | - | +4.668 | - | -0.1698
Table 3:  Reduced transition probabilities in Weisskopf units (W.u.) for the magnetic dipole (M1) and electric quadrupole (E2) transitions, and their mixing ratios $\delta$, for the transitions accompanying the axionic de-excitation under consideration. The used effective charges and $g$ factors are those listed in Table 2. Experimental values are taken from [17]. Transition |  $B$(M1) |  $B$(E2) | $\delta$
---|---|---|---
44Sc($0^{-}_{1}\rightarrow 1^{-}_{1}$) |  |  |
theo | 0.01752 | 0.00000 | 0.00000
exp | $8.71(5)\times 10^{-7}$ | - | -
57Fe($3/2^{-}_{1}\rightarrow 1/2^{-}_{1}$) |  |  |
theo | 0.03545 | 0.49 | 0.00183
exp | 0.0078(3) | 0.37(7) | 0.00223(18)
139La($5/2^{+}_{1}\rightarrow 7/2^{+}_{1}$) |  |  |
theo | 0.0001795 | 0.04414 | 0.052
exp | 0.00257(4) | - | -

From Table 1 one can see that the ground states are correctly predicted by the
NSM, as also the energies of the excited states, except for the $1^{-}_{1}$
and $0^{-}_{1}$ states in 44Sc for which the computed energies are way too
high. This discrepancy in the excitation energies is clearly reflected also as
badly described EM moments of these states, as seen in Table 2. On the other
hand, the computed EM moments are quite compatible with the experimental ones
for the other considered states, in particular for those involved in the GS-
to-GS transitions of interest for this Letter.

Predicting BRs and (partial) half-lives of $\beta$ transitions using NSM-
computed wave functions calls for the use of an effective value $g^{\rm
eff}_{\rm A}$ of the weak axial coupling [23]. $g^{\rm eff}_{\rm A}$ is
related to the quenching factor $q$ as $q=g^{\rm eff}_{\rm A}/g_{A}$ [23],
where $g_{\rm A}=1.276$ [24] is the free-nucleon value. The key inference we
consider here is that for different branches of the EC decay of the parent
nucleus, $q$ is not unique. This follows from the fact that the $g^{\rm
eff}_{\rm A}$ is an artifact reflecting the incompleteness of the wave
functions involved [25]. The more complete the state, the closer the value of
q is to unity [25].

The validity of the above inference is justified when we look at the two
Gamow-Teller (GT) branches of the decay of 57Co: The GT transitions that
populate the $5/2^{-}_{1}$ and $5/2^{-}_{2}$ levels in 57Fe have BRs of
99.8(3) % and 0.174(10) %, respectively [17]. The branches are identical in a
sense that the initial and final spin-parities $J^{\pi}$ are the same. Also,
the wave function of the initial state is common for both. The $g^{\rm
eff}_{\rm A}$ for these respective branches are found to be 1.25 and 0.548 for
the GT matrix elements $M_{\rm GT}=-0.1065$ and $M_{\rm GT}=0.0572$,
respectively, when using the phase-space factors of [26]. This difference in
quenching is due to the difference in the quality of final-state wave
functions. In table 2, it can be seen that for the $5/2^{-}_{1}$ state, the M1
moment $\mu$ is in very good agreement with the experimental value, justifying
a near-unity $q$ value of 0.982. Along the same lines, in the decay of 139Ce,
the GT transition populating the $5/2^{+}_{1}$ state with BR$\approx 100\%$
[17] turns out to have $g^{\rm eff}_{\rm A}=0.914$ for $M_{\rm GT}=0.340$.
This $g^{\rm eff}_{\rm A}$ value is quite compatible with the relatively well-
predicted EM moments of Table 2. Based on all these examples, we conjecture
that the quenched effective value $g^{\rm eff}_{\rm A}$ appears as a free
parameter for all $\beta$ transitions. Since no relevant experimental BRs are
known for the GS-to-GS decays, we make estimations of BRs and partial half-
lives for a conservative interval $g^{\rm eff}_{\rm A}\in[0.5-1.27]$, found
sufficient, e.g., in the works [23, 27, 28].

An additional free parameter, namely the small relativistic vector NME (sNME)
appears for FNU decays [29, 27, 28]. In the cases where the sNMEs are fitted
to experimental BRs, it is seen that there exist two values of sNMEs that
reproduce a given BR [30, 31, 32, 28]. Since two out of three GS-to-GS decays
under study are FNU, for these cases values of sNME must be fixed as well. The
problem of the sNME is solved by looking at the variation of the GS-to-GS
half-life with the value of sNME for a given $g^{\rm eff}_{\rm A}$. It turns
out that for all $g^{\rm eff}_{\rm A}$, there is a unique sNME that maximizes
the GS-to-GS half-life and, in turn, minimizes the corresponding BR. This
opens up a way for estimation of the lower limits for the GS-to-GS BRs which
is valuable for experiments.

In order to study closer the influence of the sNME on the BRs we can use as a
testing ground the fact that the decay of the 0${}^{+}_{1}$ GS of 44Ti
populates the excited states $1^{-}_{1}$ and $0^{-}_{1}$ in 44Sc, with
branching ratios 0.7(3)% and 99.3(3)%, respectively. Both these branches are
1st FNU but the transition to the $0^{-}_{1}$ state does not depend on sNME
[33] and thus can be left out of the discussion. However, the study of the
$1^{-}_{1}$ branch can help locate the sNME that minimizes the BR relative to
the two sNMEs that reproduce the experimental BR. The relevant sNME and
$g^{\rm eff}_{\rm A}$ values are given in table 4.

Table 4: Computed sNME values (sNME1 and sNME2) which reproduce the experimental BR of the 44Ti$(0^{+}_{1})\rightarrow^{44}$Sc*$(1^{-}_{1})$ transition and the sNME value sNMEmin which minimizes the BR of this transition for the effective weak axial couplings of interest in this work. The CVC value of the sNME, sNMECVC, is given for reference. $g^{\rm eff}_{\rm A}$ | sNME1 | sNME2 | sNMEmin
---|---|---|---
0.5 | -0.00535 | -0.00094 | -0.00314
0.6 | -0.00596 | -0.00154 | -0.00375
0.7 | -0.00656 | -0.00215 | -0.00435
0.8 | -0.00717 | -0.00275 | -0.00496
0.9 | -0.00777 | -0.00336 | -0.00556
1.0 | -0.00838 | -0.00396 | -0.00617
1.1 | -0.00898 | -0.00457 | -0.00678
1.2 | -0.00959 | -0.00517 | -0.00738
1.27 | -0.01001 | -0.00560 | -0.00780
| sNME${}_{\rm CVC}=-0.01350$ |  |

It can be seen that sNMEmin is always located in between NME1 and NME2. This
means that sNME values larger or smaller than sNMEmin produce larger
branchings for all relevant $g^{\rm eff}_{\rm A}$. The CVC value of sNME,
denoted as sNMECVC in Table 4, is also unique and can potentially give
additional information on the physical value of sNME [29, 27, 28]. Yet, due to
the lack of perfectly modeled wave functions, predictions using the CVC value
usually fall short of reproducing experimental values of BRs, making the CVC
value only a good reference in searches of the proper value of the sNME in
different nuclear models [29, 27, 28]. This is also the case for the 1st FNU
transition at hand, where sNMECVC reproduces the experimental BR for $g^{\rm
eff}_{\rm A}>1.276$.

Table 5:  Values of sNMEs (sNMEmin) which minimize the GS BR (BRmin) and hence maximize the corresponding half-life ($t_{\rm 1/2,max}$) for the effective weak axial couplings of interest in this work. The transition 57Co$(7/2^{-}_{1})\rightarrow\,^{57}$Fe$(1/2^{-}_{1})$ is FU and thus independent of sNME. The lowest values of BR are highlighted, and they can serve as conservative estimates for experiments.  |  44Ti$(0^{+}_{1})\rightarrow\,^{44}$Sc$(2^{+}_{1})$ |  57Co$(7/2^{-}_{1})\rightarrow\,^{57}$Fe$(1/2^{-}_{1})$ |  139Ce$(3/2^{+}_{1})\rightarrow\,^{139}$La$(7/2^{+}_{1})$
---|---|---|---
| 2nd-forbidden non-unique | 2nd-forbidden unique | 2nd-forbidden non-unique
$g^{\rm eff}_{\rm A}$ | sNMEmin | BR${}_{\rm min}[\%]$ |  $t_{\rm 1/2,max}$[s] | BR[%] |  $t_{1/2}$[s] | sNMEmin | BR${}_{\rm min}[\%]$ |  $t_{\rm 1/2,max}$[s]
0.5 | -0.0203 | 6.15E-7 | 3.03E+17 | 1.63E-7 | 1.44E+16 | -0.136 | 1.00E-7 | 1.19E+16
0.6 | -0.0238 | 5.95E-7 | 3.13E+17 | 2.35E-7 | 1.00E+16 | -0.163 | 1.12E-7 | 1.06E+16
0.7 | -0.0272 | 5.76E-7 | 3.24E+17 | 3.19E-7 | 7.35E+15 | -0.191 | 1.25E-7 | 9.54E+15
0.8 | -0.0306 | 5.57E-7 | 3.35E+17 | 4.17E-7 | 5.63E+15 | -0.219 | 1.38E-7 | 8.61E+15
0.9 | -0.0341 | 5.38E-7 | 3.47E+17 | 5.28E-7 | 4.45E+15 | -0.246 | 1.53E-7 | 7.79E+15
1.0 | -0.0375 | 5.20E-7 | 3.59E+17 | 6.52E-7 | 3.60E+15 | -0.274 | 1.68E-7 | 7.09E+15
1.1 | -0.0409 | 5.02E-7 | 3.72E+17 | 7.89E-7 | 2.98E+15 | -0.302 | 1.84E-7 | 6.47E+15
1.2 | -0.0444 | 4.84E-7 | 3.85E+17 | 9.39E-7 | 2.50E+15 | -0.330 | 2.01E-7 | 5.92E+15
1.27 | -0.0468 | 4.72E-7 | 3.95E+17 | 1.05E-6 | 2.23E+15 | -0.349 | 2.13E-7 | 5.58E+15

For the GS-to-GS transitions, BRs are tabulated in Table 5. There we give the
value sNMEmin, minimizing the GS BR (BRmin) and maximizing the half-life
$t_{\rm 1/2,max}$, for the FNU decays of 44Ti and 139Ce, and BR and $t_{1/2}$
for the FU decay of 57Co. As can be seen, the BRmin changes smoothly with
$g^{\rm eff}_{\rm A}$ being smallest for $g^{\rm eff}_{\rm A}=0.5$ for 57Co
and 139Ce decays, whereas for 44Ti decay the smallest value is achieved for
$g^{\rm eff}_{\rm A}=1.27$. Therefore, the most conservative estimates are the
lowest BRs obtained, $4.72\times 10^{-7}$ for the decay of 44Ti$(0^{+}_{1})$,
$1.63\times 10^{-7}$ for the decay of 57Co$(7/2^{-}_{1})$, and $1.00\times
10^{-7}$ for the decay of 136Ce$(3/2^{+}_{1})$, and these are the ones
suggested for the experiments. It should be noted here that for 139Ce our
result BR${}_{\rm min}=1.00\times 10^{-7}$ is well in line with the
experimental upper limit of $5(5)\times 10^{-7}$ [6].

In summary, we give for the first time theoretical estimates for the minimum
branching ratios of the ground-state-to-ground-state second-forbidden
electron-capture transitions
44Ti$(0^{+}_{1})\rightarrow\,^{44}$Sc$(2^{+}_{1})$,
57Co$(7/2^{-}_{1})\rightarrow\,^{57}$Fe$(1/2^{-}_{1})$, and
139Ce$(3/2^{+}_{1})\rightarrow\,^{139}$La$(7/2^{+}_{1})$. The nuclear-
structure calculations have been done exploiting well-established nuclear
shell-model Hamiltonians and an advanced theory of $\beta$ decay. The obtained
minimum branching ratios are $4.72\times 10^{-7}$ for the decay of Ti,
$1.63\times 10^{-7}$ for the decay of Co, and $1.00\times 10^{-7}$ for the
decay of Ce, the last fully compatible with the available experimental upper
limit of the branching ratio. It is conjectured that in the absence of
experimental measurements of these decay branches, these estimates are of
utmost importance for terrestrial searches of solar axions since they can
potentially mimic nuclear axion-production rates in the related experiments.

###### Acknowledgements.

We acknowledge grants of computer capacity from the Finnish Grid and Cloud
Infrastructure (persistent identifier urn:nbn:fi:research-infras-2016072533 )
and the support by CSC – IT Center for Science, Finland, for the generous
computational resources.

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DESY 18-128

# Measuring the Boiling Point of the Vacuum of Quantum Electrodynamics

Anthony Hartin<EMAIL_ADDRESS>University College London, Gower
Street, London WC1E 6BT, United Kingdom  Andreas Ringwald
<EMAIL_ADDRESS>DESY, Notkestrasse 85, 22607 Hamburg, Germany
Natalia Tapia<EMAIL_ADDRESS>Departamento de Fisica, Universidad de
Santiago de Chile, Casilla 307, Santiago, Chile

###### Abstract

It is a long-standing non-trivial prediction of quantum electrodynamics that
its vacuum is unstable in the background of a static, spatially uniform
electric field and, in principle, sparks with spontaneous emission of
electron-positron pairs. However, an experimental verification of this
prediction seems out of reach because a sizeable rate for spontaneous pair
production requires an extraordinarily strong electric field strength
$|\mathbf{E}|$ of order the Schwinger critical field, ${\rm
E}_{c}=m_{e}^{2}/e\simeq 1.3\times 10^{18}\ {\rm V/m}$, where $m_{e}$ is the
electron mass and $e$ is its charge. Here, we show that the measurement of the
rate of pair production due to the decays of high-energy bremsstrahlung
photons in a high-intensity laser field allows for the experimental
determination of the Schwinger critical field and thus the boiling point of
the vacuum of quantum electrodynamics.

###### pacs:

12.20.-m, 12.20.Ds, 12.20.Fv

##  I Introduction

Quantum Electrodynamics (QED) is one of the most successful theories in
physics. Its predictions for observables accessible by an ordinary
perturbative expansion in the electromagnetic coupling $e$, such as for
example for the anomalous magnetic moment of the electron, have been verified
experimentally to a very high accuracy.

There are, however, also observables which are inaccessible by ordinary
perturbation theory and whose prediction lacks an experimental verification.
Among them, the most famous is the rate (per unit volume $V$) of spontaneous
electron-positron pair production (SPP) in a strong static electric field $\bf
E$ Sauter:1931 ; Heisenberg:1936qt ; Schwinger:1951nm  ,

$\displaystyle\frac{\Gamma_{\text{SPP}}}{V}=\frac{m^{4}_{e}}{(2\pi)^{3}}\left(\frac{{\bf{|E|}}}{\rm{E}_{c}}\right)^{2}\sum^{\infty}_{n=1}\frac{1}{n^{2}}\exp\left(-n\pi\frac{\rm{E}_{c}}{{\bf{|E|}}}\right)\,,$
(1)

where

${\rm E}_{c}\equiv\frac{m_{e}^{2}}{e}\simeq 1.3\times 10^{18}\ {\rm V/m}$ (2)

is the so-called Schwinger critical field. Clearly, this rate is non-
perturbative in $e$,

$\Gamma_{\rm SPP}\propto\exp\left(-\pi\frac{m_{e}^{2}}{e{\bf{|E|}}}\right)\,,$
(3)

as typical for a process which can occur, for ${\bf|E|}\lesssim{\rm E}_{c}$,
only via quantum tunnelling. This so-called Schwinger effect and its analogues
have been suggested to play a role in many problems of phenomenological and
cosmological interest, ranging from black hole quantum evaporation
Hawking:1975sw ; Damour:1976jd ; Gibbons:1978pt ; Gavrilov:1996pz  to particle
production in hadronic collisions Casher:1979wy ; Andersson:1983ia ;
Biro:1984cf  and in the early universe Parker:1969au ; Birrell:1982 ;
Tangarife:2017rgl , to mention only a few. Unfortunately, there is no
practical way to produce a static electric field of this strength in the
foreseeable future111One possibility considered was the field at the crossing
of two intense laser beams Bunkin:1970 ; Brezin:1970 ; Popov:1971 ; Popov:1974
. However, the required laser peak power is in the hundreds of exawatt range
(for a laser operating in the optical range, focussed to the diffraction
limit) Ringwald:2001ib  and thus still far beyond the present technology.
However, the pair production can be strongly enhanced if one superimposes the
intense field at the laser focus with a further weak high frequency
electromagnetic field Schutzhold:2008pz ; Dunne:2009gi , paving the way to a
possible detectibility at the Extreme Light Infrastructure ELI
Narozhny:2015vsb ; Turcu:2016dxm .. Therefore, a direct laboratory test of
prediction (1) seems utopic.

Figure 1: Leading order Furry picture Furry:1951zz  Feynman diagram for OPPP.
The double line pointing forward (backward) in time represents an electron (a
positron) in the background of the electromagnetic field of the laser.

As an alternative to spontaneous pair production in a strong static electric
field, we consider here laser-assisted one photon pair production (OPPP) – the
decay of a high energy photon in the overlap with an intense optical laser
beam into an electron-positron pair, cf. Fig. 1. This process is kinematically
possible because the electron-positron pair can pick up momentum from the
laser photons. Already in the 1960‘s, when first lasers where developed, this
process has been identified as an opportunity to study the transition from
stimulated to spontaneous pair production in an external electromagnetic field
Reiss:1962 ; Narozhnyi:1965 .222The OPPP process continues to attract much
modern interest with new theoretical approaches Lv:2018qxy , analyses which
take into account real interacting laser pulses Jansen:2015idl ; Otto:2016fdo
; DiPiazza:2016maj ; DiPiazza:2015xva  and experimental schemes to realise the
process Lobet:2017vug . We will show in this paper that it offers a timely way
to probe the so far elusive boiling of the vacuum of QED and to determine the
Schwinger critical field experimentally.

The paper is organised as follows. We first examine the transition rate of the
OPPP process as it varies with the laser field intensity and the photon recoil
parameter (cf. Sec. II). The rate will be shown to be well described by
asymptotic expressions, which depend on the Schwinger critical field in a
simple way, for both low high laser intensities. Next we will consider in Sec.
III the effect of generating high energy photons via bremsstrahlung from a
foil on the asymptotic features of the rate. Finally we consider in Sec. IV
real experimental parameters and the effect of the finite duration of the
laser pulse and the variability of the laser intensity throughout the
interaction region on the determined value of the Schwinger critical field. We
conclude in Sec. V.

Figure 2: The dimensionless function $F_{\gamma}(\xi,\chi_{\gamma})$, Eq. (6),
describing the probability of laser-assisted OPPP, as a function of the laser
intensity parameter $\xi$, for different values of the photon recoil parameter
$\chi_{\gamma}$ (solid lines). The dotted (dashed) line shows the analytic
result valid at small (large) values of the intensity parameter, Eq. (8) (Eq.
(9)).

##  II One photon pair production

To leading order in Furry picture Furry:1951zz  perturbation theory (for a
recent review, see Ref. Hartin:2018egj ), the rate of laser-assisted OPPP can
be written in the form

$\Gamma_{\text{OPPP}}=\frac{\alpha
m_{e}^{2}}{4\,\omega_{\text{i}}}\,F_{\gamma}(\xi,\chi_{\gamma})\,,$ (4)

where $\alpha=e^{2}/(4\pi)$ is the fine structure constant,
$k_{i}=(\omega_{\text{i}},{\mathbf{k}}_{\text{i}})$, with
$\omega_{i}^{2}={\mathbf{k}}_{\text{i}}^{2}$, is the four-momentum of the
initial state photon, and $\xi$ and $\chi_{\gamma}$ are the laser intensity
parameter and the photon recoil parameter, respectively,

$\displaystyle\xi\equiv\frac{e\left|\mathbf{E}\right|}{\omega
m_{e}}=\frac{m_{e}}{\omega}\frac{\left|\mathbf{E}\right|}{{\rm
E}_{c}}\,,\quad\quad\quad\quad\chi_{\gamma}\equiv\frac{k\cdot
k_{i}}{m_{e}^{2}}\,\xi=(1+\cos\theta)\,\frac{\omega_{i}}{m_{e}}\,\frac{\left|\mathbf{E}\right|}{{\rm
E}_{c}}\,,$ (5)

in terms of the electric field $\left|\mathbf{E}\right|$ of the laser beam,
its frequency $\omega$, and its angle $\theta$ with respect to the direction
of the incident photon. The dimensionless function
$F_{\gamma}(\xi,\chi_{\gamma})$, for the idealized case that the
electromagnetic field of the laser beam can be described as a circularly
polarized infinite plane wave (IPW),333The general problem of assisted pair
production from counter propagating laser/photon pulses was considered,
primarily from a theoretical standpoint, in Ref. Fedotov:2013uja . Three
theoretical approximations were considered, the delta pulse method involving
the overlap of in and out states applicable for $\chi_{\gamma}\gg 1$, the
locally constant field approximation for $\xi\gg 1$, and perturbation theory
for $\xi\ll 1$. Instead, for the scheme proposed in this paper, with high
energy photons produced from foil bremsstrahlung, the initial states vary
widely across energy and spatial ranges. Preliminary analysis shows that the
IPW approximation, using local values of strong field parameters, are a
suitably accurate description for the real experiments being envisaged. is
given by a sum over the effective number of laser photons $n$ absorbed by the
electron-positron pair Narozhnyi:1965 ,

$\displaystyle
F_{\gamma}(\xi,\chi_{\gamma})=\sum\limits_{n>n_{\text{o}}}^{\infty}\int_{1}^{v_{n}}\frac{\text{d}v}{v\sqrt{v(v-1)}}\left[2\,J_{n}^{2}(z_{v})+\xi^{2}(2v-1)\left(J_{n\text{+1}}^{2}(z_{v})+J_{n\text{-1}}^{2}(z_{v})-2J_{n}^{2}(z_{v})\right)\right],$
(6)

with Bessel functions $J_{n}$ and

$n_{0}\equiv\frac{2\xi\left(1+\xi^{2}\right)}{\chi_{\gamma}},\quad\quad
z_{v}\equiv\frac{4\xi^{2}\sqrt{1+\xi^{2}}}{\chi_{\gamma}}\left[v\left(v_{n}-v\right)\right]^{1/2},\quad\quad
v_{n}\equiv\frac{\chi_{\gamma}\,n}{2\xi(1+\xi^{2})}.$ (7)

In Fig. 2, we display $F_{\gamma}(\xi,\chi_{\gamma})$ as a function of $\xi$,
for three values of $\chi_{\gamma}$. Clearly, at low laser intensities,
$\xi\ll 1$, laser-assisted OPPP appears to proceed perturbatively,
$F_{\gamma}\propto\xi^{2}\propto\alpha$, as expected from the necessity to
absorb at least one laser photon to allow for photon decay kinematically. In
fact, expanding $F_{\gamma}$ for small $\xi$ yields

$F_{\gamma}(\xi,\chi_{\gamma})=2\,\xi^{2}\left[\log\left(\frac{2\chi_{\gamma}}{\xi}\right)-1\right]+{\mathcal{O}}\left(\xi^{3}\log\,{\xi}\right).$
(8)

This behaviour reproduces the full result for laser-assisted OPPP up to values
of $\xi\sim 0.1$, cf. Fig. 2. As the laser intensity $\xi$ increases, the
threshold number of absorbed photons $n_{0}$ to produce an electron-positron
pair increases, and more and more terms in the summation over the number of
absorbed laser photons in Eq. (6) drop out of the probability, resulting in
the appearance of less and less pronounced maxima in $F_{\gamma}$, see Fig. 2.
At large $\xi$, finally, the probability of laser-assisted OPPP approaches a
finite value, the latter growing with increasing $\chi_{\gamma}$. Indeed, for
$\xi\gtrsim 1/\sqrt{\chi_{\gamma}}\gg 1$, $F_{\gamma}$ behaves as Ritus:1985

$\displaystyle
F_{\gamma}(\xi,\chi_{\gamma})=\frac{3}{4}\sqrt{\frac{3}{2}}\,\chi_{\gamma}\,{\rm
e}^{\left[-\frac{8}{3\chi_{\gamma}}\left(1-\frac{1}{15}\xi^{-2}+\mathcal{O}(\xi^{-4})\right)\right]}.$
(9)

This behaviour applies to a very good accuracy already for $\xi\gtrsim 1$ and
$\chi_{\gamma}\lesssim 1$, cf. Fig. 2. Importantly, we infer from Eq. (9) that
the asymptotic value of $F_{\gamma}$ is non-perturbative in the
electromagnetic coupling $e$ and that the rate of laser-assisted OPPP
asymptotes to

$\Gamma_{\text{OPPP}}\to\frac{3}{16}\sqrt{\frac{3}{2}}\,\alpha\,m_{e}\,(1+\cos\theta)\,\frac{}{}\,\frac{\left|\mathbf{E}\right|}{{\rm
E}_{c}}\exp\left[-\frac{8}{3}\frac{1}{1+\cos\theta}\frac{m_{e}}{\omega_{i}}\frac{{\rm
E}_{c}}{\left|\mathbf{E}\right|}\right]\,,$ (10)

ressembling the rate (1) of SPP in a constant electric field444The leading
term in the exponent in Eq. 10 is independent of the laser polarisation, while
the pre-factor depends on it Ritus:1985 . This has to be expected, since large
intensity parameter, $\xi\gg 1$, corresponds to a quasi-static electric field
of the laser, $\omega\ll e\,\left|\mathbf{E}\right|/m_{e}$, cf. Eq. (5).
However, in contrast to SPP, in laser-assisted OPPP the produced electron-
positron pair, in its rest frame, experiences an electric field enhanced by
the relativistic boost factor $\omega_{i}/m_{e}$. This enhanced electric field
is of order the Schwinger critical value ${\rm E}_{c}$, if the photon recoil
parameter is $\chi_{\gamma}\sim 1$, cf. Eq. (5). Hence, the Schwinger critical
field – the boiling point of the QED vacuum – can be determined in principle
experimentally from the measurement of the rate of laser-assisted OPPP at
$\xi\gtrsim 1/\sqrt{\chi_{\gamma}}\gg 1$. Next, we consider the effect of
enhancing the OPPP rate with the use of high energy bremsstrahlung photons.

##  III Bremsstrahlung photon pair production

Figure 3: Sketch of an experiment to produce high energy photons by
bremsstrahlung conversion in a high-$Z$ thin target and to cross them with a
laser beam to let them decay into electron-positron pairs. Switching off the
laser allows for a determination of the bremsstrahlung spectrum. Removing the
target allows in addition for the study of HICS, followed by OPPP, and of the
one-step trident process.

Note, that for a laser of frequency $\omega=1\,\text{eV}$, focussed to an
intensity $I$, corresponding to555This relation assumes that the intensity is
given by the modulus of the Pointing vector, i.e.
$I=\left|\mathbf{E}\right|^{2}$ for a plane wave.

$\xi=2.370\left(\frac{I}{10^{19}\,\text{W}/\text{cm}^{2}}\right)^{1/2}\left(\frac{1\,\text{eV}}{\omega}\right)\,,$
(11)

the condition $\xi\gtrsim 1/\sqrt{\chi_{\gamma}}$ leads to a lower bound on
the energy of the high energy photon,
$\omega_{i}\gtrsim{19.6\,\text{GeV}}\left(\frac{1\,\text{eV}}{\omega}\right)\frac{\left({2.37}/{\xi}\right)^{3}}{(1+\cos\theta)}$.
Unfortunately, there are no mono-energetic photon beams with energies in the
$\mathcal{O}(10)$ GeV range available. On the other hand, there are
$\mathcal{O}(10)$ GeV electron beams, notably the ones exploited by X-ray free
electron lasers, such as LCLS Arthur:2002ap  in Stanford or the European XFEL
Altarelli:2006zza  in Hamburg. Such an electron beam can be sent to a high-$Z$
target in which it is converted by bremsstrahlung into a collimated high
energy photon beam, which can then be crossed with a high-intensity laser
beam, cf. Fig. 3. Such an experiment to study laser-assisted bremsstrahlung
photon pair production (BPPP) has been envisaged long time ago in Ref.
Reiss:1971wf  and more recently discussed in Refs. Gemini_proposal:2010 ;
Turcu:2016dxm ; Blackburn:2018ghi . Here, we show that even after integration
over the bremsstrahlung spectrum, the Schwinger critical field can be
determined experimentally from the measurement of the total rate of electron-
positron pair production at large laser intensity.

Given the energy spectrum $\text{d}N_{\gamma}/\text{d}\omega_{i}$ of photons
generated by an electron impinging on the foil, the rate of laser-assisted
BPPP is given by

$\displaystyle\Gamma_{\text{BPPP}}$ $\displaystyle=$
$\displaystyle\frac{\alpha\,m_{e}^{2}}{4}\int_{0}^{E_{e}}\frac{\text{d}\omega_{i}}{\omega_{\text{i}}}\,\frac{\text{d}N_{\gamma}}{\text{d}\omega_{i}}\,F_{\gamma}(\xi,\chi_{\gamma}(\omega_{i}))$
(12) $\displaystyle=$
$\displaystyle\frac{\alpha\,m_{e}^{2}}{4}\frac{\chi_{e}}{E_{e}}\int_{0}^{\chi_{e}}\frac{\text{d}\chi_{\gamma}}{\chi_{\gamma}}\,\frac{\text{d}N_{\gamma}}{\text{d}\chi_{\gamma}}\,F_{\gamma}(\xi,\chi_{\gamma})\,,$

where $E_{e}$ is the energy of the incident electrons and $\chi_{e}\equiv
k\cdot k_{e}\,\xi/m_{e}^{2}=(1+\cos\theta)\omega E_{e}\xi/m_{e}^{2}$ is the
electron recoil parameter.

Figure 4: Number of $e^{+}e^{-}$ pairs produced per electron bunch ($6\times
10^{9}$ electrons of energy $E_{e}=17.5\,\text{GeV}$) impinging on the
converter target (thickness $X/X_{0}=0.01$) and per laser shot (duration
$35\,\text{fs}$) crossed with the bremsstrahlung photons, as a function of the
laser intensity parameter $\xi$, for different values of $\chi_{e}$. The
dashed line shows the analytic prediction resulting from (14), valid at
$\xi\gtrsim 1/\sqrt{\chi_{e}}\gg 1$.

For a target of thickness $X\ll X_{0}$, where $X_{0}$ is the radiation length,
the bremsstrahlung spectrum can be approximated by Tsai:1973py

$\omega_{i}\frac{\text{d}N_{\gamma}}{\text{d}\omega_{i}}\approx\left[\frac{4}{3}-\frac{4}{3}\left(\frac{\omega_{i}}{E_{e}}\right)+\left(\frac{\omega_{i}}{E_{e}}\right)^{2}\right]\frac{X}{X_{0}}\,,$
(13)

if one assumes complete screening.666We have checked via Monte Carlo
simulations with GEANT Agostinelli:2002hh  that (13) is valid in the parameter
range we use it in e.g. Figs. 4, 5 and 6. For the interpretation of the
experiment itself one does not have to rely on a theoretical prediction, since
the bremsstrahlung spectrum can be measured by switching off the laser, cf.
Fig. 3. This results, at high laser intensities, $\xi\gtrsim
1/\sqrt{\chi_{e}}\gg 1$, in the non-perturbative, $e^{-8/(3\chi_{e})}$
dependence of the laser-assisted BPPP rate,

$\displaystyle\Gamma_{\text{BPPP}}\to\frac{\alpha\,m_{e}^{2}}{E_{e}}\frac{9}{128}\sqrt{\frac{3}{2}}\,\chi_{e}^{2}\,e^{-\frac{8}{3\text{$\chi_{e}$}}\left(1-\frac{1}{15\xi^{2}}\right)}\frac{X}{X_{0}}\,,$
(14)

ressembling the behavior of the laser-assisted OPPP rate, Eqs. (4) and (9), if
one replaces in the latter expression $\chi_{\gamma}$ by $\chi_{e}$.
Therefore, the Schwinger critical field can be inferred from the asymptotic
behavior of laser-assisted BPPP for high laser intensities,

$\Gamma_{\text{BPPP}}\to\frac{9}{128}\sqrt{\frac{3}{2}}\,\alpha\,E_{e}\,(1+\cos\theta)^{2}\left(\frac{\left|\mathbf{E}\right|}{{\rm
E}_{c}}\right)^{2}\exp\left[-\frac{8}{3}\frac{1}{1+\cos\theta}\frac{m_{e}}{E_{e}}\frac{{\rm
E}_{c}}{\left|\mathbf{E}\right|}\right]\frac{X}{X_{0}}\,.$ (15)

Figure 5:  Top panel: Total number of $e^{+}e^{-}$ pairs produced per electron
bunch ($6\times 10^{9}$ electrons of energy $E_{e}=17.5\,\text{GeV}$)
impinging on the bremsstrahlung target (thickness $X/X_{0}=0.01$) and per
laser shot (duration $35\,\text{fs}$, laser frequency
$\omega=1.55\,\text{eV}$, corresponding to a laser wavelength of 800 nm)
crossed with the bremsstrahlung photons at an angle of $\theta=\pi/12$, as a
function of the laser intensity. The black dots exploit the full numerical
result for specific values of the intensity. The dashed line shows the
analytic prediction resulting from (14), exploiting the relations (11) and
(16). The dotted (dot-dashed) line shows the same analytic prediction, but for
the case where the value of the Schwinger critical field ${\rm E}_{c}$
deviates by a multiplicative factor of $\kappa=0.9$ ($\kappa=1.1$) from its
nominal value (2). Bottom panel: The laser intensity parameter $\xi$ (dotted)
and the electron recoil parameter (dashed), as a function of the intensity,
cf. Eqs. (11) and (16).

##  IV Experimental considerations

###  IV.1 Expected sensitivity to critical field

For the BPPP process, high energy electrons will impinge in bunches onto the
target. The electron beam of the European XFEL, for example, contains $6\times
10^{9}$ electrons of energy up to $E_{e}=17.5\,\text{GeV}$, with small energy
spread and a good emittance Altarelli:2006zza . The high intensities of the
laser are reached conceivably in laser pulses of duration around
$35\,\text{fs}$, as in the LUXE experiment which is currently in the design
phase for a proposal to the European XFEL Facility luxe . In Fig. 4, we show
the number of pairs produced per electron bunch and per laser shot expected in
this case. The solid lines are obtained from the numerical solution of Eqs.
(6) and (12), while the dashed lines exploit the analytic asymptotics (14).
Importantly, the latter approaches the former already at $\xi\gtrsim 1$ and
$\chi_{e}\lesssim 1$. Moreover, the number of produced pairs is favorably
high, even for the most interesting parameter range of large $\xi$ and small
$\chi_{e}$. From this we conclude that this type of experiment allows for a
determination of the Schwinger critical field ${\rm E}_{c}$ by fitting the
experimentally determined number of produced $e^{+}e^{-}$ pairs to the
theoretical prediction, (14) or, equivalently, (15), using ${\rm E}_{c}$ as a
fit parameter, cf. Figs. 5 (top panel) and 6.

Figure 6:  Total number of $e^{+}e^{-}$ pairs produced per electron bunch
($6\times 10^{9}$ electrons of energy $E_{e}$) impinging on the bremsstrahlung
target (thickness $X/X_{0}=0.01$) and per laser shot (duration
$35\,\text{fs}$, laser frequency $\omega=1.55\,\text{eV}$, intensity
$I=5\times 10^{18}\,\text{W}/\text{cm}^{2}$) crossed with the bremsstrahlung
photons at an angle of $\theta=\pi/12$, as a function of $E_{e}$. The black
dots exploit the full numerical result for specific values of the electron
beam energy. The dashed line shows the analytic prediction resulting from
(14), exploiting the relations (11) and (16). The dotted (dot-dashed) line
shows the same analytic prediction, but for the case where the value of the
Schwinger critical field ${\rm E}_{c}$ deviates by a multiplicative factor of
$\kappa=0.9$ ($\kappa=1.1$) from its nominal value (2).

In practice, in an experiment as sketched in Fig. 3, it will be easiest to
change the intensity of the laser and the energy $E_{e}$ of the electron beam.
In this case, the electron recoil parameter can be expressed as

$\chi_{e}=0.1576\left(1+\cos\theta\right)\left(\frac{E_{e}}{17.5\,\text{GeV}}\right)\left(\frac{I}{10^{19}\,\text{W}/\text{cm}^{2}}\right)^{1/2}\,.$
(16)

The predicted number of electron pairs per electron bunch and per laser shot
are presented in Fig. 5, for fixed electron beam energy, as a function of the
laser intensity, and in Fig. 6, for fixed laser intensity, as a function of
the electron beam energy. We infer that the number of produced pairs per
electron bunch and laser shot rapidly grows with increasing intensity $I$ and
electron beam energy $E_{e}$ to values above 1 already at $I\sim 5\times
10^{18}$ W/cm2 and $E_{e}\sim 14$ GeV. Therefore, the asymptotic regime for
the BPPP process should be experimentally accessible with reasonable accuracy
at the European XFEL faciltiy, requiring only modest parameters for a focussed
intense laser to ensure stable operation at a strong field experimental
interaction point. This will allow a precision comparison with the asymptotic
result according to Eq. (14), which sensitively depends on the value of ${\rm
E}_{c}$, cf. Figs. 5 (top panel) and 6: A variation of ${\rm E}_{c}$ around
its nominal value (2) by 10 % results in a change in the predicted rate by
nearly an order of magnitude.

By removing the target in the experimental setup of Fig. 3, the strong-field
trident process can be studied in addition. In its two-step variant, it occurs
via high intensity Compton scattering (HICS), followed by OPPP. Exploiting in
this way the energetic photons from HICS as an alternative source of high
energy photons, the asymptotic regime of the OPPP process can again be in
principle measured. However, the rate for HICS is considerably lower and is
cut off by the stepped Compton edge, compared to that of bremsstrahlung.
Nevertheless, trident pair production is of significant interest Bula:1997eh ;
Hu:2010ye ; Ilderton:2010wr ; King:2013osa ; Dinu:2017uoj  since its
measurement in the 1990s by the SLAC E144 experiment exploiting the 46.6 GeV
electron beam777At this energy, multiple laser photons are required in order
to produce the OPPP pair. Pure photon-photon pair production could only be
achieved with a 200 GeV electron beam. In the absence of such higher energy
electron beams, an alternative way to achieve sufficient centre of mass energy
to produce a pair may be via bremsstrahlung photon interaction inside a laser-
heated blackbody radiation cavity Pike:2014wha . A similar idea, but with a
different scheme to reach the pair production threshold via high energy
photons, is proposed by Ribeyre:2015fta . In this case, two interacting beams
of MeV level photons are produced by impinging lasers on solid or gas targets.
Though these schemes potentially produce copious amounts of pairs, the
asymptotic regime of interest in this paper, $\xi\gtrsim 1$, does not appear
to be reachable in the foreseeable future. of the Stanford Linear Accelerator
Bamber:1999zt  and is an important additional strong field process that can be
measured by the experiment described in this paper.

In order to quantify the expected experimental accuracy, detailed simulations
will have to be carried out. These will include a GEANT Agostinelli:2002hh
model of the converter target to produce a large flux of energetic photons.
Macro-particles, representing a train of electron bunches varying randomly
within known beam conditions, will also be required. An accurate
representation of laser pulse shape and jitter will be needed, as well as a
full accounting for crossing angles and beam overlap. Along with GEANT, a full
strong field QED particle-in-cell code including higher order processes has
also to be developed.

###  IV.2 Effect of finite laser pulse length

A substantial amount of analytical work on strong field processes over the
last decade has concerned itself with the fact that a real experiment will
utilise a real laser pulse. The infinite plane wave approximation (IPW) which
we have exploited in order to determine the OPPP transition rate neglects the
finite length and shape of the laser pulse. Allowance for these realistic
features has the potential to significantly complicate the transition rate
expression. Nevertheless, various additional approximations have been employed
in order to produce manageable results.

If the pulse length is not too short in comparison to the laser wavelength,
the slowly varying envelope approximation (SVEA) can be employed. This
requires the assumption that the pulse envelope does not change much over the
course of one wavelength of the laser field. For LUXE, an 800 nm laser with a
35 fs pulse length is envisaged luxe . This gives $N\mathcal{\approx}12$
periods within the pulse, meaning that the SVEA may be a reasonable
assumption.

The existing SVEA calculations can be utilised with the aim of testing the
asymptotic limits of the OPPP rate. Returning to the matrix element of the
OPPP process, the IPW describes a definite number of laser photons
participating in the process expressed by a delta function. A finite pulse
however smears out the longitudinal light cone (four-vector,
$z\equiv(\vec{z}_{+},z_{-})$) momentum transfer and the delta function is
replaced with a pulse Heinzl:2010vg ,

$\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle\sum_{n}\delta^{(4)}\\!\left(z\right)\,M_{n}\quad,z\equiv
p+q-k_{i},\hfill\quad\text{(IPW)},$ $\displaystyle S_{fi}$ $\displaystyle=$
$\displaystyle\sum_{n}\delta^{(3)}\\!\left(\vec{z}_{+}\right)\frac{\sin N\pi
z_{-}\operatorname{sinc}\pi(z_{-}-n)}{\sin\pi
z_{-}}\,M_{n},\hfill\quad\text{(SVEA)}.$

For a square pulse, there is sub-threshold as well as envelope behaviour.
However, the delta comb structure is largely restored as the pulse length
increases to $N=4$, suggesting that the IPW approximation is satisfactory for
the experimentally planned pulse length of $N\approx 12$. To be sure of the
asymptotic behaviour of the OPPP process though, a full calculation of the
transition probability with a realistic pulse shape is necessary.

Such a calculation has been performed and analysed for a range of intensities
and pulse lengths in Refs. Nousch:2012xe ; Titov:2012rd ; Titov:2013kya ;
Titov:2015pre . In the range of very high and very low intensity parameter
$\xi$ the calculation of the OPPP with finite pulse length was performed
analytically. In the intermediate range, the calculation was carried out
numerically, across a range of photon energies and incident angles. The IPW
was shown to be a very good assumption by the time the pulse length increased
to $N=10$, meaning that the IPW analysis performed in this paper is suitable
for a LUXE type experiment.

###  IV.3 Effect of variable laser intensity

In a real strong field experiment, with a laser pulse focussed to a small spot
size, the infinite plane wave assumption can also be questioned in regards to
directions perpendicular to the pulse propagation. One approach is to develop
electron wave functions embedded explicitly in an external field 4-potential
with an appropriate transverse description. The equations of motion can be
solved approximately, by assuming that the electron energy is the largest
dynamical parameter, which is reasonable for experiments with ultra
relativistic electrons DiPiazza:2015xva .

The solutions which allow for the transverse focussing lead to the conclusion
that the Volkov solution with IPW assumption can be used as long as the local
value of the strong laser intensity, at the point of pair production, is taken
into account. This is necessarily the case for any strong field PIC code that
simulates the pair production in a real bunch/pulse interaction Hartin:2018egj
.

The use of a strong field PIC code in future experimental studies will allow
the full variation of the laser pulse, both longitudinally as well as
transversely, to be taken into account. Experimentally, the high intensity
laser pulse is expected to have a flat intensity in the transverse direction,
but a Gaussian shape longitudinally. A real experiment will seek to account
for the Gaussian shape with some sort of analysis filter or by making
kinematic cuts to detector signals.

Another experimental possibility is to deliberately form the pulse length so
that the laser intensity is relatively constant throughout the interactions of
interest. Since there is a relationship of the laser intensity $I$ to pulse
energy $E$ as well as pulse length $\tau$ and pulse width $w_{0}$, it is
possible to adjust the former in order to set the latter to desired values,

$I=\frac{E}{\pi w_{0}^{2}\tau}.$ (17)

A further, extensive experimental study will be necessary in order to
determine suitable operating conditions. Such a study will take into account
the angular spread of bremsstrahlung from the foil, the distance between foil
and strong laser interaction point, and the characteristics of the detector
system recording the resultant produced pairs.

##  V Conclusions and Outlook

The theory study and experiment, presented here, already promises us the first
ever measurement of the Schwinger critical field value, through an asymptotic
limit. Many additional, important strong field effects can be experimentally
tested through further theoretical and phenomenological studies.

One may ask what happens if the quantum recoil parameters continue to increase
in value with either increasing gamma energy and/or electromagnetic field
intensity Fedotov:2010ja . In such a case, our strong field experiment would
probe both smaller distances and the quantum vacuum would be increasingly
polarised. In such circumstances, there is good reason to think that higher
order, strong field processes may increasingly play a role. There is not only
the opportunity to study and search for such higher order processes in
dedicated strong field experiments, there is also the ability to question non-
perturbative quantum field theory itself. The Furry picture Furry:1951zz ,
which is the semi non-perturbative theory from which most strong field QED
predictions are made, includes the quantum recoil parameters in its effective
coupling constant. Theoretical estimates Narozhnyi:1979at ; Ritus:1985  put
the effective coupling constant at $\alpha\chi^{1/3}$, meaning that the theory
breaks down at a high enough value of the recoil parameters. Whereas this
breakdown regime is experimentally some distance away, the value of the
effective coupling constant through the higher order terms is possibly in
reach.

Acknowledgments.

A.R. would like to thank Paola Arias, Holger Gies, Axel Lindner, Gerhard
Paulus, Javier Redondo, and Andreas Wipf for early discussions on the
possibility of experiments exploiting the electron beam of the European XFEL
and a high-intensity laser to measure non-linear and non-perturbative
processes predicted by QED. We would like to thank Beate Heinemann for a
careful reading of the manuscript and useful suggestions. A.H. acknowledges
support from a Leverhulme Trust Research Project Grant RPG-2017-143 and by
STFC, United Kingdom. N.T. thanks to the Conicyt scholarship 21160064 and to
the University of Santiago de Chile.

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