Photon-photon physics at the LHC and laser beam experiments, present and future

Under certain running conditions, the CERN Large Hadron Collider (LHC) can be considered as a photon-photon collider. Indeed, in proton-proton, proton-ion, ion-ion collisions, when incoming particles pass very close to each other in very peripheral collisions, the incoming protons or ions remain almost intact and continue their path along the beam axis. Then, only the electromagnetic (EM) fields of these ultra-relativistic charged particles (protons or ions) interact to leave a signature in the central detectors of the LHC experiments. The interest is that the photon-photon interactions happen at unprecedented energies (a few TeV per nucleon pairs) where the quantum electrodynamics (QED) theory can be tested in extreme conditions and unforeseen laws of nature could be discovered. In this report, we propose a focus on a particular reaction, called light-by-light scattering in which two incoming photons interact, producing another pair of photons. We describe how experimental results have been obtained at the LHC. In addition, we discuss prospects for on-shell photon-photon interactions in dedicated laser beam facilities. Potential signatures of new physics might manifest as resonant deviations in the refractive index, induced by anomalous light-by-light scattering effects. Importantly, we explain how this process can be used to probe the physics beyond the standard model such as theories that include large extra dimensions. Finally, some perspectives and ideas are given for future data taking or experiments.


Introduction
Collisions of ions at ultra-relativistic energies, like lead(P b 82+ )-lead(P b 82+ ) collisions at the LHC, are usually studied in cases where the ions interact hadronically, producing large multiplicities and a volume in which participants to the reactions can form a domain of hot, dense quark-gluon matter [1,2,3]. However, in addition to central collisions, it is of interest to look also at very peripheral (or ultra-peripheral) collisions (UPC), which are characterized by the condition that the impact parameter b of the reaction is larger than the sum of the ions radii. Then, the two ions do not interact directly via the nuclear interaction, which is mostly short range, but mainly through their electromagnetic (EM) fields. Indeed, relativistic ions generate strong EM fields (with magnitudes of E ∼ 10 25 V/m for lead-lead collisions at the LHC with a center of mass energy per nucleon pair of 5 TeV), where the electric and magnetic fields of each ion (measured at a point of transverse distance of order b from the ion center of mass) are perpendicular and almost transverse to the direction of propagation. These EM fields may be treated as a flux of nearly-real photons and can give rise to photonuclear (photon versus ion) or photon-photon interactions. By definition, these interactions are expected to occur mainly in UPC collisions. The order of magnitude given above (E ∼ 10 25 V/m for ultra-relativistic lead) is easy to justify.
We start from the definition of the electric field measured at a transverse distance b from the ion center of mass: E ∼ Zeγ/b 2 where γ is the Lorentz factor, Z the charge number and e the elementary electric charge. Then, we convert this expression as: E ∼ Zeγ/b 2 ∼ (Z = 82)γα em E cr where α em = e 2 /(4π) 1/137 and E cr the critical (Schwinger) electric field. What we call the critical electric field E cr is defined as: eE cr 1 me = m e , which gives an order of magnitude of E cr ∼ 10 18 V/m. Thus, we find: E ∼ (Z = 82)γα em E cr ∼ 10 25 V/m >> E cr . We understand that the physics relevant to describe these processes is the quantum electrodynamics (QED) at high intensity but with an interaction time still very short δt ∼ b/γ, meaning that perturbation theory can (in principle) still be used. An important physics parameter that is characteristic of the high intensity regime of QED is the ratio: Y = E/E cr . When this ratio is approaching unity or above, the high intensity regime is reached. A broader discussion on this property is done in section (4.2.3) where the ratio Y is examined and calculated in a given laser beam experiment. This is an essential purpose of the measurements presented in this document to confirm or infirm this statement. In the following, we focus the presentation of the physics case of photon-photon interactions in UPC [4], where the impact parameter of the collision is large (larger than the sum of the sizes of the colliding ions in the transverse plane). Thus, the strong interactions are suppressed while the EM fields of the colliding ions interact (see Fig. 1).
One key point in such EM interactions, which means the interactions of the EM fields from the parameters where the strong interaction is heavily suppressed. The nuclei can produce high-energy nearlyreal photons coherently from the entire nucleus, leading to photon-photon interaction (Figure 1). Since the photon flux associated with each nucleus scales as Z 2 , the photon-photon luminosities are enhanced by a factor of Z 4 = 4.5 × 10 7 when using Pb+Pb collisions instead of pp collisions provided by the Large Hadron Collider (LHC).ṽ  [1]. Electromagnetic interaction between the ions can be described as an exchange of photons that can couple to form a given final-state X.  Therefore, photon-photon effective luminosities are then enhanced by a factor Z 4 = 4.5 10 7 for lead-lead collisions, Z 4 = 3.9 10 7 for gold-gold collisions, compared to proton-proton collisions.
However, as the instantaneous luminosity in lead-lead is about 7 orders of magnitude below the one in proton-proton, the global factor between lead-lead and proton-proton effective luminosities in γγ is not that large. In practice, with the present parameters of the LHC machine, we find that: What matters also is what we call the pile-up environment which is in general quite different for ion-ion and proton-proton collisions. We expose all this in details in the next sections. First results have been obtained in AuAu → AuAu(γγ) → AuAue + e − at the Relativistic Heavy Ion Collider (RHIC) for a center of mass energy per nucleon pair of 200 GeV [5,6]. Then, results have been obtained at the LHC first in proton-proton collisions at center of mass energies of 7 TeV, 8 TeV and 13 TeV with several final states produced: pp → pp(γγ) → ppµ + µ − , pp → pp(γγ) → ppe + e − [7,8,9,10] or pp → pp(γγ) → ppW + W − [11,12,13] and then in lead-lead collisions at a center of mass energy per nucleon pair of 5.02 TeV (2.76 TeV) with P bP b → P bP b(γγ) → P bP bµ + µ − [14] (P bP b → P bP b(γγ) → P bP be + e − [15]).
Let us mention that colliding energies coincide, making a very coherent set of data for ion-ion and proton-proton collisions. For example, a center of mass energy of 5 TeV per nucleon pair for lead-lead scattering correspond to a proton-proton center of mass energy of 5 * 208/82 ∼ 13 TeV.
Similarly, for 2.7 TeV per nucleon pair for lead-lead scattering, we get 2.7 * 208/82 ∼ 7 TeV in proton-proton. Also, as the interaction time is of the form δt ∼ b/γ, this means that the maximum energy for an interacting photon is inversely proportional to the impact parameter (ω max ∼ γ/b) and thus, for a given nucleon-nucleon energy, is maximum for proton-proton, then for proton-ion (larger for proton-argon than for proton-lead) and finally the photon energy is minimal for ion-ion (larger for argon-argon than for lead-lead). Following these statements, we show in this review how to reach an optimal use these properties in order to cover a large range of possibilities in searches beyond the standard predictions. Finally, photon-photon interactions (in lead-lead collisions) at the LHC have provided the observation of the so-called light-by-light scattering: P bP b → P bP b(γγ) → P bP bγγ [16,17,18,19,20]. We can mention already that for this reaction, the two photons that collide produce another set of two photons, which corresponds to an elastic collision of two photons and thus justifies the label of the reaction: light-by-light scattering.
In this report, we describe all these aspects of photon-photon physics at the LHC, mainly in the ultra-peripheral collision configuration. We intend to present extensively the experimental results in proton-proton, ion-ion or proton-ion collisions, with an emphasis of rare phenomenon as the recent observation of the elastic scattering of two photons in lead-lead collisions. For the latter case, with some motivations explained in the report, we extend the discussion from lead beam experiments feasible at the LHC to laser beam experiments feasible in the laboratory. For each case, we intent to describe the level of understanding, experimental and theoretical, that we have achieved. We hope this work could be useful to help interpreting correctly any deviation which may be observed by current experiments and plan ahead the future of the UPC physics program at LHC or HL-LHC (high luminosity LHC).

Position of the problem
First, let us pose the physics problem of photon-photon interactions in the case of proton-proton collisions. This is the simplest case and will have an immediate extension for proton-ion or ion-ion collisions. The theory is well known, since this is directly related to interactions in QED and a precise determination of the electromagnetic form factors of the proton. Our purpose is to pose some notations, explain how we can build the generators on existing calculations and emphasize the potential theory uncertainties. As mentioned in the introduction, calculations of the cross-section for photon-photon interactions in proton-proton collisions are based on the equivalent photon approximation (EPA) [21,22,23,24,25]. This relies on the property that the EM field of a charged particle, here a proton, moving at high velocity becomes more and more transverse with respect to the direction of propagation. As a consequence, an observer in the laboratory frame cannot distinguish between the EM field of a relativistic proton and the transverse component of the EM field associated with equivalent photons. Therefore, using the EPA, we can write the cross-section for a reaction where the two photons fuse to give a generic final state X: where f (.) are the equivalent photon flux functions and ω 1,2 represent the energies of the photons, integrated over. These functions can be obtained easily from the Poynting vector associated to the EM field produced by the ultra-relativistic proton. They read: where F (.) represents the proton form factor, including electric and magnetic components. Here γ is the Lorentz contraction factor, ω and k ⊥ represent the energy and transverse momentum of photons respectively. For each photon, the maximum energy is the energy of the incident proton √ s/2. However, there is also the constraint that the highest available energy for one photon is of the order of the inverse Lorentz contracted radius of the proton, γ/r p , where r p represents the proton radius and γ the Lorentz factor equal to √ s/(2m p ) here. Let us note that the two photon center-of-mass energy squared is W 2 γγ = 4ω 1 ω 2 , and the rapidity of the two photons system is defined as y γγ = 0.5 ln[ω 1 /ω 2 ]. In Eq. (1), the photon distributions f (.) are already integrated over the virtuality (Q 2 1,2 ) of the photons. As this dependence is of the order of 1/Q 2 1,2 , this justifies the approximation that both photons are nearly-real.
Let us mention that there is another way of deriving Eq. 1 without using explicitly the EPA.
We can write brute force the cross section for the process (for example producing two leptons in the . This gives: (3,4) dΓ (1,2) where the momenta of particles are labeled (k 1 , k 2 , k 3 , k 4 ) and the phase space integrands are written as dΓ (1,2) and dΓ (3,4) . To simplify the notations, we have included the normalization with flux factor and the conservation law (2π) 4 δ 4 (k 1 + k 2 − k 3 − k 4 ) into the transition amplitude T µν (k 1 , k 2 , k 3 , k 4 ), computable from first principles, which describes the interaction of the two incoming photons with the 6 produced outgoing particles. The functions A µ 1 (k 1 , b) and A ν 2 (k 2 ) are the EM potentials produced by the colliding protons, whose velocities are u 1 = γ(1, v, 0, 0) and u 2 = γ(1, −v, 0, 0), with the condition that the impact parameter of the reaction is b. The amplitude T µν (k 1 , k 2 , k 3 , k 4 ) is calculable from QED rules. Then, standard calculations give back an expression of the form of Eq. 1 under the approximation that the photons are nearly-real. This is why this formula seems to be quite robust and is a good starting point for discussion.
We can remark that for practical issues, the situation may be more complex. Indeed, each proton can either survive and, then, is scattered at a small angle, as considered above. This is the case of elastic emission. Elastic two-photon processes yield very clean event topologies at the LHC: two very forward protons measured away from the interaction point and a few centrally produced particles (forming the final state X). But, it is also possible that one or both protons dissociate into a hadronic state. This is the case of inelastic emission. Indeed, in case of collisions of ions the inelastic excitation will contribute also to the photon spectrum. This has been studied for the excitation of the most important excited state, called the giant dipole resonance (GDR). For the collisions of protons, it was found that the photon emission from individual quarks gives a contribution that is larger than the elastic one. We come back on these effects in the next sections when describing the measurements.

Impact parameter dependences
In posing the problem, an important remark is that basic Eq. (1) is incomplete and requires significant corrections due to proton absorptive effects. Indeed, the proton has a finite size already encoded in the form factor F (.) (in the above expression) but that will also carry other important deviations to Eq. (1). These effects are mainly related to proton-proton strong-interaction exchanges that accompany the photon-photon interaction and that lead to the production of additional hadrons in the final state. These effects are of the order of 20 % in the measurements performed at the LHC and thus cannot be neglected (see next sections). This includes both the strong proton-proton absorptive correction and the photon-proton coherence condition (the impact parameter of the reaction needs to be larger than the size of the incoming protons). It can be shown that these corrections can be accounted for making the replacement [26,27,28]: where n(b, ω) are generalized photon flux, which still depend on the energy of the photon but also of the distance b to the proton trajectory (measured in the plane transverse to this trajectory). By symmetry, this is clear that n(b, ω) depends only on the modulus of b. Based on the same logic as for the EPA, it reads [26,27,28]: In the following of the reasoning let us discuss the case of ion-ion (that we label generically as A − A) collisions. It seems very logical that all the arguments mentioned above also hold for ion-ion, after changing r p by the ion radius (labeled r A ) and with two changes in Eq. (3): (i) the charge factor is not e 2 but Z 2 e 2 and (ii) the form factors are related to the ion charge and magnetic distributions.
The point (i) is not so obvious for A − A collisions. Indeed, it assumes the coherent action of all the charges (whose number is Z) in A. For this condition to be correct, the wavelength of the photon needs to be larger than the size of A, such that it does not resolve the individual nucleons but sees the coherent action of them. This means that the coherence condition limits the virtuality Q 2 = −q 2 of the photon to very low values: Q 2 1/R 2 A , where R A is the charge radius of A. For protons, R p 0.7 fm and for lead, R P b 7 fm. We have already mentioned above that the (main) dependence in 1/Q 2 of functions n(.), which itself comes from the rapid decrease of the nuclear electromagnetic form factor for high Q 2 values, justifies this assumption and the interacting photons can be considered as nearly-real. In this report, we restrict the discussion to coherent photon exchange only.
At this level, as a consequence, we can set more accurately the bounds on the energies of interacting photons and also on their transverse momenta. The four-momentum of one emitted photon (from the ion or the proton moving at velocity v) reads: q µ = (ω, q ⊥ , q 3 = ω/v). With Q 2 = ω 2 γ 2 + q 2 ⊥ , we obtain the bounds: ω < ω max ≈ γ R A and q ⊥ 1 R A . At the LHC, this gives ω max 1.8 TeV (1.1 TeV) in proton-proton collisions at a center of mass energy of 13 TeV (8 TeV) and ω max 70 GeV in lead-lead collisions at a center of mass energy per nucleon pair of 5 TeV. In particular, this means that all processes are not accessible to lead-lead collisions at √ s N N = 5 TeV. The reaction P bP b → P bP b(γγ) → P bP bW + W − is forbidden and thus the exclusive production of pairs of W bosons from photon-photon interaction is only possible in proton-proton collisions at the LHC, for example at a center of mass energy of 8 TeV.

Getting the general expression of the cross section
Here, we have all the elements to pose the cross section for the photon-photon reaction in the case of ion-ion collisions: AA → AA(γγ) → AAX. This gives: From this expression, we deduce the cross section scales as Z 4 , as stated in the introduction. This expression also holds for proton-proton collisions with A ≡ p and Z = 1. As a detail, the condition can be written differently using what is called an opacity function, but this does not make any practical difference and the interpretation is still that it represents a condition for non overlap. Then, this is not difficult to finish the calculations once we know the functions n(.) for ions, that derive from the EM form factors of the relativistic ions. They are measured using the elastic scattering of electrons from ions. In general n(.) reads: where J 1 (.) is the Bessel function of the first kind. It is clear that there exists simple form of the form factor that provides simple calculations. For example, introducing monopole form factor: <r 2 > , the expression of n(.) becomes: where K 1 (.) is the modified Bessel function of the second kind. The interest of this expression is that it can be easily encoded in computer computations. In a second step, this is not complicated to consider a more realistic form for the form factor F (.) using the charge distribution ρ(r) in the ion. For example, a quite general function for ρ(r) can be defined as: ρ(r) = ρ 0 1+exp((r−r 0 )/a) . Where r 0 is the characteristic size of the ion and a characteristic length scale. Then, the form factor: q ρ(r) sin(qr)dr can be directly integrated and encoded in the photon flux above. Of course, all the above discussion is useful if the elementary cross section σ(γγ → X), where X is a pair of leptons or photons, is computable in QED. This is a reasonable question as the natural expansion parameter Zα em is not a small parameter for ions. However, what matters also is the fact that the Lorentz factor is very large at LHC energies and then 1/ ln(γ 2 L ) becomes a small parameter [28]. Finally, it has been shown that calculations can be done in the framework of perturbation theory [28] and it follows that Eq. (4) can be used to compute the observed process at the LHC in ion-ion collisions, as well as in proton-proton or proton-ion collisions. These photon equivalent distributions including impact parameter dependences are included in several Monte-Carlo (MC) simulations used at the LHC. For ion-ion an proton-proton collisions, all UPC physics processes can be simulated using Superchic [28], Starlight [29,30] and Fpmc [31,32], while Lpair [33] is restricted to the proton-proton case. For all these MC simulations, several possibilities for the form factors of the ions or protons can be used.
Let us note also that the nuclear break-up probability can easily be incorporated into Eq.  3 On the validity of this formalism at the LHC The purpose of the present section is to prove experimentally that this formalism is reasonable in the LHC era. This means that it allows a correct description of the data using the formalism encoded in different MC mentioned above [28,29,30,31,32,33], depending on the precise process under investigation. We start the discussion by the heaviest final states, the exclusive production of pairs of W bosons, then pairs of leptons. At the end of this part, this will bring us to the physics case we want to emphasize in this report, namely the light-by-light scattering.

Experimental observation
The exclusive production of pairs of W bosons, pp → pp(γγ) → ppW + W − have been observed at the LHC [11,12,13]. Below we describe the measurements and results by the ATLAS and CMS experiments [11,12,13], performed using 20.2 fb −1 of proton-proton collisions at 8 TeV recorded during 2012. Due due a kinematic requirement that follows from the formalism described above, this process can only been observed in proton-proton collisions and is forbidden in lead-lead. The initial idea for the analysis is to consider the opposite-charge different-flavor µ ± e ∓ final states. This garanties that the exclusive production of W + W − has a clean signature, since no extra activity is produced in addition to the two (identified) leptons. Also, it is a key process to probe theW W γγ quartic gauge couplings (next subsection). Without tagging the initial state protons combined with having neutrinos in the final states, it is not possible to separate the elastic from the dissociative events, where a proton is outgoing in an excited state of low mass (labeled as p * ). As already discussed in another physics case in a previous section, we identify single dissociative events (SD): Therefore, the elastic, SD and DD processes are together considered as signal. We cannot discriminate one with respect to the other. In a second step, the dissociative contribution is estimated from data, using an exclusive sample of pairs of muons with an invariant mass above 160 GeV. The datadriven scale factor f γ is given by the ratio of the number of data after subtracting off the number of background events to the number of elastic events predicted by a purely elastic process simulation.
So the total expected signal for analysis is the prediction for the elastic W + W − production scaled by f γ : The events selected for analysis satisfy any of the following triggers: (i) a single-muon trigger with p µ T > 24 GeV, (ii) a single-electron trigger with p e T > 24 GeV or (iii) an electron-muon trigger with p e T > 12 GeV and p µ T > 8 GeV. The pileup conditions in 2012 are quite high. Therefore, a new track-based strategy for selecting exclusive events has been developed for this analysis. The first analysis requirement is to select two leptons with p T ≡ p eµ T > 30 GeV and invariant masses above 20 GeV. Then, the event vertex is reconstructed as the average of the longitudinal impact parameters (z 0 ) of the leptons. Also, in order to suppress most of the inclusive background, the two leptons are required to be within 1 mm in z (along the beam line) of each other. At the next step, the longitudinal distance between the lepton vertex and other extra tracks (∆z 0 ) is computed. Finally, the exclusivity selection is defined as the condition that only events with zero extra track within a 1 mm longitudinal distance from the lepton vertex (also written as ∆z iso 0 = 1 mm) . Finally, 23 candidates are observed in data, while 9.3 signal (elastic+SD+DD types of events -see above-) and 8.3 background events are expected. The observed significance over the background only hypothesis is 3σ, constituting an evidence for exclusive production of pairs of W bosons in proton-proton collisions. Fig. 4 shows the distributions in the signal region (pp → pp(γγ) → ppW + W − +SD+DD) of the transverse momentum of the electron-muon pair p eµ T and its azimuthal separation ∆φ eµ . The simulation describes correctly the data. From this measurement, the cross section for γγ → W + W − → µ ± e ∓ + X (for proton-proton collisions at 8 TeV) can be determined in the signal region (for the fiducial phase space measured) and then extrapolated to the full W + W − → µ ± e ∓ + X phase space.    Figure 5: Kinematic distributions in the exclusive W + W − signal region comparing the simulations to data from CMS [11]. The exclusive W + W − signal is pictured in light blue.

fb
Then, the cross-section times branching ratio is found to be: For the same energy in the center of mass, the CMS experiment is obtaining a very comparable result with a cross section of 10.8 ± 5.1 ± 4.1 fb [11,12]. Kinematical distributions in the signal region (pp → pp(γγ) → ppW + W − +SD+DD) of the invariant mass of of the electron-muon pair M (eµ) and its acoplanarity 1 − |∆φ(eµ)|/π are presented in Fig. 5. We observe that the descriptions of the data (ATLAS and CMS) by the simulations, exclusive signal plus backgrounds, is quite good within the experimental and theoretical uncertainty.

Anomalous quartic couplings
In the measurement described above, the contribution of the elastic part of the signal region comes from the reaction γγ → W + W − which is possible in the SM through its Lagrangian density: Where A µ is the photon field and W µ is the W boson field. This is basically what we have pictured as the exclusive signals ( Fig. 4 and 5) once all experimental requirements and efficiencies are taken into account. At this point, this is possible to push the discussion a bit forward in exploring potential non standard physics that could be hidden in the measurement. Indeed, this is well established that these measurements are good candidates to search for anomalous gauge couplings (aQGC), which would reveal a sign of new physics. From the conclusion of the experimental observation, we know in advance that we will not conclude to any sign of deviation. However, the exercise is interesting to derive upper limits on the aQGC coupling parameters.
The most general Lagrangian densities leading to aQGC γγW W (and γγZZ ) can be written as: Where Z α stands for the Z boson field. Λ is a new scale which is introduced such that the Lagrangian density has the correct dimension and can be interpreted as a scale of new physics. Finally, a 0 , a C represents the new coupling constants. Also, the W W and ZZ photon-photon cross sections rise quickly at high energies when any of the anomalous parameters are taken to be non-zero. Therefore, the cross section rise has to be regulated by a form factor which vanishes in the high energy limit. A possibility is then to modify the coupling parameters by form factors that have the desired behavior, Figure 6: The p eµ T distribution for data [13] compared to the standard model prediction for events satisfying all the exclusive W + W − selection requirements apart from the one on p eµ T itself. Also shown are the various predictions for aQGC coupling parameter values.
i.e. they modify the coupling at small energies only slightly but suppress it when the center-of-mass energy W γγ increases. Here, we consider a functional form as a → a (1+W 2 γγ /Λ 2 ) n , with n = and Λ given below in the figures.
The experimental requirements needed to select quartic anomalous gauge coupling W W events are similar as the ones we mentioned in the previous section, in the domain p eµ T > 120 GeV, where the aQGC contributions are expected to be important and standard model backgrounds are suppressed [13]. The p eµ T distributions is shown in Fig. 6 for data compared to the standard theoretical predictions together with various aQGC scenarios. We observe that aQGC predictions enhance the exclusive signal at high p eµ T , while the background is negligible when p eµ T > 80 GeV. The 95 % Confidence Level limits on the couplings a W C Λ 2 and a W 0 Λ 2 can then be extracted with a likelihood test using the one observed data event as a constraint [13], see Fig. 7 where ATLAS and CMS results are shown.
Essentially, no anomalous effect is visible with existing data as foreseen, but the limits themselves are an interesting basis for these kinds of non standard predictions.

Proton-proton collisions
It is simpler to start with proton-proton collisions and then to continue the discussion with ion-ion as it was done in the description of the formalism. The CMS and ATLAS experiments have measured the exclusive production of pairs of muons in ultra-peripheral collisions: pp → pp(γγ) → ppµ + µ − [7,8,9,10]. Below, we present results for collisions at a center of mass energy of 7 TeV recorded in 2010 by the CMS experiment for an integrated luminosity of 40 pb −1 [7].
An important initial task is to record correctly the events of interest. They are selected by triggers requiring the presence of two muons with a minimum p T of 3 GeV with some vetoes to avoid a too large fraction of dissociative events. Then, at the analysis level, by construction, the signal is characterized by the presence of two muons, no additional tracks, and no activity above the noise threshold in the calorimeters. Of course, the presence of additional interactions in the same bunch crossing would spoil this signature by producing additional tracks and energy deposits in the calorimeters. This is why the selection requires a valid vertex, reconstructed with exactly two muons and no other associated tracks, and this vertex is required to be separated by more than 2 mm from any additional tracks in the event. This value depends of the distribution of additional vertices and thus depends on the running conditions. Then, in order to enhance the signal efficiency,  Figure 4: Distribution of p T (µ + µ ) for the selected sample. Data are shown as points with statistical error bars. The histograms represent the simulated signal (yellow), single (light green) and double (dark green) proton dissociative backgrounds, and DY (red). The yields are determined from a fit using the distributions from simulation.

Maximum likelihood fit
The elastic pp ! pµ + µ p contribution is extracted by performing a binned maximum-likelihood fit to the measured p T (µ + µ ) distribution. Shapes from Monte Carlo simulation are used for the signal, single-proton dissociation, double-proton dissociation, and DY contributions, with all corrections described in Section 4.4 applied.
Three parameters are determined from the fit: the elastic signal yield relative to the LPAIR prediction for an integrated luminosity of 40 pb 1 (R El El ), the single-proton dissociation yield relative to the LPAIR single-proton dissociation prediction for 40 pb 1 (R diss El ), and an exponential modification factor for the shape of the p T distribution, characterized by the parameter a. The modification parameter is included to account for possible rescattering effects not included in the simulation, as described in Section 3. Given the small number of events expected in 40 pb 1 , the double-proton dissociation and DY contributions cannot be treated as free parameters and are fixed from simulation to their predicted values. The contribution from exclusive gg ! t + t production is estimated to be 0.1 events from the simulation, and is neglected.
The p T (µ + µ ) distribution in data is shown overlaid with the result of the fit to the shapes from Monte Carlo simulation in Fig. 4. The result from the best fit to the data is: data-theory signal ratio: R El El = 0.83 +0.14 0.13 ; single-proton dissociation yield ratio: R diss El = 0.73 +0.16 0.14 ; modification parameter: a = 0.04 +0.23 0.14 GeV 2 , with asymmetric statistical uncertainties computed using MINOS [25]. The corresponding value mass [GeV] µ µ 20 40 60  Figure 7: Muon pair invariant mass spectrum (left) and acoplanarity (right), with all selection criteria applied and the simulation normalized to the best-fit value. Data are shown as points with statistical error bars, while the histograms represent the simulated signal (yellow), single (light green) and double (dark green) proton dissociative backgrounds, and DY (red).   a. The relative difference of the data-theory signal ratio between the modified and the nominal fit result is taken as a systematic uncertainty.

Pileup correction systematic uncertainties
Charged tracks from pileup interactions more than 2.0 mm from the dimuon vertex may induce a signal inefficiency, if they are misreconstructed to originate from within the 2.0 mm veto window. The h-dependent single-track impact parameter resolution in CMS has been measured to be less than 0.2 mm in the transverse direction, and less than 1.0 mm in the longitudinal direction [20]. The track-veto efficiency is studied in zero-bias data by varying the nominal 2.0 mm veto distance from 1.0 to 3.0 mm. The maximum relative variation is found to be 3.6%, when some kinematic requirements are also imposed: the transverse momenta p T of the identified muons is required to be greater than 4 GeV, with a restriction in pseudo-rapidities as |η| < 2.1, the invariant mass of the pair of muons is required to be greater than 11.5 GeV, the acoplanarity A = 1 − ∆φ/π is taken to be smaller than 0.1 (following the same arguments that in the lead-lead case) and, finally, the scalar difference in the p T of the two muons, |∆p T |, is required to be smaller 1 GeV. The results are presented in Fig. 8. The Lpair MC [33], including absorptive corrections, is used to produce the simulated samples shown in Fig. 8. We can observe that the dominating contribution comes from the exclusive production of pairs of muons: pp → pp(γγ) → ppµ + µ − , as discussed in the previous section.
However, there is also a non negligible fraction of single dissociative events, namely when one proton is producing a low mass excited state, p ( * ) , after the collision: pp → pp ( * ) µ + µ − . For such processes (single proton dissociation), the cross section calculations depend on the proton structure functions and how the proton is fragmented into a particular low mass excited state [7]. On the principle, this follows the ideas exposed in the section on the formalism, this time with an inelastic photon distribution. We note also in Fig. 8 that a small contribution comes from double dissociative events: pp → p ( * ) p ( * ) µ + µ − . All this taken into account, the data are correctly described by the simulations.
In order to illustrate more clearly the effect of the absorptive corrections when compared to a (bare) simulation without such corrections, we present a compilation of results from the ATLAS and CMS of signal events (N i excl. ) and correction factors (C i ) are also shown. The measurements are listed together with the statistical ( stat. ), and total systematic ( syst. ) uncertainties. In addition, the contributions from the individual correlated and uncorrelated systematic error sources are provided. The last row lists d /dm µ + µ in the total fiducial region. The uncertainties in N i excl. correspond to the combined statistical and systematic uncertainties. These are correlated across m µ + µ bins. ! µ + µ di↵erential fiducial cross-section measurements as a function of dimuon invariant mass m µ + µ . (b) Comparison of the ratios of measured and predicted cross-sections to the bare EPA calculations as a function of the average dimuon invariant mass scaled to the pp centre-of-mass energy used. Data (markers) are compared to various predictions (lines). Full circle markers represent the four mass points presented in this paper, while open circle, up-triangle and down-triangle depict the previous results obtained with m µ + µ > 11.5 GeV [2], m µ + µ > 20 GeV [3] and m µ + µ > 45 GeV [7] requirements on the dimuon invariant mass. The inner error bars represent the statistical uncertainties, and the outer bars represent the total uncertainty in each measurement. The yellow bands represent the theoretical uncertainty in the predictions. The bottom panel in (a) shows the ratio of the predictions to the data. since at mid-rapidity (y µ + µ ⇡ 0) one has x 1 ⇡ x 2 . Figure 5 (b) shows the evolution of the survival factor as a function of the average dimuon invariant mass scaled by a given pp centre-of-mass energy. Exclusive two-photon production of muon pairs in pp collisions at the LHC has been studied by the CMS experiment at p s = 7 TeV for m µ + µ > 11.5 GeV [2].
The ATLAS experiment measured exclusive production of muons at p s = 7 TeV in the region m µ + µ > 20 GeV [3]. Recently the production of exclusive ! µ + µ at p s = 8 TeV was also studied by ATLAS in the context of exclusive ! W + W measurement [7]. The probed invariant mass region in this case is m µ + µ > 45 GeV. The hm µ + µ i for di↵erent measurements is calculated using the Herwig generator and corresponding fiducial region definitions. The deviations from unity of the ratios of experiments in Fig. 9. In particular, Fig. 9 (a) illustrates cross section measurements of exclusive production of pairs of muons using data in proton-proton collisions collected at a centre-of-mass energy of 13 TeV during 2015, for an integrated luminosity of 3.2 fb −1 [10] . Then, Fig. 9 is grouping all experimental results obtained at the LHC [7,8,9,10] as a ratio of the measured cross section to the (bare) EPA prediction. We observe the effect of order 20 % of the absorptive corrections (also called finite-size correction), already mentioned in the formalism part of this report.

Lead-lead collisions
We have already mentioned in the introduction that first results in Au − Au collisions and exclusive production of electron-positron pairs have been obtained at RHIC [5,6]. Below, we discuss only the recent high statistics ATLAS analysis. Indeed, the ATLAS experiment has measured cross sections for exclusive production of pairs of muons in ultra-peripheral lead-lead collisions at a center of mass energy of 5.02 TeV per nucleon pair, for invariant masses of the pairs of muons larger than 10 GeV:   Fig. 3. The cross-sections for exclusive dimuon production, Pb+Pb (γγ) → Pb ( * ) +Pb ( * ) µµ, as a function of (a) pair mass and (b) pair rapidity [3]. A possible EM excitation of the outgoing ions is denoted by ( * ). The data are indicated by the symbols while the MC predictions are shown by solid histograms. Error bars indicate statistical uncertainties while the grey bands indicate the combined systematic uncertainties. entire detector. Events are then required to have exactly two photons with transverse energy greater than 3 GeV, pseudorapidity |η| < 1.37 or 1.52 < |η| < 2.37, and a diphoton invariant mass greater than 6 GeV.
Since the analysis requires the presence of low-energy photons, which are not typically used in ATLAS analyses, detailed studies of photon reconstruction and calibration are also performed. In particular, exclusive dielectron pairs from the reaction γγ → ee are used for various aspects of the analysis, for example to validate the EM calorimeter energy scale and resolution. Figure 4 presents the transverse momentum of the exclusive dielectron system for selected ee pairs, where a good agreement between the data and simulation is observed.
Possible backgrounds can arise mainly from misidentified electrons from the γγ → ee process, as well as from the central exclusive production of two photons from the fusion of two gluons (CEP gg → γγ). In order to reduce the γγ → ee background, a veto on the presence of any charged-particle tracks is imposed. To reduce other fake-photon backgrounds (for example, cosmic-ray muons), the transverse momentum of the diphoton system is required to be below 2 GeV. Diphoton acoplanarity distribution for events satisfying abovementioned selection criteria is shown in Figure 5a. The shape of this distribution is different for the signal and background processes, therefore an additional requirement on diphoton acoplanarity (Aco < 0.01) is imposed.
Diphoton invariant mass distribution for events satisfying all selection criteria is shown in Figure 5b. In total, 13 events are observed in data whereas 7.3 signal events and 2.6 background events are expected. The statistical significance against the background-only hypothesis is found to be 4.4 standard deviations.
After background subtraction and detector corrections, the cross section for the Pb+Pb (γγ) → Pb ( * ) +Pb ( * ) γγ process is measured in a fiducial phase space defined by the photon transverse energy E T > 3 GeV, photon absolute pseudorapidity |η| < 2.4, diphoton invariant mass greater than 6 GeV, diphoton transverse momentum lower than 2 GeV and diphoton acoplanarity below 0.01. The measured fiducial cross section is σ f id = 70 ± 24 (stat.) ± 17 (syst.) nb, which is in agreement with the predicted values of 45 ± 9 nb [5] and 49 ± 10 nb [6] within uncertainties. The first essential point in this kind of analysis is to record the needed events during the data taking, what we also call trigger the events. In this sense, events have been recorded using a trigger specially designed to detect events with a muon and little additional activity in the detector. The primary requirement is for the presence of a region of interest associated with a track in the muon spectrometer, with no lower bound on p T in order to be sensitive to relatively low transverse momentum muons. In order to keep the statistics acceptable on tape, the events are rejected at this level if the calorimeter system contained 50 GeV or more transverse energy, integrated over the full calorimeter acceptance. Also, events are rejected if more than one hit was registered in either of the inner rings of two MBTS arrays, scintillator slats positioned between the inner detector and the calorimeter. This avoids to keep too many events where the ions dissociate, while the search is for photon-photon interactions for which the ions stay mainly intact. In addition, in order to ensure the presence of a muon, the event has to contain at least one track reconstructed with p T > 400 MeV.

Conclusion
This is on this basis, with recorded events, that the analysis is conducted. An event of this kind is pictured in Fig. 11 [35]. At this level, events are selected using a single-muon trigger, in association with an otherwise low-multiplicity event. The events are then required to have a primary vertex formed solely from two oppositely-charged muons, each having transverse momentum p T > 4 GeV and pseudo-rapidity smaller that 2.4. The kinematic variable distributions are then corrected for muon trigger and reconstruction efficiency, and for vertex efficiency. The nuclear EM form factors ensure that the muons are emitted back-to-back, resulting in a small acoplanarity, defined as A = 1 − ∆φ/π, where ∆φ is the azimutal separation of the two outgoing photons. The simulation based on Starlight [29,30] shows that over 99.9% of pairs have A < 0.008, while in the data non negligible tails for the acoplanarity of the pairs of muons are also observed. This is the reason why the fraction of measured events with A ≥ 0.008 and found to be around 5%. Two assumptions -that the acoplanarity tail is all background, and that it is all signal due to higher-order QED effects -are The studied di-muon kinematic range is limited to 8 < m µµ < 60 GeV and rapidity |y µµ | < 2.4. In Starlight MC, only muon pairs from the LO γγ scattering are generated, and the calculation is performed by integrating over the entire impact parameter space for ultra-peripheral collision events.
No differential impact parameter dependence of initial photon p T is considered in Starlight MC. Figure 12 shows the corrected acoplanarity α distributions of µ + µ − pairs in lead-lead collisions for different neutron multiplicity classes. Each α spectrum is made up of a narrow core close to zero and a long tail, where the core component mostly originates from the LO γγ → µ + µ − scattering while the tail component mostly originates from high-order electromagnetic interactions processes.
These high-order processes include, for example, soft photon radiations off the produced lepton and scattering of multiple photons. The average acoplanarity ( α core ) is determined from fits to the α distribution. The neutron multiplicity dependence of α core is shown in Fig. 13. Likewise, the mean m µµ dependence on the neutron multiplicity is shown in Fig. 13. A clear dependence of either of these variables as a function of the neutron multiplicity is observed. This observation demonstrates the transverse momentum and energy of photons emitted from relativistic ions have impact parameter dependence.  electromagnetic (EM) fields. The EM fields generated by the relativistic ion can interact with the other nucleus or its EM fields. Therefore, besides nuclear hadronic interactions, EM interactions also occur in ultra-relativistic heavy-ion collisions. These EM interactions can be studied in so-called ultra-peripheral collisions (UPC) which occur when the distance between two nuclei in the transverse plane is larger than two times the nuclear radius, and hadronic interactions are thus suppressed [572].
A broad range of processes can be studied with interactions in UPC. In the following, a few examples of photon-induced processes are considered at the HL-LHC: exclusive production of µ + µ or pp pairs, a rare process of light-by-light (LbyL) scattering and a potential of searches for axion-like particles (ALP).
[GeV] Exclusive production of di-muon pairs ( ! µ + µ ) in UPC can offer a precision measurement of photon fluxes associated with ion beams, and as such can be used to constrain predictions for the other processes covered in this section. The cross section at high pair mass is also sensitive to the nuclear

Light-by-light scattering at the LHC and beyond
After having shown that the formalism presented in the first section is able to describe different exclusive processes at LHC energies, we can address a more subtle problem, with the observation of light-by-light scattering. Indeed, this reaction P bP b → P bP b(γγ) → P bP bγγ, pictured on the left of Fig. 16 adds one level of complexity to the understanding due to the fact the prediction needs a box calculation. Then we can expect that the rate of production for exclusive photon-photon final states is thus weaker than the di-lepton final states by a factor α 2 em . Moreover, as explained below, in the optical domain, light-by-light scattering in its full glory using laser beams is even more difficult to observe, if not impossible. The purpose of this section is to show in which conditions the observation of light-by-light scattering has been obtained at the LHC and how this result can inspire some ideas for the optical domain.

Experimental observations
A search for the reaction P bP b → P bP b(γγ) → P bP bγγ (at a center of mass energy per nucleon pair of 5.02 TeV) has been performed by the ATLAS and CMS experiments, leading to the observation of the γγ → γγ process (light-by-light scattering) [17,18,19,20]. Below, we give some precisions for the analysis of the ATLAS measurement in the field [17,18,19], and then we complete this presentation with the CMS results [20] and the recent update by ATLAS [17,18,19].
Again, prior to any analysis, the first task is to record the events. For the study described here, At a first level of selection, the total transverse energy registered in the calorimeter after noise suppression is required to be between 5 and 200 GeV. Then at a next step, events are rejected if more than one hit was found in the inner ring of the MBTS (MBTS veto) or if more than ten hits were found in the pixel detector. This means that events are recorded if they present a moderate activity in the calorimeter (covering the pseudo-rapidity range |η| < 4.9) with little additional activity in the 25 entire detector. dimuon rapidity, and are found to be in agreement with leading-order QED calculations. The Pb+Pb (γγ) → Pb ( * ) +Pb ( * ) γγ process is evidenced with a significance of 4.4 standard deviations, and the corresponding fiducial cross section is also found to be in agreement with QED predictions. Once this is done, the analysis selection can be performed with all the needed corrections. Then, events are required to have exactly two photons with transverse energy greater than 3 GeV, pseudo-

Introduction
Elastic light-by-light (LbL) scattering, gg ! gg, is a pure quantum mechanical process that proceeds, at leading order in the quantum electrodynamics (QED) coupling a, via virtual box diagrams containing charged particles (Fig. 1, left). In the standard model (SM), the box diagram involves contributions from charged fermions (leptons and quarks) and the W ± boson. Although LbL scattering via an electron loop has been indirectly tested through the highprecision measurements of the anomalous magnetic moment of the electron [1] and muon [2], its direct observation in the laboratory remains elusive because of a very suppressed production cross section proportional to a 4 ⇡ 3 ⇥ 10 9 . Out of the two closely-related processesphoton scattering in the Coulomb field of a nucleus (Delbrück scattering) [3] and photon splitting in a strong magnetic field ("vacuum birefringence") [4,5]-only the former has been clearly observed [6]. However, as demonstrated in Ref. [7], the LbL process can be experimentally observed in ultraperipheral interactions of ions, with impact parameters larger than twice the radius of the nuclei, exploiting the very large fluxes of quasireal photons emitted by the nuclei accelerated at TeV energies [8]. Ions accelerated at high energies generate strong electromagnetic fields, which, in the equivalent photon approximation [9][10][11], can be considered as g beams of virtuality Q 2 < 1/R 2 , where R is the effective radius of the charge distribution. For lead (Pb) nuclei with radius R ⇡ 7 fm, the quasireal photon beams have virtualities Q 2 < 10 3 GeV 2 , but very large longitudinal energy (up to E g = g/R ⇡ 80 GeV, where g is the Lorentz relativistic factor), enabling the production of massive central systems with very soft transverse momenta (p T . 0.1 GeV). Since each photon flux scales as the square of the ion charge Z 2 , gg scattering cross sections in PbPb collisions are enhanced by a factor of Z 4 ' 5 ⇥ 10 7 compared to similar proton-proton or electron-positron interactions. Many final states have been measured in photon-photon interactions in ultraperipheral collisions of proton and/or lead beams at the CERN LHC, including gg ! e + e [12][13][14][15][16][17][18][19][20][21], gg ! W + W [22][23][24], and first evidence of gg ! gg reported by the ATLAS experiment [25] with a signal significance of 4.4 standard deviations (3.8 standard deviations expected). The final-state signature of interest in this analysis is the exclusive production of two photons, PbPb ! gg ! Pb (⇤) ggPb (⇤) , where the diphoton final state is measured in the otherwise empty central part of the detector, and the outgoing Pb ions (with a potential electromagnetic excitation denoted by the (⇤) superscript) survive the interaction and escape undetected at very low q angles with respect to the beam direction (Fig. 1, left). The dominant backgrounds are the QED production   Fig. 4. Dielectron transverse momentum distribution for Pb+Pb (γγ) → Pb ( * ) +Pb ( * ) ee event candidates [7]. Electrons are required to have a transverse energy E T > 2.5 GeV and pseudorapidity |η| < 2.47 with the calorimeter transition region 1.37 < |η| < 1.52 excluded. Data (points) are compared to simulation (histograms). The statistical uncertainties on the data are shown as vertical bars. dimuon rapidity, and are found to be in agreement with leading-order QED calculations. The Pb+Pb (γγ) → Pb ( * ) +Pb ( * ) γγ process is evidenced with a significance of 4.4 standard deviations, and the corresponding fiducial cross section is also found to be in agreement with QED predictions. photon-photon invariant mass after applying the acoplanarity requirement. Data (points) are compared to MC predictions (histograms) [17]. The statistical uncertainties on the data are shown as vertical bars. rapidity |η| < 1.37 or 1.52 < |η| < 2.37, and a photon-photon invariant mass greater than 6 GeV.
Since the analysis requires the presence of low-energy photons, which are not typically used in ATLAS analyses, detailed studies of photon reconstruction and calibration are also performed. In particular, exclusive pairs of electron-positron from the reaction γγ → e + e − are used for various aspects of the analysis, for example to validate the EM calorimeter energy scale and resolution. Fig. 15 presents the transverse momentum of the exclusive electron-positron pairs. A good agreement between the data and simulation is obtained. Possible backgrounds can arise mainly from misidentified electrons from the γγ → e + e − process, as well as from the central exclusive production of two photons from the fusion of two gluons (CEP gg → γγ ). The signal and the two main backgrounds are pictured in Fig. 16.
In order to reduce the γγ → e + e − background, a veto on the presence of any charged-particle tracks is imposed. To reduce other fake-photon backgrounds (for example, cosmic-ray muons), the photon-photon transverse momentum is required to be below 2 GeV. The photon-photon acoplanarity distribution for events satisfying above mentioned selection criteria is shown in Fig. 17. The shape of this distribution is different for the signal and background processes, therefore an additional requirement on photon-photon acoplanarity below 0.01 is imposed.
The photon-photon invariant mass distribution for events satisfying all selection criteria is shown 27 in Fig. 17. In total, 13 events are observed in data whereas 7.3 signal events and 2.6 background events are expected. The statistical significance against the background-only hypothesis is found to be 4.4 standard deviations. After background subtraction and detector corrections, the cross section for the P bP b → P bP b(γγ) → P bP bγγ process is measured in a fiducial phase space defined by the transverse energy of photons above 3 GeV, the photon absolute pseudo-rapidity below 2.4, the photon-photon invariant mass greater than 6 GeV, the transverse photon-photon momentum lower than 2 GeV and the photon-photon acoplanarity below 0.01. The measured fiducial cross section is found to be: This is in agreement with the predicted values of 45 ± 9 nb and 49 ± 10 nb within uncertainties.
A similar analysis has been done by the CMS experiment [20], also using data collected during the same period in 2015 in lead-lead collision ( √ s N N = 5.02 TeV) and corresponding to an integrated luminosity of 390 µb −1 . The fiducial kinematic domain of the measurement is slightly different to the one described above for the ATLAS measurement. In particular the photon-photon invariant mass is requested to be greater than 5 GeV and the transverse energy of photons are requested to be above 2 GeV. Kinematic distributions are presented in Fig. 18. Then, the fiducial cross section is found to be: σ fid = 120 ± 46(stat.) ± 28(syst.) ± 4(theo.) nb.
This is consistent with the prediction of 138 ± 14 nb.
Recently, by combining the 2015 and 2018 lead-lead collisions data sets (at a center of mass energy per nucleon pair of 5.02 TeV), the ATLAS collaboration has been able to produce differential cross sections for light-by-light scattering [17,18,19]. In practice the differential fiducial cross sections are determined using an iterative Bayesian unfolding method. The idea is to correct for bin migrations between particle and detector-level distributions due to detector resolution effects, and applies reconstruction efficiency as well as fiducial corrections. The reconstruction efficiency corrects for events inside the fiducial region that are not reconstructed in the signal region due to detector inefficiencies; the fiducial corrections take into account events that are reconstructed in the signal region, but originate from outside the fiducial region. Results are presented in Fig. 19 and compared to MC predictions. A reasonable agreement is obtained. acoplanarity. It is found to be # gg = (20.7 ± 0.4)%, mostly driven by the inefficiencies of the single photon reconstruction and identification, and of the trigger (# g,reco+ID , # gg,trig. ⇡ 70%). The quoted uncertainty here is statistical only, reflecting the finite size of the LbL scattering

Discussion on the light-by-light scattering 4.2.1 LHC versus the optical domain
Interestingly, we can convert the cross section obtained in lead-lead collisions to the pure light-by-light cross section for the reaction γγ → γγ at a center of mass energy √ŝ for the initial photon-photon state which is not fixed in this case (what is fixed is the ion-ion collision energy). However, it is easy to determine that √ŝ ∼ 20 GeV. Then, we obtain: This cross section value can be compared to the predicted value in the optical domain: This is interesting to put in perspective these two cross section values for the reaction γγ → γγ.
In Quantum Electrodynamics (QED) the γγ → γγ reaction proceeds at lowest order in the fine-

Light-by-light scattering in the optical domain
In the optical domain, typically with laser beams, the conditions are reversed with E << E c (and similarly for the magnetic field) and also with fields that vary slowly over a length equal to the reduced electron Compton wavelength. This means that in this physics case, only an effective form of the photon-photon interaction needs to be considered [36]: where a and b are parameters (of similar units) that can be determined using the limit of the QED calculations in the low energy (optical) domain (with 4b = 7a). We can formulate easily the expression (9) in terms of the EM fields (E, B): This expression leads to the prediction of Eq. (8) once we compute the cross section from the Lagrangian density above: where ω is the energy of one incoming photon and φ the scattering angle in the center of mass frame. This gives: where K is a numerical constant (K = 973 10125π ) and r e = e 2 /(4πm e ) the classical radius of the electron. Interestingly, we can rewrite Eq. (11) as: with ω expressed in eV. Regarding the value obtained in the optical domain (prediction of Eq. (8)), we understand that this will be extremely difficult to observe light-by-light scattering in its full glory through collisions of laser beams [37,38,39,40]. In principle, the experiment could be done by focusing two laser beams on the same point in space. Then, a third laser beam focused on the same point would stimulate the diffusion in a fourth beam, with the conservation of energy and momentum. From the expression of the Lagrangian density (9) or (10), it is not complicated to compute the value of the electric field for the fourth beam in function of the three others. For any polarisations of the beams, we obtain an expression of the form: where χ 3 depends linearly of the parameters a and b of Eq. (9). We find an expression of the form: where p is dimensionless form factor of order of unity, which depends on polarisation states of the laser waves. Then, by measuring the number of emitted photons within the fourth beam, which is proportional to |χ 3 | 2 , we could (in principle) get an evidence of light-by-light scattering in a laser beam experiment. The smallness of |χ 3 | 2 which is exactly equivalent to the smallness of σ(γγ → γγ, ω ∼ 1 eV) makes this objective in its full glory difficult to reach.
However, it is possible to find a subtle way that could be achieved in practice. Below, we study the interaction of two counter-propagating EM waves along the (z) axis produced by high intensity laser beams. We are quite precise in the development of these classical calculations because we reuse them in a next section (4.3.2) for a non-standard case and obtain a non trivial result. The two laser beams considered here are supposed to be produced in high finesse cavities. For simplicity, we expose the calculations for plane waves of different intensities (linearly polarized), defined as: with E 0 E 1,x , E 1,y and different angular frequencies ω 0 = ω. Without loss of generality, we can pose: E 0 = constant and E 1,x = E 1,x (z), E 1,y = E 1,y (z), where the z dependence for E 1,x and E 1,y will be a consequence of the bi-quadratic terms of the EM fields in expression (9). Introducing a transverse profile to the EM fields would not modify the conclusions exposed below. We obtain E 1,x (z) and E 1,y (z) by solving the non-linear Maxwell equations derived from the Lagrangian density (9), namely: with the vectors D and H defined as: where K = 16π  (14) and (15) in order to derive the vector fields D and H. Finally, by solving equation (13), we obtain: The interaction of the two counter-propagating EM wave through the creation of virtual electronpositron pairs, leading to light-by-light scattering, generates a birefringence of the vacuum. Then, the non-linear coupling of the fields induces an increase of the vacuum refractive index. In the physics case considered here, the refractive index along the strong electric field is found to be: n x = 1 + 16K|E 0 | 2 , while the index in the perpendicular direction is: n y = 1 + 28K|E 0 | 2 .
Experimentally, let us consider that the wave E 1 is initially polarized with an angle α w.r.t. the (x) axis and then interact with the counter propagating intense wave E 0 over a length L. After the propagation over L: E 1 (L, t) = (Ae x cos αe inxωL + Ae y sin αe inyωL )e iωt + c.c.
There will be a field component perpendicular to the initial polarization and the intensity of this mode is proportional to sin 2 2α sin 2 θ with θ = π(n y − n x )L/λ.
Then, we can write (up to numerical factors denoted by k and k ): Numerically, one can consider a field of high intensity of about I = 10 24 W/cm 2 focused on a cross section area of s = 10 −6 cm 2 . This intensity corresponds to a beam pulse of energy e = 10 kJ with a duration τ = L, then I = e/(τ s). With λ = 0.5 µm, we conclude from the previous calculations that: θ ∼ 10 −8 rad. It seems that these values are non unrealistic.
Let us note that the last form in θ ∼ α em cr L/λ is very general for any experiment using properties linked to the vacuum birefringence in intense EM fields. The intense field can be a static magnetic field, a static electric field or the EM field of a counter propagating wave as in the physics case discussed above. The important feature is that the prediction for a typical observable characterizing the birefringence (here θ) will always be of the form of Eq. (17).

Other studies of non-linear QED
The first collaboration to have reported a direct observation of strong-field effects in QED is the Experiment-144 (E-144) at the Stanford Linear Accelerator Center (SLAC) [41]. In this experiment, The Ω 0 pulses are directed along a path that is approximately antiparallel to that of the e − beam (see Fig. 21). This geometry exploits the extremely relativistic nature of the electron beam. In particular, the Lorentz transformation of the electric field E of the laser pulse in the laboratory frame yields, in the e − rest frame, a field strength on the order of 10 16 V/cm, an amplitude that is otherwise unattainable in a laboratory setting. Indeed, in the e − rest frame, the electric field of the laser beam is aboutĒ = 2γ L E lab 2 10 5 E lab ∼ 10 16 V/cm (where γ L is the Lorentz factor).
Then multiple photon effects (strong-field QED) are possible, like non-linear Compton scattering, in which an electron absorbs multiple photons from the laser field, but radiates only a single (high energy) photon:

34
In this reaction, ω 0 represents a photon from the strong EM wave, n ≥ 1 indicates the number of such photons absorbed and γ represents the (high energy) outgoing photon, see Fig. 22. Within the E-144 configuration [41], n is shown to be above 3. Also, the energy of the photon γ can be easily computed. In the rest frame of the electrons, we can write: .
WhereĒ ω 0 = 2γE ω 0 is the energy of the laser photons (in the rest frame of the electrons). This gives

35
the value E γ in the laboratory frame: Then E γ 30 GeV [41], which proves the statement above on the the energy of the outgoing photon.
An important dimensionless quantity can be defined as: Here, E rms is the electric field value of the laser beam (in the laboratory frame), namely: With an intensity of the laser beam that can vary from 3 10 −16 W/cm 2 to 47 10 −16 W/cm 2 at E-144, depending on the focal area. In the weak field regime of QED, η << 1 while in the strong-field regime it is expected that η ∼ 1. With the parameters of E-144 [41], η can be varied from 0.1 to 0.4 (see below). We can relate η to another dimensionless quantity that incorporates the critical field of QED: Where we recall thatĒ ω 0 = 2γE ω 0 is the energy of the laser photons (seen in the rest frame of the electrons). This parameter can reach 0.2 − 0.3 at E-144.
Let us describe the experimental observation and why this is consistent with the detection of non-linear Compton scattering and thus with the appearance of the strong-regime of QED. First, results for non-linear Compton scattering are best done in terms of rates rather than cross sections, as the later are not well defined for initial states involving multiple (unknown) laser photons. We can formulate in a very concise form the rates for the reaction of non-linear Compton scattering with n laser photon absorbed scattering (Fig. 22) as: Where C is the case for 1 photon absorbed, thus linear Compton scattering. This relation is correct in the limit η << 1. The total scattering rate is therefore: Also valid in the domain η ∼ 1 upon the condition that the sum is not truncated. Data have been recorded for circularly infrared IR (λ = 1053 nm) and green (λ = 527 nm) laser beams. Then, nonlinear Compton scattering can be studied by measuring the scattered electrons, as well as observing the forward high energy photons (see Fig. 22). Fig. 23 shows the differential yield 1 Nγ dN dP for electrons scattered from the IR laser beam at six different laser intensities (similar results have been obtained for green laser beam). The observed yield (Fig. 23)  Once the non-linear Compton scattering reaction has occurred, the high energy photon can propagate through the laser field and thus can interact to produce an electron-positron pair (see Fig. 24left) as γ + nω 0 → e + e − or light-by-light scattering (see Fig. 24-right). The reaction γ + nω 0 → e + e − is the one that is studied at E-144 in order to study non-linear effects in QED, see Fig. 25. The key observation is to identify positrons produced after each laser pulse. Fig. 25 shows the yield R e + of positrons per laser shot as a function of η. The line is a power law fit to the data and gives R e + ∝ η 2n with n = 5.1 ± 0.2 ± 0.8. Thus, the observed positron production rate is highly non-linear. This is in good agreement with the theoretical rate of multi-photon reactions involving n laser photons in η 2n . and with the kinematical requirement that five photons are needed to produce a pair (electron-positron) near threshold. The detailed simulation indicates that, on average, 1.5 photons are absorbed from the laser field and 4.7 in the reaction γ + nω 0 → e + e − , thus in agreement with the predicted value. These results are therefore a first observation of inelastic photon-photon scattering with real photons. Interestingly, a new experiment is being build at DESY with the idea to use the high-quality and high-energy electron beam of the European XFEL and a powerful laser beam [42]. Indeed, as in the E-144 experiment, high-energy electrons, accelerated by the European XFEL linear accelerator, and high-energy photons, produced via Bremsstrahlung of those beam electrons, colliding with a laser beam will give access to the strong regime of QED. A potential reach is shown in Fig. 26

ALPs of masses of a few GeV
It is well known that central exclusive production of photon pairs in lead-lead collisions at the LHC, namely P bP b → P bP b(γγ) → P bP bγγ, has an interesting discovery potential for axion-like particles (ALP) [32,43,44,45,46,47]. Let us remind that ALPs, for masses accessible through this reaction at the LHC, are pseudo-scalars weakly coupled to Standard Model (SM) fields that appear in theories with spontaneously broken global, approximate, symmetries as pseudo Nambu-Goldstone bosons.
At the LHC, ALPs (below labeled as the field a(.)) of masses above a few GeV could be produced as intermediate states in the reaction: γγ → a → γγ (see Fig. 27), thus modifying the photon-photon invariant mass distribution. The purpose of an experimental analysis would then be to detect these potential tiny deviations in these invariant mass spectrum, knowing that the standard (light-by-light scattering) spectrum has itself some non negligible irreductible uncertainties. d ions used at the LHC) by a factor 82 4 compared to pp collisions. , evidence for the scattering of light-by-light was presented by the and CMS collaborations in lead-lead collisions [12,13], paving the r future analyses in heavy-ion collisions.

xion-like particles in light-by-light scattering
Ps of masses above a few GeV and weakly coupled to other particles SM can be probed at the LHC. Their presence would be seen as a on on the elastic scattering of light-by-light [13,14,15]. In this , ALPs could be produced as intermediate states in the reaction:  exists maximum energy for the incoming photons (emitted by the relati incident P b) which is of the form /R P b , where is the Lorentz factor R P b the transverse size of the incident ions. Indeed, the need for a mini distance separation between the two hadrons for a UPC to take place an upper bound on the invariant mass of the final two photons, whi inversely proportional to the size of the incident particles. Thus, large photon masses could be probed with ions of smaller sizes, or even bett pp collisions [15]. Experimentally, such measurements in pp collision achievable if protons are tagged in dedicated forward proton spectrom (very far from the interaction point of the reaction), otherwise the dete As in Ref. [32], we parameterize the photons-ALP interaction by a Lagrangian density of the form: where a is the massive scalar ALP field (of mass m a ), (E, B) the electric and magnetic components of the EM field, and 1/f the coupling of the interaction (the unit of f is the energy). This means in particular that the equation of motion of the field a reads: In the high energy limit, for the narrow width approximation, in which we suppose that σ γγ→a→γγ to be non-zero only when the invariant mass of the two photons is equal to m a plus/minus the width of the resonance, this cross section can be shown to be of the form: σ γγ→a→γγ ∝ 1 f 2 B a→γγ , where B a→γγ is the branching ratio of the ALP into photons. In the extraction of the limit for the coupling in 1/f (as a function of m a ), we pose B a→γγ = 1. Indeed, we consider that ALPs predominantly couple to photons and have a negligible coupling strength to a photon and a Z 0 boson. Then, the invariant mass distribution can be used as the discriminating variable, with bin widths comparable to the expected resolution of a narrow resonant signal and exclusion limits can be derived for the coupling 1/f [32]. Results and presented in Fig. 28 and 29.

ALPs of masses of a few eV
We understand that in lead-lead collisions at the LHC, ALPs of masses of a few GeV could be accessible in light-by-light scattering because the photons energies are also of a few GeV. The principle can be extended at smaller energies using the laser beam experiment described above using counterpropagating EM waves. In this context, we can study how the results obtained above are modified in the presence of ALPs, also considered as a pseudo-scalar field a(.) of mass m a with a coupling to the EM fields of the form: L int = −η aE · B, where we have redefined η ≡ 4/f . The coupling of the fields a, E and B is then encoded in two equations: In Eq. (22), we see that the right-hand side of the equation contains terms in e ±i(ω 0 +ω)t and terms in e ±i(ω 0 −ω)t , which correspond to possible time dependence for the ALP field. Then, the equation of motion for a (22) gives: Following this, we can write: with: Here, it is possible to inject this expression into equation (23) and derive the modified refractive index due to the presence of the ALP field a. We find the index along the (x) axis is left unchanged while n y is modified as: In the experimental conditions considered above, λ = 0.5 µm ∼ 2.5 eV, we obtain (4ω 0 ω) 1 2 ∼ 5 eV. Then, there is a resonant effect for ALPs of this mass, that will dominate the non-linear contribution in K|E 0 | 2 . For deep red of λ = 0.7 µm ∼ 1.75 eV, the resonance will be for m a = 3.5 eV. This means that this experimental configuration is interesting in order to obtain a sensitivity (through a resonant effect) to ALP of masses of the order of eV, when there is the possibility to scan several values of laser wavelengths. This strategy follows what has been developed for the LHC era: the search for ALPs as a resonant deviation in the light-by-light scattering. In order to quantify this sensitivity, we need to compare the second term in equation (25), namely 4η 2 m 2 a |E 0 | 2 m 4 a −(4ω 0 ω) 2 with the light-by-light term: 28K|E 0 | 2 . Thus, we need to compare: √ K ∼ 10 −5 GeV −1 eV to η ma A, where A is the amplification factor due to the resonant effect: the better is the finesse of the cavities, the largest is A.
The existing and potential limits in the search for ALPs of masses of the eV or much below [52,53,54,55,56] are of the order η ma ∼ 10 −10 GeV −1 eV for ALPs masses of order of the eV. Therefore, we need an amplification factor A of order 10 5 so that one may obtain competitive sensitivity using this technique, which is an experimental challenge. An amplification factor of 10 3 to 10 4 is already feasible with high quality cavities where the laser beams propagate. Also, using astrophysical probes, some prospects have been extended for ALP in the eV range down to photon-ALP coupling of η ma ∼ 10 −12 GeV −1 eV [57]. However, the analysis using light-by-light scattering stays a complementary approach, that could already provide first results in a near future [37].

Other exotic intermediate states in light-by-light scattering
In Fig. 27 (b), the exchanged field a(.) have been considered to be a pseudo-scalar ALP (of spin 0).
However, nothing prevents to consider the exchange of spin 2 states with the condition that such fields would be acceptable in theory. Such fields can be considered: they are called the states of Kaluza-Klein (KK) graviton, hereafter labeled as g KK , have spin 2 and appear in large extra dimension (LED) extensions of the standard model [48,59,60]. They can be exchanged in the s-channel (as in Fig. 27 (b)) or in the t-channel. In this context, only proton-proton collisions at the LHC make sense as the masses of g KK are expected to be of a few TeV (see below), with the consequence that the light-by-light M γγ spectrum could be modified only at very large masses typically above 500 GeV.
In this section, we use the previous experimental results to obtain some constraints on the graviton. This is not a trivial translation and that is the reason why we give some explanations on the ideas that bring to the results at the end.
Where M S is the fundamental Planck energy scale of the higher dimensional theory. Indeed, the action in 4 + n dimensions reads: Here, V n (2πR) n represents the volume of the extra dimensions space and the integral is taken on the standard four dimensions space-time. This relation has to be matched with the action in four dimensions: Which concludes. Then, if we assume in addition that the SM fields also live in the extra dimensions, another relation can be obtained. Indeed the SM action that we take only for gauge fields for simplicity can be written as: Where the indices (µ, ν) run over 0 to 3 (standard space-time). This implies, after matching the gauge coupling with the four dimensions physics case: R ∼ 1/M P l . In this case, any effect of extra dimensions is out of reach. However, we can also assume that the SM fields are localized in a brane, here in the four dimensions (standard) space-time. In this case, only gravity propagates in the higher dimensions. We can also consider a reasonable value for M S ∼ 1 TeV. In this view, the largeness of the four dimensional Planck scale M pl could be attributed to the existence of extra dimensions of volume R n . If R is large enough to make M S on the order of the electroweak symmetry breaking scale (∼ 1 TeV), the hierarchy problem (gravity is exceedingly weak compared to other forces) would also be naturally solved. For example, with M S ∼ 1 TeV, we obtain from Eq. (26) a set of possible couples (n, R): n = 1 for R ∼ 10 13 cm, which is ruled out, n = 2 for R ∼ 1 mm, also ruled out by dedicated experiments, or n ≥ 3 for R < 1 nm (but still R >> 1/M P l ), which lets the opportunity to detect graviton signal at colliders like the LHC.
The next problem is how to couple graviton to SM fields (like photons). It can be shown that the graviton field corresponds to the fluctuation of metric around the standard Minkowski η µν [59,60,61,62]. Then, the action that describes the coupling is of the form: Here h µν represents the graviton field, T µν is the energy-momentum tensor and the integral is taken on the four dimensions (standard) space-time. The coordinates x µ are also labelling the four dimensions space-time. Let us recall that we have assumed the SM fields are living on this four dimensions space-time. Also, T µν is simply a re-writting of the standard model action as: Here g µν is the entire metric tensor including the graviton fluctuations: In particular Eq. (27), with Eq. (28) and Eq. (29), takes a simple form in some models [59,60,61,62]: With 1/f 2 a coupling factor in units of 1/energy 4 . From this expression, it is possible to extract the contribution of the denumerable set of g KK to light-by-light scattering. We obtain: where κ is the coupling strength of the order of unity in the form given above andk is related to the extra dimensional curvature and gives the mass of the first mode for the g KK (m KK ). It can be shown thatk m KK /3.8 [59,60,61,62]. Interestingly, the interaction described by the Lagrangian density (30) can be transformed into a modification for the differential cross section of the light-by-light scattering as: where s, t are the usual Mandelstam variables and a is a constant a = 1. 56  and by definiteness we consider real scalar DM. In cosmology, we need to consider the radion (scalar field) also. Besides the interaction through KK gravitons, we also take into account that the SM and DM fields can interact with the radion field. Then, it is possible to obtain the correlation between the effective Planck scale and the mass of the first KK mode in a given cosmological scenario [64].
We can extend this derivation by the observation that Eq. (30) contains the most general dimension 8 operators that are needed for four-photons interaction. Thus, we can write the same relation but with another view: where ζ 1 and ζ 2 are parameters of units 1/GeV 4 . Then, this Eq (32) the most general effective Lagrangian density that describes light-by-light scattering and any possible theory would lead to a definite expression of the parameters ζ 1 and ζ 2 , in the same way as it has been done above the the KK gravitons. Here also, the contribution of Eq (32) to light-by-light scattering would manifest at large M γγ and we can thus consider this form (Eq (32)) and use the results of section 4.1 in order to extract upper limits on the general parameters ζ 1 and ζ 2 .

A few perspectives and prospects
In proton-proton collisions, there is a clear experimental possibility in order to increase the precision of the above measurements. Indeed the detection of the outgoing scattered proton that survives the quasi-real photon exchange is already a realistic technology both in ATLAS and CMS experiments.
Then, the proton that survives this interaction, owing to its small momentum loss, can be separated In particular, the CMS and TOTEM Collaborations have presented the physics results of (semi)exclusive production of lepton pairs in proton-proton collisions at 13 TeV, where at least one of the protons were tagged with the PPS system [66]. For an integrated luminosity of 9.4 fb −1 , a total of 12 µ + µ − and 8 e + e − pairs with at least one proton with matching kinematics are found. The dilepton pair has invariant mass of more than 110 GeV, and the estimated number of background events is 1.49 ± 0.07 (stat) ± 0.53 (syst) and 2.36 ± 0.09 (stat) ± 0.47 (syst) for the di-muon and di-electron channels, respectively. The fractional momentum loss of the proton ξ(RP) has values between 2% and 10% for candidate events. The kinematic correlation of candidate events between the central di-lepton system and the forward proton(s) is shown in Fig. 32. The study proves the feasibility of operating a set of RP detectors during standard luminosity runs at the CERN LHC, which can be used to study photon-exchange interactions with very low production rates. Similar results have been obtained by the ATLAS experiment using the same technology [65]. Using protonproton collisions recorded in 2017 with a center of mass energy of 13 TeV, corresponding to an integrated luminosity of 14.6 fb −1 , cross sections in the fiducial detector acceptance are measured to be: σ ee = 11.0 ± 2, 6 ± 1.2 ± 0.3 fb and σ µµ = 7.2 ± 1.6 ± 0.9 ± 0.2 fb in the di-electron and di-muon channels respectively. Fig. 33 shows the difference between the momentum loss measured from the di-lepton pairs or with the RP detectors. This figure is equivalent to Fig. 32 (for CMS).
In addition, recently, the CMS and TOTEM Collaborations have reported a preliminary search for exclusive two-photon production via photon exchange in proton-proton collisions using intact protons [67]. The data correspond to an integrated luminosity of 9.4 fb −1 collected in 2016 at 13 TeV, when the LHC operated at standard instantaneous luminosity. A preselection of events requires each photon to have p T > 75 GeV (where the trigger is efficient), pseudorapidity |η| < 2.5 with a veto on the CMS electromagnetic calorimeter barrel-endcap transition region, and a diphoton mass of m γγ > 350 GeV. The diphoton acoplanarity is defined as a ≡ 1 − |∆φ γγ /π|, where ∆φ γγ is the azimuthal separation of the two photons. This is one of the discriminating variables used in the analysis, with exclusive diphoton candidates satisfying a < 0.005 (back-to-back). The invariant mass distribution, up to this stage of event selection, is shown in Fig. 34. The kinematics of an oppositearm, two-proton system is converted into missing mass and rapidity of the central system kinematics through m pp = √ sξ + ξ − , and y pp = (1/2) log(ξ + /ξ − ). In the case of exclusive diphoton production, both systems are correlated through m pp = m γγ and y pp = y γγ . In this search, a 2σ uncertainty window is used in matching the mass and rapidity between the central two photons and the twoproton systems within their uncertainties. Only two events remain with an expected background prediction of 2.11 +0.96 −0.66 (stat) when no matching criteria is applied. Of these events, none of them contains a pair of forward proton tracks. In the 2σ and 3σ matching windows, the background prediction is respectively 0.23 +0.08 −0.04 (stat) and 0.43 +0.14 −0.08 (stat) events. No diphoton candidates with exclusive kinematic features are observed in either of the windows. A 95% confidence level observed upper limit of 3 fb is quoted on the light-by-light cross section within the fiducial region. This result permits the derivation of upper limits on anomalous four-photon coupling parameters, which are found to be |ζ 1 | < 3.7 × 10 −13 GeV −4 and |ζ 2 | < 7.7 × 10 −13 GeV −4 at 95% CL. A similar procedure is used to derive the two-dimensional limits on the ζ 1,2 parameters. The resulting two-dimensional 95% confidence region is shown in Fig. 35.
The results by ATLAS and CMS demonstrate the feasibility of operating near-beam RP detectors in standard luminosity runs at the LHC for further photon-photon physics studies. A perspective for these techniques is then to obtain measurements of the light-by-light scattering using proton tagging, as it has been started with the study mentioned above. We have shown in this review that with photon-photon invariant mass above 1 TeV would allow to study physics at the TeV scale beyond the SM. In particular, we have explained how large extra dimension theories could be examined in this context. The limits we have extracted in this report could be improved by a factor 10 using these new techniques and a luminosity of 200 fb −1 . This would be a breakthrough in this domain.
Let us also recall that in the context light-by-light scattering, we are at the edge of observing the process in its full glory through collisions of laser beams [37,38,39,40]

53
In a longer future, huge improvements in the study of exotic states can be expected, for example with the Future Circular Collider (FCC). This machine is designed such that it could start as an e + e − collider (FCC-ee) for a center of mass energy ranging from 90 GeV to 365 GeV. In the second stage, the FCC would be turned into a 100 TeV proton-proton machine (FCC-hh) with high-field magnets of up to 16 T, also suitable for heavy-ion collisions. With the addition of an energy recovery electron linac (ERL) of 60 GeV, also electron-proton interactions could be explored providing additional input to achieve the ultimate precision of the FCC-hh. Using similar strategies as exposed in the previous sections, ALPs studies can be performed leading to improved constraints [68].