Light-by-Light Scattering Constraint on Born-Infeld Theory

The recent measurement by ATLAS of light-by-light scattering in LHC Pb-Pb collisions is the first direct evidence for this basic process. We find that it requires the mass scale of a nonlinear Born-Infeld extension of QED to be $\gtrsim 100$~GeV, a much stronger constraint than those derived previously. In the case of a Born-Infeld extension of the Standard Model in which the U(1)$_{\rm Y}$ hypercharge gauge symmetry is realized nonlinearly, the limit on the corresponding mass scale is $\gtrsim 90$~GeV, which in turn imposes a lower limit of $\gtrsim 11$~TeV on the magnetic monopole mass in such a U(1)$_{\rm Y}$ Born-Infeld theory.

Over 80 years ago, soon after Dirac proposed his relativistic theory of the electron [1] and his interpretation of 'hole' states as positrons [2], Halpern [3] in 1933 and Heisenberg [4] in 1934 realized that quantum effects would induce light-by-light scattering, which was first calculated in the low-frequency limit by Euler and Kockel [5] in 1935. Subsequently, Heisenberg and Euler [6] derived in 1936 a more general expression for the quantum nonlinearities in the Lagrangian of Quantum Electrodynamics (QED), and a complete calculation of light-by-light scattering in QED was published by Karplus and Neuman [7] in 1951. However, measurement of light-by-light scattering has remained elusive until very recently. In 2013 d'Enterria and Silveira [8] proposed looking for light-by-light scattering in ultraperipheral heavy-ion collisions at the LHC, and evidence for this process was recently presented by the ATLAS Collaboration [9], at a level consistent with the QED predictions in [8] and [10].
In parallel with the early work on light-by-light scattering in QED, and motivated by a 'unitarian' idea that there should be an upper limit on the strength of the electromagnetic field, Born and Infeld [11] proposed in 1934 a conceptually distinct nonlinear modification of the Lagrangian of QED: where β is an a priori unknown parameter with the dimension of [Mass] 2 that we write as β ≡ M 2 , andF µν is the dual of the field strength tensor F µν . Interest in Born-Infeld theory was revived in 1985 when Fradkin and Tseytlin [12] discovered that it appears when an Abelian vector field in four dimensions is coupled to an open string, as occurs in models inspired by M theory in which particles are localized on lower-dimensional 'branes' separated by a distance 1/ √ β = 1/M in some extra dimension 1 . Depending on the specific brane scenario considered, M might have any value between a few hundred GeV and the Planck scale ∼ 10 19 GeV. For the purposes of this paper, we consider only the relevant terms of fourth order in the gauge field strengths in (1).
Until now, there has been no strong lower limit on the Born-Infeld scale β or, equivalently, the brane mass scale M and the brane separation 1/M . A constraint corresponding to M 100 MeV was derived in [14] from electronic and muonic atom spectra, though the derivation has been questioned in [15]. Measurements of photon splitting in atomic fields [16] were considered in [17], where it was concluded that they provided no limit on the Born-Infeld scale and it was suggested that measurements of the surface magnetic field of neutron stars [18] might be sensitive to M = √ β ∼ 1.4 × 10 −5 GeV. More recently, measurements of nonlinearities in light by the PVLAS Collaboration [19] are somewhat more sensitive to the individual nonlinear terms in (1), but are insensitive to the particular combination appearing in the Born-Infeld theory, as discussed in [20] where more references can be found.
Here we show that the agreement of the recent ATLAS measurement of light-by-light scattering with the standard QED prediction provides the first limit on M in the multi-GeV range, excluding a significant range extending to entering the range of interest to brane theories. This limit is obtained under quite conservative assumptions, and plausible stronger assumptions would strengthen our lower bound to M 200 GeV. One may also consider a Born-Infeld extension of the Standard Model in which the hypercharge U(1) Y gauge symmetry is realised non-linearly, in which case the limit (2) is relaxed to where we have used B µ Y = cosθ W A µ EM − sinθ W Z µ and sin 2 θ W 0.23, with θ W the weak mixing angle. As a corollary of this lower limit on the U(1) Y brane scale, we recall that Arunasalam and Kobakhidze recently pointed out [21] that the Standard Model modified by a Born-Infeld U(1) Y theory has a finite-energy electroweak monopole [22,23] solution M, whose mass they estimated as M M 4 TeV + 72.8 M Y . Such a monopole is less constrained by Higgs measurements than electroweak monopoles in other extensions of the Standard Model [24], and hence of interest for potential detection by the ATLAS [25], CMS and MoEDAL experiments at the LHC [26]. However, our lower limit M Y 90 GeV (2) corresponds to a 95% CL lower limit on the mass of this monopole M M 11 TeV, excluding its production at the LHC. Following the suggestion of [8] 2 , we consider ultraperipheral heavy-ion collisions in which the nuclei scat-ter quasi-elastically via photon exchange: Pb + Pb (γγ)→Pb ( * ) + Pb ( * ) + X, as depicted in Fig. 1, effectively acting via the equivalent photon approximation (EPA) [27] as a photon-photon collider. The EPA allows the electromagnetic field surrounding a highly-relativistic charged particle to be treated as equivalent to a flux of on-shell photons. Since the photon flux is proportional to Z 2 for each nucleus, the coherent enhancement in the exclusive γγ cross-section scales as Z 4 , where Z = 82 for the lead (Pb) ions used at the LHC. This is why heavyion collisions have an advantage over proton-proton or proton-lead collisions for probing physics in electromagnetic processes [8]. Photon fusion in ultra-peripheral heavy-ion collisions has been suggested as a way of detecting the Higgs boson [28,29] and, more recently, the possibility of constraining new physics beyond the Standard Model (BSM) in this process was studied in [30,31].

FIG. 2:
Comparison between the angular distributions (with arbitrary normalisations) as functions of cos θ in the centreof-mass frame (where θ is the polar angle) for the leadingorder differential cross-sections in U(1)EM Born-Infeld theory and QED, plotted as solid blue and dashed red lines, respectively.
As already mentioned, the possibility of directly observing light-by-light scattering at the LHC was proposed in [8], and this long-standing prediction of QED was finally measured earlier this year with 4.4σ significance by the ATLAS Collaboration [9] at a level in good agreement with calculations in [8,10]. The compatibility with the Standard Model constrains any possible contributions from BSM physics. Born-Infeld theory is particularly interesting in this regard, as constraints from low-energy optical and atomic experiments have yet to reach the sensitivity of interest for measuring light-by-light scattering [19,20].
The leading-order cross-section for unpolarised lightby-light scattering in Born-Infeld theory in the γγ centre-of-mass frame is given by [17,32]: where m γγ is the diphoton invariant mass and the differential cross-section is We recall that the parameter β = M 2 enters as a dimensionful parameter in the Born-Infeld theory of non-linear QED defined by the Lagrangian (1). If this originates from a Born-Infeld theory of hypercharge then the corresponding mass scale is M Y = cos θ W M . We plot in Fig. 2 the angular distributions as functions of cos θ in the centre-of-mass frame (where θ is the polar angle) for the leading-order differential cross-sections in both Born-Infeld theory and QED (with arbitrary normalisations), as solid blue and dashed red lines, respectively. We see that the Born-Infeld distribution is less forward peaked than that for QED. For the latter, we used the leading-order amplitudes for the quark and lepton box loops in the ultra-relativistic limit from [33], omitting the percent-level effects of higher-order QCD and QED corrections, as well the W ± contribution that is negligible for typical diphoton centre-of-mass masses at the LHC.
The total exclusive diphoton cross-section from Pb+Pb collisions is obtained by convoluting the γγ → γγ cross-section with a luminosity function dL/dτ [34], We have introduced here a dimensionless measure of the diphoton invariant mass, τ ≡ m 2 γγ /s N N , where √ s N N = 5.02 TeV is the centre-of-mass energy per nucleon pair in the ATLAS measurement. The luminosity function, derived for example in [34], can be written as an integral over the number distribution of photons carrying a fraction x of the total Pb momentum: where the distribution function f (x) depends on a nuclear form factor. We follow [34] in adopting the form factor proposed in [29], while noting that variations in the choice leads to ∼ 20% uncertainties in the final crosssections [8]. A contribution with a non-factorisable distribution function should also be subtracted to account for the exclusion of nuclear overlaps, but this is not a significant effect for the relevant kinematic range, causing a difference within the 20% uncertainty [31] from the photon luminosity evaluated numerically using the STARlight code [35]. For √ s N N = 5.5 TeV and m γγ > 5 GeV we obtain a QED cross-section of σ QED excl. = 385 ± 77 nb, in good agreement with [8]. The ATLAS measurement is performed at √ s N N = 5.02 TeV and for m γγ > 6 GeV, for which we find σ QED excl. = 220 ± 44 nb. This total γγ → γγ cross-section is reduced by the fiducial cuts of the ATLAS analysis, which restrict the phase space to a photon pseudorapidity region |η| < 2.4, and require photon transverse energies E T > 3 GeV and the diphoton system to have an invariant mass m γγ > 6 GeV with a transverse momentum p γγ T < 2 GeV and an acoplanarity Aco = 1 − ∆φ/π < 0.01. We simulate the event selection using Monte-Carlo sampling, implementing the cuts with a 15% Gaussian smearing in the photon transverse energy resolution at low energies and 0.7% at higher energies [9,36] above 100 GeV. Since the differential cross-section does not depend on φ we implement the acoplanarity cut as a fixed 85% efficiency in the number of signal events after the p γγ T selection, following the ATLAS analysis [9]. The total reduction in yield for the QED case is a factor ∼ 0.30, which results in a fiducial cross-section σ QED fid. = 53 ± 11 nb for √ s N N = 5.02 TeV, in good agreement with the two predictions of 45 and 49 nb quoted by ATLAS [9].
Following this validation for the QED case, we repeat the procedure for the Born-Infeld cross-section. Since the Born-Infeld γγ → γγ cross-section grows with energy, the dominant contribution to the cross-section comes from the τ 0.2 part of the integral, compared with τ 10 −4 for the QED case. We show in Fig. 3 the distributions of the σ(γγ → γγ) cross-section multiplied by the photon flux luminosity factor -normalised by the total exclusive cross-section -as functions of the invariant diphoton mass distribution, for the QED case in the left panel and in Born-Infeld theory with M = √ β = 200 GeV in the right panel.
We see that the invariant-mass distribution in the Born-Infeld case extends to m γγ > M , where the validity of the tree-level Born-Infeld Lagrangian may be questioned because the Taylor expansion of the square root in the non-polynomial Born-Infeld Lagrangian (1) could break down. With this in mind, we use two approaches to place plausible limits on M = √ β. In the first and most conservative method we consider γγ scattering only for m γγ ≤ M , while in the second approach we integrate the γγ cross-section (4) up to the diphoton invariant mass where the unitarity limit σ BI ∼ 1/m 2 γγ is saturated, beyond which we assume that the cross-section saturates the unitarity limit and falls as ∼ 1/m 2 γγ . We find fiducial efficiencies for the cut-off and unitarization approaches to be ∼ 0.39 and 0.14, respectively. Whilst the E T and η cuts have much less effect than for QED, as expected from the difference in the angular distributions visible in Fig. 2, the larger invariant masses appearing in the Born-Infeld case are much more affected by the p γγ T requirement. Our calculations of the corresponding U(1) EM Born- Infeld fiducial cross-sections are plotted in the left panel of Fig. 4 as a function of M = √ β: the green curve is for the more conservative cut-off approach, and the blue curve assumes that unitarity is saturated. These calculations are confronted with the ATLAS measurement of σ fid. = 70 ± 24 (stat.) ± 17 (sys.) nb [9], assuming that these errors are Gaussian and adding them in quadrature with a theory uncertainty of ±10 nb. We perform a χ 2 fit to obtain the 95% CL upper limit on a Born-Infeld signal additional to the 49 nb Standard Model prediction 3 . This corresponds to the excluded range shaded in pink above σ 95%CL fid.
∼ 65 nb in the left panel of Fig. 4, which translates into the limit M = √ β 100 (190) GeV in the cut-off (unitarized) approach, as indicated by the green (blue) vertical dashed line in Fig. 4, respectively. √ β in the U(1)EM Born-Infeld theory is shown as a solid green (blue) line for a hard cut-off (unitarized) approach, respectively, as discussed in the text. The lower diphoton invariant mass cut-off is set at 6 GeV (25 GeV) on the upper (lower) plot. This is compared with the 95% CL upper limit obtained from the ATLAS measurement [9] by combining the statistical and systematic errors in quadrature as well as a 10 nb theoretical uncertainty in the cross section predicted in QED [8,10]  These limits could be strengthened further by considering the m γγ distribution shown in Fig. 3(b) of [9], where we see that all the observed events had m γγ < 25 GeV, in line with expectations in QED, whereas in the Born-Infeld theory most events would have m γγ > 25 GeV. Calculating a ratio of the total exclusive crosssection of QED for m γγ > 6 GeV and > 25 GeV as σ mγγ >25 GeV excl.
∼ 0.02, we estimate a 95% CL upper limit of ∼ 2 nb for m γγ > 25 GeV. The corresponding exclusion plot is shown in the right panel of Fig. 4, where we see a stronger limit M = √ β 210 (330) GeV in the cut-off (unitarized) approach with the same colour coding as previously.
Our lower limit on the QED Born-Infeld scale M = √ β 100 GeV is at least 3 orders of magnitude stronger than the lower limits on M = √ β obtained from previous measurements of nonlinearities in light [14-17, 19, 20]. Because of the kinematic cuts made in the ATLAS analysis, our limit does not apply to a range of values of M 10 GeV for which the nonlinearities in (1) should be taken into account. However, our limit is the first to approach the range of potential interest for string/M theory constructions, since models with (stacks of) branes separated by distances 1/M : M = O(1) TeV have been proposed in that context [37]. Our analysis could clearly be refined with more sophisticated detector simulations and the uncertainties reduced. However, in view of the strong power-law dependence of the Born-Infeld crosssection on M = √ β visible in (4), the scope for significant improvement in our constraint is limited unless experiments can probe substantially larger m γγ ranges. In this regard, it would be interesting to explore the sensitivities of high-energy e + e − machines considered as γγ colliders.
As mentioned in the Introduction, Arunasalam and Kobakhidze have recently pointed out [21] that the Standard Model modified by a Born-Infeld theory of the hypercharge U(1) Y contains a finite-energy monopole solution with mass M M = E 0 + E 1 , where E 0 is the contribution associated with the Born-Infeld U(1) Y hypercharge, and E 1 is sssociated with the remainder of the Lagrangian. Arunasalam and Kobakhidze have estimated [21] that E 0 72.8 M Y , where M Y = cos θ W M , and Cho, Kim and Yoon had previously estimated [23] that E 1 4 TeV 4 . Combining these calculations and using our lower limit M 100 GeV (2), we obtain a lower limit M M 11 TeV on the U(1) Y Born-Infeld monopole mass 5 . Unfortunately, this is beyond the reach of MoEDAL [26] or any other experiment at the LHC [25], but could lie within reach of a similar experiment at any future 100-TeV pp collider [38], or of a cosmic ray experiment.
In this paper we have restricted our attention to possible Born-Infeld modifications of U (1) gauge factors and their constraints in light-by-light scattering only. We plan to examine in the future the experimental constraints from measurements at the LHC on possible Born-Infeld extensions of the SU (3) C and SU (2) L gauge symmetries of the Standard Model.