Heavy photophilic scalar at the LHC from a varying electromagnetic coupling

We investigate the phenomenology of a heavy scalar $\phi$ of the type involved in Bekenstein's framework for varying electromagnetic coupling theories, with the difference that the scalar in our model has a large mass. The model has only two free parameters, the mass $M_{\phi}$ of the scalar and the scale $\Lambda$ of new physics. The scalar is dominantly produced through photon-photon fusion at the LHC and leads to a diphoton final state. It can also be produced by quark-antiquark fusion in association with a photon or a fermion pair. Its dominating decay is to diphotons, but it also has a large three-body branching to a fermion pair and a photon, which can provide an interesting search channel with a dilepton-photon resonance. We derive exclusion limits on the $M_{\phi}-\Lambda$ plane from the latest 13 TeV LHC diphoton resonance search data. For a benchmark mass of $M_{\phi}\sim 1$ TeV, we find a lower limit on $\Lambda$ of 18 TeV. We discuss the more complex possibility of varying couplings in the full electroweak theory and comment on the possibility that the new physics is related to extra dimensions or string theory.


I. INTRODUCTION
In this paper we perform the first ever study of the collider physics of a model for space-time-varying gauge couplings. The model we study was constructed almost thirtyfive years ago by Jacob Bekenstein [1,2] and introduces a new scalar field associated with a variation in the electromagnetic (EM) coupling constant (α EM ). This model has, to the best of our knowledge, previously only been studied in the context of cosmology with a massless, or very light, scalar, and bounds on the model have been considered based on low-energy physics and astrophysics, see e.g. [3]. Here we propose that the Bekenstein model can be relevant for particle physics experiments, and that the scalar can have a mass on the TeV scale and therefore be accessible at the LHC. Bekenstein's model is the first consistent such model -it is Lorentz, gauge and timereversal invariant and respects causality. The original motivation was to accommodate possible variations of the fine-structure constant over cosmological scales, but despite careful searches such variations have not been detected [3].
In string theory all couplings are associated with scalar fields called moduli, and are thus subject to variations. The excitations of these fields are typically very heavy and as a consequence the couplings will be locked to essentially constant values, given by the specific compactification scenario, see e.g. [4] for a discussion. The Bekenstein model is not derived from string theory, but we consider it the simplest consistent scenario for associating couplings with scalar fields.
Considering variations of α EM , this implies the existence of a new scalar field φ which couples to photons, i.e., it is photophilic. This would provide a discovery potential at the LHC through the decay φ → γγ, which gives a striking signal of high-energy photons pairs with invariant mass M γγ = M φ that may extend to several TeV. The possibility of such a discovery signal was illustrated by the excesses in the diphoton invariant mass distribution around 750 GeV reported [5,6] by both ATLAS and CMS based on the first 13 TeV data with ∼ 3 fb −1 integrated luminosity, even though no excess was observed when the larger dataset of ∼ 16 fb −1 had later been collected [7,8].
Although no signal for such a scalar is presently observed, it is well worth investigating this theory for Beyond Standard Model (BSM) physics which is fundamentally different from other BSM theories considered at the LHC. Therefore, we will here derive an explicit model for such a new scalar field φ and study the general phenomenology at the LHC.
In particular, we use the available data to constrain the model parameters. The model is economic in the sense that it introduces only one new field and two new parameters, i.e., the scalar φ with mass M φ , and the new energy scale Λ. With M φ around the electroweak (EW) or TeV scale, α EM remains constant throughout most of the history of the universe and no deviations would have been detectable so far. 1 The photophilic nature of this new scalar favors its production through photon-photon (γγ) fusion rather than through gluon-gluon (gg) fusion as is more commonly considered for new scalars in BSM theories. Although the scalar in our model can be produced in a pure s-channel process, the basic new vertices also give production modes together with a photon or a dijet or dilepton pair, which imply characteristic predictions of the model to be tested against data. This paper is organized as follows. In Section II we introduce the model and derive the interactions of the new scalar. We also discuss briefly the generalization to the whole EW sector of the SM. In Section III we discuss the phenomenology of the model, and through detailed numerical comparison with LHC limits we extract bounds on the model parameters in Section IV. In Section V we conclude by considering the results in a larger context.

II. THE MODEL
In the Bekenstein model [1,2] the variation of the coupling e is derived from an action that reduces to electromagnetism for constant coupling. It is assumed that the space-time variation of the coupling is given by e = e 0 ǫ(x), where ǫ(x) is a scalar field with dynamics given by the kinetic term 1 2 where Λ is an energy scale. It is assumed that the field ǫ multiplies the electric charge e everywhere in the Lagrangian of the model. Specifically, this means that eA µ is everywhere replaced by e 0 ǫA µ . Gauge invariance, and invariance with respect to a rescaling of 1 The new scalar should also be investigated as a candidate for the inflaton. The quadratic contribution to the potential, with a TeV scale mass cannot be used for inflation, but one might consider higher order terms providing a useful shape.
ǫ, then requires that the field strength tensor be given by To keep our notation as explicit as possible we define F µν = ∂ µ A ν − ∂ ν A µ , and introduce a scalar field ϕ such that ǫ = e ϕ . We will be assuming small fields, working at lowest order, and therefore write ǫ ≃ 1 + ϕ, and keep only terms linear in ϕ. We then find, for the kinetic term of the field, that F µν F µν = F µν F µν + 4 ∂ µ ϕ A ν F µν . Finally, we define a new field φ = ϕΛ so that all fields have their usual mass dimensions. In this way we find a Lagrangian for electromagnetism plus the scalar field given by Because of the definition of e, the new scalar field φ will in addition to the interaction in Eq. (3) couple to all electrically charged fields. The coupling of the EM field to charged fermions is obtained from the covariant derivative, which is given by where Q is the charge of the coupled field, and e = e 0 ǫ so that If we now define D µ = ∂ µ −ie 0 QA µ as the more familiar covariant derivative of electromagnetism, the gauge invariant kinetic term for fermions is given by The final interaction of φ to consider is the coupling to W ± bosons, which is obtained by inserting eA µ = e 0 ǫA µ in the EW gauge kinetic term written in terms of the mass eigenstates. The resulting terms, to lowest order in the electric coupling, are shown below in Eq. (5). There are also five-point vertices of the type φV V W + W − coming from the quartic gauge boson couplings. These couplings contain a factor e 2 0 /Λ 2 and are suppressed with respect to the displayed couplings. For instance, there are four-body decays of φ, such as φ → W + W − γγ where the five-body vertices contribute. The amplitude from the vertices in Fig. 1 is proportional to e 2 0 /Λ, while the amplitude from the φγγW + W − vertex is proportional to e 2 0 /Λ 2 . For this reason we neglect the higher-order terms in this paper.
It is shown in [1], see also [9], that one through a partial use of the equation of motion for F µν can map the theory to a new theory where the scalar only couples through (1/2Λ)φ F µν F µν , without any coupling between the scalar and the fermions. These theories are equivalent, and we here consider the original action due to Bekenstein.
As we have described above, this model has been used as a framework for a spacetime varying α EM with a massless or very light scalar field φ. We will now add one new ingredient: a mass term for the scalar field with M φ around the EW scale or up to several TeV. The energy cost of moving φ away from its minimum will then be very large for energies M φ , so that α EM will have negligible variation.
To get the Lagrangian in its final form, let us now drop the hats on the above expressions, so that from now on, F µν is the standard field strength tensor. Collecting all terms, we then have the Lagrangian of our model where L SM represents the ordinary Standard Model Lagrangian, The vertices are shown in Fig. 1 and the corresponding Feynman rules are where in (6) and (8) the numbered indices refer to the particles in the order listed.
Above, we have only let the EM coupling e vary. This has the consequence that only the photon and the fields carrying electric charge couple to φ. It is possible to instead let the U(1) Y or SU(2) L or both couplings vary, which will couple the φ not only to the photons but to the weak gauge bosons as well. The Bekenstein model of varyingα EM theory has been generalized [10] to vary the SU(2) L and U(1) Y couplings of the electroweak (EW) theory, either with both couplings varying in the same way (one scalar field associated with both variations), or independently of each other. In the latter case, there will be two scalar fields associated with the variations of the two gauge couplings.
Let us therefore here briefly consider the extension of the model to the entire electroweak gauge group. Consider, therefore, two scalars S and S ′ associated with the variation of g and g ′ , where g and g ′ are the SU(2) L and U(1) Y gauge couplings respectively. The interaction Lagrangian (before EWSB) can be expressed as (for simplicity using the equivalent formulation discussed in connection with Eq. (3)) where B µν and W µν are the field-strength tensors and Λ and Λ ′ are the scales for the SU(2) L and U(1) Y gauge groups respectively. In Eq. (9), we now replace where s w and c w are the sine and cosine of the Weinberg angle, respectively. After this replacement, we obtain The fields S and S ′ need not be the mass eigenstates, which in general could be linear combinations of S and S ′ . We therefore define the mass eigenstates φ and φ ′ as the linear where α is the mixing angle determined from the free parameters of the scalar potential (which we will not present here). We may identify φ as the lightest scalar and φ ′ as a heavier mass eigenstate by properly arranging various free parameters in the scalar potential. In the following, we only focus on the lightest scalar, which may reasonably be expected to first show up at the LHC, and assume that signatures of the heavier will be detected later. 3 In this general set-up, both φ and φ ′ will decay to γγ, γZ, ZZ and W W modes. In case Λ ′ ≫ Λ ∼ O(1) TeV, the BR φ → W W becomes suppressed compared to other BRs of φ. Neglecting the small phase-space suppression for the heavy gauge bosons in the final state, the two-body branching ratios (BRs) of φ are in the following proportions, Therefore, for any values of α and Λ, the BRs of φ becomes (assuming Λ ′ ≫ Λ) One can see that the γγ decay of φ dominates and the γZ and ZZ decays are suppressed and therefore not observable with presently available data samples. In this first study, it is therefore well motivated to find the most essential phenomenology of this varying coupling theory by the simplification to only vary the EM coupling and concentrate on final states with photons that provide clean experimental signals. Thus, we leave the more complex study of the full theoretical framework for future work [11].

III. DECAYS AND PRODUCTION AT THE LHC
In this section, we study the phenomenology of our model and derive limits on the two model parameters M φ and Λ from the relevant 8 and 13 TeV data from the LHC. In particular, we use γγ [7,8], γZ [12,13] and ZZ [14] resonance searches from the LHC.  Pythia8 [18] for parton shower and hadronization. Detector simulation is performed for ATLAS and CMS using Delphes3 [19] which uses the FastJet [20] package for jet clustering using the anti-k T algorithm [21] with clustering parameter R = 0.4.
Since φ originates from the variation of the fine-structure constant, it directly couples to photons through an effective dimension-5 operator of the type φF 2 µν . The only possible two-body decay of φ is the diphoton mode (Fig. 2a), which is a tree level decay -not a loop-induced decay through a charged particle as for the SM Higgs. There are subdominant three-body decays of φ mediated through an off-shell photon as shown in Figs. 2b and 2c. The partial decay width into two photons, shown in Fig. 2a, is given by The analytical expressions for the subdominant three-body decay modes are more complicated due to massive particles in the three-body phase space, and we compute partial widths of those modes numerically in MadGraph5. In Table I, we show the partial widths and branching ratios (BR) of φ into its two and three body decay modes for M φ = 1 TeV and Λ = 2 TeV. It is important to note that the BRs of φ only depend on its mass but are independent of the scale Λ. This is because all partial widths and hence the total width scale as Λ −2 and the Λ dependence cancels in the ratios. From Table I, we can see that φ → γγ is the dominant decay mode and this mode has a branching ratio of about 65%, so BRs of φ to other modes are non-negligible. In our analysis we always use the total decay width including all contributions coming from the three body decays.
In Fig. 3, we show a few sample Feynman diagrams of the main production channels of φ at the LHC. Unlike the gg initiated SM-like Higgs boson production, these channels are induced by γγ, γq and qq initial states. The dominant production channel of φ is the γγ fusion where the initial photons come from the photon distribution of the proton. In the γγ fusion process, the dominant contribution comes from the inelastic scattering, where the proton would break up. On the other hand, elastic collisions, where the protons remain intact, are subdominant but provide a much cleaner channel that can be identified with forward detectors. This exclusive production of φ, i.e. pp → φpp, where two forward protons are tagged, is an interesting process to search for (this production mode has been considered in [22][23][24], for models very different from ours).
In Table II, In our analysis, we include elastic, semi-elastic and inelastic contributions (as shown in Fig. 3) in the γγ fusion process. In order to include these three contributions properly without double counting, we use the MLM matching algorithm [25] to match matrix element partons with the parton shower to generate inclusive pp → φ signal events. Our    inclusive signal includes up to two jets and we generate it by combining the following processes, where we set the matching scale Q cut = 125 GeV and the curved connection above two photons signify that they come from the decay of φ. To determine the appropriate Q cut for the process, we check the smooth transition in differential jet rate distributions between There is another channel pp → φ → γℓℓ (where ℓ = {e, µ}) which might also be interesting to search for as the BR of φ to γℓℓ mode is relatively big and also very clean final state. But since the two leptons are coming from an off-shell photon, they are not spatially very separated and therefore a dedicated analysis is required to observe this channel. Similar analyses for the SM Higgs have been done previously for the LHC [26].

IV. EXCLUSION FROM THE LHC DATA
Run-I and run-II LHC data on the diphoton resonance searches set strong upper limits on σ × BR. It should be remembered that all these searches are generally optimized for an s-channel resonance (spin-0 or spin-2) produced through gg fusion and decaying to two photons. For exclusive searches, one demands exactly two selected photons with no selected jet, whereas for inclusive searches, one keeps events with at least two selected photons and any number of selected jets. In order to derive a limit on Λ by recasting the σ × BR upper limit from an experiment, we need to properly take care of the cut efficiencies. The cut efficiencies can significantly change for different selection criteria and also for different production mechanisms. For instance, in our case, the scalar is dominantly produced through γγ fusion and the signal cut efficiency can be different from gg fusion production. This can be properly taken care of by using the following relation: where N s is the number of events for the signal considered for luminosity L and σ s is the corresponding signal cross section. The corresponding signal cut efficiency is denoted by ǫ s . When different types of signal topology and/or final state contribute to any experimental observable, N s can be expressed by the sum ( i σ i × ǫ i ) × L. Here, i runs over all contributing signal processes to any observable.
We roughly employ the selection cuts used by ATLAS [7] and CMS [8] for their 13 TeV analyses as listed below. Although they have not mentioned any jet selection, we include basic jet selection cuts in the following list to demonstrate how our signal cut efficiencies change with different number of selected photons and jets.
• Selection cuts for ATLAS 13 TeV analysis:  Table III, we show cut efficiencies for different selection criteria on the number of photons and jets for the inclusive pp → φ → γγ and pp → φγ → 3γ channels at the 13 TeV LHC.
From Table III, it is evident that the cut efficiencies are highly dependent on the selection criteria. For example, if we demand exactly two selected photons and any number of jets (i.e., 2γ+ ≥ 0j), the cut efficiency is 68% for CMS (φ → γγ). On the other hand, this becomes 28% when we select two photons and at least one jet (i.e., 2γ+ ≥ 1j). This is in contrast to the case of an s-channel production of SM-like scalar through gg fusion decays to two photons, where we should not expect drastically different cut efficiencies for 2γ+ ≥ 0j and 2γ+ ≥ 1j selection categories. This is because due to the presence of gluons in the initial state, radiation of jets will be more compared to photon initiated process and therefore in most of the events one would expect at least one additional jet. In our model, φ is dominantly produced from γγ fusion, but in many BSM theories, a TeV-scale scalar can be dominantly produced from gg fusion, just like the SM Higgs. If it is produced from gg fusion, due to the presence of colored particle in the initial state, the jet activities is expected to be different from that in γγ fusion production. Therefore, some jet observables like jet multiplicity, pseudorapidity, or transverse momentum distributions of jets can act as good discriminating variables to distinguish different production mechanisms. In Fig. 4a, we show the jet multiplicity distribution for φ produced from γγ fusion for the 2γ+ ≥ 0j selection category. In Figs. 4b and 4c, we show the η and p T distributions of the hardest jet for the 2γ+ ≥ 1j category. The issue to distinguish different production mechanisms by analyzing various kinematic distributions are discussed in Refs. [27][28][29][30][31][32][33][34] in connection to the 750 GeV diphoton excess. Since the excess is no longer there, we do not pursue this issue here.
Now we turn to the 8 TeV LHC data related to the diphoton resonance searches [35][36][37]. All these experiments search for either a spin-0 or spin-2 object produced through gg fusion decaying to two photons. We collect the observed upper limit on the cross sections and the corresponding efficiencies for resonance mass around 1 TeV for each experiment.
We estimate the cut efficiencies for the process pp → φ → γγ for these experiments by employing selection cuts in the detector simulator Delphes3. From this information, one can estimate the lower limit on Λ from the formula given in Eq. (17). The overall result from ATLAS and CMS set an upper limit of the cross section roughly 1 fb for 1 TeV resonance mass. Using this we can extract a lower limit on Λ for our model.
This depends, however, on how the data is characterized in terms of number of photons and jets, since that affects the relevant cut efficiencies. Without reporting all numerical details here, we conclude that for the interpretation of the data as 2γ+ ≥ 0j, we obtain a lower limit of Λ in the range 8.5 − 9 TeV for M φ = 1 TeV. Choosing instead 2γ+ ≥ 1j would, however, give a lower limit Λ in the range 4.5 − 5 TeV. There is another relevant experiment at the 8 TeV LHC by ATLAS with L = 20.3 fb −1 which is important to mention in this context. In [38], a triphoton resonance is searched for by ATLAS. This analysis is limited up to a resonance mass of 500 GeV. In our model, there is a possibility of a three photon final state originating from pp → φγ → 3γ (although not a triphoton resonance), and therefore, triphoton resonance searches can also be used to set limits on our model parameters in the future, if the analysis extends the resonance mass range beyond 1 TeV.
The latest 13 TeV LHC data on the diphoton resonance searches also set strong upper limit on σ × BR ∼ 1 fb for a diphoton resonance mass of around 1 TeV.  The shaded regions are excluded from the observed data.
Λ ∼ 18 TeV for M φ = 1 TeV. On the other hand we get slightly smaller Λ ∼ 12 TeV for the selection category 2γ+ ≥ 1j. In Fig. 5, we show the derived limits on Λ as functions of M φ using latest ATLAS and CMS diphoton resonance search data at 13 TeV.
Finally we wish to briefly discuss the general varying EW theory, which we introduced in Section II, in light of the LHC data. In Table I, we see that the BR for φ → γγ is about 65% for M φ = 1 TeV when only the variation of the EM coupling is considered. For the varying EW theory, this BR will reduce due to the appearance of other two-body decay modes. For instance, for the case Λ ′ ≫ Λ, the BR for φ → γγ becomes 39% (this comes from 0.65 × 0.6 ≈ 0.39 where 0.6 is the γγ BR as shown in Eq. (13)). One should note that this BR reduction does not lower Λ drastically, since the production cross section scales as Λ −2 . Therefore, the derived Λ, in this case, will reduce by a factor √ 0.6 ≈ 0.77.
For example, Λ 18 TeV for M φ = 1 TeV as shown in Fig. 5 will become 14 TeV.
It is also necessary to check how the latest 13 TeV LHC data on the γγ, γZ and ZZ resonance searches place bounds on Λ for the varying EW theory. The 13 TeV data set rough upper limits on the cross sections for γγ, γZ and ZZ resonance searches around mass 1 TeV as 1 fb [7,8], 10 fb [12,13] and 20 fb [14], respectively. Since the limits on the cross sections for γZ and ZZ resonance searches are less strong than the γγ limit, they cannot put stronger bounds than that derived from γγ data as the γγ BR is the largest for the choice Λ ′ ≫ Λ in Eq. (13).

V. CONCLUSIONS
In this paper, we present the first ever study of the phenomenology of a heavy scalar associated with the variation of the fine-structure constant. This variation introduces a new scalar field in the theory as originally proposed by Jacob Bekenstein. We introduce a TeV-scale mass of the scalar which can therefore be accessible at the LHC. This model predicts that the scalar dominantly couples to photons. Therefore, the dominant production channel is γγ fusion and it dominantly decays to a photon pair. The scalar can also be produced together with an additional real or virtual photon, which, if virtual, gives rise to a pair of jets or leptons. This gives another prediction: the existence of an additional photon or jets in the events, which are not part of the resonance. These predictions can be searched for in the future at the LHC.
The model we study here is very economical and has only two new parameters, the mass M φ of the scalar and the energy scale Λ. We use latest 13 TeV LHC diphoton resonance search data to derive exclusion regions on the M φ − Λ plane. In particular, for the mass M φ ∼ 1 TeV, we obtain the lower limit Λ 18 TeV. We also discuss how different selection criteria can affect the exclusion limits and derive limits from relevant LHC data for different selection categories. In this first paper, we primarily consider varying only α EM in the SM and the resulting phenomenology with photons as clean experimental signals at the LHC. The variations of gauge couplings in the full electroweak theory leads to more complex possibilities, which we will study in a forthcoming paper. Already here, however, we briefly discuss this and derive limits for a specific benchmark point.
In a broader perspective, one should note that the interaction terms for the new scalar are non-renormalizable such that the theory needs to be UV-completed with new physics at the energy scale Λ. The scalar in our model could then be interpreted as a moduli field parametrizing a varying electromagnetic coupling. Natural candidates for such new physics are extra dimensions or string theory, with effects that may be within reach of the LHC.