E6 inspired composite Higgs model and 750 GeV diphoton excess

In the E6 inspired composite Higgs model (E6CHM) the strongly interacting sector possesses an SU(6)\times U(1)_B\times U(1)_L global symmetry. Near scale f\gtrsim 10 TeV the SU(6) symmetry is broken down to its SU(5) subgroup, that involves the standard model (SM) gauge group. This breakdown of SU(6) leads to a set of pseudo--Nambu--Goldstone bosons (pNGBs) including a SM--like Higgs and a SM singlet pseudoscalar A. Because of the interactions between A and exotic fermions, which ensure the approximate unification of the SM gauge couplings and anomaly cancellation in this model, the couplings of the pseudoscalar A to gauge bosons get induced. As a result, the SM singlet pNGB state A with mass around 750 GeV may give rise to sufficiently large cross section of pp\to \gamma\gamma that can be identified with the recently observed diphoton excess.


Introduction
The discovery of the Higgs boson with mass m h 125 GeV allows one to estimate the values of parameters of the Higgs potential. In the standard model (SM) the Higgs scalar potential is given by The 125 GeV Higgs mass corresponds to m 2 H ≈ −(90 GeV) 2 and λ ≈ 0. 13. At the moment current data does not permit to distinguish whether Higgs boson is an elementary particle or a composite state. Although the discovered Higgs boson can be composed of more fundamental degrees of freedom, the rather small values of |m 2 H | and the Higgs quartic coupling λ indicate that the Higgs field may emerge as a pseudo-Nambu-Goldstone boson (pNGB) from the spontaneous breaking of an approximate global symmetry of some strongly interacting sector.
The minimal composite Higgs model (MCHM) [1] includes weakly-coupled elementary and strongly coupled composite sectors (for a recent review, see [2]). The weakly-coupled elementary sector involves all SM fermions and gauge bosons. The strongly coupled sector gives rise to a set of bound states that contains Higgs doublet and massive fields with the quantum numbers of all SM particles. These fields are associated with the composite partners of the quarks, leptons and gauge bosons. The elementary states couple to the composite operators of the strongly interacting sector leading to e-mail: roman.nevzorov@adelaide.edu.au mixing between these states and their composite partners. In this framework, which is called partial compositeness, the couplings of the SM states to the composite Higgs are set by the fractions of the compositeness of these states. The observed mass hierarchy in the quark and lepton sectors can be accommodated through partial compositeness if the fractions of compositeness of the first and second generation fermions are quite small. In this case the flavour-changing processes and the modifications of the W and Z couplings associated with the light SM fermions are somewhat suppressed. At the same time, the top quark is so heavy that the right-handed top quark t c should have sizeable fraction of compositeness.
The strongly interacting sector of the MCHM possesses global SO(5) × U(1) X symmetry that contains the SU(2) W × U(1) Y subgroup. Near the scale f the SO(5) symmetry is broken down to SO(4) so that the SM gauge group remains intact, resulting in four pNGB states which form the Higgs doublet. The custodial global symmetry SU(2) cust ⊂ SO(4) allows one to protect the Peskin-Takeuchî T parameter against new physics contributions. Experimental limits on the parameter S imply that m ρ = g ρ f 2.5 TeV, where m ρ is a scale associated with the masses of the set of spin-1 resonances and g ρ is a coupling of these ρ-like vector resonances. This set of resonances, in particular, contains composite partners of the SM gauge bosons. Even more stringent bounds on f come from the non-observation of flavor changing neutral currents (FCNCs). In the composite Higgs models, adequate suppression of the non-diagonal flavour transitions can be obtained only if f is larger than 10 TeV. This bound on the scale f can be considerably alleviated in the models with additional flavour symmetries FS. In the models with FS = U(2) 3 = U(2) q × U(2) u × U(2) d symmetry, the bounds that originate from the Kaon and B systems can be satisfied even for m ρ ∼ 3 TeV. In these models the appropriate suppression of the baryon number violating operators and the Majorana masses of the left-handed neutrino can be achieved if global U(1) B and U(1) L symmetries, which ensure the conservation of the baryon and lepton numbers to a very good approximation, are imposed. Thus the composite Higgs models under consideration are based on The couplings of the elementary states to the strongly interacting sector explicitly break the SO(5) global symmetry. As a consequence, the pNGB Higgs potential arises from loops involving elementary states. This leads to the suppression of the effective quartic Higgs coupling λ.

E 6 inspired composite Higgs model
In the E 6 inspired composite Higgs model (E 6 CHM) the Lagrangian of the strongly coupled sector is invariant under the transformations of an SU(6)×U(1) B ×U(1) L global symmetry. The E 6 CHM can be embedded into N = 1 supersymmetric (SUSY) orbifold Grand Unified Theories (GUTs) in six dimensions which are based on the E 6 × G 0 gauge group [3]. (Different aspects of the E 6 inspired models with low-scale supersymmetry breaking were recently considered in [4]- [19].) Near some high energy scale, M X , the E 6 × G 0 gauge group is broken down to the SU  (6) can remain an approximate global symmetry of the strongly coupled sector at low energies if the gauge couplings of this sector are considerably larger than the SM ones. As in most composite Higgs models, the global SU(6) symmetry in the E 6 CHM is expected to be broken below scale f . Here we assume that it gets broken to SU(5) subgroup, so that the SM gauge group is preserved. Since E 6 CHM does not possesses any extra custodial or flavour symmetry, the scale f must be much larger than the weak scale, i.e. v f . In particular, the adequate suppression of the FCNCs requires f 10 TeV. The SU(6)/SU(5) coset space includes eleven pNGB states that correspond to the broken generators Tˆa of SU (6). These pNGB states can be parameterised by where the SU (6) generators are normalised so that TrT a T b = 1 2 δ ab and In the leading approximation the Lagrangian, that describes the interactions of the pNGB states, can be written as The field φ 0 is real and does not participate in the SU , form a fundamental representation of the unbroken SU(5) subgroup of SU (6). The components H ∼ (φ 1 φ 2 ) transform as an SU(2) W doublet. Therefore H corresponds to the SM-like Higgs doublet. Three other components ofH, i.e. T ∼ (φ 3 φ 4 φ 5 ), transform as an SU(3) C triplet. In the E 6 CHM neither H nor T carry baryon and/or lepton number. The pNGB effective potential V e f f (H, T, φ 0 ) is induced by the interactions of the elementary states with their composite partners, which break SU(6) global symmetry. The analysis of the structure of this potential including the derivation of quadratic terms m 2 H |H| 2 and m 2 T |T | 2 in the composite Higgs models, which are similar to the E 6 CHM, shows that there is a considerable part of the parameter space where m 2 H is negative and m 2 T is positive [20]- [21]. In this parameter region the SU(2) W × U(1) Y gauge symmetry gets broken to U(1) em , associated with electromagnetism, while SU(3) C colour is preserved. Because in the E 6 CHM the scale f 10 TeV, a significant tuning, ∼ 0.01%, is required to get the appropriate value of the parameter m 2 H that results in a 125 GeV Higgs state. Since in the E 6 CHM all states in the strongly interacting sector fill complete SU (5) representations the corresponding fields contribute equally to the beta functions of the SU(3) C , SU(2) W and U(1) Y interactions in the one-loop approximation. As a consequence the convergence of the SM gauge couplings is determined by the matter content of the weakly-coupled sector. In this case, approximate gauge coupling unification can be achieved if the right-handed top quark, t c , is entirely composite and the weakly-coupled elementary sector involves the following set of multiplets (see also [22]): where α = 1, 2 runs over the first two generations and i = 1, 2, 3 runs over all three. In Eq. In the E 6 CHM the lightest exotic fermion state has to be stable. Indeed, the baryon number conservation implies that the Lagrangian of the E 6 CHM is also invariant under the transformations of the discrete Z 3 symmetry which can be defined as Here B is the baryon number of the given multiplet Ψ and n C is the number of colour indices (n C = 1 for 3 and n C = −1 for 3). This symmetry is called baryon triality [20]. All states in the SM have B 3 = 0. At the same time exotic fermion states carry either B 3 = 1 or B 3 = 2. As a result the lightest exotic state with non-zero B 3 can not decay into SM particles and should be stable. Since models with stable charged particles are ruled out by various experiments [23]- [24], the lightest exotic fermion in the E 6 CHM must be neutral. It is also worth noting that the coupling of this neutral Dirac fermion to the Z-boson have to be extremely suppressed. Otherwise this stable exotic state would scatter on nuclei resulting in unacceptably large spin-independent cross sections. Thus, only a Dirac fermion, which is mostly a superposition of η andη, can be the lightest exotic state in the E 6 CHM.

750 GeV diphoton resonance
The SM singlet pNGB state φ 0 can be identified with the 750 GeV diphoton resonance recently reported by ATLAS and CMS. It is important that no 750 GeV resonance has been observed in other channels like pp → tt, WW, ZZ, bb, ττ and j j. This may be an indication that the detected signal is just a statistical fluctuation. At the same time, if these observations are confirmed this should set stringent constraints on the new physics models that may lead to such a signature. For example, in the E 6  field. In particular, the couplings of the SM singlet pNGB state φ 0 = A to the top quarks is induced by Because of the almost exact CP-conservation the mixing between the Higgs boson and pseudoscalar A is forbidden. The Lagrangian that describes the interactions between A and exotic fermions can be written in the following form [25] In the most general case the couplings κ i and λ i in Eq. (8) and the exotic fermion masses μ i , induced below scale f , i.e.
are entirely independent parameters, which are not constrained by the SU(6) and SU(5) symmetries. In order to get the cross section σ(pp → γγ), which corresponds to the production and sequential diphoton decays of the pseudoscalar A, of about 5 − 10 fb we assume that μ D , μ Q , μ L , μ E and μ η are larger than 375 GeV. As a result the on-shell decays of A into the exotic fermions, that result in the strong suppression of the branching ratios of the decays of this pNGB state into photons, are not kinematically allowed. Integrating out the exotic fermion states one obtains the effective Lagrangian that describes the interactions of the pseudoscalar A with the SM gauge bosons [25] L where B μν , W a μν , G σ μν are field strengths for the U(1) Y , SU(2) W and SU(3) C gauge interactions, G σμν = In Eq. (11) GeV is the mass of the SM singlet pNGB state A, α Y = 3α 1 /5 while α 1 , α 2 and α 3 are (GUT normalised) gauge couplings of U(1) Y , SU(2) W and SU(3) C interactions. Using Eqs. (10)-(11) one can obtain an analytical expression for the coupling of the pseudoscalar A to the electromagnetic field F μν where θ W is the weak mixing (Weinberg) angle and F μν = 1 2 μνλρ F λρ . Since at the LHC the pseudoscalar A is predominantly produced through gluon fusion the cross section σ γγ = σ(pp → A → γγ) can be presented in the following form [26]

QUARKS-2016
where C gg 3163, √ s 13 TeV, Γ A is a total width of the pseudoscalar A while partial decay widths Γ(A → γγ) and Γ(A → gg) are given by First of all it is worthwhile to identify the scenario that leads to the suppression of the decay rates A → tt, WW, ZZ, γZ because no indication of the 750 GeV resonance has been observed in the channels associated with these decay modes. The analytical expressions for the corresponding partial decay widths can be presented in the following form Here we set Λ t 80 TeV. The partial decay widths (16)-(18) become substantially smaller than Γ(A → γγ) if |c 2 | |c 1 |. The appropriate suppression of |c 2 | can be achieved when the exotic fermions that form SU(2) W doublets are considerably heavier than the SU(2) W singlet exotic states. On the other hand the non-observation of any new coloured particles with masses below 1 TeV at the LHC implies that the exotic coloured fermions in the E 6 CHM should be rather heavy. Thus to simplify our numerical analysis we assume that μ D = μ Q = μ L = μ 0 μ E and κ D = κ Q = λ L = λ E = σ. For μ 0 μ E the decay rates for A → tt, WW, ZZ and Zγ are very suppressed. However μ 0 cannot be too large, otherwise the LHC production cross section of the pseudoscalar A becomes smaller than 5 − 10 fb. In our numerical analysis we vary μ 0 from 1 TeV to 5 TeV. In this case A mainly decays into a pair of gluons. As a consequence Γ A ≈ Γ(A → gg) and the cross section (13) Fig. 1 demonstrates that σ(pp → A → γγ) decreases very substantially when μ E increases. For μ E = 400 GeV and σ 1.5 the cross section (16) can be about of 5 fb, even when all other exotic fermions are rather heavy μ 0 5 TeV. At the same time for μ E 700 − 800 GeV the corresponding diphoton production cross section becomes sufficiently large only if μ 0 1 TeV. When μ 0 changes from 5 TeV to 1 TeV the ratio Γ A /m A increases from 10 −5 to 10 −4 that corresponds to the variation of the total LHC production cross section of the pseudoscalar A from 100 fb to 1 pb.
In Fig. 2 the dependence of the branching ratios of the pseudoscalar A on μ 0 is explored for μ E 400 GeV and σ = 1.5. From this figure it follows that A → gg is the dominant decay channel. Its branching fraction is always close to 100%. When μ 0 5 TeV the branching ratio BR(A → γγ) is the second largest one. BR(A → WW), BR(A → ZZ), BR(A → Zγ) and BR(A → tt) are substantially smaller than BR(A → γγ). This might be a reason why the decays A → WW, ZZ, Zγ, tt have not been detected yet. The decays A → gg can be rather problematic to observe because the total LHC production cross section of the pseudoscalar A is quite small. The branching fractions BR(A → ZZ) and BR(A → WW) increase while BR(A → γγ) and BR(A → tt) decrease with decreasing μ 0 .