Constraining nonlinear corrections to Maxwell electrodynamics using $\gamma\gamma$ scattering

The recent light-by-light scattering cross-section measurement made by the ATLAS\ Collaboration is used to constrain nonlinear corrections to Maxwell electrodynamics parametrized by the Lagrangian $L=F+4\alpha F^{2}+4\beta G^{2}$. The ion's radiation is described using the equivalent photon approximation, and the influence of four different nuclear charge distributions is evaluated. A special attention is given to the interference term between the Standard Model and the nonlinear corrections amplitude. By virtue of the quadratic dependence on $\alpha $ and $\beta $, the nonlinear contribution to the Standard Model $\gamma \gamma $ cross-section is able to delimit a finite region of the parameter's phase-space. The upper values for $\alpha$ and $\beta$ in this region are of order $10^{-10}\text{GeV}^{-4}$, a constraint of at least $12$ orders of magnitude more precise when compared to low energy experiments. We also give our predictions for the Standard Model cross-section measured at ATLAS for each distribution and analyze the impact of the absorption factor. We finally give predictions for the future measurements to be done with upgraded tracking acceptance $\left\vert \eta \right\vert<4$ by the ATLAS Collaboration.


I. INTRODUCTION
Maxwell electrodynamics is one of the most successful theories in physics. Since its publication in the mid-XIX, it has been the source of notable predictions, such as electromagnetic waves, and served as a keystone for the proposal of new theories, such as Einstein's special relativity. The efforts to quantize the theory of electrodynamics helped to lay the foundations of quantum field theory. Its quantized version is capable of matching experimental results up to 10 parts per billion [1] making it one of the most precise theories available. Despite all these achievements, the increasing ingenuity of new experiments, at both low and high energy domains, imposes the necessity to keep testing, whether to validate the theory or to find new sources of physics.
Historically, Maxwell's equations were derived phenomenologically. It's interesting, however, to look at them from another point of view. Following a "top-down" approach, its Lagrangian can be derived imposing a Lorentz invariant gauge theory with U(1) symmetry and second-order linear equations of motion for the potentials [2]. In this way, generalizations of Maxwell electrodynamics can be obtained by breaking at least one of the restrictions mentioned above. Indeed, Proca and Podolsky electrodynamics arise by breaking the internal U(1) symmetry -introducing a mass term -, and allowing higher order equations of motion, respectively [3][4][5]. On the other hand, by allowing nonlinear equations of motion, an interesting class of electrodynamics arises which are generically called nonlinear electrodynamics or simply NLED [6]. The most well-known examples of NLED are * pniau7@fisica.ufrn.br † leogmedeiros@ect.ufrn.br Euler-Heisenberg [7] and Born-Infeld [8,9] theories, both proposed in the 1930s with very different purposes. While the first one emerges as a direct consequence of Dirac's relativistic theory of the electron, the second arises as an attempt to solve the divergence of a point-like particle potential.
In the present paper, we focus our study on nonlinear corrections to Maxwell electrodynamics, which includes NLED in regimes where their Lagrangians can be correctly described by the first terms of their respective MacLaurin series. Consequences of these corrections are well known and are expected if QED proves to be right [10]. For this reasons, several groups are currently working on proposing feasible tests based on these phenomena. Low energy experiments, such as PVLAS [11] and BMV [12], are built to detect the presence of magnetic birefringence by measuring the ellipticity acquired by a linearly polarized beam after traversing a magnetic field. While their current results are compatible with zero, they can be used to restrict a region of the parameter-space constraining nonlinear corrections, such as done in [13]. These experiments, however, are sensitive to specific combinations of the parameters, and thus cannot completely constrain the phasespace by themselves.
The hydrogen atom -and more generally hydrogen-like atoms -form a neat low energy laboratory to test for nonlinear corrections. High precision measurements of their transition energies are readily found in the literature [14]. Through perturbation theory, it is possible to analyze the modification of the energy spectrum by the inclusion of several terms to the Lagrangian. In particular, this framework can be used to study how the modification of Coulomb's potential by NLED affects the ground-state energy. Comparison with experimental results constrains the magnitude of these corrections and, consequently, the parameters of the theory [15][16][17]. It is noteworthy that the complete Lagrangian is needed in this proce-arXiv:1809.01296v1 [hep-ph] 5 Sep 2018 dure, which imposes a particular analysis for each theory.
The equations of motion that describe classical electrodynamics are linear and, as such, cannot predict the interaction between electromagnetic waves in vacuum. Light-by-light scattering is a purely quantum process, which arises as a consequence of vacuum polarization and occurs at leading-order via O α 4 virtual one-loop diagram consisting of charged particles. This phenomenon had already been measured indirectly through the electron's and muon's anomalous magnetic moment [18,19]. Recently, in 2013, d'Enterria and Silveira suggested that the observation of light-by-light scattering would be achievable at LHC energies in ultra-peripheral collisions with heavy-ions [20]. As a consequence, the AT-LAS Collaboration announced the first direct detection in 2016 [21].
If the vacuum is invariant by C, P, and T transformations, the first order nonlinear corrections can be described by the addition of Lorentz invariants F 2 and G 2 to Maxwell's Lagrangian (see Eqs. (1) and (2) for definitions). When compared to the free Lagrangian, these terms dominate at high energy regimes where their effects become relevant. For this reason, the light-by-light scattering cross-section may be used to obtain today's most precise constraints for nonlinear corrections to Maxwell electrodynamics. This idea has already been used to constrain Born-Infeld's parameter [22].
In this work, we completely constrain the phase-space of nonlinear parameters associated with the F 2 and G 2 invariants. Using the equivalent photon approximation, we compare the results obtained with 4 different charge distributions, study the impact of the absorption factor [23,24], and the relevance of the interference term arising between nonlinear corrections and the Standard Model amplitudes. This paper is organized in the following way. In section II, imposing a series of requirements, the general form for the Lagrangian describing nonlinear corrections to Maxwell electrodynamics is presented, and followed by a brief discussion of its consequences. Considering corrections up to quadratic order in the invariants, we deduce the differential and total cross-sections for the elastic non-polarized γγ scattering. In section III, we detail the necessary ingredients for the theoretical description of the experiment. Four different distributions are proposed to describe the nuclear charge. Using the equivalent photon approximation, the ions are then treated as high-energy photon sources. We also review the experimental cuts and describe the γγ cross-section according to the Standard Model and nonlinear corrections. Next, in section IV, we present our calculations based on the Standard Model for the cross-section measured at ATLAS. We compare the results obtained from each charge distribution, with and without the absorption factor. Using the cross-section measured by the AT-LAS Collaboration, we derive an expression which fully constrains the phase-space accessible to the parameters. Finally, we end the section with our prediction for the cross-section to be measured at LHC with extended acceptance tracking. Our conclusions are given in section V.

II. NONLINEAR CORRECTIONS
The form of the Lagrangian for a generic electrodynamic theory is greatly restricted by the imposition of the Lorentz group symmetry. The only relativistic invariants available are F and G which can be defined as where F µν is the electromagnetic field strength andF µν ≡ 1 2 ε µναβ F αβ its dual. We focus on nonlinear corrections coming from theories whose Lagrangian are expressible through analytic functions. In this way, theories can be described as a MacLaurin series in the invariants [10], Since the features of these theories arise in intense field regimes, all of them must be, in the weak field limit, indistinguishable from Maxwell electrodynamics. Thus, the coefficients must be chosen to be c 00 = c 01 = 0 and c 10 = 1.The Lagrangian can be restricted furthermore by assuming an invariant vacuum by C, P and T transformations. Since the Lorentz invariant G acquires a sign through parity and time reversal transformation, the Lagrangian can only be a function of its squared power G 2 . Therefore, any coefficient c i j with an odd value of j must be zero. The first terms of the Lagrangian expansion are then where the first term is Maxwell's Lagrangian. Due to the analyticity of the Lagrangians, their power series must always converge inside a convergence radius or energy regime. For this to be true, given a fixed value of the field strength, each term must consistently be less relevant when compared with the ones with lower degree. Thus, any NLED that satisfies all previous requirements can be described by (4) and, for weak enough fields, can be correctly approximated by the first terms of the series. For the purposes of this investigation, we consider as relevant terms up to second order in the invariants, where α and β are parameters with dimension of energy to the inverse fourth power. It is important to remember that each nonlinear theory possesses a particular energy regime at which their effects become relevant and thus, the validity of the expansion (5) needs to be verified for each one them separately. It is possible to study from (5) the behavior of several theories by simply matching the coefficients. As an example, to recover Born-Infeld theory -and in general, Born-Infeldlike theories [25][26][27] -, we must choose α = β = 1 8b 2 , where b represents the maximum value of the electric field 1 . The presence of these nonlinear corrections has profound consequences. Classically, they may be interpreted, through the classical constitutive equations, as giving rise to dielectric properties of the vacuum. The electric permittivity and the magnetic permeability are now tensors and depend on the electromagnetic field itself. As a result of this, several nonlinear processes emerge in the presence of an external electromagnetic field. From this point of view, Euler-Heisenberg effective theory classically describes the effects of vacuum polarization due to electron-positron pair creation in energy regimes well below the electron's mass [10].
On the other hand, the quantization of (4) gives rise to the direct auto-interaction of photons without resorting to any intermediate virtual particle. The interaction between 4 photons can be made explicit by rewriting the Lagrangian (5) as The gamma matrix is defined as where α and β are the same parameters found in (5), and with and ≡ ε a 1 a 2 a 3 a 4 ε a 5 a 6 a 7 a 8 .
We have used a block notation A i j ≡ a i a j to emphasize the matrices invariance through their permutation. For example, permuting A 12 ↔ A 34 indicates that we need to permute a 1 ↔ a 3 and a 2 ↔ a 4 simultaneously. The matrix γ G 2 , is also symmetric by the simultaneous permutation of A 12 ↔ A 56 and A 34 ↔ A 78 . With the help of these properties, we are able to derive the probability amplitude for the elastic γγ → γγ scattering, which can be written as where p and ε (p, i) generically represents the 4-momentum and the polarization vector of the photons. In (10), the totally symmetric permutation operator P A 12 A 34 A 56 A 78 acts on γ and indicates that we must add together all possible γs with permutated indexes. As it should be expected, the substitution of (7) into (10) gives the total amplitude as the sum of the amplitudes due to each of the squared invariants terms and thus is a linear function of the parameters α and β. As a result, the parameters can be easily extracted from the interference term between (10) and the leading-order amplitude from the Standard Model, easing up numerical computations. The non-polarized square of (10) can be expressed in a simple and reference-independent way in terms of Mandelstam's variables as: Particularizing to the center-of-mass frame allows us to write the differential cross-section as and the total cross-section as where m γγ is the total energy in center-of-momentum frame or the invariant mass of the diphoton system. From a dimensional point of view, the cross-section's dependence on the sixth power of the invariant mass can be expected from the linear dependence of (10) on the parameters. From these expressions, we readily recover known results for Born-Infeld and Heisenberg-Euler by choosing α BI = β BI = 1 8b 2 , and α HE = 4 7 β HE = 4 90 α 2 m 4 , respectively [28,29].

III. γγ SCATTERING IN THE EQUIVALENT PHOTON APPROXIMATION
The LHC has been optimized for proton collisions and therefore most of the physics coming from it is based on that kind of experiment. However, for a short period of the year -for one or two months -, the LHC is dedicated to heavyion collisions. Analyzing 480µb −1 of 208-lead-ion collision data collected in 2015, the ATLAS Collaboration announced the detection of light-by-light scattering with a cross-section of 70 ± 24(stat.)±17(syst.) nb 2 [21]. This process can be produced in ultra-peripheral collisions (UPC) of charged particles, where they cross each other with impact-parameter greater than the sum of the ion's radii ( Figure 1). This kind of collision has the advantage of avoiding strong interaction from nuclear overlap and thus cleaning the signal's background.
Any charged particle accelerated at high energies produce an intense electromagnetic field [30]. Comparison of the electromagnetic energy flux with the photon flux in the frequency space allows estimating the distribution of photons emitted by the ion. This is the essence behind the semi-classical equivalent photon approximation (EPA) [31][32][33]. In this way, 2 Most of the systematic uncertainty comes from photon reconstruction and identification efficiency uncertainty. charged particles taking part in electromagnetic processes can be replaced by their respective photon distribution. As observed by d'Enterria and Silveira [20], heavy-ions allow observing light-by-light scattering due to the coherent production of radiation by their nucleons. The luminosity is enhanced by a factor of Z 4 ∼ 10 7 compensating the low crosssection of order O α 4 ∼ 10 −9 . On the other hand, the electromagnetic radiation produced by the nuclear charge distribution interfere destructively when wavelengths are of the order of the ion's radius R. This limits the upper value of the energy spectrum to ω max ≈ γ R ≈ 80GeV for the lead-208 ion [20]. In the EPA, the production of photons by the quasi-elastic scattering of ions in UPC AA → A * A * + γγ can be described convoluting the subsystem γγ → γγ cross-section with the effective photon flux [34]: with where m γγ = √ 4ω 1 ω 2 is the diphoton invariant mass, Y = 1 2 (y 1 + y 2 ) is the rapidity of the diphoton in the lab reference, and b 1 and b 2 are the impact-parameters 3 . N (ω, b) represents the flux of photons with energy ω emitted by the ion at a distance b in the plane perpendicular to the motion. The absorption factor S 2 (b) encodes the ion's probability of survival when scattering with impact parameter b, ensuring that only UPC are considered and can conveniently described in a first approximation as where R 7.1fm is the lead radius. The connection between the impact-parameters is given by the expressions: 3 The labels 1 and 2 refer to the ions and the photons each one produces. Table I. Charge distributions and corresponding form factors. Yukawa and Gaussian distribution parameters, Λ = 0.088GeV and Q 0 = 0.060GeV, are such as to obtain the lead root mean square radius [23,37]. Fermi 2P distribution parameters a = 0.549fm and c = 6.642fm describe the diffuseness and the radius of the lead ion respectively [38]. The homogeneously charged sphere distribution is characterized by the ion's radius R = 7.1fm [39].

Model Charge Distribution Form Factor
More details on this framework can be found in [35]. The photon flux in the impact-parameter space can be written as: where the function φ (ω, b), given by is associated with the intensity of the electric field produced by the ion, γ is the ion's Lorentz factor, and J 1 (x) is the Bessel function of the first kind. The main ingredient of the photon flux is the form factor F (u) given by the Fourier transform of the charge distribution: for radially symmetric charge distributions. There are several parameterization for the charge distribution, and they introduce an important theoretical uncertainty [20]. More realistic, and therefore complex, parameterization carry more details of the charge distribution introducing proximity effects absent in others. However, at large impact-parameters, all parameterizations should be equivalent. We compare the results obtained with the four charge distributions shown in Table I. While a Yukawa charge distribution is considered rather unrealistic, it has the advantage of allowing an analytical expression for φ (ω, b), A second charge distribution is parameterized using a Fermi with 2 parameters model [38]. The constant ρ 0 is such that its form factor is normalized to 1 at the origin. This model is considered much more realistic but has no closed form for its corresponding form factor. An expression, however, can be obtained in terms of a series [36]. Two widely used distributions in the literature, Gaussian and of a homogeneously charged sphere, have simple form factors and are included for comparison [37]. A normalized plot of these charge distribution is shown in Figure 2.  Table I Several triggers and cuts are used during the operation of the detector and along the analysis of the data. At the LHC, up to 600 million collisions per second can occur, each one producing several events. Triggers are chosen in order to reduce this huge amount of information to roughly 400 events per seconds that the ATLAS detector is capable of recording. Then, cuts are applied to clean the signal from its background and to take advantage of the detector's components efficiencies. As a consequence, in order to correctly predict the measurements, this cuts must be included. During the analysis of the γγ scattering, the main cuts used by the AT-LAS Collaboration to select events were: individual photon transverse momentum p t > 3GeV, pseudorapidity |η| < 2.37 (excluding the electromagnetic calorimeter transition region 1.37 < |η| < 1.52) and invariant diphoton mass m γγ > 6GeV. To include this cuts, we replace the total cross-section with the differential distribution σ γγ → dσ γγ d p t d p t , and perform a change of variables transforming the invariant mass m γγ and the diphoton rapidity Y into the rapidities of the outgoing photons y 1 and y 2 using Photons produced by the ions may have transverse momentum up to q ⊥ 1/R 28MeV but are assumed to be emitted along the beam in order to derive (21). This assumption is part of the EPA scheme and connects the center-of-mass reference frame to the laboratory-frame through a simple boost in the z-direction. Other cuts, such as on the diphoton transverse momentum p γγ t < 2GeV and on the acoplanarity 1−∆φ γγ /π < 0.01, are imposed as a total fixed cut of 15% estimated using Table 1 in [21]. The γγ → γγ scattering has contributions from several mechanisms. In the Standard Model, the leading-order contribution proceeds via virtual one-loop box diagram ( Figure  3). The elementary particles that can compose the loop are charged fermions (leptons and quarks) and bosons (W ± ). The main contribution from each one of those particles is around three times their masses. Thus, for the LHC energy regime and the physical limitations established by the ion's charge distribution, contributions coming from the W ± bosons and t quark are negligible. Another process by which photons fluctuate into vector mesons, called VDM-Regge mechanism, has contributions in the experiment energy range. However, the photons produced by this mechanism are very-forwarded (|η| 5 or θ = 0.77 • with the beam) and cannot be detected by the ATLAS detector. Furthermore, their transverse momentum is such that applied cuts would completely kill their contribution [34]. Therefore, only leptons and light quarks are considered.
Extensions of the Standard Model introduce all sorts of contributions through hypothetical charged or neutral virtual particles. In this sense, the light-by-light scattering can be used as a way of probing the quantum vacuum. Nonlinear electrodynamics introduce interaction vertexes allowing photon fusion. In particular, the lowest order correction terms F 2 and G 2 allow direct interaction between 4 photons, as depicted in Figure 4. The amplitude and cross-sections due to these terms are given as a function of the parameters α and β by equations (10), (12) and (13). It is noteworthy that, in the non-polarized case, it isn't possible to distinguish the contribution from F 2 and G 2 .
In order to constrain the contribution to the total γγ crosssection from nonlinear corrections, we assume the total amplitude of the process to be the sum of the Standard Model mechanisms mentioned above and (10). Hence, the theoretical cross-section, to be compared with ATLAS' result, is composed of pure contributions due to the Standard Model and γ γ γ γ nonlinear corrections, and an interference term 4 : As discussed in the previous section, the interference term is a linear function of α and β. Therefore, writing them out explicitly, (23) can be rewritten as whereσ NL ,σ α andσ β are given in Table II, σ AT LAS = 70 ± 24(stat.)±17(syst.) nb and σ S M is given in Table III for each distribution. The Standard Model cross-section was obtained using Fey-nArts 3.10 [40] to generate the diagrams and build the amplitude, FormCalc 9.6 [41] for algebraic simplifications and numerical computations, and LoopTools 2.15 [41] for loop calculations. For the interference term, we also used Feyn-Rule 2.3 [42] package. Numerical results for purely nonlinear corrections were confronted with those obtained in section II.

IV. RESULTS
In Table III we list the result of our calculations using the Standard Model for the cross-section measured at ATLAS. The results were obtained for each one of the four distributions presented in section III. Besides, for comparison purposes, we include the corresponding cross-sections obtained by neglecting the absorption factor (16). As a consequence, without the absorption factor, the integration over a wider range of the phase-space overestimates the cross-section by around 20% 5 . It's an interesting fact that the cross-sections obtained with the Gaussian and homogeneously charged sphere distributions differ from the one derived using Fermi 2P by less than 0.1%. On the other hand, cross-sections obtained with Yukawa distribution are, in every cases, 10% larger than those 4 In this case, the symbol means that both sides must be compatible. 5 Strong interaction due to nuclear overlap was not taken into account.
obtained with Fermi 2P, in agreement with [34]. Theoretical uncertainties are mainly due to lack of knowledge in the ion's charge and are considered to be of order 20% of the total cross-section [20].
To constrain the parameters α and β, we deduct the Standard Model cross-section prediction σ S M (second column of Table III) from the experimental result obtained by the AT-LAS Collaboration σ AT LAS and treat the remaining value as being produced by the nonlinear corrections alone (see Eq. 24). The theoretical, statistical and systematic uncertainties are added in quadrature. Using 3σ of confidence level, we are able to impose an upper limit on the nonlinear correction contribution given by the expression: with coefficients given in Table II for each distribution. Gaussian and homogeneously charged sphere distributions give similar results to the Fermi 2P distribution. The inequation (25) describes a region delimited by an ellipse whose major axis is parallel to the line β = α. The effect of first-degree monomials on the ellipse equation is to shift its center and correct the length of the axis. It can be shown that for (25), the translation of the center from the origin due to the interference term is less than 1% of the major axis length and the corresponding axis correction is of order 0.001%. Therefore, any contribution coming from the interference terms is completely clouded by the theoretical uncertainty and may be neglected. The phase-space volume accessible to the parameters α and β is presented in Figure 5. We show the outer bounds for Yukawa and Fermi 2P distributions (the Gaussian and homogeneously charged sphere are similar to the latter) as well as the line β = α representing Born-Infeld-like theories. As a consequence of the quadratic dependence on the parameters  Table IV. Upper values of the parameters for each distribution. The second column shows the upper values when α = β. The third column shows the upper values of α when β = 0 (due to the α ↔ β symmetry of (25) these also correspond to β when α = 0.
of the cross-section due to nonlinear corrections, we are able to completely constrain a finite region of the phase-space with one experimental data. This is not always possible, as is the case of experiments that measure the magnetic birefringence or Lamb shift effect [13]. Additionally, due to causality and unitarity principles, the parameters must be positive [43]. We note that Yukawa distribution is more restrictive than the others. This is due to the fact that it leads to an overestimation of the Standard Model cross-section (Table III), therefore assigning a smaller contribution to nonlinear corrections. As a result, values accessible with the Fermi 2P, Gaussian, and sphere distributions can be up to 15% larger.  Figure 5. Phase-space accessible to the parameters α and β derived from (25). The more restrictive blue region is obtained with a Yukawa distribution. The broader yellow region is obtained with all three distributions: Fermi with 2 parameters, Gaussian and homogeneously charged sphere. The green line β = α are the values accessible to Born-Infeld-like theories.
As an example, we use the upper limits in order to constrain the parameter of Born-Infeld-like theories defined by α = β = 1 8b 2 (see Table IV). The lower bound obtained using the Yukawa distribution is b Y 3.0×10 4 GeV 2 1.3×10 28 V m −1 , while for Fermi 2P and the others is b F 2.8 × 10 4 GeV 2 1.2 × 10 28 V m −1 . Similarly, we may define a mass M ≡ √ b, for which we obtain M 170GeV for all four distributions, in accordance with [22].
Finally, with the future project of the ATLAS Collaboration to measure the γγ → γγ scattering with extended tracking acceptance |η| < 4 in mind, we calculate the Standard Model prediction using the same remaining cuts to be σ Y = 52 ± 10 nb, with the Yukawa distribution, and σ F,G,S = 45 ± 9 nb with the Fermi 2P, Gaussian and homogeneously charged sphere distributions.

V. CONCLUSION
The recent measurement of the γγ scattering by the AT-LAS Collaboration has opened a new possibility to test QED and constraining with great precision the phase-space where nonlinear corrections live. In this work, using the equivalent photon approximation, we calculate the Standard Model prediction for this phenomenon measured at the ATLAS detector using four nuclear charge distributions (see Table III). These results are in accordance with the literature [20,21,34]. We investigated leading-order nonlinear corrections to Maxwell electrodynamics parameterizing the square of the invariants F 2 and G 2 and obtained an analytic expression for the nonpolarized squared amplitude for the γγ scattering (11) as well as the differential (12) and total (13) cross-sections.
To constrain the parameters, we deducted the Standard Model cross-section prediction from the measured value by the ATLAS Collaboration and interpreted the remaining cross-section as coming from the nonlinear corrections. As a consequence of the functional dependence on α and β, a finite region from the parameters phase-space could be completely constrained (Eq. 25). The interference term between the Standard Model and nonlinear correction amplitudes was analyzed and found to be negligible. Lastly, we have shown the upper bound and its dependence on the nuclear charge distribution (see Figure 5).
The constraints obtained in this paper for the nonlinear corrections derived using light-by-light scattering cross-section measurement are much more precise than those obtained with any other experiment. When confronted with those obtained in low-energy experiments, as in [13], our constraints are up to 20 orders of magnitude lower for α = β. In [17], where the effects of Born-Infeld-like theories were analyzed using the Hydrogen's ionization energy, the lower bound b ≥ 1.07 × 10 21 V m −1 , corresponding to α = β ≤ 8.1 × 10 4 GeV −4 , is 14 orders of magnitude larger. Lastly, we get 12 orders of magnitude of precision were obtained when comparing the upper bound for the Born-Infeld parameter in [15]. Also, defining the energy regime in Born-Infeld theory in terms of its parameter as M ≡ √ b, we obtain the lower bound M 170GeV, which is compatible with [22].
The ATLAS Collaboration is aiming to improve the tracking acceptance from |η| < 2.5 para |η| < 4. With this in mind, we calculated the cross-section to be measured to be 45 ± 9 nb using the realistic Fermi with 2 parameters nuclear charge distribution.
The first direct observation of the γγ scattering made by the ATLAS Collaboration is, without any doubt, a great achieve-ment made possible by theoretical and experimental efforts. This mechanism proves to be an elegant and efficient way to probe the quantum vacuum which allows constraining a great variety of Beyond Standard Model theories. While this first measurement is compatible with QED predictions, its 40% absolute uncertainty is still an obstacle to overcome. Future measurements, with greater precision, wider phasespace, and higher energy regimes, will allow to analyze with greater detail several contributions that compose the mechanism. As commented in [35], forthcoming light-by-light scattering measurements could be used to constrain nuclear charge distributions.
Finally, the increasing energy scales and experiment precision will impose more sophisticated theoretical analysis. In the scope of nonlinear corrections to Maxwell electrodynamics, in order to obtain more precise constraints in these scenarios, a future investigation would be to include higher-order terms from the Lagrangian expansion.