pionic atoms with atomic numbers $Z$ up to 92, and suggested tuning of the
photon-energy spread are presented in Ref. [280]. These estimates indicate
that $p_{max}$ can reach up to $\sim 10^{10}$ atoms per second, which offers
orders-of-magnitude improvement compared with other production methods, e.g.,
producing pionic atoms by capturing free pions as in Ref. [281], where
$\approx 10^{5}$ pionic helium atoms per second are produced at a 590 MeV
proton facility.

For the estimates presented in Ref. [280], cross sections for producing free
pions with final nuclei in the ground state are needed. While there exist such
experimental data for light nuclei [282, 283, 284, 285, 286], we are not aware
of such data for heavy nuclei. There are studies of charged-pion
photoproduction from heavy nuclei [287, 288, 289] using bremsstrahlung-photon
beams and radiochemical measurements, where the measured cross section
corresponding to the final nucleus in the ground state actually include
contributions of decays from excited states. To measure the cross section for
producing free pions with final nuclei in the ground state, monochromaticity
of the GF photons would be useful, especially for heavy nuclei where the
energy difference between the nuclear ground state and the first excited state
is typically smaller than in light nuclei. We can tune the energy of the
narrow-band gamma rays at the GF precisely so that excited nuclear states
cannot be produced. These cross section data would be invaluable for
estimating the rates of bound-pion photoproduction discussed in this section.

We note that gamma rays at the GF can be utilized to study coherent
photoproduction of $\pi^{0}$ [290, 291] near threshold including the cross
section dependence on $A$, the mass number of the nucleus. An advantage is
that, with narrow-band gamma rays tuned close to the threshold, contributions
other than coherent photoproduction (i.e., incoherent production and breakup
reactions [291]) are energetically suppressed. Coherent pion photoproduction
is further discussed in the following sections.

####  4.8.2 Neutron-skin measurements in coherent ($\gamma,\pi^{0}$) reactions

The all-important nuclear equation of state (EOS) relates objects with orders-
of-magnitude different sizes, describing the structure and stability of
nuclear matter as well as the properties of neutron stars [292]. In
particular, the authors of Ref. [293] used the EOS to likely rule out the
neutron-star nature of an object with 2.6 solar masses observed via detecting
gravitational waves [294]. The recent LIGO observation of a neutron-star
merger constrains the tidal neutron-star polarizability [295].

In turn, tidal polarizability crucially depends on the linear slope $L$ of the
density dependence of the symmetry energy. The neutron skin $R_{n}-R_{p}$, the
difference of the radii of the neutron and proton distributions within a
nucleus, is also a sensitive probe of isovector interactions within nuclei and
is strongly correlated with $L$ [296], as shown in Fig. 15.

Figure 15: Correlation of the neutron skin of 208Pb with the slope $L$ of the
density dependence of symmetry energy. Blue circles indicate predictions of
various models. The precision of the experimental measurements of the neutron
skin of 208Pb (but not the central values) is indicated by the error bars at
the green diamond [297, 298] and the red/magenta circle [292, 299]. The plot
is adopted from Ref. [296], and is copyrighted by the American Physical
Society.

While proton distributions are known with high precision across the nuclear
chart [300], information on neutron distributions is scarce. The idea of
accessing neutron skins with parity-violating electron scattering on nuclei
relies on the fact that the neutral weak boson $Z^{0}$ couples predominantly
to the neutrons. Recent measurements at the Jefferson Lab [297, 298] furnished
precise information on the neutron skin of lead, leading to tighter
constraints on the parameters of the EOS [301]. Further experiments to extract
neutron skins from electron scattering are under analysis [302, 303] or
planned [292, 299].

Additional ways to access neutron skins include measuring PV in atoms [304],
and coherent $\pi^{0}$ photoproduction that was used by the A2 collaboration
at Mainz to investigate neutron skin of 208Pb using photons in the energy
range of 180 to 240 MeV [305].

Figure 16: Coherent neutral pion photoproduction in the $\Delta(1232)$ region.

The reaction mechanism is mainly the $\Delta(1232)$ resonance excitation, as
shown in Fig. 16, by isospin symmetry the same on protons and neutrons, so
that the cross section can be brought in correspondence with the baryon
density $Z\rho_{p}(r)+N\rho_{n}(r)$. Using precise knowledge of nuclear charge
(i.e. proton) densities [300], one can then deduce the neutron density and the
neutron skin $R_{n}-R_{p}$, using the definition of the proton and neutron
radii,

$R_{n,p}^{2}=\int d^{3}\vec{r}\,r^{2}\rho_{n,p}(r).$ (19)

The GF with its unique characteristics can allow one to measure coherent
$\pi^{0}$ photoproduction on various nuclear targets with an unprecedented
precision and over a wide range of photon energies and momentum transfers. In
order to reconstruct a $\pi^{0}$ from the two decay photons one will need a
$4\pi$-detector similar to the Crystal Ball used in the Mainz experiment
[305]. Importantly, the 100% polarization of the photon beam in GF will allow
to suppress $\pi^{0}$’s coming from incoherent processes which do not give
access to the neutron skin. These incoherent contributions constitute a
slowly-varying background that extends over a wide range of momentum transfer,
limiting the applicability of the form factor fit beyond the second
diffraction minimum. As a result, they contribute a significant part of the
systematic uncertainty (see, for example, Ref. [306] which discusses
additional theory inputs that may affect the interpretation of the
experimental data). A measurement of coherent $\pi^{0}$ photoproduction at GF
thus offers a promising avenue for extracting accurate information about
neutron skins alternative to parity-violating electron scattering and atomic
parity violation.

####  4.8.3 Pion decay constant from the Primakoff effect

The same process, coherent $\pi^{0}$ photoproduction, but measured at very
forward angles, provides a direct way to measure the radiative $\pi^{0}$ width
$\Gamma_{\pi^{0}\gamma\gamma}$ via the Primakoff effect, see Fig. 17.

Figure 17: Neutral pion photoproduction via the Primakoff effect.

The differential cross section for this process is given by

$\frac{d\sigma^{\rm
Prim}}{d\Omega}=\frac{8\alpha\Gamma_{\pi^{0}\gamma\gamma}}{m_{\pi}^{3}}Z^{2}F_{\rm
Ch}^{2}(t)\frac{\nu^{4}\beta_{\pi}^{2}}{t^{2}}\sin^{2}\theta_{\pi},$ (20)

with $m_{\pi}$ being the pion mass and $Z,\,F_{\rm Ch}$ the charge and charge
form factor of the nucleus, respectively. Furthermore, $\nu$ is the photon
energy, $\beta_{\pi}=\sqrt{1-m_{\pi}^{2}/\nu^{2}}$ the pion velocity, $t$ the
momentum transfer to the nucleus and $\theta_{\pi}$ the angle the pion makes
with respect to the incoming photon momentum. The most recent PrimEx-II
experiment at JLab [307] combined with older experimental determinations leads
to $\Gamma(\pi^{0}\to\gamma\gamma)=7.806(052)(105)$ eV, with the first and the
second uncertainty referring to the statistical and systematic one,
respectively. This 1.5% determination improved over its predecessor PrimEx-I
[308] by almost a factor of 2. Historically, the chiral anomaly
$\pi^{0}\to\gamma\gamma$ has served as an important milestone for establishing
the chiral perturbation theory (ChPT), the low-energy effective theory of
quantum chromodynamics (QCD), and has been considered the perfect place for
comparing ChPT predictions to experimental results ever since. Currently, the
most precise next-leading order (NLO) ChPT prediction for the $\pi^{0}$
radiative width [309, 310] reads $\Gamma(\pi^{0}\to\gamma\gamma)=8.10(8)$ eV
with a 1% uncertainty, showing a mild $2\sigma$ tension with the experimental
result, see Fig. 18.

Figure 18: Theoretical predictions for the $\pi^{0}$ decay constant: Ref.
[311] (Chiral Anomaly), Ref. [312] (IO QCD) Ref. [313] (KM NNLO) Ref. [310]
(AM NLO) and Ref. [309] (GBH NLO) in comparison with results from earlier
experiments [314, 315, 316, 317] along with the recent ones by the PrimEx
collaboration [308, 307]. The figure is adopted from Ref. [307], and is
copyrighted by Science Magazine.

The main challenge in extracting the $\pi^{0}$ radiative width from a pion-
production experiment on a nucleus is due to the presence of coherent and
incoherent strong-interaction background channels,

$\frac{d\sigma}{d\Omega}=\frac{d\sigma^{\rm
Prim+Coh}}{d\Omega}+\frac{d\sigma^{\rm Incoh}}{d\Omega},$ (21)

which need to be separated from the Primakoff signal. The latter is suppressed
with respect to those background processes by $\alpha^{2}$, but has a
characteristic $1/t^{2}$ behavior which leads to a strong peak at forward
angles. This peak resides close to the minimal possible momentum transfer
$t_{min}\approx m_{\pi}^{4}/4\nu^{2}$, while strong-interaction contributions
reside at larger values of $t$. This kinematic separation is the lever arm
that is used to reliably extract the Primakoff signal from the pion-production
data. As an example for this separation, in the PrimEx kinematics ($\nu\approx
5$ GeV) for the 208Pb target the maximum of the Primakoff peak of
$d\sigma/d\theta_{\pi}$ resides at $\theta_{\pi}\approx 0.03^{\circ}$, while
the coherent nuclear contribution is peaked at $\sim 1^{\circ}$. For
comparison, for $\nu\approx 400$ MeV accessible for secondary photons in the
Gamma Factory the peak positions are $\theta_{\pi}\approx 4.5^{\circ}$ and
$\sim 11^{\circ}$, respectively. The shift of the Primakoff peak to larger
values of $t$ automatically leads to its amplitude being reduced according to
the $1/t^{2}$ dependence in Eq. (20) with respect to the background signal
that has a less steep $t$-dependence. To remedy this situation, the GF may
offer a significantly higher statistics as compared to PrimEx-II that uses a
tagged Bremsstrahlung photon source. With a higher statistics and assuming
that the systematic effects are well under control (which is likely the case
due to the lower energy), one can envision the possibility to independently
determine all the ingredients by a fit over a wide angular range. This nicely
connects the here envisioned Primakoff effect measurements with the
measurements of neutron skins, the subject of the previous subsection, within
the same GF experiment.

If a significant flux of tertiary photons can be produced with the GF, one can
go to higher energies to conduct a Primakoff experiment in a more conventional
setting.

###  4.9 Precision measurement of nuclear $E1$ polarizabilities

Precise knowledge of electric dipole nuclear polarizabilities (discussed in
Sec. 4.7 for light nuclei) is also important for eliminating uncertainties in
determination of the neutron skins [318, 319]. Polarizability determinations
via measuring gamma-ray transmission are part of the scientific program of the
ELI-NP facility [181]. The 208Pb$(\gamma,\gamma^{\prime}$) reactions were
studied also at the electron-bremsstrahlung source ELBE [320]. The GF will
offer a possibility of carrying out such studies on a broad range of targets
with much higher statistical sensitivity and spectral resolution. A
measurement program of nuclear electric polarizabilities will complement
coherent $\pi^{0}$ photoproduction method described in Sec. 4.8.2.

###  4.10 Delta-resonance region and continuum effects

The Delta resonance that can be excited in all nuclei is located at around 300
MeV excitation energy and has a large width close to 100 MeV. Of special
interest is the pionic decay of this resonance. First of all, pion production
on loosely bound nucleons can provide unique information on the wavefunction
of exotic nuclei, especially if it would be possible to follow the evolution
of this process along a chain of nuclei, from stable to unstable isotopes.

There is another non-trivial interest in the detailed study of the Delta-
resonance region. A Mainz experiment in the late 1990s [321] claimed the
existence of narrow peaks on the background of the broad Delta resonance (see
Fig. 19). Later, this result was put into doubt, and the experimental
situation remains unclear. Simple theoretical arguments [322, 323] show that
in the energy region of the Delta resonance there are several possible nuclear
Delta + nucleon hole states with the same quantum numbers which are strongly
coupled to the pion channel.

Figure 19: Narrow resonances on top of the broad Delta resonance as claimed in
Ref. [321]. Figure adapted from Ref. [323].

This is a typical situation of several intrinsic states interacting both
through internal nuclear forces and through the virtual decay to the continuum
channel. In such cases, a strong continuum coupling leads to the phenomenon of
superradiance when a special collective combination (state) acquires a broad
width (short superradiant lifetime), while other states with the same quantum
numbers become long-lived narrow resonances. The theory of such many-body
superradiance, an analog of the theoretically predicted and later discovered
electromagnetic Dicke superradiance in optics, was developed earlier [324,
325, 326], supported by numerical simulations for various systems with quantum
signal propagation and interaction through the continuum, as reviewed in Ref.
[327]. The simplest explanation, according to P. von Brentano [328], is in the
fact that interaction of quantum states through a common decay mode, in
contrast to the usual Hermitian perturbation, leads to the width (imaginary
part of the complex energy) repulsion and vastly different lifetimes.
Therefore, the presence of narrow pionic resonances on the background of a
“superradiating” Delta peak is not excluded theoretically and should be
further explored experimentally. Narrow resonances of similar nature are
possible also at higher energy as a result of specific quark structures.

Detailed studies of the fine structure of the Delta resonance possible with an
intense gamma beam of appropriate energy with different nuclear targets would
open a new branch of nuclear physics of relatively high energy where particle
physics is combined with details of the nuclear many-body problem.

###  4.11 Parity-violation in photo-nuclear processes

PV signatures in the interaction of real photons with nucleons and nuclei
arise due to the SM weak interaction among quarks. This underlying mechanism,
however, needs to be matched onto hadronic observables. As a result of this
matching, a number of effective PV hadron-hadron couplings are generated.
Those involving the lightest mesons $\pi,\rho,\omega$ are generally expected
to govern low-energy processes, and a minimal set was introduced by
Desplanques, Donoghue and Holstein (DDH) [329]. Among these couplings the PV
$\pi N$ coupling $h^{1}_{\pi}$ is expected to play a major role since it is
the only non-derivative pion-nucleon coupling. Naive dimensional analysis
leads to an expectation $h^{1}_{\pi}\sim G_{F}\Lambda_{\chi}F_{\pi}\sim
10^{-6}$ with $\Lambda_{\chi}\approx 1$ GeV the chiral symmetry breaking
scale, and $F_{\pi}=92.4$ MeV the pion decay constant. The “DDH preferred
value” reads $h^{1}_{\pi}\sim 5\times 10^{-7}$ [329].

Isotope | Transition | $T_{1/2}$ | Type | Admixture | $T_{1/2}$ | Type |  $\Delta E$ (keV) | 104 $A^{\rm PVTC}$ | Ref.
---|---|---|---|---|---|---|---|---|---
18F ($1^{+}$) | $0^{-}(1081)\to{g.s.}$ | 19 ps | $E1$ | 0${}^{+}(1042)\to g.s.$ | 1.8 fs | $M1$ | 39 | $-10\pm 18$ | [330]
19F (${1}/{2}^{+}$) | ${1}/{2}^{-}(110)\to g.s.$ | 0.6 ns | $E1$ | ${1}/{2}^{+}-{1}/{2}^{-}$ |  |  | 110 | $-0.68\pm 0.18$ | [331, 332]
21Ne (${3}/{2}^{+}$) | ${1}/{2}^{-}(2789)\to g.s.$ | 81 ps | $E1/M2$ | ${1}/{2}^{+}(2795)\to g.s.$ | 5.5 fs | M1 | 5.7 | $24\pm 29$ | [333]
|  |  |  |  |  |  |  | $8\pm 14$ | [334]
${{}^{180}}\textrm{Hf}\,(0^{+})$ | $8^{-}(1142)\to 6^{+}(641)$ | 5.53 h | $M2/E3$ | $8^{+}(1085)\to 6^{+}(641)$ | 2 ps | $E2$ | 57 | $-148\pm 26$ | [335]
Table 5: Gamma transitions with an enhanced PV component in isotopes of
fluorine, neon and hafnium. Energies are given in keV. Parameters are from the
ENSDF database [32].

The hierarchy of DDH couplings has recently been questioned from a new
perspective that combines chiral effective theory with the large-$N_{c}$
approach [336], where $N_{c}$ is the number of colors. Within this new
paradigm, couplings associated with isovector transitions involving spin (or
isoscalar transitions without spin) are large, while the opposite spin-isospin
correlation is suppressed. In meson-nucleon interactions without PV, this
pattern is supported by a large $\pi N$ (spin-isovector), vector $\omega N$
(no spin-isoscalar) and magnetic $\rho N$ (spin-isovector) couplings. From
this perspective, $h^{1}_{\pi}$ (no spin-isovector) should be suppressed. The
first non-zero determination of $h^{1}_{\pi}$ in the $\vec{n}+p\to d+\gamma$
reaction [105] obtained $h^{1}_{\pi}=(2.7\pm 1.8)\times 10^{-7}$, about half
the DDH best value, leaving the question, whether or not the large-$N_{c}$
hierarchy is realized in nature, open.

In Sec. 3 low-energy, nonresonant Compton processes with a PV signature
enhanced by low-lying parity doublets is considered, with primary laser
photons backscattered off the ion beam, with the limitation $\omega\lesssim
60$ keV in the rest frame of the ion. When using secondary photons on a fixed
target, higher energies will be achieved with a possibility to tune the photon
energy to resonance transitions of interest (see also Sec. 2). Parameters of
relevant resonance transitions with a significant enhancement due to parity
doublets are listed for several nuclei in Table 5. The resonance will de-
excite by emitting photons which will need to be counted in a $\approx 4\pi$
integrating photon detector, and the dependence of the photon flux in the
detector on the circular polarization of the incident photon beam will give
access to the PV asymmetry. Due to the resonance kinematics, the enhancement
is much stronger than in the nonresonant case, compare to Table 3. On the
other hand, lower rates will be achieved: the necessity to excite a narrow
resonance will reduce the rate accordingly, due to the energy-angle resolution
correlation of the secondary-photon GF beam. Still, much larger asymmetries
will dominate over lower rates in FOM$=\mathrm{Rate}\times A^{2}$. While there
is a substantial overlap with Table 3, the PV asymmetries listed here are
significantly enhanced due to the resonant nature of the process. In the past,
PV in 19F and 21Ne (see the respective entries in Table 5) was studied with
nuclear, rather than photon, polarization. As shown in Ref. [337], the two
observables are fully analogous and, when integrated over the full solid angle
of the emitted photon, are equal.

In general, a global analysis of PV observables, typically performed with
nuclear systems, is complicated by the need to embed these PV meson-nucleon
couplings into PV nuclear potentials and solve nonrelativistic equations which
may bear significant uncertainties. As a result, different couplings are
intertwined with complicated nuclear effects and may be difficult to
disentangle. An alternative way is to move to higher energies where one can
directly produce light mesons. Tuning the energy of the photons to the
relevant domain one may hope to enhance–if not isolate–the dominant
contribution.

Threshold $\pi^{+}$ photoproduction on the proton with circularly polarized
photons was proposed in Ref. [338] as a way to access $h^{1}_{\pi}$. The PV
asymmetry is practically isotropic and is given by

$A_{\gamma}\approx\frac{\sqrt{2}F_{\pi}(\mu_{p}-\mu_{n})}{|g_{A}|M_{N}}h^{1}_{\pi}\approx
0.52h^{1}_{\pi},$ (22)

with $g_{A}\approx-1.27$ being the nucleon axial coupling and $\mu_{p,n}$
being the proton and neutron magnetic moment, respectively. The sensitivity to
the value of $h^{1}_{\pi}$ compares well to that in the nuclear $\vec{n}+p\to
d+\gamma$ process for which the asymmetry is
$A_{\gamma}\approx-0.11h^{1}_{\pi}$. To estimate the rate and FOM, we note
that the total cross section for $\pi^{+}$ photoproduction off the proton at
180 MeV photon energy is $\approx 100\,\mu$b. Since the asymmetry is roughly
energy-independent, no particularly high energy resolution is necessary. Let
us assume $10^{-3}$ out of $10^{17}$ photons arrive on target; hydrogen target
is probably not the best option, so let us assume it is carbon – then we will
be looking at $\pi^{\pm}$ in the detector. The asymmetry is roughly the same
but the cross section is $Z^{2}=36$ times that on the proton, for each pion
species. Assuming a 1 cm3 target, its cross section will have $\approx
10^{23}$ 12C nuclei to interact with. So, we get
$10^{14}\times(3.6\times{10^{-27}}$ cm${}^{2})/1\,$cm${}^{2}\times
10^{23}=3.6\times 10^{10}$ $\pi^{+}$ per second. For asymmetries $\approx
1.4\times 10^{-7}$ we get FOM$\sim 7\times 10^{-4}\,$s-1.

A similar process was proposed with a semi-inclusive $\pi^{+}$
electroproduction at threshold $p(\vec{e},\pi^{+})e^{\prime}n$.
Photoproduction has an advantage over electroproduction in that there is no
background from $Z$-exchange, and all the signal is due to $h^{1}_{\pi}$.

As one moves to the resonance region, a direct PV photoexcitation of a nucleon
and $\Delta$ resonances become possible. Most notably the PV $\gamma N\Delta$
coupling $d_{\Delta}^{+}\,(d_{\Delta}^{-})$ [339] responsible for a $E1$
$\gamma p\to\Delta^{+}$ ($\gamma n\to\Delta^{0}$) transition, respectively,
gives rise to the asymmetry

$A_{\gamma}^{\pm}\approx-\frac{2d_{\Delta}^{\pm}}{C_{3}^{V}}\frac{M_{N}}{\Lambda_{\chi}},$
(23)

with $C_{3}^{V}=1.6$ being the usual parity-conserving coupling inducing the
M1 $\gamma N\Delta$ transition. A measurement of $\pi^{-}$ electroproduction
on the $\Delta$ resonance on a deuterium target by the G0 collaboration [340]
obtained

$A_{\gamma}^{-}=-(0.36\pm 1.06\pm 0.37\pm 0.03)\,{\rm ppm},$ (24)

with the three uncertainties being the experimental statistical, theoretical,
and experimental systematic, respectively. Accordingly, a loose bound
$d_{\Delta}^{-}=(3\pm 10)\times 10^{-7}$ was extracted. It can be expected
that the GF will be able to measure these small asymmetries $\sim 10^{-8}$ due
to high intensity and $100\%$ photon polarization, together with relatively
high resonant cross sections $\approx 300\,\mu$b.

Another promising task for the GF is the resonant PV $\Sigma$ hyperon
production $\vec{\gamma}+p\to\Sigma^{+}$ with a subsequent decay to the $\pi
N$ final state. Until now, the reverse $\Sigma^{+}\to\vec{\gamma}+p$ decay
with a branching ratio $1.23(5)\times 10^{-3}$ has been accessible, with the
scope of studying the asymmetry $a_{\gamma}=0.76\pm 0.08$ [341]. The competing
process is the nonresonant $\gamma p\to\pi N$ process which, at the
$\approx$200 MeV incident beam energy, is well understood, and the asymmetry
for the inclusive reaction is due to the PV coupling $h^{1}_{\pi}$ leading to
$10^{-8}$ asymmetries. While the contribution of $\Sigma$ production is
suppressed with respect to nonresonant pion photoproduction, the figure of
merit introduced in Sec. 3 favors $\Sigma$ photoproduction: it depends on the
cross section linearly, and quadratically on the asymmetry. With a precise
tune of the photon energy to the $\Sigma$ hyperon position, the large
asymmetry more than compensates for low rate.

##  5 Experimental considerations with fixed targets

###  5.1 Thermal-load issues

In designing fixed-target experiments at the GF, it is important to realize
that the secondary photon beam carries significant power. With the maximum
gamma-ray energy and a total photon flux (at all energies), the power carried
by the photons is 3 MW. To make such energy load manageable for the target, it
may be beneficial to position the target as far away as possible from the
interaction region. With $\gamma\approx 10^{3}$ and neglecting the initial PSI
angular spread, positioning a 2 m diameter target at a distance of 1 km would
match the target size to the diameter of the “spot” produced by the gamma rays
emitted in the $\approx 1/\gamma$ cone.

###  5.2 Parallel spectroscopy with spatially resolved detection

The strong correlation of the photon energy with its angle of propagation (see
Sec. 1.1) suggests a possibility of conducting experiments utilizing the
entire flux of the GF secondary photons. This requires a detector providing
spatial resolution (Fig. 20). With this arrangements, a photon with a given
energy will hit a ring-shaped area on the target.

Figure 20: The parallel spectroscopy configuration.

Parallel spectroscopy offers the advantage of efficient utilization of the GF
running time as opposed to scanning arrangements, where a narrow slice of the
gamma-ray energies is selected via collimation and the energy is tuned, for
instance, by simultaneously adjusting the PSI $\gamma$ factor and tuning the
primary-photon energy.

###  5.3 Gamma-ray/X-ray and gamma-ray/gamma-ray pump-probe spectroscopy

Pump-probe techniques are ubiquitous in atomic, molecular, and condensed
matter physics, and are used in many variants, both in time and spectral
domain. Here the system under study is first subject to the “pump” photons
that cause the system to undergo a transition, for example, an excitation of
an atomic or molecular state or melting a crystalline lattice. The changes in
the system are then monitored via its interactions with the “probe” photons.

Pump-probe spectroscopy, in novel regimes, is also possible with the GF (Fig.
21). The pump and probe pulses of secondary GF photons can be produced using
two laser pulses. This gives an option for both the pump and the probe being
gamma rays. We can, for example, have two primary laser pulses interacting
with the same ion bunch. The energy of the pump and probe can be tuned within
$\approx 10^{-3}$ (the energy spread of the PSI that can be maintained in the
ring) by selecting different parts of the ion energy distribution. One can
also do “gross” tuning by choosing different atomic transitions in the PSI. It
could be possible to imprint a temporal structure onto the gamma-ray pulse in
order to perform pump-probe experiments with sub-ps resolution. The duration
and timing of the gamma-ray pulses will be determined by geometric overlap
between the sufficiently short laser pulse and the ion beam. With a $\approx
10\,\mu m$-wide ion beam in a typical LHC interaction point, the minimum
duration of the gamma-ray pulse could be $\approx 30\,$fs, which comes from
the 10 $\mu$m divided by the speed of light, assuming that the laser beam
propagates perpendicularly to the ion beam. This requires a correspondingly
short laser pulse, but generation of such pulses is routine with modern
lasers. Different-energy photons could come in overlapping pulses (down to 30
fs) or be separated by a fraction of a ns if they are derived from the same
ion bunch, or can come from different ion bunches and have correspondingly
long separation.

Some of the options for GF based pump-probe spectroscopy include:

  * •

The light pulses could be tuned to two different transitions of the PSI;

  * •

Pump and probe can be of the same energy (depending on the experiment);

  * •

The pump and probe can be produced from different PSI bunches;

  * •

Pump and probe pulses do not necessarily need to be temporally separated;

  * •

Pump-probe spectroscopy with nuclear isomers. If the produced nuclear states
are sufficiently long-lived, one could move the production target [Fig. 21(a)]
to probe produced nuclear states using gamma rays with different energy at
different spatial locations;

  * •

Because the photons emitted at the periphery of the $\approx 1/\gamma$ cone
are of low energies, one can design a pump-probe experiment with a gamma-ray
pump and an X-ray probe [Fig. 21(b)] using the well-developed technology of
grazing-incidence X-ray mirrors [342]. An attractive feature of this scheme is
that the X-ray energy can be tuned by tilting the mirror; a potential downside
for some experiments would be the inherent time delay (due to the difference
in the optical path length) between the pump and probe pulses.

Figure 21: The gamma factory configurations suitable for the pump-probe
experiments: a) sufficiently large angle of collision between the laser and
the ion beam allows for the laser temporal profile to be imprinted into the
gamma-ray pulses; b) X-ray probe pulse can be selected and energy-tuned using
a small-angle X-ray mirror. Figure 22: Pump-probe spectroscopy with a) 103Rh
and b) 115In isomers. The branching ratio after exciting the nucleus to a
higher energy level, $I^{P}$ and $T_{1/2}$ of energy levels are shown
(according to the NuDat 2 database [55]).

Two specific examples of a possible application of pump-probe spectroscopy are
shown in Fig. 22. These examples involve isomers from Table 7. Pump-probe
spectroscopy as discussed here could be used to prove the feasibility of
producing the isomers at the GF and to study reaction cross sections and
production rates.

###  5.4 Highly monochromatic gamma beams

Energy bandwidth is a key parameter when aiming for exciting narrow nuclear
resonances with typical intrinsic widths of a few eV. Exciting such a
resonance with a $\gamma$ beam of a few tens to hundreds of keV, as currently
available from existing facilities, will predominantly result in background
generated by the large amount of off-resonant incident photons. So aiming for
clean $\gamma$-spectroscopic conditions requires reducing the energy bandwidth
of the $\gamma$ beam ideally to the physical limit set by (thermal) Doppler
broadening.

The highly brilliant $\gamma$-ray source of the Gamma Factory will deliver a
$\gamma$ beam with a divergence in the range of milliradians, with a photon
flux of up to $10^{17}$ $\gamma$ rays per second, and a possibility of
monochromatization on the order of $\Delta E_{\gamma}/E_{\gamma}\approx
10^{-3}$ over a wide energy range. Due to pile-up problems, this intense beam
would put serious constraints on detection systems if it were used without
further preparation. A substantial fraction of foreseen experiments at this
facility will be based on the concept of nuclear resonance fluorescence (NRF;
see Sec. 4.1). For NRF experiments the optimal monochromatization should be
equal to the width of the excited nuclear resonances, which is expected to be
of the order of $\Delta E_{\gamma}/E_{\gamma}\approx 10^{-6}$, due to thermal
motion of the target atoms. Therefore, further monochromatization of the
$\gamma$ beam would on the one hand provide an improved signal-to-noise ratio
for these experiments as well as solve eventual detection problems. Therefore,
when aiming for an ultimate quality of nuclear $\gamma$ spectroscopy, the
unprecedented intensity of the GF photon beam can serve as an asset to trade
beam intensity for spectral resolution. $\gamma$-ray optics allowing to
further monochromatize the $\gamma$ beam is thus of high importance for a
future high-intensity $\gamma$-beam facility like the GF.

The most accurate method for the absolute determination of $\gamma$-ray
wavelengths relies on crystal diffraction from highly perfect flat crystals of
silicon or germanium, allowing to obtain a resolving power which is unequalled
by any other $\gamma$-ray spectroscopic methods. Over many years, the concept
of crystal-based photon diffraction has been pioneered and optimized at the
GAMS facility at the Institut Laue-Langevin (ILL) in Grenoble [343, 344, 345]
as a technique to monochromatize $\gamma$ beams [346].

Diffraction of photons is governed by Bragg’s equation

$n\cdot\lambda=n\frac{hc}{E_{\gamma}}=2d~{}sin\theta_{B}.$ (25)

Here $h$ represents Planck’s constant, $c$ is the speed of light, $n$ is the
diffraction order, $d$ the lattice spacing of the diffracting crystals and
$\theta_{B}$ the diffraction angle. While this equation provides information
on the energy dependence of $\theta_{B}$, information on resolution power is
obtained from dynamical diffraction theory [347], predicting the dependence of
diffracted intensity $I$ on the diffraction angle $\theta$. The result is
summarized by the following simplified expression

$I(\theta)\propto\frac{sin^{2}\left(A\sqrt{1+y^{2}}\right)}{1+y^{2}},~{}~{}y\propto\frac{hc}{E_{\gamma}}\left(\theta-\theta_{B}\right),~{}~{}A\propto\frac{hc}{E_{\gamma}}.$
(26)

Hence the diffracted intensity consists of an oscillating term surrounded by a
Lorenztian envelope and its width, corresponding to the ‘acceptance width’ of
the monochromator can be estimated to scale with $hc/E_{\gamma}$. Comparing
the scaling of the acceptance width to Eq. (25) results in

$\frac{\Delta
E_{\gamma}}{E_{\gamma}}=\frac{\Delta\theta}{\theta_{B}}\approx\frac{C}{n},$
(27)

where $C$ is a constant. From Eqs. (25) and (26), it follows that the
achievable energy resolution is independent of the incident photon energy
$E_{\gamma}$. The acceptance width of a perfect crystal can be as small as a
few nanoradians. Thus the number of $\gamma$ rays accepted from the incident
beam for diffraction is small. In fact, a perfect crystal acts for $\gamma$
rays as an excellent collimator, accepting almost exclusively the non-
divergent part of the beam and diffracting it in a different direction with
respect to the incoming beam. Since the initial beam is typically several
orders of magnitude more divergent, this selection results in a drastically
attenuated diffracted beam. For the divergence expected for the Gamma Factory,
one expects a loss factor of about nine orders of magnitude. In order to fully
exploit the properties of a perfect crystal for the production of
monochromatic $\gamma$ rays it would be required that the divergence of the
incident photon beam were comparable to the acceptance width of the crystal.
This can be achieved by combining two crystals to form a double-crystal
monochromator. The first crystal will produce a multitude of monochromatic
beams, where each energy is diffracted in a low-divergence beam directed at
its particular Bragg angle. A particular energy may then be selected with the
second crystal. This scheme is realized in the GAMS facility at the ILL
(Grenoble), as shown in Fig. 23 [345].

Figure 23: Layout of a double-crystal monochromator as realized at the GAMS
facility of the ILL (Grenoble) [346]. The consecutive action of the two
crystals is seen. Both crystals are mounted on interferometer arms. The
positions of these arms with respect to a movable spectrometer table are
controlled with optical interferometers. The diffracted beam is separated with
a movable collimation system away from the intense direct beam. Figure adapted
from [346].

A double-crystal spectrometer can be operated in two geometries. In the so-
called non-dispersive alignment mode, the two crystals are positioned in
parallel, which means that all $\gamma$ rays diffracted by the first crystal
are accepted by the second crystal. In this case, no energy selection is made
and there is no monochromatization. However, this geometry delivers a
measurement of the intrinsic instrument resolution as it measures the
convolution of two single-crystal intensity profiles $I(\theta)$ of Eq. (26).
In the dispersive geometry, the spectrometer realizes a dedicated Bragg angle
$\theta_{B}$ between the crystals and therefore selects specific energies. A
detailed review on the two geometries can be found in Ref. [348].

A typical measurement procedure involves rocking of the second crystal (and
detector) such that its orientation is $+\theta_{B}$ and $-\theta_{B}$
relative to the first diffracted beam. Except for small ($\approx 10^{-7}$)
corrections due to a finite vertical divergence, the angular separation of
these two diffracted beams is 2$\theta_{B}$. The diffraction angles are
measured by polarization sensitive Michelson interferometers which have a
sensitivity of $\approx 10^{-9}$ rad [344] and which are calibrated using an
optical polygon [345]. The left panel of Fig. 24 shows the $\gamma$-ray energy
dependence of the Bragg diffraction angle $\theta_{B}$ for a Si crystal cut
along the (220) direction, typically (depending on diffraction order and
photon energy) ranging between 1 mrad (at 1 MeV) to about 10 mrad for
$E_{\gamma}=$10 MeV. The single-crystal rocking curve, i.e. the distribution
of the diffracted $\gamma$-beam intensity for the same type of (220) Si
crystal is shown in the right panel of Fig. 24, exhibiting diffraction angles
of the order of 10 nrad.

Figure 24: Left: Energy dependence of the Bragg diffraction angle $\theta_{B}$
for a (220) silicon crystal. Right: Rocking curve of a (220) Si crystal, i.e.
distribution of the diffracted intensity as a function of the diffraction
angle.

Such a double-crystal monochromator can reach high resolution and instrumental
diffraction widths nearly equal to those predicted by dynamical diffraction
theory [347] have been obtained for energies up to about 6 MeV.

From a practical perspective, to realize an ultimate $\gamma$-ray energy
resolution a three-stage measurement chain is needed to link the $\gamma$-ray
wavelengths of interest to visible wavelengths [343]. In a first step, the
lattice spacing of a Si crystal is measured in terms of the wavelength of an
iodine-stabilized HeNe laser operating near 633 nm. This step employs
simultaneous X-ray and optical interferometry and yields a calibrated Si-
crystal sample. In the second step, the lattice spacings of various other
crystals are compared to the calibrated Si crystal to yield a family of
crystals whose lattice spacings are known relative to the optical wavelength.
This method employs an X-ray crystal comparator. In the final step, which
comprises the double-crystal monochromator setup, $\gamma$ rays are diffracted
by the crystals calibrated in the second step and the diffraction angles are
accurately measured. By combining the measured lattice spacing and diffraction
angles, $\gamma$-ray wavelengths are determined with high accuracy.

The difficulty in making sub-ppm wavelength measurements via this prescription
may be seen by considering the accuracy required in the measurement of the
lattice-spacing parameter $d$ and the diffraction angle $\theta$. The crystal-
lattice constant $d$ must be determined with an accuracy of $\sim 10^{-17}$ m.
The Bragg angle ($\sim 10^{-3}$ rad at 5 MeV) must be determined with an
angular precision of $\sim 10^{-10}$ rad. In addition, the angular scale upon
which the Bragg angle is measured must have an accuracy of $\sim$ 1 in 107
[344]. Such properties have been demonstrated at the GAMS facility and can
thus serve as a ‘role model’ for a potential monochromator to be integrated
into the GF experimental device suite with an envisaged relative energy
resolution $\Delta E_{\gamma}/E_{\gamma}\sim 10^{-6}$.

##  6 Photophysics with a storage ring for radioisotopes

Figure 25: A sketch of a possible arrangement of the low-energy storage ring
at the GF. The photon beam from the GF comes from the left and interacts with
the stored secondary ion beam. The photon beam leaves the ring on the right
and can be employed in a next experimental station. An internal gas target can
be implemented as an option.

The secondary beam of the GF can be used to irradiate solid-state targets of
stable or long-lived nuclides. In general, such experiments have been
performed in the past and are planned at future photon facilities, though with
photons at different energies and/or lower intensities as compared to the GF.
However, the ultimate interest and the highest discovery potential lie in
investigations of exotic nuclei with large proton-to-neutron asymmetries.
Nuclei far from the line of stability are currently at the center of interest
in nuclear structure physics. They offer insight into the physics of loosely
bound finite drops of Fermi liquid, and are intricately related to
astrophysical processes of formation of stable nuclei with their observed
abundances on Earth and in the Universe. Estimates predict the existence of
more than 7000 nuclei stable with respect to particle decay [349]; the
lifetime of those outside of the valley of stability is determined by weak
decays in the direction of this valley. Experimental access to these species
is rendered possible by several new-generation facilities under construction
including FAIR in Germany, SPIRAL2 in France, HIAF in China and FRIB in the
USA.

Exotic nuclei are inevitably short-lived and their production rates in nuclear
reactions are small. For instance, halo phenomena, where one or a few valence
nucleons are weakly bound, occur in short-lived nuclei lying close to the
limits of nuclear existence, the so-called driplines, see Sec. 4.8. The
“classic” halo nucleus 11Li has a half-life of merely 8.75(14) ms [55], and no
target can be produced out of it. As known from first detailed studies [350],
the size of external weakly bound neutron orbitals in 11Li is essentially the
same as the size of the strongly bound doubly magic isotope 208Pb. There is
still no general agreement on the mechanism leading to the binding at such
large distances. The wavefunction of halo nucleons looks like that of one or a
few Cooper pairs bound to a normal core by some special forces [351, 352].
There are also competing mechanisms: the collective modes of the core coupled
through the continuum may create loosely bound states, as for example in the
heaviest oxygen isotopes [353].

The location of the GF at CERN has a unique advantage of having in its close
proximity a state-of-the-art radioactive-ion-beam facility, the isotope
separator on-line device (ISOLDE) [354]. The ISOLDE facility offers a large
variety of secondary beams. As of today, an intense 1.4 GeV proton beam
impinges on a thick, several 10 g/cm2, production target, such that the
reaction products stop in the target material. Target spallation and fission
are the major nuclear reactions giving access to exotic nuclei of interest. A
variety of schemes have been developed to achieve clean secondary beams
delivered to various experimental stations. In the present context, the major
option is HIE-ISOLDE [355] (HIE stands for high intensity and energy), which
provides post-accelerated beams at energies of up to 10 MeV/nucleon in high
atomic charge states after charge breeding in a dedicated REXTRAP/REXEBIS
system [356, 357].

To bring exotic nuclei in collisions with photons, their storage is
indispensable. About a decade ago, a proposal was put forward at ISOLDE to
install a dedicated storage ring [358]. Such storage ring is a part of the
upgrade program of the ISOLDE, the EPIC project (Exploiting the Potential of
ISOLDE at CERN) [59]. On the one hand, its installation will enable a broad
range of unique physics experiments at HIE-ISOLDE, see [358]; on the other
hand, the research scope can be dramatically extended if merging with the
photons from the GF is achieved. Since the photons cannot be transported to
ISOLDE, the storage ring needs to be constructed next to the GF. A beamline
from the ISOLDE to the GF site is required to transport secondary ion beams at
energies of a few MeV/nucleon from the ISOLDE hall to the GF. Dependent on the
exact location of the GF, such beamline can be 100-300 m long.

The interactions of the GF photons with antiprotons can also be envisioned.
Various technical solutions can be considered. A straightforward approach is
to construct a beamline connecting the Antiproton Decelerator (AD) facility
and the proposed storage ring at the GF. The PUMA project (antiProton Unstable
Matter Annihilation [359]), aims at transporting trapped antiprotons from AD
to ISOLDE to study antiproton collisions with exotic nuclei. A similar
methodology may as well be applied.

The design of the storage ring will be based on the experience gained in
operation of the low-energy storage rings TSR in Heidelberg [358] and
CRYRING@ESR in Darmstadt [360]. It will have a circumference of about 40 m.
Secondary ions stored at energies of a few MeV/nucleon will have typical
revolution frequencies of a few hundred kHz. The ring will have four straight
sections, where two of them will be used for injection/extraction and an
electron cooler. The experimental straight section will be aligned with the
photon beam, see Fig. 25. In order to reduce heat load and stress on vacuum
windows, an evacuated pipe can be used to connect SPS/LHC and the storage
ring.

Dependent on the specific physics case, coasting as well as bunched beams will
be employed. If nuclear and storage lifetimes allow, accumulation of beam
currents up to about 1 mA is possible [358]. The electron cooled ion beam has
a momentum spread on the order of $10^{-4}$ or better and the transverse size
is about 1 mm. These parameters constrain the interaction region and thus
offer excellent conditions for high energy and angular resolution in
experiments. The beam is stored at a few MeV/nucleon energy which facilitates
the detection of the beam-like recoils and reaction products. Any product of a
charge-changing (atomic or nuclear) reaction will be deflected differently by
the dipole magnets as the primary beam and can thus be intercepted by a
particle detector. It should be emphasized that all tools developed at various
storage rings [361, 362] can be available here as well, which allow, for
example, to prepare the beam in a specific, well-defined atomic or nuclear
state by employing internal targets or dedicated laser beams.

The photon beam from the GF will pass through the interaction region of the
storage ring and can be used in a different experiment downstream the storage
ring.

##  7 Colliding-beam opportunities

The LHC collider (see Fig. 26) has two counter-rotating beams which are
normally separated by 20 cm horizontally and brought to collision at several
interaction points around the ring. Superconducting magnets guiding both beams
share the same cryostat. Due to the magnet design, the strength of the
magnetic field guiding each beam is the same (rigidity of both beams is the
same). The LHC can operate in proton-proton, ion-ion, or ion-proton collision
mode.

Figure 26: The LHC layout. Adapted from [363]. The separation between the
beams is exaggerated here (in reality it is 20 cm). IP: interaction points.
Figure 27: The typical LHC interaction point (IP) configuration compared to a
$1/\gamma$ cone of gamma-radiation. The diameter of the beams in this picture
is $4\sigma_{x,y}$ (at the IP $\sigma_{x,y}\approx 19~{}\mu\mathrm{m}$). The
beams are separated by $10\sigma_{x}\approx 0.2$ mm horizontally in order to
avoid stripping of the PSI due to collisions with counter-propagating ion or
proton beam.

The secondary photons produced at the GF can themselves be directed onto a
relativistic beam, for instance the counter-circulating beam of the LHC (see
Fig. 27), thus benefiting from another Lorentz boost of photon energy (for
secondary photons in the frame of the counter-circulating ions/protons) or two
Lorentz boosts (for tertiary photons in the lab frame). This possibility, that
may be realized at a later stage of the GF program, may open additional
physics opportunities. We briefly discuss some of them here.

###  7.1 Photoabsorption structure functions

Via scattering secondary photons from the GF head-to-head off a proton or ion
beam, one can study polarized and unpolarized inclusive structure functions at
high energies.

For secondary photons with energy $\omega$ in the laboratory frame and a
proton beam with a relativistic factor $\gamma_{p}$ up to $\approx 7000$ from
the LHC, the highest invariant mass amounts to

$\sqrt{s}=\sqrt{4M_{p}\gamma_{p}\omega}\leq 108\,{\rm
GeV}\sqrt{\frac{\gamma_{p}}{7000}\cdot\frac{\omega}{400\,{\rm MeV}}}\,.$ (28)

The total photoabsorption cross section can be written in terms of inclusive
structure functions familiar in the context of inelastic electron scattering
[341]. Those surviving for real photons are $F_{1,3}$ and $g_{1,5}$, and
denoting the photon circular polarization $\xi=\pm 1$ and the proton helicity
$h=\pm 1/2$, we write

$\displaystyle\sigma(\xi,h)$
$\displaystyle=\frac{8\pi^{2}\alpha}{s-M^{2}}\left[F_{1}-2h\xi
g_{1}+\frac{\xi}{2}F_{3}+2hg_{5}\right].$ (29)

The structure functions $F_{i},\,g_{i}$ are functions of $s$. The first two
terms in Eq. (29) conserve parity, while the last two terms are parity-
violating.

High-energy behavior of scattering amplitudes is governed by Regge theory and
is economically described by $t$-channel exchanges. The leading contribution
to the spin-averaged structure functions $F_{1}$ at asymptotically high
energies is identified in QCD with exchanges of gluons which combine into
colorless compounds, while quark-antiquark (meson) exchanges are suppressed
[364]. The well-known pomeron that has quantum numbers of the vacuum and
couples equally to particles and antiparticles ($C$-parity even), is
identified with the colorless two-gluon exchange. An exchange of a colorless
three-gluon state leads to a $C$-odd odderon which couples to particles and
antiparticles with an opposite sign. Predicted nearly five decades ago [365],
this elusive kind of interaction of hadrons has just recently been observed in
Tevatron/LHC data [366]. While not accessible with the inclusive
photoabsorption cross section by Furry’s theorem (structure functions are the
imaginary part of the forward Compton amplitude), the odderon can contribute
to exclusive channels, e.g. photoproduction of axial vector mesons, or in
parity-violating asymmetries, as outlined below. This would be the first
observation of the odderon in electromagnetically induced scattering
processes.

Data on $F_{1}$ with real photons exist up to $\approx 200$ GeV [341],
nonetheless the new data at GF are expected to significantly improve the
precision. Using ion beams one can study nuclear shadowing: it is observed
that a high energy probe does not see all nucleons within a nucleus, but only
part of them [367]. GF can allow to study nuclear shadowing at highest
attainable energies where only the leading pomeron exchange gives a sizable
contribution (asymptotic regime), or below where other contributions are non-
negligible (sub-asymptotic regime).

Measuring the structure function $F_{3}$ entails shining a beam of circularly
polarized photons onto an unpolarized proton/ion beam. No data on the purely
electromagnetic $F_{3}$ exist. Data on this structure function in electron
scattering are almost exclusively sensitive to interference of the
$Z^{0}/\gamma$ exchange between the electron and the proton. A
superconvergence relation
$\int_{0}^{\infty}\frac{d\omega}{\omega^{2}}F_{3}(\omega)=0$ was derived in
Ref. [368], but has never been verified experimentally. GF can provide the
input to this sum rule from the inelastic threshold up to highest achievable
energies. The structure function $F_{3}$ violates parity but conserves $CP$,
therefore it is also $C$-parity odd. This feature removes the restriction of
Furry’s theorem and allows for the odderon direct contribution to $F_{3}$ at
asymptotic energies: the odderon leads to slowly (e.g., logarithmically [365])
growing cross section, while Regge meson exchanges lead to a $\sim 1/\sqrt{s}$
behavior.

The spin structure function $g_{1}$ with polarized proton/ion beams is
relevant for the Gerasimov-Drell-Hearn sum rule [264, 265] that equates the
squared anomalous magnetic moment of the proton to an energy-weighted integral
over $g_{1}$. Data on $g_{1}$ with real photons exist up to $\sqrt{s}\approx
2$ GeV, allowing to experimentally verify this important sum rule [369].
However, the higher-energy part of the integral is estimated using a model-
based parametrization of existing deep-inelastic data at low $Q^{2}$ [370,
371, 372, 373] and their extrapolation to the real photon point [374, 375]. A
direct measurement at the GF would serve as an explicit check of that model.

The parity-violating spin structure function $g_{5}$ requires a polarized
proton/ion beam, and can be obtained from a single-spin asymmetry upon
averaging over photon polarization. This structure function has generally been
elusive even in deep-inelastic electron scattering. No data with real photons
exist. Together with $F_{3}$, a measurement of $g_{5}$ at asymptotic and sub-
asymptotic energies will be completely new in terms of addressing PV in the
Regge domain: Regge theory operates with $t$-channel exchanges which have
well-defined parity. In this context, it is worth mentioning the current
$3\sigma$ tension in the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM)
matrix in its top row [341]. This tension came about due to a recent re-
evaluation of the electroweak radiative $\gamma W$-box correction to the
neutron and nuclear $\beta$ decay rate in Refs. [376, 377]. The mechanism
found in those Refs. is the Regge exchange contribution to the parity-
violating structure function $F_{3}$.

###  7.2 Production of ultrahigh-energy gamma rays

####  7.2.1 Scattering of secondary photons off stored relativistic ions

Consider a secondary photon beam generated by the GF that is back-scattered
(if possible, resonantly) from relativistic ions. Nuclear levels have energies
of several MeV. We take 15 MeV as reference, as it is done in Sec. 2.4.3. This
places the energy of the secondary photons to $\approx 15$ MeV$/2\gamma$,
between about 2.6 keV and 40 keV using the $\gamma$-factor range for the LHC
and assuming head-on collisions.

An example of a configuration that yields parameters not too far from these is
the following. Assume the LHC at injection energy with Li-like Xe beam
($\gamma\approx 190$) using the 120 eV $2s-2p_{1/2}$ transition (similar to
the transition proposed for the GF Proof-of-Principle experiment at the SPS
[7]). The maximum energy of emitted secondary X-rays in this case will be 46
keV. In the frame of reference of the counter-rotating ion beam these photons
will appear to be of 17 MeV energy suitable for excitation of some nuclear
levels. The re-emitted gamma radiation from the nuclear level will be Lorentz
boosted to 7 GeV in the lab frame. The lifetime of the upper state of the
$2s-2p_{1/2}$ transition in Li-like Xe is 186 ps [378]. This extends the 46
keV X-ray emission region to 10 m which is about 10 times longer than the
focused ion beam region in the IP (see Fig. 27). Longer $2s$ lifetimes of
lighter ions limits their efficiency for such a scheme.

A more efficient although much more expensive approach would be to use a
dedicated X-ray FEL facility generating the required X-ray pulses.

If a 15 MeV nuclear resonance decays via photons, with highest accessible
relativistic factors, one would produce gamma rays with energies of up to
$\approx 90$ GeV in the lab frame. For most efficient gamma production, one
would choose a resonance that decays in the photon channel with a high
branching ratio. How many such photons can be produced this way strongly
depends on the specific system and the details of experimental arrangements;
preliminary estimates show that on the order of 106 high-energy photons per
second could be achievable in favorable scenarios.

####  7.2.2 Scattering of secondary photons off a proton beam

To achieve the highest possible photon energies, one may shine secondary GF
photons head-on onto the proton beam at the LHC.151515Earlier it was also
suggested to use X-ray FEL for that purpose [379]. With the inverse Compton
scattering process, the energy of the resulting photons will be close to the
LHC proton-beam energy 6.5 TeV. However, the cross-section of this process is
low and we can expect only several such scattering events per hour with the GF
beam intensity [380]. Another promising scenario with a much higher cross-
section would be to excite the $\Delta(1232)$ resonance at $\sim$300 MeV in
the rest frame of the proton. It will be producing $\pi^{0}$’s and $\pi^{+}$’s
which decay in flight:

$p\ +\ \gamma\to\Delta\to\ p\ +\ \pi^{0}\to p\ +2\gamma.$ (30)

The incident photon energy must be tuned to $\omega\approx 300\,{\rm
MeV}/2\gamma_{p}\approx 22$ keV for the relativistic factor of the LHC proton
beam $\gamma_{p}=7000$. Such low-energy secondary photons might originate from
a lower-energy PSI beam (for example, at the SPS) or by directing the
secondary photons on the proton beam at a large relative angle. Either way,
the secondary photon beam should be produced not too far from the point at
which the collision with the proton beam occurs to have as many secondary
photons interacting with the proton beam as possible. Note that since the
width of the $\Delta(1232)$ is 120 MeV one can use a fairly broad secondary
photon spectrum, say $\omega=260\pm 60$ MeV. The $\pi^{0}$ will be emitted
with the energy $\omega_{\pi}\sim 2\gamma_{p}\,(230\pm 60)\,$MeV $\sim(3.2\pm
0.8)$ TeV. These pions will decay within $\gamma_{p}\tau_{\pi^{0}}\approx
6\times 10^{-13}\,$s and all this energy will be shared between two photons
going within a small angle $\sim m_{\pi}/\omega_{\pi}$. The cross section for
$\pi^{0}$ production on top of the resonance is $\sigma\approx 300\,\mu$b. The
background processes are associated with charged particles, such as pair
production (with a cross-section of $\approx 10\,$mb for 300 MeV photons) and
$\pi^{+}$ production, where almost equal amount of $\pi^{0}$’s and $\pi^{+}$’s
are produced (the latter live much longer and do not produce photons).

Pion production rate can be estimated as follows. Assuming $10^{15}$ photons
per second (to account for the divergence of the photon beam) colliding with
the proton bunch of $3\times 10^{10}$ protons [13] in a bunch of $(16\,\mu
m)^{2}$ cross section, we have

$10^{15}\times 3\cdot 10^{10}\times\frac{3\cdot
10^{-28}\,\mathrm{cm}^{2}}{(16\cdot 10^{-4}\,\mathrm{cm})^{2}}\sim 4\cdot
10^{3}\,\mathrm{s}^{-1}\,.$ (31)

The number of photons is double that number. The photon energy distribution in
the lab frame is expected to be broad and monochromatization via collimation
may not be possible in this case (this will be studied in more detail in
future work). The large width of the $\Delta$ resonance of $\approx 120$ MeV
is of advantage for the number of pions and photons produced: we do not need
to tune the secondary photons too precisely and will be integrating over the
whole $\Delta$ peak. The highest attainable energy photon beam of 4 TeV is
unprecedented (the fixed-target program COMPASS is limited to below $\approx
100$ GeV photons). Neutral pions are routinely produced, for example, in $pp$
collisions at the LHC. However, because of the symmetric kinematics, pions and
photons are spread over $4\pi$ solid angle. With the asymmetric kinematics
considered here, we will obtain a high-energy photon beam with small
divergence.

The very high energy gamma rays (VHEGR) are of interest in astrophysics, so
having an intense source of VHEGR with energy up to 4 TeV will help
calibrating the detectors for the VHEGR detection and to study VHEGR
interactions with matter.

##  8 Production of isotopes and isomers for medicine, dark matter search and
gamma lasers

Isotope and isomer production via photonuclear reactions is among the
important applications of gamma sources; see, for example, Refs. [381, 382]
for a detailed discussion. The advantages of the GF are, first of all, high
photon flux, but also high monochromaticity allowing one, in some cases, to
take advantage of resonant cross section enhancement [381]. We consider the
production of useful nuclear isotopes and isomers via ($\gamma,n$) or
($\gamma,\gamma^{\prime}$) reactions from fixed targets containing long-lived
nuclei (half-life longer than one year).

###  8.1 Production of medical isotopes via ($\gamma,n$) reactions

Z |  A+1X |  AX | $I^{P}$ | $T_{1/2}$ | Decay mode |  $S_{n}$(keV)
---|---|---|---|---|---|---
29 |  65Cu |  64Cu | 1+ | 12.7 h | $\%\varepsilon+\%\beta^{+}=61.50$ | 9910.4
|  |  |  |  | $\%\beta^{-}=38.50$ |
42 |  100Mo |  99Mo | 1/2+ | 66.0 h | $\%\beta^{-}=100.00$ | 8294.2
46 |  104Pd |  103Pd | 5/2+ | 17.0 d | $\%\varepsilon=100.00$ | 10009.2
68 |  170Er |  169Er | 1/2- | 9.4 d | $\%\beta^{-}=100.00$ | 7256.9
75 |  187Re |  186Re | 1- | 3.7 d | $\%\beta^{-}=92.53$ | 7360.7
|  |  |  |  | $\%\varepsilon=7.47$ |
77 |  193Ir |  192Ir | 4+ | 73.8 d | $\%\beta^{-}=95.24$ | 7772.0
|  |  |  |  | $\%\varepsilon=4.76$ |
79 |  197Au |  196Au | 2- | 6.2 d | $\%\varepsilon+\%\beta^{+}=93.00$ | 8072
|  |  |  |  | $\%\beta^{-}=7.00$ |
Table 6: Examples of medical isotope production via the
${}^{A+1}_{Z}$X($\gamma,n$)${}^{A}_{Z}$X reaction [381, 383]. For produced
medical isotopes ${}^{A}_{Z}$X, their ground-state nuclear spins $I^{P}$,
half-lives $T_{1/2}$ and decay modes are provided. Here $\%\varepsilon$ and
$\%\beta^{+}$ represent probabilities of nuclear decay via electron capture
$(\varepsilon)$ and $\beta^{+}$ decay, respectively. $S_{n}$ is the neutron-
separation energy of the initial nucleus ${}^{A+1}_{Z}$X. The data are from
Ref. [55].

Production of medical isotopes via $(\gamma,n)$ can be realized by either
resonantly exciting specific nuclear energy levels beyond the neutron
separation threshold or exciting the giant dipole resonance (see Sec. 4.3)
using photons in a broad band. The former approach could be cleaner by taking
advantage of photon monochromaticity achievable at the GF and involve less
power deposited on the target. The photon bandwidth can be adjusted to match
the total width of the excited state predominantly arising from neutron
emission. But the resonant cross section is suppressed due to the small gamma
emission branching ratio of the excited state. The latter approach could be
more effective due to a larger peak cross section and a broader width of the
GDR (together resulting in a larger integral cross section), allowing for more
photons to be absorbed. This was studied in, for example, Ref. [383] employing
gamma photons produced by laser photons Compton back-scattered off
relativistic electrons at the Canadian Light Source (CLS); see also Table 13.
With orders of magnitude higher photon fluxes at the GF, we can expect a
significant improvement of production rates, thus making $(\gamma,n)$ reaction
practical for producing medical isotopes. Radioisotopes which may have better
application in nuclear medicine but are not available due to low production
rates accompanied with high cost (see, for example, Ref. [381]) might become
accessible with the advent of the GF. A number of isotopes useful for medicine
[383] can be produced via $(\gamma,n)$ reactions; see the examples given in
Table 6.

We present an example of producing the ${}^{99}\textrm{Mo}$ medical isotope at
the GF via the GDR. The GDR in heavy nuclei have resonance energies
$E_{0}\approx 77\times A^{-1/3}\approx 14\,$MeV with widths
$\Gamma_{tot}\approx 23\times A^{-1/3}\approx 5\,$MeV. Here we can tune the
maximal energy of secondary photons at GF to
$E_{\gamma}=E_{0}+\Gamma_{tot}\approx 19\,$MeV and use photons emitted within
the $1/\gamma$ cone, i.e., photons with energy between 9.5 MeV and 19 MeV (see
Sec. 1.1), by suitable collimation. The average background photon-attenuation
cross section in this energy range is $\sigma_{bg}\approx 6\,$b, dominated by
electron-positron pair production ($\approx 4.4\,$b) and Compton scattering
off electrons ($\approx 1.7\,$b) [341, 384]. The peak cross section of the
${}^{100}\textrm{Mo}(\gamma,n)^{99}\textrm{Mo}$ reaction is $\sigma_{0}\approx
0.15\,$b [383]. Therefore, the maximal production rates of the
${}^{99}\textrm{Mo}$ isotope can be estimated as

$p\approx
j\frac{\Gamma_{tot}}{E_{0}+\Gamma_{tot}}\frac{\sigma_{0}}{\sigma_{0}+\sigma_{bg}}\approx
6\times 10^{14}\,\textrm{s}^{-1}.$ (32)

This production rate is reached when almost all photons (after collimation)
are absorbed in the target. Therefore, the thickness of an enriched 100Mo
target or multiple thin targets should be greater than the absorption length
$l\approx 1/(\sigma_{bg}n)\approx 2.8\,$cm, where $n$ is the number density of
${}^{100}\textrm{Mo}$ in the target with a density of approximately 10 g/cm3.
After a week of irradiation, ${}^{99}\textrm{Mo}$ isotopes with activities of
$\approx 500$ TBq can be produced. Such activities are a significant
improvement compared to those obtained at other facilities including some
nuclear reactors [385, 386]. The worldwide demand of 99Mo isotopes is
approximately 9000 6-day Ci 99Mo per week [387], where 6-day Ci refers to
activities measured 6 days after the end of target processing. Assuming one
day for the target processing after the one-week irradiation, $\approx 2300$
6-day Ci 99Mo per week, nearly one quarter of the global supply, can be
obtained at the GF. The 99Mo isotope having a half-life of $T_{1/2}\approx
65.9\,$h decays via $\beta^{-}$ to 99mTc ($T_{1/2}\approx 6.0\,$h). Therefore,
99Mo isotopes are mainly used for the production of the 99mTc, the most
frequently used isomer in nuclear medicine. More examples of producing
shorter-lived isomers or isotopes from decay of longer-lived mother isotopes,
so-called generators, are listed in Ref. [381]. The worldwide demand for 99Mo
is nowadays mainly satisfied by a small number of nuclear reactors via the
${}^{235}\textrm{U}(n,f)^{99}\textrm{Mo}$ reaction [386]. Some nuclear
reactors have suffered shutdown for maintenance or breakdowns, leading to
shortages or interruptions in the supply of 99mTc. Therefore, looking for
alternative methods of producing ${}^{99}\textrm{Mo}$ isotopes has become
important.

Production rates for other medical isotopes such as 192Ir and 196Au of
$\approx 10^{15}$ per second can analogously be derived. Similar numbers of
neutrons are also produced from the $(\gamma,n)$ reaction, thus offering
tertiary neutron beams; see also Sec. 12.4. One convenience of exploiting the
GDR is that we may use the same experimental setup for producing many
different medical isotopes, since the dependence of resonance energy and width
on $A$ is weak, both proportional to $A^{-1/3}$. Multiple thin targets
consisting of different atoms can also be used to produce various isotopes
simultaneously.

It is noted that specific activity is an important quality criteria of medical
isotopes. To reach high specific activity via photoproduction, photons with
high flux density $\Phi$ (not just high fluxes) are required, which can be
estimated from $\sigma\Phi\approx\ln{2}/T_{1/2}$ [381], where $\sigma$ is the
reaction cross section of transmuting the target isotope into the product
isotope and $T_{1/2}$ is half-life of the product isotope. Since the secondary
GF photons with high fluxes are mainly emitted within a small cone (see Fig.
1), high flux density can be obtained by putting a target close to the gamma
source. When high specific activities are required, one may need to use thin
targets to avoid significant photon attenuation. Besides using photon beams
with high flux density, high specific activity can also be acquired via
effectively separating and concentrating the product isotope, for example,
through the magnetically activated and guided isotope separation method
(MAGIS) [388], which can also be applied to preparation of targets containing
enriched isotopes.

Not only isotopes, but also nuclear isomers would be copiously produced at the
GF via $(\gamma,n)$ reactions, since nuclei will partially decay to isomeric
states after neutron emission. The isomer ratio, i.e., ratio of produced
nuclei in the isomeric state to those in the ground state, has been studied
with many nuclear reactions including $(\gamma,n)$ via GDR (see, for example,
Ref. [389]) using bremsstrahlung photons. The knowledge of isomer ratios of
different isotopes and their energy dependences could help to reveal the
nuclear structure and mechanism of nuclear reactions. But currently available
data are scarce and have significant discrepancies [389]. Secondary photons at
the GF would offer significant improvement in measuring such data compared
with bremsstrahlung photons, as a result of better energy resolution and
higher photon fluxes.

###  8.2 Production of nuclear isomers via ($\gamma,\gamma^{\prime}$)
reactions

Radioactive isomers relevant to medicine can be produced via
$(\gamma,\gamma^{\prime})$ reactions. Using the highly monochromatic GF
photons, one can selectively excite transitions from the stable or long-lived
nuclear ground state to certain higher-energy levels serving as gateway states
which will partially decay to the isomeric state directly or by cascade, which
is similar to the isomer-depletion process discussed in Sec. 2.1; see Fig. 3.
Table 7 lists some isomers relevant for medicine, together with candidate low-
lying gateway levels.

To give an example, production of the 115mIn can be realized by selectively
exciting the 1078 keV transition with a radiative width $\Gamma_{rad}\approx
4.6\times 10^{-4}\,$eV and a branching ratio $f\approx 0.16$ of decay to the
isomeric state [55]. Doppler broadening of this transition leads to
$\Gamma_{tot}\approx 1\,$eV at room temperature. The photon-attenuation
background is dominated by Compton scattering as $\sigma_{bg}\approx 10\,$b is
much larger than the effective cross section
$\sigma_{eff}\approx\sigma_{0}\Gamma_{rad}/\Gamma_{tot}\approx 0.5\,$b with
$\sigma_{0}\approx 10^{3}\,$b being the resonant photon-absorption cross
section. The production rate of 115mIn can be approximated as

$p\approx
j\frac{\Gamma_{tot}}{E_{\gamma}}\frac{\sigma_{eff}}{\sigma_{eff}+\sigma_{bg}}f\approx
j\frac{\Gamma_{\gamma}}{E_{\gamma}}\frac{\sigma_{0}}{\sigma_{bg}}f\approx
10^{9}\,\textrm{s}^{-1}.$ (33)

Here, as before, $E_{\gamma}$ is the maximal energy of secondary photons at
the GF; it is tuned to $E_{\gamma}\approx 1078\,$keV in this case.

We note that for exciting narrow resonances in cases where Doppler width is
larger than the radiative width but the effective cross section $\sigma_{eff}$
is still larger than $\sigma_{bg}$, Doppler broadening could allow more
photons ($\Gamma_{tot}/E_{\gamma}\approx 10^{-6}$ at room temperature) to be
resonantly absorbed leading to higher nuclear isomer production rates as
$p\approx jf\Gamma_{tot}/E_{\gamma}$ [see Eq. (33)].

Z |  AX |  AmX | $I_{g}^{P}$ | $E_{e}(\textrm{keV})$ | $I_{e}^{P}$ | $E_{m}(\textrm{keV})$ | $I_{m}^{P}$ | $T^{m}_{1/2}$ | $E_{d}(\textrm{keV})$ | $\lambda L$ | Decay mode
---|---|---|---|---|---|---|---|---|---|---|---
43 |  99Tc |  99mTc | 9/2+ | 1207.26 | (7/2-) | 142.683 | 1/2- | 6.0 h | 2.173 | $E3$ | $\%\textrm{IT}=99.996$
|  |  |  |  |  |  |  |  | 142.63 | $M4$ | $\%\beta^{-}=0.004$
45 |  103Rh |  103mRh | 1/2- | 357.396 | 5/2- | 39.753 | 7/2+ | 56.1 min | 39.755 | $E3$ | $\%\textrm{IT}=100$
|  |  |  | 651.716 | (3/2)+ |  |  |  |  |  |
49 |  113In |  113mIn | 9/2+ | 1024.28 | 5/2+ | 391.699 | 1/2- | 99.5 min | 391.698 | $M4$ | $\%\textrm{IT}=100$
|  |  |  | 1131.48 | 5/2+ |  |  |  |  |  |
49 |  115In |  115mIn | 9/2+ | 933.780 | 7/2+ | 336.244 | 1/2- | 4.5 h | 336.241 | $M4$ | $\%\textrm{IT}=95.0$
|  |  |  | 941.424 | 5/2+ |  |  |  |  |  | $\%\beta^{-}=5.0$
|  |  |  | 1078.16 | 5/2+ |  |  |  |  |  | $\%\beta^{-}=5.0$
68 |  167Er |  167mEr | 7/2+ | 264.874 | 3/2- | 207.801 | 1/2- | 2.3 s | 207.801 | $E3$ | $\%\textrm{IT}=100$
|  |  |  | 531.54 | 3/2+ |  |  |  |  |  |
|  |  |  | 667.900 | (5/2)- |  |  |  |  |  |
|  |  |  | 745.32 | 7/2- |  |  |  |  |  |
|  |  |  | 810.52 | (5/2)+ |  |  |  |  |  |
77 |  191Ir |  191mIr | 3/2+ | 658.90 | (3/2-) | 171.29 | 11/2- | 4.9 s | 41.89 | $E3$ | $\%\textrm{IT}=100$
Table 7: Examples of medical isomer production in AX($\gamma$, $\gamma^{\prime}$)AmX reactions that proceed via an intermediate excited state of energy $E_{e}$ [381]. The indices “g”, “e”, “m” denote the ground, intermediate excited and final isomeric states, respectively, for which the nuclear spin and parity $I^{P}$ are provided. The isomeric state is characterized by the energy $E_{m}$ and half-life $T^{m}_{1/2}$, with direct radiative decay via a gamma-ray of energy $E_{d}$ and multipolarity $\lambda L$. The last column presents decay modes of the isomeric states, which are for the listed cases either isomeric transition (IT), i.e., the transition to a lower-lying level of the same nucleus, or $\beta^{-}$ decay. The data are from Refs. [55, 390].  Z |  AX |  AmX |  $E_{m}$(keV) | $I_{m}^{P}$ | $T_{1/2}^{m}$ |  $E_{d}$ (keV) | $\lambda L$ | Decay mode
---|---|---|---|---|---|---|---|---
17 |  34Cl |  34mCl | 146.36(3) | 3+ | 32.0 min | 146.36(3) | $M3$ | %IT = 44.6
|  |  |  |  |  |  |  | $\%\varepsilon+\%\beta^{+}=55.4$
25 |  52Mn |  52mMn | 377.749(5) | 2+ | 21.1 min | 377.748(5) | $E4$ | %IT = 1.78
|  |  |  |  |  |  |  | $\%\varepsilon+\%\beta^{+}=98.22$
35 |  80Br |  80mBr | 85.843(4) | 5- | 4.42 h | 48.786(5) | $M3$ | %IT = 100
36 |  81Kr |  81mKr | 190.64(4) | 1/2- | 13.1 s | 190.46(16) | $E3$ | %IT = 99.9975
|  |  |  |  |  |  |  | $\%\varepsilon=2.5\times 10^{-3}$
43 |  94Tc |  94mTc | 76(3) | (2)+ | 52.0 min | 76(3) |  | $\%\varepsilon+\%\beta^{+}=100$
|  |  |  |  |  |  |  | $\%\textrm{IT}<0.1$
50 |  117Sn |  117mSn | 314.58(4) | 11/2- | 14.0 d | 156.02(3) | $M4$ | $\%\textrm{IT}=100$
|  |  |  |  |  | 314.3(3) |  |
72 |  178Hf |  178mHf | 1147.416(6) | 8- | 4.0 s | 88.8667(10) | $E1$ | $\%\textrm{IT}=100$
78 |  193Pt |  193mPt | 149.78(4) | 13/2+ | 4.3 d | 135.50(3) | $M4$ | $\%\textrm{IT}=100$
78 |  195Pt |  195mPt | 259.077(23) | 13/2+ | 4.0 d | 19.8 | $M4$ | $\%\textrm{IT}=100$
|  |  |  |  |  | 129.5(2) |  |
Table 8: Further examples of isomers for medical applications [391]. In these
cases, specific gateway levels for production of these isomers have not yet
been identified. However, they may potentially be produced via the excitation
of PDR or GDR. See Table 7 for explanations of the notations. As additional
decay modes, some of these isomers undergo electron capture $(\varepsilon)$ or
$\beta^{+}$ decay.

Alternatively, a broad excitation via the PDR or GDR with decay to the
isomeric state can be pursued; see examples in Table 8. Isomer production via
the GDR using $(\gamma,n)$ reactions is briefly discussed at the end of Sec.
8.1. Photoproduction of isomers via $(\gamma,\gamma^{\prime})$ reactions was
investigated employing photons Compton back-scattered off relativistic
electrons, and the obtained cross sections are larger for photons in the PDR
energy range compared to GDR [382]. There exists a maximum cross section
between 5 and 10 MeV on the order of mb with a broad width on the order of
MeV, which lies in the PDR energy range; see Sec. 4.2. It is demonstrated in
Ref. [382] that production of 99mTc, 103mRh, 113mIn, and 115mIn isomers with
activities of more than 10 mCi can be realized by use of gamma rays from
laser-electron Compton scattering with photon fluxes of $10^{13}$ per second
and 6-hour irradiation. The achieved activity is generally sufficient for
medical imaging but is still inadequate for therapeutic applications. With
nearly four orders of magnitude improvement in photon fluxes at the GF,
isomers with considerably higher activities can be produced thus meeting the
requirement for therapeutic applications. We can tune the maximal energy of
secondary photons at the GF to a few hundred keV above the neutron-separation
threshold to cover the PDR. Using photons emitted within $\approx 1/\gamma$
cone (so roughly 1/2 of all secondary photons, see Sec. 1.1), production rates
of these isomers could reach $\sim 10^{13}$ per second.

Returning to the example of 115mIn production discussed above, it can also be
realized by use of photons in the PDR energy range, about 5 to 10 MeV. In this
energy range, the average background photon-attenuation cross section is
$\sigma_{bg}\approx 6\,$b and the integrated cross section of the
${}^{115}\textrm{In}(\gamma,\gamma^{\prime})^{115m}\textrm{In}$ reaction is
around $5\times 10^{-3}$ b MeV [382]. Following an estimate similar to that
for isotope-production rates in Sec. 8.2, we find that production of $\approx
10^{13}$ per second of 115mIn can be achieved at the GF, when almost all
photons (after collimation) are absorbed in a ${}^{115}\textrm{In}$ target or
multiple thin targets with total thickness greater than the absorption length
$l\approx 4.4\,$cm.

In this case, the production rate of the isomer of interest is higher when
using broadband excitation rather than the narrow resonance, by roughly four
orders of magnitude. The isomer-production rate corresponds to activities of
produced nuclear isomers after irradiating for a long time comparable to
$T_{1/2}$ of the isomeric state.

Besides medical applications, nuclear isomers were recently proposed as
detectors for certain kinds of dark matter which could induce collisional
deexcitation of the isomers; see Table 9 [392]. We note that isomers listed in
Table 9 are not the only candidates for dark matter search. The GF can produce
a range of nuclear isomers with larger quantities accessible, including
heavier isomers whose larger nuclear radii allow deexcitation to be induced by
smaller momentum exchange during collision with dark matter particles [392].
Nuclear isomers have also been suggested for building gamma-ray lasers (see
Sec. 9).

Z |  AX |  $E_{m}$(keV) | $I_{m}^{P}$ | $T^{m}_{1/2}$ | $E_{d}$ | $\lambda L$ | Decay mode
---|---|---|---|---|---|---|---
56 |  137Ba | 661.659(3) | 11/2- | 2.55 min | 378.0(4) | $E5$ | %IT = 100
|  |  |  |  | 661.657(3) | $M4$ |
71 |  177Lu | 970.175(3) | 23/2- | 160.4 d | 115.868(3) | $E3$ | %$\beta^{-}=78.6(8)$
|  |  |  |  |  |  | %IT = 21.4(8)
72 |  178Hf | 2446.09(8) | 16+ | 31 y | 12.7(2) |  | %IT = 100
|  |  |  |  | 309.50(15) | $M4(+E5)$ |
|  |  |  |  | 587.0(1) | $E5$ |
73 |  180Ta | 77.1(8) | 9- |  $>1.2\times 10^{15}$ y |  |  |
Table 9: Isomers for dark matter detection [392, 390]. See Table 7 for
explanations of the notations.

Heating due to the high flux of gamma rays is expected to be a tractable
problem with above reactions, since there are several applicable heat-
dissipation techniques, for example, using a stack of multiple thin targets
[381].

##  9 Induced gamma emission and gamma lasers

Realization of stimulated photon emission in the gamma range has been an
outstanding challenge to the community for many years. Many different
proposals have been contemplated [393]; however, only a handful of schemes can
be considered realistic (and none have been realized up till now). The advent
of the GF motivates an examination of whether this facility may bring a gamma
laser closer to reality.

There are two main groups of gamma-ray laser (graser) proposals. One suggests
building a recoilless gamma-ray laser [393, 394] using nuclei incorporated in
a crystal. The reason to use a crystal is that the Mössbauer effect may
mitigate the issue of the Doppler broadening of the transition which could
significantly suppress the stimulated emission cross section. But this meets
the so-called graser dilemma, where high intensity of pumping required for
creating population inversion, i.e., more nuclei in the upper lasing state
than in the lower lasing state, could be destructive to other conditions for
stimulated emission gain, including the Mössbauer and Bormann effects. In
order to relax the population inversion requirements, an approach was proposed
[395, 396] to suppress resonant absorption for the Mössbauer nuclear
transitions of the ions placed in a lattice. This can be done via coherent
optical driving of electronic transitions of these ions and exploiting the
hyperfine coupling between the nuclear and electronic degrees of freedom.
These concepts are reminiscent of the physics discussed in Sec. 2.3. We note
that Mössbauer effect works for transitions with energy below $\approx
180\,$keV. Candidate low-energy gamma transitions may be selected from a list
in Ref. [397].

We highlight here two examples of graser proposals benefiting from the
Mössbauer effect. It has been proposed to build a graser based on collective
excitations of 57Fe Mössbauer nuclei, also known as nuclear excitons, in a
nuclear forward scattering setup [398]. There is also a proposal for a VUV
laser based on the 8 eV nuclear transition between the ground and isomeric
states of 229Th ions doped in a VUV transparent crystal [399]. While pumping
of the isomer can be done with a conventional VUV laser, Zeeman splitting
induced by an external magnetic field or electric quadrupole splitting in the
crystal result in population inversion for the lasing transition between
specific hyperfine sublevels.

The other group of proposals suggest building a graser with the assistance of
recoil [400, 401], which can eliminate the requirement of population inversion
due to reduced overlap between the photon-absorption lineshape and photon-
emission lineshape. The idea is that if the recoil energy $E_{R}$ is larger
than the Doppler width of the transition, emitted photons would not be
absorbed by the nucleus in the ground (initial) state, but they would still
stimulate deexcitation of an excited nucleus; see Fig. 28. However,
implementation of this scheme requires cooling to suppress Doppler broadening
in order to get acceptable stimulated emission gain. In the two-level pumping
scheme with hidden population inversion [401], the photon intensity required
for pumping is estimated to be $\geq 10^{30}$ ph/(cm2 s keV) using samples
cooled down to 10 $\mu$K [402].

Figure 28: The influence of recoil on stimulated emission and resonant
absorption. $E_{1}$ and $E_{2}$ are the level energies. The resonant gamma-ray
energy for spontaneous and stimulated emission $E_{e}$ is lower than
$E_{2}-E_{1}$ by the recoil energy $E_{R}$, while the gamma-ray energy
required for resonant absorption $E_{a}$ is higher than the internal energy
difference, again, by the recoil energy. Thus, the emitted photons are detuned
from resonance and are not readily absorbed.

Many graser proposals involve using nuclear isomers; see, for example, Table
10. A detailed compilation of nuclear isomers with half-lives $\geq$ 10 ns is
provided in Ref. [390], though possible usages of these isomers are not
underlined. Nuclei potentially valuable for realizing a gamma-ray laser based
on the concept of hidden population inversion are tabulated in Ref. [403].
High photon fluxes from the GF will help produce copious amount of candidate
isomers. High tunability and high resolution of the gamma beams could help
characterize the samples during the isomer-separation process before
incorporating concentrated isomers into a crystal. The GF may also assist in
finding optimal candidate nuclei such as nuclei with small energy difference
between the isomeric state and an upper lasing state, where an X-ray or even
optical laser can be used for pumping, see the energy level structure
suggested in Fig. 29. Here, an isomeric state with a lifetime long enough to
enable accumulation of isomers and their incorporation into a host crystal is
used. The isomer is then excited via a single-photon or a multiphoton laser-
driven transition to an upper lasing state. A likely difficulty with the
latter is that multiphoton transitions are strongly suppressed; see Sec. 4.3.
Two examples, 144Pr and 152Eu, exhibiting such energy level structures are
shown in Fig. 30. Here, the required excitation energy is relatively low
(about 40 keV for 144Pr and 20 keV for 152Eu); photons at this energy may soon
become available at XFEL facilities. However, the main drawback remains that
such transitions have narrow radiative widths of the upper lasing state. This
holds particularly true for 152Eu. Therefore, it may be difficult to realize
effective pumping from the isomeric state to the upper lasing state.

Figure 29:  A gamma-ray laser scheme [394]. After a large number of isomers
are produced, the isomeric state is excited to an upper lasing state via a
single-photon or multiphoton process indicated by the blue arrows.
Deexcitation from the upper lasing state to the lower lasing state is shown as
the red arrow.  Figure 30: Examples of nuclear isomers for a prospective
gamma-ray laser; see also Fig. 29.

For graser schemes where the required transition energy cannot be reached by
X-ray lasers, the GF may be utilized for pumping, such as transferring isomers
to an upper lasing state. We note that secondary photons with total fluxes of
$\approx 10^{17}$ photons per second, uniformly distributed over the entire
energy spectrum, at the GF consist of pulses having fluxes of $10^{10}$
photons per pulse at a repetition rate of $\approx 10\,$MHz. The pulse
duration is typically $\approx 100$ ps and can be reduced to $30\,$fs limited
by the transverse size of the ion beam. The strong energy-angle correlation of
GF photons, see Sec. 1.1, may allow application of the Borrmann effect [394],
i.e., minimized photon absorption when photons are incident at special angles
satisfying the Bragg law, to alleviate heating problems.

Z |  AX |  $E_{m}$ (keV) | $I_{m}^{P}$ | $T^{m}_{1/2}$ | $E_{d}$ | $\lambda L$ | Decay mode
---|---|---|---|---|---|---|---
27 |  60Co | 58.59(1) | 2+ | 10.5 min | 58.603(7) | $M3+(E4)$ | %IT = 99.75(3)
|  |  |  |  |  |  | %$\beta^{-}$ = 0.25(3)
70 |  169Yb | 24.1999(16) | 1/2- | 46 s | 24.20(2) | $E3$ | %IT = 100
72 |  178Hf | 2446.09(8) | 16+ | 31 y | 12.7(2) |  | %IT = 100
|  |  |  |  | 309.50(15) | $M4(+E5)$ |
|  |  |  |  | 587.0(1) | $E5$ |
72 |  177Hf | 2740.02(15) | 37/2- | 51.4 min | 214 | $E3$ | %IT = 100
72 |  179Hf | 1105.74(16) | 25/2- | 25.0 d | 21.01(12) | $M2$ | %IT = 100
|  |  |  |  | 257.37(15) | $E3$ |
73 |  180Ta | 77.1(8) | 9- |  $>1.2\times 10^{15}\,$y |  |  |
77 |  192Ir | 56.720(5) | 1- | 1.45 min | 56.71(3) | $E3$ | %IT = 99.9825
|  |  |  |  |  |  | %$\beta^{-}$ = 0.0175
Table 10: Isomers potentially useful for gamma-ray lasers [404, 405]. See
Table 7 for explanations of the notations.

To summarize, the GF can help facilitate research on induced emission and
lasing with nuclear transitions, although, a clear path to a graser does not
seem to be in sight as yet.

##  10 Gamma polarimetry

###  10.1 Polarimetry with narrow resonances

Narrow–band GF photons in combination with polarized nuclear–target technology
open a possibility to perform polarimetry using the target as a polarization
filter. To this end, the energy of the GF photons should be tuned in resonance
with a gamma transition in the target nucleus.

Suitable transition can be found in the 13C ($I^{P}=1/2^{-}$ for the stable
ground state; see also Fig. 8), which can be made into a polarized target
[406]. For example, there is an $E1$ transition to an $I^{P}=1/2^{+}$ state at
3.09 MeV that has a radiative width of 0.43 eV and radiatively decays back to
the ground state. There is also an $M1$ transition to an excited
$I^{P}=1/2^{-}$ state at 8.86 MeV that has a width of 150 keV. The upper state
of this transition lies above the neutron-separation energy for 13C (4.9 MeV)
and predominantly decays by neutron emission to the ground state of 12C (the
ground state is the only possible state of the resultant 12C because the
excited states of this nucleus lie sufficiently high).

The resonant cross section for a closed transition is (cf. Eq. 43)

$\sigma\simeq\frac{\lambda^{\prime 2}}{2\pi},$ (34)

where $\lambda^{\prime}$ is the wavelength in the ion frame. For the 3.09 MeV
$E1$ transition, $\sigma\approx$260 b (1 b=10$\,{}^{-24}$ cm2) is large enough
that it is feasible to construct a target comprising multiple absorption
lengths for resonant photons. The expected asymmetry for absorption of
circularly polarized photons on a $1/2\rightarrow 1/2$ transition is 100%, and
thus such a system can be an efficient circular analyzer, perhaps only limited
in precision by the counting statistics of the gamma rays. One should note in
this respect that the resonance will be narrower than the spectral width of
the gamma beam, so only a fraction of the photons will, in fact, be resonant,
see Appendix B.

This should not be a problem for the broader 8.86 MeV $M1$ resonance; however,
the resonant cross section is some five orders of magnitude smaller because
the width is dominated by decay via neutron emission rather than gamma
transition and because of the smaller photon wavelength.

Due to these factors, polarimetry may need to rely, instead of photon-
transmission measurements, on the detection of reaction products (neutrons or
non-collinear photons).

###  10.2 Other polarimetric techniques

Polarization measurements of gamma and X-rays have a long tradition in many
branches of modern physics. For example, a number of polarization–sensitive
studies were performed in astrophysics [407, 408], atomic, nuclear and plasma
physics [409, 410, 411]. Depending on the energy of the photons, the
photoelectric effect, Compton effect, and electron–positron pair production
are typically employed in these experiments. In addition, in the recent years,
Bragg scattering was successfully used for high–precision polarization
measurements in the keV energy range [412, 413, 414].

During the last two decades, a number of experiments were performed also to
study linear polarization of photons, emitted in relativistic collisions at
ion storage rings [415, 410, 416]. These measurements usually employ the
polarization sensitivity of Compton scattering of light by (quasi–free)
electrons. Namely, the angular distribution of Compton–scattered photons is
defined by the direction of the linear polarization of incident radiation, as
predicted by the well–known Klein–Nishina formula [415]. In order to measure
this angular distribution and, hence, to determine the linear polarization one
usually employs solid–state position sensitive detectors. These Compton
detectors allow (linear) polarization measurements in the energy range of
about 10 keV to 10 MeV [417]. Above 10 MeV, pair-production polarimetry can be
used [418, 419].

In contrast to linear polarization measurements, less progress has been made
in the development of detectors for the measurement of circular polarization
of X- and gamma rays. Due to the absence of a polarimetry technique which
could be combined with gamma-ray imaging, for example, no circular
polarization measurements have been reported in gamma-ray astronomy. In atomic
and nuclear physics, however, the experimental approach based on Compton
scattering off magnetized solid targets has been applied during the last
decades. Moreover, a combined measurement of Compton scattered photons and
subsequent bremsstrahlung of the recoiled electron has been recently proposed
as a novel approach to circular polarimetry [420]. It is expected that this
novel approach will allow accurate studies of circular polarization of X- and
gamma rays with energies up to several tens of MeV.

The $\Delta(1232)$ resonance is a dominant feature of the nucleon-excitation
spectrum for photon energies $\omega\approx 300$ MeV (in the photon-proton
center of momentum frame). It can be used to analyze the circular photon
polarization. It is common to describe the photoexcitation of $\Delta(1232)$
by the value of the resonance photoabsorption amplitudes with the parallel and
the antiparallel helicity configuration, $A_{3/2,1/2}$. For these, Ref. [341]
gives the values $A_{3/2}=-0.255(7)$ GeV-1/2 and $A_{1/2}=-0.135(7)$ GeV-1/2
[341]. Note that the uncertainties are relatively large due to averaging over
different theoretical analyses which individually have much smaller
uncertainties, comfortably within 2-3% for $A_{1/2}$ and about half that value
for $A_{1/2}$. The model dependence stems from the separation of the
experimental data into the resonant and the non-resonant parts, a procedure
that is model-dependent. Their ratio amounts to $|A_{3/2}|/|A_{1/2}|\approx
1.89$, resulting in the asymmetry
$\frac{|A_{3/2}|^{2}-|A_{1/2}|^{2}}{|A_{3/2}|^{2}+|A_{1/2}|^{2}}\approx 0.56$.
The helicity-dependent total cross section integrated over the full spectrum
is relevant for the GDH sum rule, see Sec. 4.7.

Development of precision polarimetry and spectropolarimetry (a combination of
polarimetry and spectroscopy) at the GF, would open novel possibilities for
fundamental-physics experiments such as studying PV in the vicinity of nuclear
gamma transitions (Sec. 4.11) and measurement of vacuum-birefingence effects
(Sec. 11).

##  11 Quantum vacuum effects

The quantum vacuum amounts to a highly nontrivial state, characterized by the
omnipresence of fluctuations of virtual particles. While the microscopic
theory of quantum electrodynamics (QED) does not provide for a direct tree-
level interaction among photons, quantum vacuum fluctuations induce effective
nonlinear couplings among electromagnetic fields [421, 422, 423, 424]. At zero
field, the quantum vacuum is characterized by translational invariance and the
absence of any preferred direction. Conversely, an external electromagnetic
field generically introduces a preferred direction for charged particles, and,
if inhomogeneous, also breaks translational invariance. Via the charged
particle-antiparticle fluctuations coupling to the external electromagnetic
field, this preferred direction can also impact probe photon propagation and
give rise to nonlinear QED effects such as vacuum birefringence and photon
splitting; see Fig. 31 for the corresponding Feynman diagrams at one-loop
order. See also the recent reviews [425, 426] and references therein.

Figure 31: Lowest-order Feynman diagrams of typical quantum vacuum effects,
giving rise to (a) vacuum birefringence and (b) photon splitting. The blue
wiggly lines are probe photons and the red wiggly lines ending at crosses
denote couplings to the prescribed electromagnetic field.

Here, we briefly discuss the perspectives of studying quantum vacuum effects
at the Gamma Factory, using the prominent signatures of vacuum birefringence
(in magnetic and laser fields) and photon splitting (in atomic fields) as
illustrative examples. Both effects arise from an effective four-photon
interaction mediated by a electron-positron fluctuation [427, 428, 429, 430];
cf. Fig. 31. The former scales quadratically and the latter linearly with the
background field. Besides, we comment on quasi-elastic photon scattering.

###  11.1 Vacuum birefringence

Linearly polarized probe photons (energy $\omega$) traversing a strong pump
field ($\vec{\cal E}$, $\vec{\cal B}$) can pick up an ellipticity if their
polarization vector has a nonvanishing overlap with the two distinct
polarization eigenmodes imprinted on the quantum vacuum by the pump field
[431]. See Fig. 31(a) for the corresponding Feynman diagram.

So far, this fundamental effect induced by quantum vacuum fluctuations has
never been directly verified in laboratory experiments using macroscopic
fields; see the review [432] and references therein. Typical proposals for
measuring vacuum birefringence in a laboratory experiment envision the effect
to be induced either by (A) quasi-constant static magnetic field, or (B) a
counter-propagating high-intensity laser pulse.

Given that the following conditions hold [433],

$\left\\{\Bigl{(}\frac{e{\cal
E}}{m_{e}^{2}}\Bigr{)}^{2},\Bigl{(}\frac{\Omega}{m_{e}}\Bigr{)}^{2},\frac{\omega\Omega}{m_{e}^{2}},\Bigl{(}\frac{e{\cal
E}\omega}{m_{e}^{3}}\Bigr{)}^{2}\right\\}\ll 1,$ (35)

where $\hbar=c=1$, the vacuum-birefringence phenomenon can be studied on the
basis of the leading contribution to the renowned Heisenberg-Euler effective
Lagrangian [422, 423]. Here, ${\cal E}={\rm max}\\{|\vec{\cal E}|,|\vec{\cal
B}|\\}$, $\Omega$ denotes the typical frequency scale of variation of the pump
field, $e$ is the elementary charge and $m_{e}$ is the electron mass.

In the same parameter regime, the polarization-sensitive absorption
coefficients associated with the principle possibility of electron-positron
pair production in the presence of the background field are exponentially
suppressed with $m_{e}^{2}/(e{\cal E})\gg 1$ for $\omega\ll m_{e}$ and
$m_{e}^{3}/(e{\cal E}\omega)\gg 1$ for $\omega\gg m_{e}$; cf., e.g., Refs.
[434, 435].

####  11.1.1 Quasi-constant magnetic field

First, we consider the effect of vacuum birefringence induced by a quasi-
constant static magnetic field $\vec{\cal B}$. In this case, probe photons
polarized parallel $\parallel$ (perpendicularly $\perp$) to the plane spanned
by their wave vector and the direction of $\vec{{\cal B}}$ experience
different refractive indices. Given that the conditions (35) are met, for
probe photons propagating perpendicularly to $\vec{\cal B}$ these refractive
indices are given by [436]

$\left\\{\begin{array}[]{c}n_{\parallel}\\\
n_{\perp}\end{array}\right\\}\simeq 1+\frac{\alpha}{\pi}\Bigl{(}\frac{e{\cal
B}}{m_{e}^{2}}\Bigr{)}^{2}\frac{1}{90}\left\\{\begin{array}[]{c}7\\\
4\end{array}\right\\}\,,$ (36)

with fine-structure constant $\alpha\simeq 1/137$.

Figure 32: Graphical depiction of the two scenarios envisioned for the
detection of QED vacuum birefringence at the Gamma Factory as discussed in the
text. In scenario (A) the birefringence phenomenon is induced by the quasi-
constant static magnetic field provided by LHC dipole magnets. In scenario (B)
the effect is driven by a counter-propagating focused high-intensity laser
pulse.

Letting initially linearly polarized photons having an equal overlap with both
polarization eigenmodes ($\parallel$,$\perp$) traverse a constant magnetic
field extending over a length $l$, an ellipticity characterized by the phase
difference $\Phi=\omega l(n_{\parallel}-n_{\perp})$ between the two
polarization components is induced. The explicit expression for the
accumulated phase difference is [437]

$\Phi\simeq\omega l\,\frac{\alpha}{30\pi}\Bigl{(}\frac{e{\cal
B}}{m_{e}^{2}}\Bigr{)}^{2}\,.$ (37)

Equation (37) shows that the effect scales quadratically with the magnetic
field strength and linearly with the energy of the probe photons and their
propagation distance in the magnetic field. At the same time, the
birefringence property of the polarized quantum vacuum results in signal
photons scattered into a mode polarized perpendicularly to the incident probe
light: for the case of a static quasi-constant magnetic field as considered in
this subsection, the number of polarization flipped signal photons is given by
$N_{\perp}\simeq(\Phi/2)^{2}N$, where $N$ denotes the number of gamma photons
available for probing the effect.

The GF will enable such a vacuum-birefringence experiment with a probe-photon
energy as high as $\omega\simeq 400\,{\rm MeV}$. The driving magnetic field
could be provided by a sequence of LHC dipole magnets, delivering a magnetic
field of ${\cal B}\simeq 8.3\,{\rm T}$ over a length of $L\simeq 14.3\,{\rm
m}$ each [438]; the effective diameter $d$ of the bore for traversing light is
about $d\simeq 45\,{\rm mm}$. In this case, we have
$(\Omega/m_{e})^{2}\simeq\omega\Omega/m_{e}^{2}\simeq 0$, $(e{\cal
B}/m_{e}^{2})^{2}\simeq 3.54\times 10^{-18}$ and $(e{\cal
B}\omega/m_{e}^{3})^{2}\simeq 2.17\times 10^{-12}$ fully compatible with Eq.
(35).

To achieve gamma-photon energies up to $\omega\simeq 400\,{\rm MeV}$, the
Lorentz factor $\gamma$ which effectively governs the generation of the high-
energy gamma beam in the GF needs to be as large as $\gamma\approx 3000$; cf.
Sec. 1.1 above. As the opening angle of the gamma beam is given by $\approx
1/\gamma$, the bore diameter of the magnet implies a maximum length $l_{\rm
max}\approx(d/2)\gamma\simeq 67.5\,{\rm m}$ of the magnetic field provided by
LHC magnets through which the full gamma beam could travel. In turn, we can
envision the use of up to four LHC dipole magnets resulting in $\omega l\simeq
1.16\times 10^{17}$, see Fig. 32(A) for an illustration. For these parameters
we obtain a small (but potentially measurable, see Sec. 10) value of
$\Phi\simeq 3.18\times 10^{-5}$.

We note that this value is about an order of magnitude larger than the one
predicted to be accessible in the head-on collision of a state-of-the-art
petawatt-class high-intensity-laser and free-electron-laser (FEL) pulses of
$\omega\simeq{\cal O}(10)\,{\rm keV}$ [439, 440, 441, 442, 443]. For X-rays of
$\omega\simeq{\cal O}(10)\,{\rm keV}$ the possibility of measuring such tiny
ellipticities has been demonstrated experimentally [412, 413, 414]. While
these x-ray techniques cannot be used at $400\,{\rm MeV}$, in the latter
parameter regime pair-production polarimetry may be used [418, 419]. See
Refs.[444, 445] for proposals to measure magnetic-field-induced vacuum
birefringence with gamma photons adopting somewhat different experimental
parameters.

The advantage of using a static magnetic field to drive the vacuum
birefringence phenomenon is that essentially all gamma photons traverse the
magnetic field. In turn, the associated number of polarization-flipped signal
photons $N_{\perp}$ scales directly with the total number of photons
constituting the gamma beam, and thus is ultimately limited (among other
factors) by the repetition rate of the gamma pulses. Conversely, in the
scenario utilizing a laser pulse to induce the birefringence phenomenon the
repetition rate of the high-intensity laser is the limiting factor; see below.

####  11.1.2 High-intensity laser pulse

Alternatively, the birefringence signal could be driven by a high-intensity
laser field [446, 447]; see Fig. 32(B). In such a scenario, the birefringence
signal is predominantly induced in the interaction region where the gamma
probe collides with the focused high-intensity laser pulse reaching its peak
field strength. Outside the focus, the field strength of the high-intensity
pump drops rapidly.

State-of-the-art high-intensity lasers of the petawatt-class typically deliver
pulses of energy $W={\cal O}(10)\,{\rm J}$ and duration $\tau={\cal
O}(10)\,{\rm fs}$ at a wavelength of $\lambda={\cal O}(1)\,\upmu{\rm m}$ and a
repetition rate of ${\cal O}(1)\,{\rm Hz}$. These pulses can be focused to a
waist radius of $w_{0}\gtrsim\lambda$. In turn, the maximum frequency scale of
variation of the pump field is given by $\Omega=2\pi/\lambda$. For our
explicit example, we choose the parameters characterizing a commercial 300 TW
Titanium Sapphire laser system, such as the one installed at the Helmholtz
International Beamline for Extreme Fields (HiBEF) at the European XFEL:
$W=10\,{\rm J}$, $\tau=30\,{\rm fs}$, $\lambda=800\,\upmu{\rm m}$ and a
repetition rate of $1\,{\rm Hz}$ focused to $w_{0}=1\,\upmu{\rm m}$. The
associated Rayleigh range is ${\rm z}_{\rm R}=\pi w_{0}^{2}/\lambda$. Assuming
the high-intensity laser field to be well-described as pulsed paraxial
fundamental Gaussian beam, the electric peak field strength ${\cal E}_{0}$ in
its focus can be expressed in terms of the pulse energy, pulse duration and
waist radius as [448]

${\cal E}_{0}^{2}\simeq 8\sqrt{\frac{2}{\pi}}\,\frac{W}{\pi w_{0}^{2}\tau}\,.$
(38)

For the laser parameters given above and $\omega\simeq 400\,{\rm MeV}$, the
dimensionless quantities in Eq. (35) are $(\Omega/m_{e})^{2}\simeq 9.2\times
10^{-12}$, $(e{\cal E}_{0}/m_{e}^{2})^{2}\simeq 1.46\times 10^{-7}$,
$\omega\Omega/m_{e}^{2}\simeq 2.4\times 10^{-3}$ and $(e{\cal
E}_{0}\omega/m_{e}^{3})^{2}\simeq 0.09$. These values suggest that in this
parameter regime the size of the attainable vacuum birefringence signal can
still be reliably estimated from the leading contribution to the Heisenberg-
Euler effective Lagrangian.

At the Gamma Factory, the collision point of the gamma beam with the high-
intensity laser beam should be sufficiently separated from the source of the
gamma photons such that the radius $w_{\gamma}$ of the gamma beam in the
interaction region with the high-intensity laser pulse generically fulfills
$w_{\gamma}\gg w_{0}$; cf. also Fig. 32(B) and Ref. [449]. Even though this
implies that only a fraction of the gamma photons will actually see the high-
intensity laser focus, focusing the latter less tightly to $w_{0}\approx
w_{\gamma}$ is not an option: as the ellipticity is proportional to ${\cal
E}_{0}^{2}\sim 1/w_{0}^{2}$ [see Fig. 32(a) and Eq. (39) below], an increase
of $w_{0}$ would immediately reduce the effect. Moreover, the GF is expected
to provide gamma pulses of duration $T\gtrsim 160\,{\rm fs}$.

Given that $w_{\gamma}\gg w_{0}$ and the pulse duration of the gamma beam
meets the criterion $T\gg\\{\tau,{\rm z}_{\rm R}\\}$, which is true for the
parameters of the GF, the phase difference accumulated by the gamma beam can
be expressed in a form similar to Eq. (37), yielding [433]

$\Phi\simeq\omega{\rm z}_{\rm R}\frac{2^{1/4}\alpha}{30}\Bigl{(}\frac{e{\cal
E}_{0}}{m_{e}^{2}}\Bigr{)}^{2}\,{\rm e}^{\frac{1}{2}(\frac{8{\rm z}_{\rm
R}}{\tau})^{2}}\,{\rm erfc}^{1/2}\bigl{(}\tfrac{8{\rm z}_{\rm
R}}{\tau}\bigr{)}\,,$ (39)

with complementary error function ${\rm erfc}(.)$. We emphasize that this
parameter regime even seems to be particularly beneficial for high-intensity
laser driven vacuum-birefringence experiments: given that the conditions
$T\gg\tau$ and $w_{\gamma}\gg w_{0}$ are met, the experiment is essentially
insensitive to the shot-to-shot fluctuations inherent to high-intensity laser
systems resulting in spatio-temporal offsets of ${\cal O}(w_{0})$ of the
position of the high-intensity laser focus. Variations of this order just
change the location of the high-intensity laser focus within the forward cone
of the gamma probe and thus do not impact the signal; cf. also Fig. 32(B).

Similarly to the result (37) for a constant magnetic field, the phase
difference (39) scales quadratically with the field strength of the pump
field, and linearly with both $\omega$ and the typical extent of the pump
field along the propagation direction of the probe. This extent is typically
set by the Rayleigh range ${\rm z}_{\rm R}$ of the high-intensity laser.

Plugging the above parameters into Eq. (39), we obtain $\Phi\simeq 0.13$,
which is much larger than the analogous value obtained for the case of a
static magnetic field discussed in Sec. 11.1.1. Assuming in addition that the
pulse duration and beam radius of the gamma beam in the interaction region are
given by $T\simeq 160\,{\rm fs}$ and $w_{\gamma}\simeq 20\,\upmu{\rm m}$, we
find $N_{\perp}/N\simeq 2.07\times 10^{-6}$ for the ratio of the polarization-
flipped signal photons and the number of gamma photons $N$ available for
probing the effect; note that in the present scenario
$N_{\perp}/N=(\tau/T)(w_{0}/w_{\gamma})^{2}(\Phi/2)^{2}$ [433].

However, we emphasize once again that in contrast to the case of a static
magnetic field, for the scenario involving a high-intensity laser pulse, only
a fraction of the total number of gamma photons provided by the Gamma Factory
is available for probing the vacuum birefringence phenomenon: in this case the
repetition rate of the experiment is limited by the repetition rate of ${\cal
O}(1)\,{\rm Hz}$ of the high-intensity laser.

Higher laser intensities and/or larger gamma photon energies would even allow
for experimental probes of vacuum birefringence in the parameter regime
characterized by $(e{\cal E}\omega/m_{e}^{3})^{2}\gtrsim 1$, violating the
condition (35). The theoretical study of quantum vacuum signatures in this
parameter regime requires insights beyond the Heisenberg-Euler effective
Lagrangian. Vacuum birefringence in this parameter regime [450, 451, 452, 453]
can be reliably studied resorting to the photon-polarization tensor evaluated
in the background of the pump field.

Finally, we note that aside from the prospect of directly verifying a
fundamental QED prediction for the first time, a precision measurement of the
ellipticity constituting the signal of vacuum birefringence in Eqs. (37) and
(39) at the Gamma Factory would also constitute a sensitive probe for New
Physics beyond the standard model of particle physics. The latter may leave an
imprint on the refractive index of the vacuum resulting in deviations from the
standard model prediction; see, for example, the recent review [454] and
references therein. For a survey of the potential of the Gamma Factory for
searches of axion like particles, see Ref. [455].

###  11.2 Photon Splitting

In contrast to vacuum birefringence, the effect of photon splitting in atomic
fields mediated by an electron-positron fluctuation [456, 457, 458, 459] has
already been successfully observed in a dedicated laboratory experiment
employing probe photons in the energy region of $120-450\,{\rm MeV}$ [460].
See Fig. 31(b) for the corresponding Feynman diagram. At present, the
experiment and the theory are consistent within the achieved experimental
accuracy [459].

The high flux of gamma photons at the Gamma Factory as well as the possibility
of a precise tuning of their energy will allow for detailed experimental
studies of this nonlinear QED process at high statistics and accuracy. This
will provide a sensitive test of theory at unprecedented precision.

For completeness, we note that the photon splitting process can in principle
also be triggered by constant electromagnetic fields and laser fields [461,
462, 463, 464, 465, 466, 467, 468, 469, 470], but is typically suppressed in
these cases.

###  11.3 Photon Scattering

In the scenario detailed in Sec. 11.1.2, also quasi-elastic scattering of
gamma photons off the optical high-intensity laser pulse would constitute a
signature of quantum vacuum nonlinearity. For the specific scenario considered
there, the maximum value for the total number of quasi-elastically scattered
gamma photons is given by $N_{\rm tot}=(196/9)N_{\perp}\simeq 21.8N_{\perp}$
[471]. However, for kinematic reasons, these signal photons are scattered into
the forward opening angle of the gamma beam [433], and thus generically cannot
be discerned from the large background of the gamma photons constituting the
probe and traversing the interaction region essentially unmodified. In turn,
aiming at measuring the quasi-elastic scattering signal, collimators would be
needed to reduce the divergence of the gamma photons before the interaction
with the high-intensity laser pulse [472, 473, 474]. The same is true for
scenarios envisioning the collision of gamma photons with the photons
constituting the initial laser beam.

Experimental bounds on elastic photon-photon scattering from direct searches
with optical and X-ray beams are discussed in Refs. [475, 476], while Refs.
[477, 478, 479, 480] present recent experimental evidence of light-by-light
scattering with almost real photons in the ATLAS and CMS experiments at CERN.

Finally, we note that for gamma-photon vs. laser-photon collisions, the Gamma
Factory will enable center-of-mass energies in the range of $\sqrt{s}\simeq
1\ldots 120\,{\rm keV}$ at a luminosity of up to $L\simeq 10^{40}\,{\rm
cm}^{-2}{\rm s}^{-1}$ [455]. For center-of-mass energies below the electron-
positron pair-production threshold, i.e., $\sqrt{s}<2m_{e}\simeq 1\,{\rm
MeV}$, no real electron-positron pairs are produced in such collisions. On the
other hand, for prospective gamma-photon vs. gamma-photon collisions, one
could in principle probe center of mass energies in the range $\sqrt{s}\simeq
1\ldots 800\,{\rm MeV}$ at a luminosity of up to $L\simeq 10^{27}\,{\rm
cm}^{-2}{\rm s}^{-1}$. Note however, that in this parameter regime the
signature of light-by-light scattering is likely obscured by electron-positron
pair production via the Breit-Wheeler process [481] and the corresponding
secondary scattering processes; cf. also Ref. [482].

##  12 Nuclear physics with tertiary beams

###  12.1 Tertiary beams at the GF

The GF beam of gamma rays can be used to produce tertiary beams in collisions
with an external stationary target. The photon-based scheme represents a
change of the present, canonical paradigm for the production of such
derivative beams in which the particles are produced in strong-interaction-
mediated collisions. The GF tertiary-beam production scheme is based on the
peripheral, small-momentum-transfer electromagnetic interactions of the
photons with atoms of the target material.

An example of a Feynman diagram for peripheral production of lepton pairs is
shown in Fig. 33.

Figure 33: Feynman diagram for lepton-pair production in peripheral photon-
nucleus collisions. Below the muon-pair production threshold, only electron-
positron pairs can be produced in this process. The cross section for
production of muon pairs rises rapidly with the energy of the gamma rays. For
example, for a copper target it rises from 0.2 $\mu$b, to 10 $\mu$b for the
photon energy rising from $\approx$400 MeV—which is presently accessible at
the LHC—to 1600 MeV, achievable by the high-energy (HE-LHC) upgrade [483]. The
cross section also rises rapidly with the atomic number on the target nucleus:
from 0.016 $\mu$b for hydrogen to 30.2 $\mu$b for lead, for 1 GeV photons.
Muon photoproduction is discussed in detail in Ref. [484].

In the case of peripheral electromagnetic processes, a large fraction of the
wall-plug power delivered to the stored PSI beam can be transmitted to a
chosen type of the tertiary beam as opposed to the proton-beam-driver schemes.
A photoproduction scheme may considerably reduce the target heat-load at a
fixed intensity of the produced beam, facilitating the target design and
circumventing the principal technological challenges which limit the
intensities of the proton-beam-driven beams.

As an example, a 1 MW 300 MeV photon beam, producing a beam of $4\times
10^{15}$ collected positrons per second deposits 140 kW of heat power in a
one-radiation-length (1X0) graphite target–only 14% of the wall-plug power
dissipated in such a case [485].

###  12.2 Polarized electron, positron and muon sources

The high-intensity beam of gamma rays from the GF can be converted into a
high-intensity beam of positrons and electrons. If the photon-beam energy is
tuned above the muon-pair-production threshold, such a beam also contains a
small admixture of $\mu^{+}$ and $\mu^{-}$.161616The cross-section of muon
pair production is numerically small until the energy of the photon reaches
about $4\times m_{\mu}c^{2}$ [484]. At the present LHC beam energy, crossing
this effective muon-production threshold requires laser photons in the EUV
region, down to 100 nm wavelength. While the photon pulses in this wavelength-
range can be produced–owing to the recent developments in laser technology–a
substantial progress in the mirror technology is required to amplify the
average power of the laser pulses, by stacking them in a Fabry-Perot cavity,
to the level of $\approx$10-100 kW (0.5-5 mJ pulses with the 20 MHz repetition
rate). Such an amplification is required to significantly improve the
intensity of the presently operating muon sources. Two fall-back options can
be considered: (1) doubling the LHC beam energy, as proposed in the HE-LHC
project [483], and (2) using a free electron laser (FEL) rather than a
conventional laser technology.

Beams of different lepton flavors can easily be separated using both their
respective and distinct kinematic characteristics and the time-of-flight
method, since the produced electrons and positrons move with nearly the speed
of light while the GF muons are non-relativistic.

The target intensity of the GF source of electrons/positrons is $10^{17}$
leptons per second, assuming the present CERN accelerator infrastructure and
presently available laser technology. Such intensity, if achieved, would be
three orders of magnitude higher than that of the KEK positron source [486],
and would largely satisfy the positron/electron source requirements for the
proposed future high-luminosity and high-energy $ep$ ($eA$) collider project
[487].

If the secondary GF photons are circularly polarized–for example, by using
circularly polarized laser photons colliding with spin-0, helium-like
partially stripped ions–the first generation electrons, positrons, or muons,
are polarized parallel to the photon-beam direction.

The intensity of the polarized muons which is reachable at the LHC, using the
GF photon-conversion scheme, is $10^{10}$ muons per second.171717The ratio of
cross sections for producing muon pairs and electron-positron pairs scales
approximately as the square of the ratio of their masses:
$m_{e}^{2}/m_{\mu}^{2}$ in the $E_{\gamma}\gg 2m_{\mu}$ limit. If achieved, it
would be two orders of magnitude higher than that of the $\pi E4$ beam at the
Paul Scherrer Institute [488]. With the HE-LHC upgrade, the intensity of the
polarized muon source is expected to increase to $10^{12}$ muons per second.

The intensity of the photon-conversion-based GF polarized-muon source does not
satisfy the muon-beam intensity requirements for the neutrino factory and muon
collider project [489]–($10^{13}$ muons of each sign per second). Therefore,
another production scheme, suitable for generating the requisite muon fluxes,
is being developed [485]. Preliminary calculations show that the requisite
production rate of polarized muons can be reached for 1 MW photon beams. The
principal advantage of the GF scheme, with respect to proton-beam-driver
schemes, is that the product of the muon-source longitudinal and transverse
emittances can be improved by more than three orders of magnitude. There are
many applications of intense muon beams, including studies of the basic
symmetries of nature via searches for standard-model-forbidden muon-decay
modes with unprecedented precision [490], studies of the nuclear properties
via spectroscopy of muonic atoms [491], and revisiting the feasibility of the
muon catalyzed nuclear fusion [492, 493]. For the latter two applications,
requiring high flux of negatively charged muons, the GF muon source has an
important advantage with respect to proton driven sources producing
predominantly positively charged muons. The GF tertiary, high-brilliance beams
of polarized positrons and muons can open the paths to new type of fixed-
target deep inelastic scattering (DIS) experiments requiring high-intensity
lepton beams.

A detailed comparison of production rates of $e^{+}e^{-}$ and $\mu^{+}\mu^{-}$
pairs is of interest for tests of lepton flavor universality, for example, in
the context of the “proton radius puzzle” [494]. A first observation of di-
muonium (i.e., a bound state of $\mu^{+}\mu^{-}$) production could also be of
interest [495].

###  12.3 High-purity neutrino beams

Low-emittance muon beams provided by the GF source can be accelerated and
stored in specially designed storage rings–which preserve polarization of
stored muons–to produce muon-neutrino, electron-antineutrino, muon
antineutrino and electron neutrino beams of precisely controlled fluxes.

The relative fluxes of neutrinos and antineutrinos of each flavor can be
precisely controlled by a frequent change of the sign of of the stored muons.

Thanks to the muon polarization and ($V-A$)-type of the weak currents, the
relative flux of muon-neutrinos (muon-antineutrinos) coming from negative-muon
(positive-muon) decays, and the flux of electron-antineutrino (electron-
neutrino) can be precisely controlled on the basis of their respective angular
distributions. For more details see, for example, Ref. [496].

The fluxes of both the neutrino and antineutrino beams should be equal and
they can be predicted to a tenth of a percent accuracy, limited predominantly
by the measurement of the muon-beam current in the muon storage ring.

###  12.4 Neutron and radioactive ion sources

The energy of the GF photons can be tuned to excite the GDR (see Sec. 4.3) or
fission resonances (Sec. 4.5) of large-$A$ nuclei, providing abundant sources
of: (1) neutrons with the target intensity reaching $10^{15}$ neutrons per
second (first-generation neutrons), (2) radioactive, neutron-rich ions (coming
from fission of heavy nuclei) with the target intensity reaching $10^{14}$
isotopes per second [497] (see also Sec. 8).

The above fluxes would approach those of other European projects under
construction, such as the European Spallation Source (ESS), FAIR and the
future EURISOL facilities. The advantage of the GF sources is their high
efficiency – almost $10\%$ of the LHC RF power can be converted into the power
of the neutron and radioactive-ion beams. The high-flux neutron source could
provide new opportunities for investigation of the basic symmetries of nature,
for example, via searches for the permanent electric dipole moment (EDM) of
the neutron (see Ref. [498] and references therein) or neutron-antineutron
oscillations [499].

Finally, the tertiary beams of neutrons and radioactive neutron-rich isotopes
could open a wide spectrum of industrial and medical applications in the
domains of: (1) muon catalyzed cold fusion [492, 493]; (2) energy-amplifier
(EA) research [500]; and (3) production of ions for positron emission
tomography (PET) and for the selective cancer therapy with alpha emitters
[501].

###  12.5 Production of monoenergetic fast neutrons

Monoenergetic neutrons can be produced in ($\gamma,$n) reactions near narrow
resonances like the ones in 13C (see Appendix B). If the gamma beam is
polarized, the polarization will be transferred to the neutrons.

As an example, consider neutron production using the 8.86 MeV resonance of 13C
(see Appendix B.1). Neutrons will be produced in the reaction [502]

$\gamma+{}^{13}\textrm{C}\rightarrow{}^{12}\textrm{C}+n;\quad
Q_{m}=-4.95\,\textrm{MeV}.$ (40)

Here the energy release of the reaction is $Q_{m}\approx T_{n}-E_{\gamma}$,
neglecting the recoil energy of 12C. With this, we find that the kinetic
energy of the neutron $T_{n}\approx$3.9 MeV is lower than the energy of the
first excited state of 12C (4.44 MeV). Therefore, only ground-state 12C will
be produced and the neutrons will be nearly monoenergetic.

The neutron-production rate is essentially the same as the gamma-absorption
rate (Appendix B.1). For a $1\times 1\times 1$ cm3 target containing 1 g of
13C, we estimate the neutron-production rate as $5.4\times 10^{10}$ s-1.

A potential problem for a source is re-absorption of neutrons: the produced
neutrons may further react with the target according to

$n+{}^{13}\textrm{C}\rightarrow{}^{14}\textrm{C}+\gamma;\quad
Q_{m}=8.18\,\textrm{MeV}.$ (41)

Given $T_{n}\approx 3.9$ MeV, we get $E_{\gamma}\approx 12.1$ MeV. The cross
section for this neutron-capture process can be estimated as [503]

$\sigma_{tot}=2\pi(R+\lambdabar)^{2},$ (42)

where $\lambdabar$ is the reduced wavelength of the neutron, $R=2.5$ fm,
$\lambdabar=\hbar/p=\hbar c/\sqrt{2mc^{2}T_{n}}=2.30$ fm ($p$ is the neutron
momentum), yielding $\sigma_{tot}\approx 1.4\,$b. With a $1\times 1\times
1$cm3 target containing 1 g of 13C, fewer than 10$\%$ of the neutrons will be
reabsorbed, so the issue appears tractable.

As for the 7.55 MeV resonance (Appendix B.2), the production rate of neutrons
with $T_{n}\approx 2.6$ MeV is $9.0\times 10^{9}$ s-1. The cross section for
the neutron-capture process is 2.1 b. Again, fewer than 10$\%$ of the neutrons
will be reabsorbed.

Narrow gamma resonances in conjunction with the GF, will thus allow producing
monoenergetic neutrons at a set of well defined energies (e.g., 2.6 MeV, 3.9
MeV,… for the case of a 13C target). Such neutrons may be useful for measuring
cross sections of processes (e.g., neutron capture) relevant to astrophysics.

It is important to note that the angular distribution of the neutrons will be
nearly isotropic, suggesting a geometry for the target to study the
interaction of the produced neutrons: it could be a “ball” surrounding the
neutron-production target.

###  12.6 Metrology with keV neutrons

Tertiary neutrons produced in $(\gamma,n)$ reactions can be used to improve
the operation quality of the GF. The gamma-beam energy above the neutron
threshold (typically, around 7 MeV) can be measured with high accuracy by
measuring the time-of-flight (TOF) of the neutrons. By subtracting the large
fixed neutron binding energy, resulting in rather slow neutrons with keV
energies together with a sharp start signal for the TOF measurement, a high
resolution at the $10^{-7}$ level can be achieved [504]. This enables a fast
measurement of the average gamma-beam energy and the width of the gamma beam
without perturbing the GF. This could provide feedback for adjusting the ion-
beam energy, thus facilitating experiments requiring precise control of the
ion-beam or gamma-ray energy.

##  13 Nuclear physics opportunities at the SPS

The Proof-of-Principle (PoP) GF experiment is proposed at the SPS [7]. Here,
we have a much lower relativistic factor, $\gamma\lesssim 220$. Since it is
nontrivial to get low-energy gamma rays with $\gamma\geq 200$ at the LHC, the
operation at the SPS extends the available energy range of the secondary
photons. Note that for X-rays, there exist intense coherent sources, for
example, the European XFEL [99] producing X-rays with energies $\lesssim
30\,$keV.

At the SPS, with $\lesssim 10$ eV primary laser photons, secondary photon
energies of up to 1.6 MeV will be available. The use of an FEL primary-photon
source (primary photon energies of up to a few hundred eV) can extend the
energy range, partially overlapping with the range that should be available at
the LHC with conventional lasers. Already with the secondary-photon energies
available with the currently planned PoP experiment [7], $\lesssim$44 keV,
interesting new results could be obtained. Table 2 contains information on
nuclear transitions in the range $0.008-60$ keV. We have also discussed
nuclear Raman transitions accessible with the secondary photon beams (or even
with primary beams if an FEL is used) at the SPS in Sec. 2.2.

The SPS experiments could also serve as a platform for development of laser
cooling of the PSI and ion sources, for example, 201Hg and investigation of
its 1.5 keV gamma resonance (see Table 2).

##  14 Speculative ideas and open questions

###  14.1 Applying the Gamma Factory ideas at other facilities

The ideas of the Gamma Factory can also be implemented at other facilities. In
fact, back-scattering of laser photons from hydrogen-like ions and laser
cooling of PSI were considered for RHIC [10]. A survey of the existing and
future accelerator facilities where GF concepts may be implemented and the
projected parameters of the corresponding gamma sources are surveyed in Table
11.

Facility | Lab. | p+ energy | Max. photon energy
---|---|---|---
LHC | CERN | 6.5 TeV | Pb${}^{81+}+12.6~{}\mathrm{eV}\rightarrow 373$ MeV
SPS | CERN | 450 GeV | Ti${}^{21+}+11.7\,\mathrm{eV}\rightarrow 2.1$ MeV
RHIC | BNL | 255 GeV | Cl${}^{16+}+11.8\,\mathrm{eV}\rightarrow 0.74$ MeV
NICA | JINR | 12.6 GeV | Li${}^{2+}+10.2\,\mathrm{eV}\rightarrow 0.83$ keV
SIS 100 | GSI | 29 GeV | B${}^{4+}+10.2\,\mathrm{eV}\rightarrow 6.4$ keV
SIS 300 | GSI | 87 GeV | Ne${}^{9+}+12.1\,\mathrm{eV}\rightarrow 86$ keV
SC-SPS | CERN | 1.3 TeV | Kr${}^{35+}+11.4\,\mathrm{eV}\rightarrow 15$ MeV
HE-LHC | CERN | 13.5 TeV | U${}^{91+}+7.8\,\mathrm{eV}\rightarrow 0.96$ GeV
FCC-hh | CERN | 50 TeV | U${}^{91+}+2.1\,\mathrm{eV}\rightarrow 3.5$ GeV
Table 11: Survey of the existing (first three lines) and future relativistic
heavy-ion facilities. The maximum photon energy is given assuming hydrogen-
like ions interacting with a primary laser beam in the optical range (down to
100 nm). With a dedicated FEL used as a primary source of higher energy
photons (+ heavier ion) the maximum secondary photon energy can be increased
further.

###  14.2 Nuclear waste transmutation