Constraining Gluonic Quartic Gauge Coupling Operators with $gg \to \gamma \gamma$

Gluon-gluon to photon-photon scattering $gg \to \gamma\gamma$ offers to the LHC experiments a uniquely powerful probe of dimension-8 operators in the Standard Model Effective Field Theory (SMEFT) that are quadratic in both the electromagnetic and gluonic field-strength tensors, such as would appear in the Born-Infeld extension of the Standard Model (SM). We use 13-TeV ATLAS data on the production of isolated photon pairs to set lower limits on the scales of dimension-8 operators $M \gtrsim 1$~TeV, and discuss the prospective sensitivities of possible future hadron colliders.

Introduction -A model-independent way to constrain possible extensions of the SM with high-scale new physics that decouples at low energies is provided by the SMEFT [1], which employs a systematic expansion in the effective mass dimensions of the new operators generated by high-scale physics beyond the SM. Apart from the dimension-5 operators that may contribute to neutrino masses and oscillations [2], the most prominent operators are those of dimension 6, whose coefficients scale as 1/Λ 2 , where Λ represents a generic new-physics scale. There have been many studies of the constraints on dimension-6 operator coefficients imposed by current and potential future collider data [3]. Some attention has also been paid to the experimental constraints on operators of dimension d 8, whose effects at low energies are suppressed by O(E/Λ) d−4 , particularly those involving four electroweak gauge field strengths, see, e.g., [4,5].
We focus here on dimension-8 operators in the SMEFT that are quadratic in the field strength tensors of both the gluon fields of QCD, G a µν : a = 1, ..., 8 and electroweak gauge fields, either the W i µν : i = 1, 2, 3 of SU(2) or the B µν of U(1). As we review in more detail below, there are 8 independent such dimension-8 operators, 4 involving pairs of the W i µν and 4 involving pairs of B µν . Since the electromagnetic field strength tensor F µν is a specific combination of W 3 µν and B µν , the gg → γγ scattering process is sensitive to just 4 combinations of these dimension-8 operators.
One of these combinations is of particular interest, as it arises in the Born-Infeld (BI) extension of the SM with the following Lagrangian L BISM : where β ≡ M 2 is the BI nonlinearity scale and the index λ runs over the 12 generators of the SM SU(3)×SU(2)×U(1) gauge group. Born and Infeld proposed a similar nonlinear extension of QED in 1934 [6], motivated by a 'unitarian' idea that there should be an upper limit on the strength of the electromagnetic field. However, this theory remained largely a curiosity until Fradkin and Tseytlin [7] showed in 1985 that it appears in models inspired by M-theory, e.g., in which vector fields are coupled to matter particles that are localized on lower-dimensional 'branes' [8]. We note also that it has recently been shown that BI theories have uniquely soft scattering amplitudes in the infrared limit [9]. It was pointed out in [10] that a measurement of lightby-light scattering in heavy-ion collisions at the LHC by the ATLAS Collaboration [11] imposes a constraint on the BI extension of QED that is orders of magnitude stronger than that available from previous, lower-energy experiments [12], corresponding, e.g., in the context of M-theory to an upper limit on the separation between branes 1/(100 GeV). The purpose of this Letter is to show that an ATLAS measurement of gg → γγ scattering in proton-proton collisions [13] strengthens this limit by almost another order of magnitude in the context of a BI extension of the SM. This bound penetrates significantly the parameter space of variants of M-theory with large extra dimensions [8], and we show how future hadron colliders would offer even greater sensitivity.
Dimension-8 Gluon/Photon Operators -Constructing the effective operators that contribute to gg → γγ scattering needs two gluon fields and two photon fields [14]. Since fermions and massive vector bosons are absent in the external states of this process, the candidate operators involve only these fields, appearing via the gluon field strength G a µν and the photon field strength F µν , which is a combination of the B µν of U (1) Y and the W 3 µν of SU (2) L with coefficients s W ≡ sin θ W and c W ≡ cos θ W , respectively. The dimension-8 operators arXiv:1802.02416v2 [hep-ph] 11 Jul 2018 relevant to gg → γγ scattering require two gluon field strengths G a µν and two electroweak field strengths B µν or W i µν . The two colour indices a must be contracted, as must the two SU (2) L indices i. Lorentz invariance allows 4 different ways of contracting the 8 space-time indices. Thus there are 8 independent gluonic quartic gauge coupling (gQGC) operators, and the relevant dimension-8 part of the effective Lagrangian may be writ- where the scales √ β i ≡ M i represents the scales of the physics beyond the SM that induces these effective operators. Operators of form similar to O gT,3 and O gT, 7 have not been discussed in the context of electroweak QGCs [4]. The effective Lagrangian for gg → γγ scattering may be written in the formL gT = 3 i=0 , i = 0, 1, 2, 3 and theÔ gT,i have the same forms as the O gT,i , i = 4, 5, 6, 7 but with B αβ replaced by F αβ , etc.
The first term in the BI extension (1) of the SM generates O gT,0 and O gT,4 , and hence alsoÔ gT,0 . The second term in (1) yields a quartic interaction with the Lorentz structure ( to be equal at the common BI scale M 0 = M 4 , but subject to different renormalization below that scale, and similarly for i = 1, 3 and 5, 7. However, since the constraint we find on the BI scale is not very different from the electroweak scale, this effect is small and we neglect it in our analysis. In this approximation, the experimental constraints on M 0 that we derive below are proxies for the corresponding constraints on the BI scale M = √ β. gg → γγ Scattering -The different Lorentz structures in (2) yield different cross sections for gg → γγ process [15]: where s ≡ (p γ1 + p γ1 ) 2 , t ≡ s 2 (cos θ − 1) and u ≡ −t − s are Mandelstam variables. The four different Lorentz structures have different dependences on the scattering angle θ in the centre-of-mass frame, independent of s, as shown in the upper panel of Fig. 1. Once an excess beyond the SM background is seen, the Lorentz structures can be identified by fitting the θ distribution. These angular dependences can be contrasted with that of the SM qq → γγ background: dσ/d cos θ ∝ cot 2 θ that vanishes for θ = π/2 in the massless limit. A cut on the angular distribution | cos θ| ≤ 0.8 or 0.9 (equivalent to a cut in pseudorapidity) would be effective for suppressing the SM background.
Each gauge field contributes a momentum factor to the amplitudes generated by the dimension-8 gQGC opera- tors (2), so the total cross sections scale as s 3 , The cross sections for O gT, (1,5) , O gT, (2,6) , and O gT, (3,7) are roughly one order smaller than those for O gT,(0,4) , respectively. Considering the s 4 W and c 4 W coefficients, the contributions of the eight gQGC operators have a hierarchical structure: σ gT,4 ≈ 10σ gT,(0,5,6,7) ≈ 100σ gT, (1,2,3) for identical scales of M i . A characteristic energy of gg → γγ scattering at LHC-13 TeV is O(1) TeV, which places a natural limit on the applicability of gQGC operators. If √ s exceeds M i , the O gT,i operators would eventually cease to be a good approximation, violating the unitarity constraint. To preserve unitarity, we assume that the cross section falls with the diphoton invariant mass, σ ≈ 1/s = 1/m 2 γγ , above the scales √ s i where unitarity is saturated: corresponding to ratios √ s i /M i = 4.71, 6.21, 6.51, 6.88, 3.49, 4.60,4.82, 5.10 for i = 0, · · · , 7. The cross section increases as s 3 below and decreases as 1/s above the saturation point. The cross section for the SM background qq → γγ also falls with energy: at leading order: σ SM ≈ q Q 4 q [log(1/(1 − cos θ cut )) − 2]/24πs, where Q q is the electric charge and θ cut is a cut | cos θ| < cos θ cut = 0.9 on the scattering angle in the center-of-mass frame. Higherorder QCD corrections increase this by a slowly-varying K-factor [16], yielding results in agreement with with the ATLAS measurements [13]. The lower panel of Fig. 1 compares this background with the gQGC signals for M i = 1 TeV. The SM background lies far below the scale of unitarity saturation, and falls below the potential gQGC signals when √ s ∼ 2 to 3 TeV. In addition to exploiting the different angular distribution, one may also suppress the SM background by cutting low √ s events. Constraints from the ATLAS Data -The ATLAS Collaboration searched for new physics with high-mass diphoton final states [13]. Comparing their searches for spin-0 and -2 resonant and non-resonant signals, the nonresonant case is the closest in spirit to the gQGC contribution to gg → γγ that we consider here.
The ATLAS analysis uses the fiducial region |η| < 2.37, excluding the blind transition region 1.37 < |η| < 1.52. In the non-resonant signal search, the photon transverse energies receive a cut E T > 55 GeV. For non-resonant Kaluza-Klein signal, the geometric acceptance increases from 58% at M S = 3.5 TeV to 65% at M S = 5 TeV where M S is the cutoff scale of the Kaluza-Klein spectrum.
In our analysis, we assume a constant geometric acceptance of 60%, which is to be combined with the efficiency for reconstruction and identification that is approximately constant at 77%, yielding an overall signal event selection efficiency of 46%. The upper panel of Fig. 2 shows the expected signal event rates at ATLAS with 36.7 fb −1 and 13 TeV for the different gQGC operators with M i = 1 TeV, as functions of the invariant mass m γγ [17,18]. For comparison, we also show the background rate extracted from the background-only fit in the Fig. 2b of [13]. With cutoff scale M i = 1 TeV, the background and gQGC signals cross around m γγ = 1 to 2 TeV as expected from Fig. 1. Above m γγ = 1.5 TeV, the signal rate keeps rising before saturating unitarity at m γγ 4 TeV, depending on the model. In our estimations of the SM background and signals we use the cut m γγ < 2 TeV, well below these unitarity saturation scales.
With this cut, we make a binned analysis of the ATLAS data [13] to quantify the sensitivity √ N i to these operators, where S i and B i are the predicted total signal and background while N i is the number of events actually measured in the i-th bin. We plot the significances, evaluated using the ∆χ 2 distributions, as functions of the nonlinearity scale M i in the lower panel of Fig. 2. The significances decrease very rapidly with M i . Since the gQGC operator coefficients are suppressed by 1/M 4 i , the cross sections fall as 1/M 8 i and small changes in the M i can affect the significances dramatically. The hierarchical structure of the cross sections generated by the eight gQGC operators is manifest in the 95% C.L. Sensitivities at Future Hadron Colliders -As discussed analytically above, the most effective cut for suppressing the SM background is that on the scattering angle in the centre-of-mass frame. Fig. 3 shows the cross sections obtained with different cuts on the angular distributions, applying the fiducial cut |η(γ)| < 2.37 in all cases. In comparison, the black curves were obtained with the ATLAS E T cut, E T (γ) > 55 GeV. Across the whole invariant-mass region, the scattering angle cut can reduce significantly the SM background, with much smaller effects on the signals. Therefore, the scattering angle cut in the center-of-mass frame is more suitable than E T cut for the search for dimension-8 gQGC operators. In following discussion of the sensitivities at future colliders, we apply cuts |η(γ)| < 2.37 and | cos θ(γ)| < 0.8.
The upper panel of Fig. 4 shows how the SM background (thinner lines) and the gQGC signals (thicker lines) change with collider energy. For illustration, we only plot O gT,0 whose features are shared by other gQGC operators. For comparison, we use the same luminosity 36.7fb −1 for different collider energies, noting that the event rates would be much larger for the expected luminosities at HE-LHC at 27 TeV [20], FCC-hh [21] and SppC [22]. At higher energies the signal event spectra are significantly enhanced, especially in the high m γγ range. For √ s = 100 TeV, the event rate can be as large as 10 4 /20 GeV for M i = 1 TeV. Although the SM background also increases, its contribution at high m γγ range is still negligibly small. At higher collider energies √ s, the crossing point between the SM background and the gQGC signal curves decreases. The crossings for √ s = (13, 27, 50, 100) TeV happen at m γγ = (1.60, 1.32, 1.17, 1.06) TeV. We increase the upper cut on the invariant mass roughly proportionally to the collider energy: m γγ ≤ (3,5,9,14) TeV, respectively.
The lower panel of Fig. 4 shows the significances For FCC-hh and SppC at 100TeV, the sensitivity would be another order of magnitude better than the current ATLAS analysis with 36.7fb −1 at 13 TeV, well into the range of potential interest to string models.
Conclusions -The ATLAS data on light-by-light scattering in heavy-ion collisions can exclude the QED BI [6] scale 100 GeV [10]. In this paper we have shown that the ATLAS data on gg → γγ scattering enhances the sensitivity by an order of magnitude, to 1 TeV for the analogous dimension-8 operator scales containing other combinations of gluon and electromagnetic fields. This constraint on the BI extension of SM is very interesting in view of its connections with string theory [7] and particularly models in which branes are separated by distances 1 TeV −1 [8]. Moreover, similar searches for γγ production at possible future proton-proton colliders could be sensitive to BI scales in the multi-TeV scale, complementing the searches via dimension-6 SMEFT operators [23].