Probing the anomalous γγγZ coupling at the LHC with proton tagging

The sensitivities to the anomalous quartic gauge boson coupling γγγZ are estimated via γZ production with intact protons in the forward region at the LHC. Proton tagging proves to be a powerful tool to suppress the background, which allows consideration of the hadronic decays of the Z boson in addition to the leptonic ones. We discuss the discovery potential for an integrated luminosity of 300 fb−1 and 3000 fb−1. The sensitivity we obtain at 300 fb−1 goes beyond the one expected from LHC bounds on the Z → γγγ decay by about three orders of magnitude. The γZ channel provides important discriminatory information with respect to the exclusive γγ channel, as many particles beyond the Standard Model (such as a radion or Kaluza Klein gravitons) predict a signal in the latter but not the former.


Introduction
The Large Hadron Collider currently performs collisions at the unprecedented center-ofmass energy of 13 TeV, looking for a signal of physics beyond the Standard Model (SM). One route to such discovery is to study with high precision the interactions of the Standard Model, by looking for small deviations in the SM couplings. Such precision measurements, for instance in the sector of fermion-gauge interactions or more recently of Higgs interactions, have been a focus of intense activity and have proven powerful to constrain New Physics. In this work, our interest lies in the sector of pure gauge interactions. Our focus is the γγγZ interaction, that we study through γZ production via photon fusion with intact outgoing protons in the forward region. Similar photon-induced processes with intact protons have been studied in the past for W + W − , γγ, and other final states in refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].
The measurement we study crucially relies on the forward proton detectors in the ATLAS and CMS-TOTEM experiments [20,21]. We will work in the effective Lagrangian formalism, and the anomalous γγ → γZ cross section induced by effective operators is given in section 2. The contributions to the effective γγγZ interaction from loops of charged particles and tree-level exchange of neutral particles are computed in section 3. Turning to JHEP06(2017)142 pp collisions, a brief description of photon-induced processes and their measurement in the exclusive channel is discussed in section 4, and the event generation setup is described in section 5. Our analysis in Z hadronic and leptonic channels and the subsequent expected sensitivities are then given in section 6 and 7.

The γγγZ interactions in the SM and beyond
The γγγZ interaction is induced at one-loop level in the SM via loops of fermions and W bosons. The amplitudes for the SM γγ → γZ process have been first computed in ref. [22]. This process will be neglected in our study as, just like for SM light-by-light scattering, it is greatly reduced in the acceptance of the forward detectors (see sections 4.2, 5.2). The rare SM decay Z → γγγ is another process sensitive to the anomalous γγγZ interaction. It has a branching ratio predicted to be B SM (Z → γγγ) = 5.4 · 10 −10 . The fermion loops have beeen computed in refs. [23,24] and the W loop contribution in [25], the latter is found to be subdominant.
In the presence of New Physics with a mass scale Λ heavier than the experimentally accessible energy E, all New Physics manifestations can be described using an effective Lagrangian valid for Λ E. In this low-energy effective field theory (EFT), the γγγZ interactions are described by two dimension-eight γγγZ operators L γγγZ = ζO γZ +ζÕ γZ = ζF µν F µν F ρσ Z ρσ +ζF µνF µν F ρσZ ρσ , (2.1) withF µν = 1 2 µνρσ F ρσ . The O γZ 2 = F µν F νρ F ρσ Z σµ operator can sometimes be encountered in the literature. It is related to the above basis via the identity 4O γZ 2 = 2O γZ +Õ γZ . The O γZ ,Õ γZ provides a somewhat clearer mapping onto the properties of the underlying physics.
These operators can be seen as arising from a SU(2) × U(1) Y effective Lagrangian with operators such as B µν B µν B ρσ B ρσ , where B denotes the hypercharge gauge field. The SU(2) L × U(1) Y effective Lagrangian contains ten such operators, see e.g. [3,4,26,27]. The coefficients of these operators, once expressed with the same Lorentz structures as shown in eq. (2.1), must be positive to avoid superluminal excitations in the theory [28]. The SU(2) × U(1) Y effective Lagrangian also generates 4γ, γγZZ interactions, as described in [4]. Because of the large number of effective operators in the SU(2) × U(1) Y Lagrangian, anomalous interactions in the broken phase can be considered as independent.
The operators of eq. (2.1) induce an anomalous Z → γγγ decay [29], with a partial width that in our notation reads An anomalous γγ → γZ reaction is also induced, which is the focus of this work. We find the unpolarized differential cross section to be 1 where s, t, and u are the usual Mandelstam variables and β = 1 − m 2 Z /s for the γZ final state.
As the EFT is nonrenormalizable, a breakdown of unitarity is expected at high energies. Using the well-known partial wave analysis [30] we can estimate for what values of ζ,ζ and s the theory remains unitary. By imposing unitarity on the S-wave of the EFT amplitudes and neglecting the Z boson mass one finds the conditions (see [4] for details on similar amplitudes) |ζ +ζ|s 2 < 4π , |ζ −ζ|s 2 < 12π 5 . (2.4) As most of the recorded γZ events have √ s below 1 TeV, we expect the EFT to remain unitary for couplings up to ζ,ζ < (10 −12 -10 −11 ) GeV −4 . (2.5) The sensitivities we will derive in section 7 are much lower than these unitarity bounds. However, as a caveat, we stress that unless the underlying New Physics model is very strongly coupled, the EFT typically breaks down before unitarity is violated.

Contributions from New Physics
Loops of heavy particles charged under SU(2) L × U(1) Y contribute to the γγγZ couplings. These loop contributions only depend on the mass and quantum numbers of the particle in the loop and can thus be given in full generality. Denoting hypercharge by Y , sine and 1 It has been noted in [29] that the operators O± = O γZ ±Õ γZ do not interfere. This property provides a cross check of our result eq. (2.3), as in this basis we get ζ± = ζ ±ζ, (3ζ 2 + 3ζ 2 − 2ζζ) = ζ 2 + + 2ζ 2 − and 4(ζ 2 +ζ 2 − ζζ) = ζ 2 + + 3ζ 2 − , hence a vanishing interference.

JHEP06(2017)142
cosine of the Weinberg angle by s w and c w and labeling the SU(2) L representation by its dimension d, we can write [4] (ζ,ζ) = (c s ,c s ) where s denotes the spin of the heavy particle running in the loop. 2 Beyond perturbative contributions to ζ,ζ from charged particles, non-renormalizable interactions of neutral particles are also present in common extensions of the SM. Such theories can contain scalar, pseudo-scalar and spin-2 resonances, respectively denoted ϕ, ϕ, h µν , that couple to the photon as and generate the γγγZ couplings by tree-level exchange as Interestingly, the f γZ couplings vanish if the neutral particle couples universally to the W I and B kinetic terms of the SU(2) L × U(1) Y Lagrangian. This happens in particular when the neutral particle couples to gauge bosons via the stress-energy tensor. It is the case of the Kaluza Klein (KK) graviton present in models of warped extra dimensions with gauge fields on the IR brane, as well as the radion and the KK graviton in bulk gauge field scenarios with small IR brane kinetic terms [4,6].
This peculiar feature of the γγγZ coupling becomes very interesting when put together with the measurement of the 4γ interaction. Indeed, if a γγ signal was observed, the γZ channel would then provide a clear test whether or not the underlying exchanged particle is universally coupled to the gauge kinetic terms.

Photon coherent flux
We use the equivalent photon approximation [31,32] to describe the pp → γZ pp process via photon exchanges. In this approximation, the almost real photons (with low virtuality Q 2 = −q 2 ) are emitted by the incoming protons producing a state X, pp → pXp, through photon fusion γγ → X. The photon spectrum of virtuality Q 2 and energy E γ is proportional to the fine-structure constant α em and reads: where E is the energy of the incoming proton of mass m p , the photon minimum virtuality allowed by kinematics and F E and F M are functions of the electric and magnetic form factors G E and G M . In the dipole approximation the latter read 2) The magnetic moment of the proton is µ 2 p = 7.78 and the fitted scale Q 2 0 = 0.71 GeV 2 . Since the electromagnetic form factors fall steeply as a function of Q 2 , the cross section can be factorized into the matrix element of the photon fusion process and the two photon fluxes. In order to obtain the production cross section, the photon fluxes are first integrated over Q 2 The result is given for instance in ref. [1]. These fluxes can then be used to introduce an effective differential luminosity dL γγ /dW obtained by integrating where W is invariant mass of the diphoton system. Using the effective photon luminosity, the total cross section for the pp → pXp process reads where σ γγ→X denotes the cross section of the sub-process γγ → X, dependent on the invariant mass of the two-photon system. In addition to the photon exchange, there might be additional soft gluon exchanges that might destroy the protons. To take into account this effect, we can introduce the so-called survival probability that the protons remain intact in photon-induced processes [33,34]. In this paper, we assumed a survival probability of 90%.

Proton detectors
The pp → pγZp proccess can be probed via the detection of two intact protons in the forward proton detectors at CT-PPS or AFP [20,35]  of 0.015 < ξ 1,2 < 0.15, where ξ 1,2 is the proton fractional momentum loss which is reconstructed by the forward proton detector. This leads to an acceptance in the central mass m γZ between 300 and 1900 GeV. The average number of multiple proton-proton collisions per bunch crossing, µ, sets a huge background environment on the search for exclusive events. Intact forward protons arising from the pile-up together with an uncorrelated non-exclusive process in the central detector can mimic the signal exclusive events. Usually µ ranges from 30 to 50 interactions per bunch crossing at the current LHC luminosity. The pile-up is expected to go up to µ = 200 interactions at the High Luminosity LHC which will pose a challenge on the search for New Physics. The detection and characterization of the outgoing protons provides the complete kinematic information on the event, which in turn allows us to exploit the fourmomentum conservation via rapidity and mass matching methods. These methods provide a strong background rejection which is the key feature of the forward proton detectors in exclusive processes. Further background rejection can be achieved with the use of timing detectors. Timing detectors are expected to be installed and operating in both CT-PPS and ATLAS to measure the time-of-flight of protons with a precision of ∼ 15 ps, which would allow to determine the interaction vertex of the protons with a 2.1 mm precision, thus allowing a large background rejection by a factor of ∼ 40 [36]. In this work, we will not make use of this potential future improvement.

Central detection of Zγ
In ATLAS and CMS, photons can be reconstructed in the central detectors instrumented with electromagnetic calorimeters which cover the pseudorapidity range |η| < 2.5 and provide excellent resolution in terms of energy (less than a percent at p T > 100 GeV) and position (0.001 units of η and 1 mrad on the azimuthal angle φ) for photons and leptons with p T ranging from a few GeV up to the TeV scale. Photon identification efficiency is expected to be around 75% for p T > 100 GeV. In addition, about 1% of the electrons and jets are misidentified as photons [37].
The decay of the Z boson is widely studied in CMS and ATLAS and is used for calibration of the detectors [38]. In this study we consider both leptonic (electrons and muons) and hadronic decays. For ATLAS, the fiducial acceptance corresponds to leptons with tranverse momenta p T > 25 GeV and absolute rapidity η < 2.5. For Z boson production, the dilepton invariant mass m ¯ is required to be between 66 < m ¯ < 116 GeV. Similar requirements are made in CMS. For the Z boson decay into hadrons, the jet is reconstructed by clustering particles deposited in the electromagnetic and hadronic calorimeters. The energy of photons is obtained directly from the electromagnetic calorimeter measurement. The energy of a charged hadron is determined from a combination of the track momentum and the corresponding electromagnetic and hadronic calorimeter energies. The energy of a neutral hadron is obtained from the calibrated energies in electromagnetic and hadronic calorimeters. The typical jet energy resolution is between 5-10% for jets with p T > 200 GeV. Commonly, the anti-k T jet clustering-algorithm with a radius parameter of R = 0.5 is used in CMS and ATLAS [39,40].

JHEP06(2017)142 5 Event generation
The anomalous γγ → γZ process with intact protons in the final state has been implemented in the Forward Physics Monte Carlo (FPMC) generator [41]. Contributions and simulations of the various backgrounds are discussed in the subsections below. In order to model systematic uncertainties on final states, we have included Gaussian smearings on the photon, lepton and hadron energies of 1%. In addition, we apply Gaussian smearings of 15% in the individual reconstructed jet energy, as well as 0.1% for the pseudorapidity and 1 mrad for the azimuthal angle.

Pile-up backgrounds
The largest background to the pγZp final state originates from γZ detection in the central detectors simultaneously with the detection of two intact protons from pile-up.
Background contribution in the jjγ channel is dominated by W ± γ and γZ in association with pile-up protons if we restrict ourselves to two-jet final states. Around 1% of the electrons are misidentified as photons, thus the background qqe in association with pile-up is also considered in our study. The non-exclusive background processes were simulated in PYTHIA8 [42] at leading order. Jets are reconstructed with the anti-k t clusteringalgorithm with FastJet [43] using R = 0.5 and p T min = 10 GeV at the hadron level, which is close to the standard CMS and ATLAS parameter choice for jet reconstruction. We assign a 15% resolution to the reconstructed jets energy, and apply smearings on the φ j , p T j and η j on top of the gaussian smearings applied to the individual particles that form the jet.
In the ¯ γ channel, the dominant background is the leptonic decay of the non-exclusive γZ production in association with pile-up protons, as can be seen in figure 2. We also consider misidentification of jets and electrons as photons as part of the background.
The pile-up events were simulated as follows. For each non-exclusive background event generated on PYTHIA, the number of pile-up interactions in the event is drawn from a Poisson distribution with mean µ = 50, 200 respectively for the low and high luminosity scenarios. We draw the tag probability for the protons arising from the pile-up interactions from a uniform distribution and compare it with the single, double or no tag probabilities, which were computed by propagating single and double diffractive protons generated on PYTHIA8 along the beamline up to the proton detectors within their acceptance (see [36,44] for more details). We assign the fractional momentum loss ξ = ∆p/p, which would be reconstructed by the forward detectors in AFP or TOTEM, by randomly sampling the distribution f (ξ) = 1/ξ (which roughly resembles the ξ distribution measured by the proton detectors) defined in 0.015 < ξ < 0.15 via its inverse transform. When the forward detectors have at least one proton tagged in each arm, we compare the diproton mass √ ξ 1 ξ 2 s and rapidity 1 2 log(ξ 1 /ξ 2 ) with the central mass and rapidity of the γZ final state, and select the best match in such observables. Only events with two proton tags pass through the selection cuts quoted in tables 1, 2 and 3.

Exclusive backgrounds
The SM predicts Zγ production at one-loop with two intact protons via two mechanisms: elastic gluon fusion [45], with the exchange of an extra gluon to ensure the process is colourless and the protons stay intact, and photon fusion. The elastic gluon fusion contribution can be neglected at high m Zγ (within the proton taggers acceptance) since the soft gluon emission in the gluon ladder has to be suppressed in order to get an exclusive diffractive event with intact protons. In practice, a Sudakov form factor is introduced to suppress this emission which greatly reduces the cross section at high m Zγ [45]. This is true in general for central exclusive processes in QCD.
The SM cross section for γγ → Zγ is related to the γγ cross section roughly as σ SM γγ→Zγ ≈ 6σ SM γγ→γγ within the mass acceptance (see for instance figures 4 and 5 in [46]). We can use this to extrapolate the results from the diphoton studies in ref. [14] (from table 2, last row). The expected number of events for exclusive Zγ in the jjγ ( ¯ γ) channel is ∼ 0.25 (∼ 0.04) for an integrated luminosity of 300 fb −1 after selection cuts, which is found to be less than 10% of the total background in the exclusive channel as seen in section 6. For 3000 fb −1 , we expect less than 0.25 background events in ¯ γ, less than 12% of the total background estimated in section 6. Thus, in this study we assume that the QCD and QED exclusive Zγ contribution is small and focus on the non-exclusive background contribution, which constitutes the dominant background in this search.

Event selection
The (jj), γ ( ¯ ), γ final states from the New Physics signal are typically back-to-back and have similar transverse momenta. In the dijet (dilepton) case, this translates to cuts on |∆φ jjγ − π| < 0.02 (|∆φ ¯ γ − π| < 0.02) and p T γ /p T jj > 0.90 (p T γ /p T ¯ ) > 0.95). As can be seen in figures 2 and 4, the signal events appear in the high mass region, allowing for further background rejection by asking m jjγ > 700 GeV (m ¯ γ > 600 GeV). The probability to detect at least one proton in each of the forward detectors is estimated to be 32%, 66% and 93% for 50, 100 and 200 additional interactions respectively. The pile-up background is further suppressed by requiring the proton missing invariant mass m pp to match the Zγ invariant mass within 10% (5%) resolution, m γjj = √ ξ 1 ξ 2 s ± 10% (m γ ¯ = √ ξ 1 ξ 2 s ± 5%) and the γZ system rapidity and the rapidity of the two protons y pp = 1 2 ln(ξ 1 /ξ 2 ) to be the same within |y γjj − y pp | < 0.10 (|y γ ¯ − y pp | < 0.03) units in rapidity for the hadronic (leptonic) channel. This is the key background rejection tool provided by the forward detector information. The number of expected signal and background events passing their respective selections can be seen in table 1, (table 2)      √ ξ 1 ξ 2 s to central mass ratio distribution (top) and rapidity difference distribution (bottom) in the jjγ channel for the signal and background within the acceptance 0.015 < ξ 1,2 < 0.15 considering two different coupling values after applying the requirement on p T , invariant mass m Zγ , p T ratios and angle separation according to table 1. The integrated luminosity is 300 fb −1 and the average pile-up is µ = 50. The signal width is due to a combined effect of the reconstructed jet energy low resolution (≈ 15%) and the ξ 1,2 resolution from the proton detectors. The asymmetry on the m pp /m Zγ distribution is due to the resolution on the jet energy.

Pile-up 200 200
Channel 5σ 95% C.L. 5σ 95% C.L. ¯ γ 1.8 · 10 −13 1.1 · 10 −13 1.25 · 10 −13 7.8 · 10 −14 Table 5. 5σ discovery and 95% C.L. exclusion limits on ζ,ζ. Sensitivities are given for 3000 fb −1 and µ = 200 average pile-up interactions, which corresponds to the High Luminosity LHC. Sensitivities for the jjγ channel are not quoted in this scenario, due to the high number of background events which compromises significantly the signal efficiency. events in ¯ γ channels for an integrated luminosity of 300 fb −1 for this coupling value after selection cuts. In addition, we include a similar study at a higher number of pile-up interactions per bunch crossing µ = 200 and an integrated luminosity of 3000 fb −1 . We consider the ¯ γ final states in this scenario, since we do not obtain much improvement in the hadronic channel in comparison to the 300 fb −1 . We optimized the cuts for this case to increase the background rejection even further, as shown in table 3.
Let us note that a significant fraction of the Z boson hadronic final states are reconstructed as a single jet, since the dijet system is boosted in the high mass regime where the signal is enhanced. The QCD background from qγ and gγ final states is very large, and contributes to O(10 3 ) background events after selection cuts. For this reason, we restricted ourselves to the dijet final states in this study, We stress however that a more evolved jet substructure analysis can in principle efficiently discrimate between large-radius jets from a Z and from QCD [47] but this goes beyond our study.

Expected sensitivities
Expected sensitivities to the ζ,ζ coefficients are shown in tables 4, 5 and in figure 6. The sensitivities are roughly of ∼ 2 · 10 −13 GeV −4 for both 300 fb −1 at low luminosity and 3000 fb −1 at high luminosity. The reach at high luminosity is limited by the large pile-up.  Figure 6. Sensitivity in the ζ −ζ plane for 300 fb −1 and µ = 50. Orange, green and light blue can be probed at 5σ, 3σ, and 95% C.L. using proton tagging at the LHC.
The leptonic final state turns out to be the cleanest channel for this search with nearly background-free events without the use of timing detectors. Interestingly, it turns out that a good sensitivity is also obtained in the hadronic channel at moderate pile-up, with only three background events after selection cuts. The remaining background events that may pass the selection cuts can be further rejected by a factor of ∼ 40 with the timing detectors to be in operation in AFP and CT-PPS, as discussed in section 4.
The B(Z → γγγ) branching ratio has been constrained at LEP [48][49][50]. More recently a stronger bound from ATLAS using 8 TeV data has been quoted in ref. [51], B(Z → γγγ) < 2.2 · 10 −6 . This bound translates as a limit ζ 2 +ζ 2 − ζζ 2 < 1.3 · 10 −9 GeV −4 (95% C.L.) . (7.1) Imagining the same search is done at 13 TeV data with 300 fb −1 in the same conditions, we expect very roughly an improvement by an order of magnitude of the bound of eq. (7.1). In addition, the current number of pile-up interactions at 13 TeV sets a challenge to the measurement of 3γ final states. This remains far away from the expected sensitivities obtained in the exclusive channel at the same luminosity by roughly three orders of magnitudes.
Finally, let us compare the γZ channel sensitivity to exclusive γγ channel sensitivity estimated in refs. [6,14]. We do so for the case of heavy neutral particles described in section 3. Using the γZ, γγ sensitivities at 300 fb −1 with no form factors, which are given respectively in table 4 of this paper and table 3  Depending on the relative strength of the f γγ and f γZ couplings, a similar reach can be obtained in the two channels.
We would like to stress that in order for these bounds to be the most sensitive probe of a new particle, one has to assume that the couplings to gluons are somewhat suppressed, as otherwise gluon fusion processes can provide stronger bounds. For instance, in the case of the KK graviton coupling universally to all SM fields, our 5σ sensitivity in eq. (7.3) translates to m KK < 1.4 TeV κ/0.1, where κ is the universal KK graviton coupling strength. This bound is slightly weaker than standard searches.
However, our method becomes more sensitive at large coupling, both because the resonance becomes too broad for standard searches, and also because we are sensitive to mass regions outside the kinematic reach of the LHC. Also note that, even if our method did not provide the primary discovery channel, Equations (7.2) and (7.3) show that the γZ channel could efficiently determine whether the underlying particle couples universally to (B µν ) 2 , (W I,µν ) 2 (see discussion in section 3). Finally, let us comment on electroweak production modes. Because of gauge invariance, the γγ and γZ couplings also imply nonzero W W and ZZ coupings. Therefore, inelastic photon fusion as well as W W and ZZ fusion are always present. We have checked that of the latter, inelastic photon fusion has the largest cross section. 3 However, all of these production modes have to compete with a much larger background, which becomes particularly problematic when the width of the resonance is large.
The sensitivity to charged particles is fairly weak, unless large d, Y or a large multiplicity are taken. For a vector in the SU(2) L adjoint for example, one has d = 3, Y = 0 and the reach on the mass is found to be m ∼ 120 GeV. For N vectors, the bound would increase as N 1/4 . On the other hand, this bound on charged particles is very generic, as it depends only on the quantum numbers of the underlying particle (see eq. (3.1)). This measurement is thus quite complementary to direct searches for charged particles, which are very model-dependent.

Conclusion
The forward proton detectors recently installed at the LHC provide the opportunity of measuring the anomalous γγγZ coupling with unprecedented sensitivity, providing another high precision probe into the SM gauge sector. We have estimated the discovery potential JHEP06(2017)142 for exclusive γZ production at a center-of-mass energy of 13 TeV, for both low and high luminosity scenarios.
We have computed the anomalous γγ → γZ rate induced by New Physics in the effective field theory framework. Contributions to the γγγZ couplings from tree-level exchange of neutral particles and from loops of particles with arbitrary electroweak charge have been calculated. Prospects for both hadronic and leptonic channels have been evaluated. The tagging of the protons drastically reduces the background in all cases, and allows studying the Z decay in the hadronic channel in addition to the leptonic channel, which is usually very challenging in the standard searches. For 300 fb −1 of integrated luminosity, similar sensitivities are found in both channels, and their combination amounts to reach the anomalous couplings down to ζ ∼ 2 · 10 −13 GeV −4 with 5σ statistical significance. This sensitivity goes beyond the one expected from the Z → γγγ decay searches at the LHC by roughly three orders of magnitude.
The combined sensitivity in the γZ channel turns out to be roughly comparable to the sensitivity expected in the exclusive γγ channel. Combining the information from both channels would provide a powerful way to pin down New Physics scenarios. In particular, a number of New Physics candidates such as the radion or Kaluza Klein gravitons contribute to the γγ final state and not to the γZ one.