New signals for vector-like down-type quark in $U(1)$ of $E_6$

We consider the pair production of vector-like down-type quarks in an $E_6$ motivated model, where each of the produced down-type vector-like quark decays into an ordinary Standard Model light quark and a singlet scalar. Both the vector-like quark and singlet scalar appear naturally in the $E_6$ model with masses at the TeV scale with a favorable choice of symmetry breaking pattern. We focus on the non-standard decay of the vector-like quark and the new scalar which decays to two photons or two gluons. We analyze the signal for the vector-like quark production in the $2\gamma+\geq2j$ channel and show how the scalar and vector-like quark masses can be determined at the Large Hadron Collider.

have not survived further scrutiny by the LHC experiment, the diphoton excess at LHC [3][4][5] brought back the attention to heavy vector-like quarks and extended scalar sectors amongst many other models. The SM is widely believed to be an incomplete theory due to the lack of explanation to several outstanding issues (e.g. neutrino masses, dark matter candidate, etc.). The Grand Unified Theories (GUTs) are known to present novel ideas in addressing the above issues in the SM while also proposing to unify the three SM gauge couplings to one at a high scale. Most of the GUT models have testable consequences at the TeV scale which are in the form of an extra gauge group such as an extra U (1) and some additional new particles with heavy masses. We look at such an example in the E 6 GUT model [6] where one gets down-type vector-like (VL) fermions charged under an extra U (1) gauge symmetry. In this work we focus on an interesting signal of the down-type vector-like quark (VLQ) at LHC.
Note that vector-like fermions exist in many BSM scenarios and a lot of phenomenological studies on the down-type VLQs exist in the literatures [7]. The current experimental bounds on the mass of down-type VLQ are obtained under certain assumption of its decay modes [8][9][10][11][12][13][14]. For a down-type VLQ the searches are based on the assumption that it decays to one of the SM final states Zb, W t and bh. The current experimental lower bound on the mass of the down-type vector-like quark which mixes only with the third generation quark is around 730 GeV from Run 2 of the LHC [8] and is around 900 GeV from Run 1 of the LHC [11]. Similarly, the current lower bound for a vector-like quark which mixes with the light quarks is around 760 GeV from Run 1 of the LHC [14]. While strong limits can be derived from these conventional search channels, the bounds get relaxed once new nonstandard decay modes are present and start dominating over the SM channels. In this work we discuss a non-standard decay channel of the VLQ and about its possible signatures in a non supersymmetric version of E 6 model. A recent work discussing detailed phenomenology of vector-like quarks in E 6 model can be found in Ref. [15]. In our case, we look at the VLQs and singlet scalars which are particles already present in the E 6 GUT, as discussed later. Using appropriate symmetry breaking pattern, one U (1) in addition to the SM gauge symmetry remain unbroken at the TeV scale or even higher. The heavy down-type quark xd, which is a color triplet and an SU (2) singlet with an electric charge of −1/3, is pair produced dominantly from two gluons via strong interactions at the LHC. Also, three such xd and xd quarks naturally appear in our model based on E 6 from three fermion families. A singlet scalar is also naturally present which is responsible in breaking the additional U (1) at the TeV scale. The pattern of symmetry breaking that we shall use gives the singlet scalar mass which is close to the xd-quark mass. Our E 6 model will be discussed in the next section. The quantum numbers of all the particles are fixed from the E 6 symmetry.
The VLQ has a dominant decay mode in the non-standard form of a SM quark and the new singlet scalar which is the focus of this study. We discuss the phenomenology of such a scenario and on the observable signatures for the vector-like down-type quarks at the LHC when the singlet scalar decays to a pair of photons or a pair of gluons. We shall have events with dijet/diphoton resonances at the same mass and these predictions can be tested as more data accumulates at the upcoming 13 TeV LHC run. This paper is organized as follows. In Section II below, we discuss our model and the formalism. In Section III, we discuss the phenomenology of our model. This gives emphasis on the prediction regarding the vector-like quarks through a new channel. The Section IV contains our conclusions and discussions.

II. THE MODEL AND FORMALISM
We work with an effective symmetry at the TeV scale where the SM is augmented with an extra U (1) . This extra U (1) is a special subgroup of the E 6 GUT [16][17][18][19][20][21][22][23]. We consider the non-supersymmetric version of E 6 . The symmetry group E 6 is special in the sense that it is anomaly free, as well as has chiral fermions. Its fundamental representation decomposes under SO(10) as The representation 16 contains the 15 SM fermions, as well as a right-handed neutrino.
The 5 contains a color triplet and an SU (2) L doublet, whereas5 contains a color antitriplet and another SU (2) doublet, while the 1 is a SM singlet. The gauge bosons are contained in the adjoint 78 representation of E 6 . SO(10) The U (1) ψ and U (1) χ charges for the E 6 fundamental 27 representation are also given in Table I. The U (1) is a linear combination of the U (1) χ and U (1) ψ The other orthogonal linear combination of U (1) χ and U (1) ψ as well as the SU (5) are broken at a high scale. This will allow us to have a large doublet-triplet splitting scale, which prevents rapid proton decay if the E 6 Yukawa relations were enforced. This will require either two pairs of (27,27) and one pair of (351 , 351 ) dimensional Higgs representations, or one pair of (27,27), 78, and one pair of (351 , 351 ) dimensional Higgs representations (detailed studies of E 6 theories with broken Yukawa relations can be found in [26,27].) For our model, the unbroken symmetry at the TeV scale is We explain our convention in some details as given in Table I (1, 1, 0, 10) In our model, S gives the Majorana masses to the right-handed neutrinos N c i after U (1) gauge symmetry breaking, i.e., the terms SN c i N c i are U (1) gauge invariant. Thus, the mixing angle in our model is given by The Higgs potential needed for our purpose giving rise to the extra U (1) symmetry breaking is Among the parameters in the potential V , σ is in general complex (σ 1 + iσ 2 ) and all others are real. Note that without the term σST 2 , there are two global U (1) symmetries for the complex phases of S and T . After S and T obtain the Vacuum Expectation Values (VEVs), we have two Goldstone bosons, and one of them is eaten by the extra U (1) gauge boson.
Thus, to avoid the extra Goldstone boson, one needs the term σST 2 to break one global symmetry. This leaves us with only one U (1) symmetry in the above potential, which is the extra U (1) gauge symmetry. Thus, after S and T acquire the VEVs, the U (1) gauge symmetry is broken, and S and T will be mixed via the λ ST |S| 2 |T | 2 and σST 2 terms.
The SM gauge boson masses are determined by the VEVs of the SU (2) doublet scalars GeV. The structure for the VEVs is given as The mass squared matrices for the scalar sectors (s 1 , t 1 ) and (s 2 , t 2 ) are respectively given by σ 1 is the real part of σ and the complex part σ 2 is assumed to be zero at tree level. These mass matrices have been obtained from the tree-level scalar potential under the assumption that there is no mixing in the (S, T ) and (H u , H d ) sector. The mass eigenstates for the CP-even sector (s 1 , t 1 ) is s h and t h . The massive scalar from the CP-odd sector (s 2 , t 2 ) is represented by a h . The relation between the gauge basis and the mass basis in for (s 1 , t 1 ) sector is given by where the mixing angle is given by The Yukawa couplings in our model are where i = 1, 2, 3. Thus, after S and T obtain VEVs or after U (1) gauge symmetry breaking, After we diagonalize their mass matrices, we obtain the mixings between XD c i and D c i , and the mixings between XL i and L i . The discussion of the Higgs potential for electroweak symmetry breaking is similar to the Type II two Higgs doublet model, so we will not repeat it here.
We note that the U (1) gauge boson couples to all the SM fields in addition to the new matter and scalar fields. The covariant derivatives for the SU (2) L doublet and the singlet scalars are respectively given by where where Y X (S) = 10 The mass square matrix for the neutral gauge boson sector in the (W 3 , B, Z ) basis is then given as where and We can clearly see that the new gauge boson mass is dependent on the VEVs of all the scalars, such that one can choose one singlet VEV to be much smaller than the other and still have a very heavy Z that evades the existing limits. Moreover, the mixings between W 3 /B and Z will be zero at tree level if v u = 2v d .
The mass matrix for the down-type quarks and the charged leptons in the (q 1 , q 2 , q 3 , xq 1 , xq 2 , xq 3 ) basis is given by where i, j = 1, 2, 3. The q i s and xq i s represent the down-type quarks for the left matrix and charged leptons for the right matrix. These mass matrices would be diagonalized by a bi-unitary transformation which would lead to a mixing between the vector-like fermions and the SM fermions. However, one should note that the mixings between the left-handed fermions and the right-handed fermions will be dictated by different set of mixing angles.
In our analysis we will allow mixings between the d quark and the 1st generation vector-like quark(xd 0 1 ) only and the mass matrix in the gauge basis (d 0 , xd 0 1 ) is given by The mixing matrices which transform the gauge eigenstates (d 0 , xd 0 1 ) to mass eigenstates(d, xd 1 ) are given by with the following left and right handed mixing angles We should also point out a few useful assumptions that we think are relevant for the analysis: 1. We have neglected any mixing between the electroweak doublet scalars and singlet scalars. 3. For simplicity, we will take all types of Yukawa couplings y A ij to be zero for i = j, where A ≡ T D, T L, SD, SL (see eq.8).

The mixing angles between the left-handed SM fermions and the vector-like fermions
are taken to be very very small, i.e., we assume the left-mixing angle θ L ∼ 0 to avoid the flavour physics constraints [31]. For the choice of the set of parameter values { ∼ 640 GeV, v t ∼ 10 4 GeV, y T D 11 ∼ 10 −5 } we get small values of mixing angles, i.e., sin θ L ∼ 10 −10 and sin θ R ∼ 10 −4 . The new VLQ will be dominantly produced via strong interaction, with subleading contributions coming from the s-channel exchange of the γ, Z and Z . In situations where the VLQ mass is less than M Z /2, then the Z mediated process can give a resonant contribution. However these contributions are found to be not very significant. We list the various production mechanisms of the VLQ in Fig.1. Note that one can in principle also produce the VLQs singly but they would be heavily suppressed as the production strength would depend on the mixing between the VLQs and SM quarks.

III. SIGNALS FOR VECTOR-LIKE QUARKS
In Fig. 2 we show the pair production cross section of the VLQ xd 1 as a function of its mass at both run-1 and current run of the LHC with √ s = 13 TeV. With a few 100 femtobarns of cross section, it would be highly unlikely for the LHC to miss the signal for VLQs when they decay directly to SM particles. These already put strong limits on the mass of the VLQs. However, a new decay mode for the VLQ can definitely alter the search strategies for these exotics even when the rates are significantly high.
With the details of the model discussed in the previous section, it is now possible to write down the interaction vertices for the VLQ and new scalars with the SM particles that we use in our calculation and analysis. We list the relevant interactions in Table III.
y SD 11 sin α cos θ R + y T D 11 cos α sin θ R 0   (Table III). As discussed in the previous section and to be safe from flavor constraints, one can impose small mixing angles, for example, sin θ L ∼ 10 −10 and sin θ R ∼ 10 −4 as mentioned for a set of parameter choices of the model. This will insure that the vector-like fermions do not decay to the SM gauge bosons and light SM fermions [29]. The decay to the SM Higgs and light down-type quark is again very suppressed, due to the coupling strength being proportional to sin θ R and mass of the down-type SM quark. The mixing in the Higgs sector has been neglected as a convenient choice to keep the number of free parameters to tune to be small.
The Z − Z mixing which is anyhow strongly constrained by electroweak data in any with the effective coupling λ sgg = α s F 1/2 (τ xd )/(16πv s ) where represents the loop function and f (τ xd ) = (sin −1 √ τ xd ) 2 with τ xd = m 2 s h /4M 2 xd < 1. Here, we have shown the contribution to the coupling from only one vector-like quark.
We plot the branching ratio for the scalar s h decaying into a pair of photons in Fig. 3 ( have been taken to be same (M x ) while the Yukawa couplings of s h to VLLs have been taken to be unity (y SL ii = 1). Note that the branching of the s h → γγ is very similar to the order at which the SM Higgs decay happens but slightly higher. This is because of the contributions of the VLLs which do not contribute to the s h → gg mode. However, the decay to γγ mode is still between 0.5% -0.6% at best while the remaining decay probability is made up by the gg channel. have been taken to 1.5 TeV. The Yukawa couplings (y SD 22 and y SD 33 ) have different values for different values of v s . Note that the s h production crucially depends on the Yukawa coupling which can be tuned to control its production rate. In fact, this possibility was earlier used by us [32,33] to arrive at the possible explanation of the now hitherto disproved diphoton excess for a 750 GeV resonance [3-5].
As pointed out earlier, the VEVs for S and T which are given by v s and v t respectively play a significant role in giving mass to Z . We choose the mass of Z to be 1.5 TeV which is still allowed by current LHC data, primarily due to the SM fermions carrying very suppressed quantum charges of the new U (1) as shown in Table II Note that after the decay of the VLQs there will be two s h and two jets in the final state.
As the s h decays to two gluons or to two photons only, with almost 99% to the gluonic jets, the resultant final states are either 2γ + 4j, 4γ + 2j or 6j. The cross section for the 4γ + 2j final state is quite small while the QCD background for the 6j final state is significantly large compared to the 2γ + 4j final state. Thus, these two channels would require large statistics to leave any imprint of their signal at the LHC. So in all likelihood the remaining channel of 2γ + 4j seems the most promising channel which we shall focus on for our analysis.
For the 2γ + 4j final state one of the xd 1 will eventually have a full hadronic decay to 3 jets. The bound on the branching ratio for the decay of a color triplet vector-like quark to three jets for different masses of vector-like quark has been obtained in [34] [34] shows that the physical region where BR(VLQ → 3j) ≤ 1 for different mass values of vector-like quark remains unconstrained except for a tiny region around 500 GeV.
To analyze the signal in our model, we note that the hardness of the jet from the decay xd 1 → s h j will depend on the mass difference between the VLQ xd 1 and the scalar s h .
This will affect the signal efficiency in the 2γ + 4j channel as well as dictate how well the mass reconstruction for the parent particles can be made. We will discuss these features by considering two benchmark scenarios with different mass gaps between the xd 1 and s h , where in one case the jet is hard while for the other case the jet would be comparatively soft. We  Table IV. We also check that the current upper limits on the cross section for the diphoton production through a narrow-width scalar resonance at 13 TeV run of LHC given by CMS Collaboration [39] is satisfied for our choice of the benchmark points. The upper limits on the cross section for the dijet production through a narrow-width resonance at the 13 TeV LHC, given by the CMS collaboration [40,41] are also satisfied.
Note that for BP1 the mass difference between xd 1 and s h is 40 GeV. This would mean that the jet coming from the decay of xd 1 is quite soft. Although at a hadronic machine such as the LHC, the jet multiplicity from parton showering would be invariably increased, we intend to focus on relatively hard jets and therefore would like to neglect soft jets in the process. So for the analysis of BP1 we consider a final state with smaller jet multiplicity given by 2γ + ≥ 2j. We demand that the jets have at least a minimum 40 GeV transverse momentum. The dominant SM background for such a final state is through all subprocesses contributing to pp → 2γ + ≥ 2j (with p T (j) > 40 GeV). For BP2 where the mass gap between xd 1 and s h is above 200 GeV, one expects the jet from xd 1 decay to be quite hard and thus the SM background is given by pp → 2γ + ≥ 4j.
We have implemented the TeV-scale U (1) extended model derived from E 6 GUT in LanHEP [42] to generate the model files for CalcHEP [43]. Using the model files we generated events for the pair production of VLQs (in LHEF format [44]) at the LHC with √ s = 13 TeV and the subsequent decays of the xd 1 and s h were included as a decay table for the model (in SLHA format [45]) with the help of CalcHEP. We then use these files to decay the unstable particles, and pass the generated parton-level events for showering and hadronization in PYTHIA 8.2 [46]. To enable detector simulation, we then linked the HepMC2 [47] libraries with PYTHIA 8.2 to translate PYTHIA 8 events into HepMC format. For simulating the background, we generated the events at leading-order accuracy using MadGraph5 [48]. Pythia 6 [49] interfaced in MadGraph5 was used for parton showering and hadronization of the background events, and to get event files in STDHEP format.
For both signal and background we include the detector effects and have reconstructed the final state objects using DELPHES 3 [50]. These are obtained in a CMS environment. Further, FastJet [51] embedded in DELPHES has been used to reconstruct the jets. In the DELPHES framework the anti-k T algorithm with a cone size 0.5, p j T > 20 GeV and |η parton | < 2.5 is used to reconstruct the jets. The phenomenological event-analysis is done with the MadAnalysis5 package using the event format ROOT.
In case of BP1 for which the mass difference between the lightest vector-like quark (xd 1 ) and the scalar (s h ) is small, we have generated pp → 2γ + 2j events as background at 13 TeV LHC. At the level of generation of events certain basic cuts have been imposed on the final state particles. All jets and photons satisfy |η| < 2.5 and each final state particle is separated from all other final state particles with an angular separation (∆R) value greater than 0.4. The transverse momenta of photons and jets satisfy p T (j) > 20 GeV and p T (γ) > 100 GeV.
The final state photons for the signal come from the decay of the s h which has 600 GeV mass and hence the probability for the photons for the signal to have higher p T values is more compared to the background. Hence a 100 GeV p T cut for photon has been used for the generation of background because the phase space with lower photon p T will be largely populated by background compared to the signal. For 13 TeV LHC, with the above cuts taken into account and at leading-order (LO) accuracy the value of the cross section for the parton-level background for BP1 is around 234 fb.
For BP2 where the mass difference between xd 1 and s h is 250 GeV, pp → 2γ + 4j events have been generated as the background. The basic cuts on the pseudo-rapidity (η) and on the angular separation (∆R) of the final state particles have been taken to be same as that of the benchmark BP1. The cut on the photon p T is taken to be the same 100 GeV as in case of BP1. We have imposed different p T cuts on the four jets and those are given by p T (j 1 ) > 80 GeV, p T (j 2 ) > 80 GeV, p T (j 3 ) > 40 GeV and p T (j 4 ) > 40 GeV. (18) With the above cuts the cross section at the parton-level for the background, for BP2 i.e.
for the process pp → 2γ + 4j, comes out to be around 12.15 fb. Similarly the signal events pp → xd 1 xd 1 have been generated using the event generator CalcHEP. The pair production cross sections are shown in Table IV. For the reconstructed events we choose the following selection criteria on the photons, jets and leptons • A jet is considered in an event if p T (j) > 40 GeV and |η(j)| < 2.5.
• An electron or a muon is considered in the lepton set if p T ( ) > 10 GeV and |η( )| < 2.5.
• A photon is considered in an event if p T (γ) > 40 GeV and |η(γ)| < 2.5.   The preselection criteria for BP1 is to consider events having 2 photons and a minimum of two jets. For BP2 the preselection criteria is to consider events with two photons and a minimum of four jets in the final state. We vetoed all the events having at least one isolated lepton of p T value greater than 10 GeV. For the two leading (in p T ) jets and the two leading photons in the signal it is expected that they come from the decay of the VLQ. As the mass of the VLQ for the two cases is above 600 GeV, the two leading jets will have a large amount of p T compared to the leading jets in the background. So for both the benchmarks we have considered the events with the leading two jets and two photons with p T value greater than 100 GeV.
To further increase the signal-to-background ratio we apply the following selection cuts to the analysis for BP1 The cut-flow table for BP1 signal and background is shown in Table V. Note that to generate the background events with large statistics we have used the preselection cuts given in Eq. (17).
Similarly for BP2 we have besides the criterion in Eq. (17), the additional preselection requirements for jets as given by Eq. (18) to generate the SM background with good statistics.
We further impose the stronger selection cuts on the events which help in improving the signal-to-background ratio for the signal events in the 2γ + 4j final state.
With these cuts and a 100 fb −1 integrated luminosity we get a statistical significance of 5σ for BP1 as can be seen from the  Similarly, to reconstruct the mass of the VLQ one can use the fully hadronic channel giving three jets or the semi-hadronic channel giving two photons and a jet. With the knowledge of the reconstructed mass for s h through the 2γ invariant mass peak, the mass for the vector-like quark can be reconstructed for both BP1 and BP2. To compare the reconstruction of the VLQ in the two channels, we first plot the 3j invariant mass distribution comprised of the leading jets in the events for both the signal and background in Fig. 5.
Although a distinct excess in the distribution exists around the mass of VLQ for both BP1 and BP2, the spread is quite wide and hence unclear as a mass resonance.
Although, the other channel with two photons and a hard jet should be a much more cleaner and precise mode to reconstruct the parent VLQ mass, it does suffer from the ambiguity of pairing the right jet with the pair of photons. In addition, for BP1 the mass splitting between the xd 1 and s h is quite small and therefore the choice of the right jet is affected by other soft jets that may originate from showering and fragmentation effects. To account for this ambiguity, we use the primary information on the kinematic characteristics of events for both BP1 and BP2 that is available to us to determine how we should combine the jets with the two photons. Owing to the small mass gap in BP1, we can safely assume that the two leading jets for the signal in BP1 would come from the decay of s h and therefore can be safely discounted in the combination. Of the remaining soft jets, all wrong combinations would only contribute in smearing the distribution for M γ 1 γ 2 j . We therefore propose to neutralize the smearing effects by averaging over all such soft jets (with p T > 40 GeV) in the invariant mass reconstruction and neglecting the first two leading jets for BP1. For BP2, the jets coming from the decay of VLQ to s h j is equally hard as the ones that come from the decay of s h themselves. Therefore, for BP2, the averaging is done including all jets with p T > 40 GeV. We plot their normalized distribution after averaging for both signal and background in Fig. 6. As the diphoton coming from the s h decay marks a kinematic edge in the distribution, this can be clearly seen to happen at the mass value of s h at 600 GeV for both BP1 and BP2 which is absent for the invariant mass distribution in the 3j hadronic channel. In addition, a much cleaner and distinct peak can be observed for the VLQ mass for BP2. In case of BP1, as the VLQ mass at 640 GeV is quite close to the scalar s h mass of 600 GeV, resolving the VLQ (although visible) mass peak from the sharp kinematic edge is difficult. However, for a larger mass gap the peak should be distinctly identifiable as in BP2.
Finally to impress upon the fact that the VLQ mass can be clearly reconstructed through the modified invariant mass variable proposed above, we show the distribution without any normalization in Fig. 7 overlaying the signal for BP2 over the SM background. It clearly shows the VLQ mass peak over the background. Thus, we find that both the s h and xd 1 can be reconstructed clearly to determine their masses for the channel under study. As BP2 pertains to a VLQ mass of 850 GeV, we conclude that a TeV mass VLQ with such non-standard decay modes, possible for BSM scenarios which have very little mixing with the SM sector, can be observed and its mass parameters determined at the LHC with a few 100 fb −1 of integrated luminosity.

IV. CONCLUSION
In this work we have considered an E 6 motivated extension of the SM where the larger symmetry groups are broken at a very high scale and a residual U (1) gauge symmetry is the only remaining symmetry beyond the unbroken SM gauge symmetry. This additional U (1) then gets broken at the TeV scale through new SM singlet scalars giving rise to a TeV scale particle spectrum with three generations of vector-like quarks and leptons and several neutral scalars. The vector-like quarks in the model have non-standard decay modes and decay into an ordinary light quark and a SM singlet scalar. Further the scalar decays either to two photons or two gluons. The current experimental limits for VLQ which do not decay directly to the SM particles are very weak and therefore allow their mass to be as light as 500 GeV. We analyzed the events from such VLQ production at the LHC with √ s = 13 TeV in the 2γ+ ≥ 2j final states and present a search strategy for observing its signals. We also studied how to reconstruct the masses for both the scalar as well as the VLQ through a modified construction of the invariant mass variable using the γγj sub-system. We saw that the mass of the scalar can be reconstructed from the invariant mass distribution of the two leading photons. With the upcoming high luminosity data at the LHC, the new signal for the VLQ, proposed in this work, could provide to be an interesting channel to search for new physics beyond the SM.