# New Constraint for Isotropic Lorentz Violation from LHC Data

David Amram, Killian Bouzoud, Nicolas Chanon, Hubert Hansen, Marcos R Ribeiro, Marco Schreck

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David Amram, Killian Bouzoud, Nicolas Chanon, Hubert Hansen, Marcos R Ribeiro, et al. New Constraint for Isotropic Lorentz Violation from LHC Data. Physical Review Letters, 2024, 132 (21), pp.211801. 10.1103/PhysRevLett.132.211801. hal-04381883

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Distributed under a Creative Commons Attribution 4.0 International License
New Constraint for Isotropic Lorentz Violation from LHC Data

David Amram,¹,*, Killian Bouzoud,²,† Nicolas Chanon,¹,‡ Hubert Hansen,¹,§

Marcos R. Ribeiro, Jr. ³,|| and Marco Schreck ⁴,¶

¹Université de Lyon, Université Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, Villeurbanne, France

²SUBATECH, Nantes Université, IMT Atlantique, IN2P3/CNRS, 4 rue Alfred Kastler, La Chantrerie BP 20722, 44307 Nantes, France

³Instituto de Física, Universidade de São Paulo, Cidade Universitária, São Paulo (SP), 05508-090, Brazil

⁴Departamento de Física, Universidade Federal do Maranhão, Campus Universitário do Bacanga, São Luís (MA), 65085-580, Brazil

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(Received 22 December 2023; accepted 10 April 2024; published 20 May 2024)

New calculations for the kinematics of photon decay to fermions in vacuo under an isotropic violation of Lorentz invariance (LV), parametrized by the standard-model extension, are presented in this Letter and used to interpret prompt photon production in LHC data. The measurement of inclusive prompt photon production at the LHC Run 2, with photons observed up to a transverse energy of 2.5 TeV, provides the lower bound $\bar{k}_{tr} > -1.06 \times 10^{-13}$ on the isotropic coefficient $\bar{k}_{tr}$ at 95% confidence level. This result improves over the previous bound from hadron colliders by a factor of 55. The calculations for the kinematics of photon decay have further potential use to constrain LV coefficients from the appearance of fermion pairs.

DOI: 10.1103/PhysRevLett.132.211801

*Introduction.*—Considering the technology and exper-
imental capabilities of the early 21st century, it is impos-
sible to probe quantum gravity directly, since experiments
would require particles whose energies lie in the vicinity of
the Planck scale. Such energies are indispensable to resolve
structures of the smallest length scales imaginable such as
strings or a spacetime foam [1,2]. Fortunately, there are
alternatives to observing quantum gravity phenomena in a
direct way. To do so, we resort to the two most successful
theories: the standard model (SM) of elementary particles
and general relativity (GR). The first is based on global
Lorentz invariance, whereby GR is governed by diffeo-
morphism symmetry as well as local Lorentz invariance. It
is these symmetries that play a crucial role in the quest for
quantum gravity effects.

The SM and GR describe particles and gravity astound-
ingly well for energies much lower than the Planck scale. A
valuable approach to quantum gravity phenomenology
consists in searching for tiny deviations from these estab-
lished theories. In the framework of effective field theory,
there are several possibilities of how to proceed. First, the
SM is extended by excitations of new degrees of freedom
described by non-SM fields. Second, contributions are
introduced that preserve the symmetries of the SM and
GR, but involve higher-order derivatives. Third, an

alternative is to consider violations of at least one of the
fundamental spacetime symmetries. Our interest lies on the
latter approach.

The gravitational field can be neglected for our purpose,
which means that we will focus on violations of global
Lorentz invariance. Such effects are motivated by a set of
articles indicating a breakdown of Lorentz symmetry in
string field theory [3–7], loop quantum gravity [8,9],
noncommutative field theory [10,11], spacetime foam
models [12–14], chiral field theories defined in spacetimes
with nontrivial topologies [15–17], and Hořava-Lifshitz
gravity [18]. A comprehensive technique established for a
broad program of experimental searches for Lorentz
violation (LV) is to use a model-independent effective
parametrization of deviations from Lorentz symmetry,
which the standard-model extension (SME) [19,20] pro-
vides. The latter respects the gauge symmetry and particle
content of the SM. Lorentz violation is parametrized by
tensor-valued background fields whose components, which
are denoted as controlling coefficients, describe the amount
of symmetry violation. These background fields are
coupled to the SM fields such that the resulting terms
are invariant with respect to coordinate boosts and rota-
tions. Within effective field theory, CPT violation was
shown to imply a violation of Lorentz symmetry [21],
which is why CPT-violating operators are also contained in
the SME.

In the SME, particles obey modified dispersion relations
and processes are described perturbatively by a set of
modified Feynman rules [22–24]. The modified kinematics
may enable unusual processes that are forbidden in the SM
due to energy-momentum conservation. For example,

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under certain conditions, a fermion subject to LV can lose energy by the emission of photons. This process is reminiscent of Cherenkov radiation in optical media with the crucial difference that it occurs in vacuo. Therefore, the latter is called vacuum Cherenkov radiation [25–49]. Another prominent process is photon decay into a fermion–antifermion pair [35,36,39,50–69], which the focus is on in this Letter.

Photon decay can occur in the CPT-even photon sector of the SME that is governed by 19 independent controlling coefficients [20,70,71]. Ten of these imply vacuum birefringence and are strongly constrained by spectropolarimetry measurements [70,72–77]. The remaining 9 coefficients are nonbirefringent at first order in LV. A subset of 8 leads to anisotropic photon propagation and a single one describes LV effects that are spatially isotropic. Searches for anisotropic effects via small-scale laboratory experiments have provided a large number of tight bounds on such coefficients [78–81], whereas astroparticle physics has led to competitive constraints on the isotropic coefficient [36,39,49,68] (cf. Table D16 in Ref. [82]).

When it comes to particle energy and propagation distance, astroparticle physics experiments have an advantage over collider experiments in constraining LV. The highest energies of astrophysical photons detected lie in the ballpark of few hundred TeV [66] to 2.5 PeV, as observed by LHAASO [83,84]. However, the drawback of astroparticle experiments is that the interpretation relies on models, e.g., for the distribution of sources and the injection of nuclei through the source environment [85]. These are necessary to describe the energy spectrum, composition, and arrival direction of astroparticles [65]. In addition, information related to the intrinsic spectrum of the source, its evolution, and emission mechanism is not necessarily available [67,86,87]. By contrast, Earth-based laboratory experiments provide a reproducible, controlled source of prompt photons, well understood in term of the SM predictions, leading to limits on LV that can be considered as less model dependent and, hence, more conservative. Such collider bounds are complementary to astrophysical constraints. In this work we will demonstrate how prompt photons measured at the LHC are capable of improving the constraint on isotropic CPT-even LV by a factor larger than 50, in comparison with the present limit derived at the Tevatron.

### Minimal-SME photon sector.
We consider a modified quantum electrodynamics with LV in the photon sector, but standard Dirac fermions:

$$S = \int d^4 x \left[ -\frac{1}{4} (\eta^{\mu\rho} \eta^{\nu\sigma} + \kappa^{\mu\nu\rho\sigma}) F_{\mu\nu}(x) F_{\rho\sigma}(x) + \frac{1}{2} \bar{\psi}(x) (\gamma^\mu i D_\mu - m_f) \psi(x) + \text{H.c.} \right], \quad (1)$$

with the electromagnetic field strength tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ of the U(1) gauge field $A_\mu$. Furthermore, $\psi$

is a Dirac spinor field and $\bar{\psi} \equiv \psi^\dagger \gamma^0$ the corresponding Dirac conjugated field. All fields are defined in Minkowski spacetime with metric $\eta_{\mu\nu}$ of signature $(+, -, -, -)$. We employ the standard Dirac matrices satisfying the Clifford algebra $\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}$ and $m_f$ is the fermion mass. The Dirac field is minimally coupled to the gauge field via the covariant derivative $D_\mu = \partial_\mu + iqA_\mu$ with the fermion charge $q$. The violation of Lorentz invariance is parametrized via $\kappa^{\mu\nu\rho\sigma}$, which transforms as a four-tensor of rank 4 under coordinate boosts and rotations, but remains fixed with respect to boosts and rotations of experiments proper. The object $\kappa^{\mu\nu\rho\sigma}$ implies preferred spacetime directions leading to a boost- and/or direction-dependent form of the laws of physics.

The physics of Eq. (1) is described by a nontrivial refractive index of the vacuum that leads to a speed of light different from the maximum velocity of Dirac particles. The refractive index may be anisotropic or even polarization dependent resulting in vacuum birefringence. Since the latter effect is tightly constrained at the level of $10^{-34}$ to $10^{-35}$ by spectropolarimetry measurements [82], we discard the birefringent part of $\kappa^{\mu\nu\rho\sigma}$ involving 10 coefficients. The remaining 9 coefficients are parametrized by the nonbirefringent ansatz [32]:

$$\kappa^{\mu\nu\rho\sigma} = \frac{1}{2} (\eta^{\mu\rho} \tilde{\kappa}^{\nu\sigma} - \eta^{\nu\sigma} \tilde{\kappa}^{\mu\rho} + \eta^{\nu\rho} \tilde{\kappa}^{\mu\sigma} - \eta^{\mu\sigma} \tilde{\kappa}^{\nu\rho}), \quad (2)$$

where $\tilde{\kappa}^{\mu\nu}$ is a symmetric and traceless $(4 \times 4)$ matrix. We study isotropic LV in the laboratory frame, which is the simplest choice for a first analysis before investigating more intricate cases. There is a single coefficient usually denoted as $\tilde{\kappa}_{\text{tr}}$ that parametrizes isotropic LV in the theory of Eq. (1). The coefficients of the matrix $\tilde{\kappa}^{\mu\nu}$ are then chosen as

$$\tilde{\kappa}^{\mu\nu} = \frac{3}{2} \tilde{\kappa}_{\text{tr}} \text{diag} \left( 1, \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)^{\mu\nu}. \quad (3)$$

Thus, the dispersion relation for photons is modified and reads

$$\omega = A|\mathbf{k}|, \qquad A = \sqrt{\frac{1 - \tilde{\kappa}_{\text{tr}}}{1 + \tilde{\kappa}_{\text{tr}}}}, \quad (4)$$

where $\mathbf{k}$ is the three-momentum of the photon. The isotropy of the result is evident. Since the phase and group velocities of photons obey

$$v_{\text{ph}} \equiv \frac{\omega}{|\mathbf{k}|} = A, \qquad v_{\text{gr}} \equiv |\nabla_{\mathbf{k}}\omega| = A, \quad (5)$$

there is no vacuum dispersion. For $\tilde{\kappa}_{\text{tr}} \in (0, 1]$ we have that $v_{\text{gr}} < 1$. The dispersion relation of Dirac fermions is the standard one for massive particles. Hence, the latter can
possibly propagate faster than light. For $\tilde{\kappa}_{tr} \in (-1, 0)$ we observe that $v_{gr} > 1$ and photons always move faster than fermions, no matter what their energy.

In the first kinematic regime, vacuum Cherenkov radiation can occur, whereas photon decay in vacuo is a characteristic process for the second regime. Both processes are governed by thresholds, i.e., they occur if the energy of the initial particle exceeds a certain minimum that depends on the fermion mass and the controlling coefficient. Since vacuum Cherenkov radiation with LV in photons and fermions, respectively, has already been studied exhaustively [25–33,35–48], we intend to investigate photon decay. Thus, in what follows, we will take $\tilde{\kappa}_{tr} \in (-1, 0)$.

**Kinematics and dynamics of photon decay.**—Now, let us look at the basics of photon decay $\gamma \to \bar{f}f$ within the isotropic sector of Eq. (1). The threshold energy $E^{th}$ of the incoming photon is best computed by considering the three-momenta of all particles as collinear. Energy-momentum conservation then implies [36]

$$E^{th} = 2m_f \sqrt{\frac{1 - \tilde{\kappa}_{tr}}{-2\tilde{\kappa}_{tr}}}. \qquad (6)$$

Several observations are in order. First, for a massless fermion, the process occurs without a threshold. Second, the expression is real only for $\tilde{\kappa}_{tr} < 0$, whereupon photon decay is forbidden for $\tilde{\kappa}_{tr} > 0$. Third, the threshold goes to infinity for $\tilde{\kappa}_{tr} \mapsto 0$, since the process is energetically not allowed when Lorentz symmetry is intact. Fourth, since the threshold is proportional to the particle mass, further decay channels into more massive fermion-antifermion pairs open up for rising photon energy.

The total decay width $\Gamma$ of photon decay for the isotropic sector was already obtained in Ref. [36]. Instead, what we will need in the forthcoming analysis are different quantities, which have not been reported in the literature, so far. If the incoming-photon energy exceeds the threshold, the surplus in energy is used to provide a nonzero angle $\theta$ between the three-momenta of the final-state particles. The latter reads

$$\cos \theta = \frac{E_f(E_\gamma - E_f) + \frac{\tilde{\kappa}_{tr}}{1-\tilde{\kappa}_{tr}} E_\gamma^2 + m_f^2}{\sqrt{[E_f^2 - m_f^2][(E_\gamma - E_f)^2 - m_f^2]}}, \quad (7)$$

where $E_\gamma$ and $E_f$ are the energies of the incoming photon and outgoing fermion, respectively. The energy of the outgoing antifermion is determined by energy-momentum conservation. Furthermore, the partial decay width with respect to the energy of the final fermion can be cast into the form

$$\frac{d\Gamma}{dE_f} = \frac{\alpha}{(1 + \tilde{\kappa}_{tr})^2 \sqrt{1 - \tilde{\kappa}_{tr}^2 E_\gamma^2}} \times \{(1 - \tilde{\kappa}_{tr})[2\tilde{\kappa}_{tr}E_f(E_\gamma - E_f) + (1 + \tilde{\kappa}_{tr})m_f^2] - \tilde{\kappa}_{tr}E_\gamma^2\}, \quad (8)$$

with the fine-structure constant $\alpha = e^2/(4\pi)$. Equations (7) and (8) are new results. Evaluating the final-particle phase space, the fermion energy lies within the following interval of a minimum and maximum energy:

$$E_f \in [E_{\min}, E_{\max}], \qquad (9a)$$

$$E_{\min} = \frac{1}{2}(E_{\gamma} - \bar{E}), \qquad E_{\max} = \frac{1}{2}(E_{\gamma} + \bar{E}), \qquad (9b)$$

$$\bar{E} \equiv \sqrt{\frac{1 + \tilde{\kappa}_{tr}}{1 - \tilde{\kappa}_{tr}} \left[ E_{\gamma}^{2} + 2 \left( \frac{1}{\tilde{\kappa}_{tr}} - 1 \right) m_{f}^{2} \right]}. \qquad (9c)$$

The angle of Eq. (7) is well defined only when $E_f$ lies in the interval set by Eq. (9).

**Search for photon decay to electrons in vacuo at the LHC.**—The proton-proton collisions at the LHC are providing a large sample of high-$E_T^\gamma$ prompt photons (where $E_T^\gamma$ is their transverse energy). In this section, we reinterpret in the context of a possible LV the distribution of the differential cross section $d\sigma/dE_T^\gamma$ of inclusive prompt photons, which was measured with the ATLAS detector at a center-of-mass energy of $\sqrt{s} = 13$ TeV [88]. This analysis, performed with an integrated luminosity of 36.1 fb$^{-1}$ at the LHC run 2, reports photons with the largest $E_T^\gamma$ observed at the LHC, binning up to 2.5 TeV for photon pseudorapidity $|\eta^\gamma| < 0.6$. The latter provides an improvement as compared to a previous result of the CMS experiment, with photons of up to 2 TeV [89].

Here, we make the hypothesis that photons can decay into $e^+e^-$. In the following, models are built for $E_T^\gamma$ distributions, based on the hypothesis of LV signals and SM backgrounds. Events of the process $pp \to \gamma + \text{jet}$ at tree-level with up to 3 additional partons at leading order in perturbative QCD and matched with parton showers are simulated with the SHERPA generator v.2.2.15 [90,91]. The events are further reweighted using tables from HEPData [92] such that no difference remains in the $E_T^\gamma$ distribution between the SHERPA sample used in this article and the SHERPA sample employed in the ATLAS paper [88]. The events are selected using RIVET [93], a framework providing publicly available codes for the analysis of parton-shower level Monte Carlo (MC) events. We require $E_T^\gamma > 125$ GeV within the acceptance region $|\eta^\gamma| < 0.6$, with criteria on photon isolation taken from the RIVET routine corresponding to those of a similar analysis performed at 8 TeV [94]. Note that the same criteria are employed in the 13 TeV analysis [88].

For each event passing the selection, the photon with the largest $E_T^\gamma$ is selected. Under the hypothesis of a nonzero $\tilde{\kappa}_{tr}$, the probability for photons to reach the ATLAS detector is computed as $e^{-\Gamma x}$, using the total photon decay width $\Gamma$, which follows from integrating Eq. (8). Moreover, we assume a distance of 33 mm between the interaction point
and the closest layer of the ATLAS pixel detector [95] at LHC run 2. As already noted [35], the photon decay process is very efficient, since only 0.038% of the photons do not decay over this distance, above the threshold $E^{th} = 2$ TeV. If photons decay, distributions based on Eqs. (7) and (9) are used to generate the fermion and antifermion four-momenta, employing uniform distributions in azimuths.

Since the electrons arising from photon decay in vacuo can be reconstructed within the detector, it must be verified whether or not they would still be identified as photons. The $e^{+}e^{-}$ system can be reconstructed as one-leg or two-leg converted photons, or as electrons. Reference [88] reports an uncertainty of approximately 1.5% on photon identification, which we take as an estimate of the probability for an electron to be reconstructed as a photon. If one electron is reconstructed as a photon, the value of its transverse energy, drawn according to the probability density $d\Gamma/dE_{f}$, is included in the distribution of $E_{T}^{\gamma}$. If two electrons are reconstructed as a photon, the largest transverse energy is included in the distribution of $E_{T}^{\gamma}$. Among the photons within a $E_{T}^{\gamma}$ bin, a fraction of about 3% would be recovered from the decay into $e^{+}e^{-}$ as reconstructed photons with a lower $E_{T}^{\gamma}$ value. The procedure described above provides models for the $E_{T}^{\gamma}$ distribution of SM prompt photon production (treated as a background), and of a SM contribution with LV (treated as a signal) under any hypothesis for the value of $\tilde{\kappa}_{tr}$. These can be compared to the spectrum of $E_{T}^{\gamma}$ measured with the ATLAS detector. Predictions are shown in Fig. 1 for several values of the threshold energy.

<figure><img src="image_3.png" /><figcaption>FIG. 1. Differential cross section for inclusive prompt photon production measured with the ATLAS detector [88], compared with the predictions from the SHERPA generator assuming several energy thresholds for photon decay in vacuo.</figcaption></figure>

The statistical treatment employed to extract bounds on $\tilde{\kappa}_{tr}$ uses the modified frequentist $CL_{s}$ method [96], based on the likelihood ratio of the SM + LV hypothesis against the SM-only hypothesis, assuming the number of events are Poisson distributed in each bin. The central value for the number of events $N_{i}$ is computed from the differential cross sections $d\sigma_{i}/dE_{T}^{\gamma}$ in each bin $i$ as

$$N_{i} = \frac{d\sigma_{i}}{dE_{T}^{\gamma}} \cdot \Delta E_{T,i}^{\gamma} \cdot \epsilon \cdot L, \qquad (10)$$

where $\Delta E_{T,i}^{\gamma}$ is the bin width, $\epsilon$ the efficiency of photon reconstruction and identification, and $L$ is the integrated luminosity. The systematic uncertainties in the ATLAS measurement, which impact the predictions for the models, are from multiple sources: background subtraction, unfolding of the detector effect, pileup, trigger and photon selection efficiencies, photon energy scale, and resolution. Other uncertainties considered are the MC statistical uncertainty in the SHERPA sample, as well as the theory uncertainties quoted for the ATLAS sample, arising from QCD renormalization and factorization scale, the parton distribution functions, strong coupling constant $\alpha_{s}$, and the parton shower. The values of the systematic uncertainties are added in quadrature in each bin, and treated conservatively as a separate Gaussian nuisance parameter for each bin of the $E_{T}^{\gamma}$ distribution. The total systematic uncertainty ranges from 2.3% at low values of $E_{T}^{\gamma}$ to 7.6% at high values.

The achieved bound on $\tilde{\kappa}_{tr}$ as a function of the $CL_{s}$ criterion is shown in Fig. 2. The conventional criterion of $CL_{s} < 0.05$ is employed, leading to the following limit at 95% confidence level on the allowed region:

$$\tilde{\kappa}_{tr} > -1.06 \times 10^{-13}. \qquad (11)$$

This corresponds to a threshold energy of $E^{th} = 2.218$ TeV. The bound quoted above represents a large improvement, of a factor of $\approx 55$, over the previous bound $\tilde{\kappa}_{tr} > -5.8 \times 10^{-12}$ [35]. The latter was set by reinterpreting D0 prompt photon data [97] where the last $E_{T}^{\gamma}$ bin was measured up to 340 GeV. The order of magnitude of the improvement can be understood, since the LV/SM separation is the largest in the last bin, reporting photons with the largest energy, and recalling that $\tilde{\kappa}_{tr} \propto (m_{f}/E^{th})^{2}$ according to Eq. (6). As a matter of fact, assuming conservatively that the largest energy observed is set by the left-hand side boundary of the last measured bin, i.e., 2 TeV, and that it corresponds to $\eta^{\gamma} = 0$, fixes the minimum threshold to $E^{th} \approx 2$ TeV.

By assuming that all of the higher-energy photons decay immediately, this in turn provides a qualitative bound $\tilde{\kappa}_{tr} > -1.3 \times 10^{-13}$. It should be noted that this method leads to a qualitative estimate only, since it does not include systematic uncertainties, and neither considers the probability of photons to survive up to the detector nor the
<figure><img src="image_4.png" /><figcaption>FIG. 2. Bound on $\tilde{\kappa}_{\text{tr}}$ as a function of the $CL_s$ value. A bound with a 95% confidence level is defined by the region $CL_s < 0.05$.</figcaption></figure>

probability of the $e^+e^-$ system to be reconstructed as a photon. The improvement of Eq. (11) by nearly 20% over this simple estimate arises from the statistical method employed, which makes full use of the statistical power of the last $E_T^\gamma$ bin. It is instructive to check the result without any systematic uncertainties, in which case the bound achieved would be $\tilde{\kappa}_{\text{tr}} > -1.045 \times 10^{-13}$. The result in Eq. (11) is dominated by the statistical uncertainties in the data.

An upper bound $\tilde{\kappa}_{\text{tr}} < 1.2 \times 10^{-11}$ was set from the LEP beam energy stability being interpreted as showing no evidence for vacuum Cherenkov radiation from electrons [35]. Such a limit cannot be improved by referring to the LHC beam stability, since the LHC collides protons. The SuperKEKB facility [98] is nowadays colliding $e^+$ and $e^-$, but does so at a lower center-of-mass energy than at LEP, which still provides the best results. Future particle colliders like the FCC-ee would collide electrons of 183 GeV [99] providing upper bounds improved by a factor of 3. The FCC-hh [100] is planned to collide 100 TeV protons; assuming that prompt photons of $\approx 20$ TeV would be produced, the lower bound would improve by 2 orders of magnitude over the results presented previously.

**Conclusions.**—This Letter provides a new evaluation of the kinematics of photon decay into fermions in vacuo under an isotropic violation of Lorentz invariance. These results were used to reinterpret a measurement of inclusive prompt photon production at the LHC run 2, with photons observed up to a transverse energy of 2.5 TeV. A new bound on the isotropic SME coefficient $\tilde{\kappa}_{\text{tr}}$ was set from LHC data, arising from the absence of an observation of photon decay to electrons in vacuo. The limit yields $\tilde{\kappa}_{\text{tr}} > -1.06 \times 10^{-13}$ at 95% confidence level, a result that is approximately 55 times more stringent than a previous bound extracted from Tevatron data. The calculations for the kinematics of photon decay could be used in the future to bound LV coefficients from the appearance of fermion pairs.

M. S. greatly acknowledges support via the grants FAPEMA Universal 00830/19 and CNPq Produtividade 310076/2021-8. Furthermore, M. S. is indebted to CAPES/ Finance Code 001.

*Appendix: Photon decay to top quarks.*—As an additional illustration of Eqs. (6)–(9), we focus on top quarks, assuming a mass $m_t = 172.5$ GeV. We highlight the difference between the kinematics of photon decay to top-quark pairs relative to that of SM pp $\to t\bar{t}$ processes. In this setting, the parameter $\tilde{\kappa}_{\text{tr}}$ must be replaced by a combination of photon and top-quark coefficients [35], $\tilde{\kappa}_{\text{tr}} - (4/3)c_t^{TT}$. Besides, as $c_t^{TT}$ is an isotropic coefficient in the quark sector where constraints are weak compared to those for $\tilde{\kappa}_{\text{tr}}$ [82], we can neglect $\tilde{\kappa}_{\text{tr}}$ in the following. The coefficient $c_t^{TT}$ induces a shift in the cross section for the process pp $\to t\bar{t}$, which would likely be attributed to a QCD effect [101,102].

However, top pairs produced from photon decay would entail a different kinematics, which is illustrated in Fig. 3 via the opening angle between top quarks for several threshold energies. This finding could be employed to set bounds on $c_t^{TT}$. It was checked that the azimuthal difference between top quarks shows a distribution of events concentrated at $\Delta\phi \approx 0$ for photons above an energy threshold close to 2.5 TeV. Therefore, this leads to $\Delta\phi(l^+, l^-) \approx 0$ in leptonic top-quark decay, as well, which is very different from the predictions in the SM [103,104]. This region is poorly understood from a theoretical point of view, though, with SM predictions of a toponium bound state [105] yet to be observed. Furthermore, it is not clear if top quarks arising from photon decay at this energy would still decay preferentially into a W boson and a b quark, as predicted in the SM [106], and a dedicated reconstruction algorithm might be needed. Using this difference of kinematics to constrain LV remains a challenging path in the present state of the theory, but represents an interesting possibility for the future.

<figure><img src="image_5.png" /><figcaption>FIG. 3. Opening angles between the final top quarks for a photon energy of 2510 GeV, at different threshold energies.</figcaption></figure>
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<footnote>*d.amram@ip2i.in2p3.fr</footnote>

<footnote>†kbouzoud@subatech.in2p3.fr</footnote>

<footnote>‡nicolas.pierre.chanon@cern.ch</footnote>

<footnote>§hansen@ipnl.in2p3.fr</footnote>

<footnote>||marcosribeiro@usp.br</footnote>

<footnote>¶marco.schreck@ufma.br</footnote>
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