Back to 1974: The $\mathcal Q$-onium

We show that the 750 GeV di-photon excess could be explained by the ${\mathcal Q}-$onium system of a new QCD-like theory with fermions vectorial under the SM. Beside the spin-0 di-photon singlet this scenario predicts almost degenerate colored scalars and spin-1 resonances analogous to the $J/\Psi$ in QCD. All these states are within the reach of the LHC. An apparent large width can be explained as due to production of excited states with splitting $\Delta m\sim \Gamma$.


I. INTRODUCTION
A very plausible explanation of the di-photon excess at M 750 GeV recently reported at the LHC [1] is provided by new confining gauge dynamics, dubbed technicolor (TC), with fermions Q that are vectorial under the SM [2][3][4][5]. Most theoretical speculations have focused on the regime where the Q are lighter than the confinement scale. The di-photon resonance is then identified with a TC pion SM singlet that couples to SM gauge bosons through anomalies.
In this letter we study the regime where the Q are heavier than the confinemant scale Λ TC , see [2,[5][6][7] for early work. The system so obtained is entirely analogous to quarkonium in QCD, bound states of cc or bb, see [9][10][11] for a review. The di-photon resonance is identified with η 1 Q , the lightest spin-0 color singlet ( 1 S 0 ) bound state. The analogous resonance in QCD, η c has a rate Γ(η c → γγ)/m ηc ∼ 2 × 10 −6 which is almost exactly what is required to reproduce the di-photon excess [2]. A model independent prediction is the existence of a scalar color octet (η 8 Q ), almost degenerate in mass and coupled to pairs of gluons [5] as well as spin-1 excitations similar to the J/Ψ.
It is not too surprising that η 1 Q is the first resonance discovered at LHC. Indeed it is the lightest resonance that couples to gluons and photons. The almost degenerate η 8 Q is also copiously produced but it can only decay to jets and should be discovered in the next run of the LHC. If the interpretation given in this letter is correct, various resonances of the Q−onium will be within the reach of the LHC.

II. SPECTROSCOPY OF ηQ AND ψQ
For concreteness we consider SU (N TC ) gauge theories with fermions Q in the fundamental representation. Under the SM they form a vectorial representation R (in Figure 1: Sketch of the Q-onium system. The spectrum resembles the one of charmonium and bottomonium in QCD. The di-photon resonance is interpreted as the lightest singlet 1 S0. Resonances of different spin are predicted and higher level excitations could account for an apparent large width of the resonance. general reducible) of the SM. The strong dynamics confines at the scale Λ TC < m Q producing QQ bounds states analogous to the charmonium with mass M ∼ 2m Q . The expected spectrum is sketched in Fig. 1. For each strong dynamics level we obtain a fine structure of SM multiplets. In order for the resonances to couple to gluons and photons the new fermions should carry color and electric charge. A SM rep R = (d 3 , d 2 ) Y will then produce Qonium states as, where the elipses denote possible further representations, depending on R. The presence of the di-photon singlet coupling to photons and gluons is always accompanied by a scalar color octet. The ground state spin-zero ( 1 S 0 ) color singlet (η 1 Q ) decays to gluons and photons and will be identified with the di-photon resonance while the spin-0 color octet (η 8 Q ) couples only to gluons. Spin-1 states, arXiv:1603.07719v3 [hep-ph] 4 Jul 2016 ( 3 S 1 ) color singlet (ψ 1 Q ) and color octet (ψ 8 Q ), couple instead to pairs of SM fermions or 3 SM gauge bosons.
Other colored states could appear for example (8,3) 0 that would couple to W bosons and gluons. If other fermions with mass above Λ TC exist more Q-onium bound states will be formed. One difference with the TC pion scenario (Λ T C > m Q ) is that these states will not significantly mix unless the masses are almost degenerate so they will appear as separate resonances. Therefore we can focus on irreducible SM reps in what follows.
The dynamical scale of the theory is given by, where we have included n light flavors. For α TC > α s the bound states are TC singlets formed due to the new strong interactions [8]. Two regimes can be distinguished. If α TC M Λ TC confinement gives small corrections and the system can be described in first approximation as a positronium-like bound state with Coulomb potential V = −C N α TC /r. For fermions in the fundamental rep C N = (N 2 − 1)/(2N ). The binding energies are given by, where n = 1, 2, . . . are the radial excitation levels and α TC should be evaluated at the scale of the bound state size r 2 1/Λ TC . For this we need the information on the radial wave-function R(r), normalized to ∞ 0 |R(r)| 2 r 2 dr = 1. In what follows we will be primarily interested in its value at the origin (|R(0)|) . In the Coulomb regime for the n−th radial excitation this is given by, We note that such a weakly coupled picture fails if applied to the lowest lying states of charmonia and bottomonia, and a recent numerical lattice QCD simulation indicates deviations from the positronium-like behavior even for QCD bound states with mass close to M [12], even though α s (M )M Λ QCD . In the opposite regime α TC M Λ TC the confinement effects modify significantly the bound state and splitting of energies becomes larger. Moreover while |R(0)| 2 /M 3 is constant in the Coulomb regime, |R(0)| 2 becomes almost independent of M when confinement effects dominate. In Table I we report the masses and wave-function values extracted for QCD η singlets. For both charmonium and bottomonium, confinement effects appear to be dominant.
In the perturbative α T C regime the mass splitting between the 1 S 0 and 3 S 1 states can be estimated analogously to the hyperfine structure of positronium or atoms Coul.
2.983 2 × 10 −6 1.5 × 10 −2 ηc(2S) 3.639 10 −6 6 × 10 −3 η b (1S) 9.398 5 × 10 −8 6 × 10 −3 η b (2S) 10 2 × 10 −8 2.5 × 10 −3 Table I: η 1 Q singlets in QCD [13]. Their widths into photons, are not measured directly, but are derived using the decay of ψ into electrons through Γ(ψQ →f f )/Γ(ηQ → γγ) = Q 2 f /(3Q 2 Q ). The value of the wave-function at the origin is extracted using the formula where the second equality is valid in the Coulomb-like limit (when α TC M Λ TC ). The mass splitting is thus extremely sensitive to the precise value of α TC . This could be expected, since in the chiral regime (m Q Λ TC ) 1 S 0 state becomes a Nambu-Goldstone boson of the approximate chiral Q flavor symmetry, while in the asymptoticly free m Q → ∞ limit (when TC interactions are not strong enough to flip the spin of Q), spin becomes a globally conserved quantum number of the TC sector. For the charmonium this splitting if 3.7%, and for bottomonium 0.7% . Within each Q-onium level SM interactions split the multiplets. Assuming that the bound state is formed due to the TC interactions this can be treated as a small perturbation and implies that the splitting is linear in α s When α TC becomes comparable with α s ≈ 0.1 it is important to include QCD effects for the bound state. For the singlet this provides an extra attractive force so that in the Coulomb regime the effective coupling that controls the bound state is replaced by C N α TC + C 3 α s . We note that recent calculations using potential models and lattice simulations estimate the irreducible QCD contribution to (|R 1 (0)| 2 /M 3 ) QCD ∼ (0.0002−0.0008) [12,14]. For the color octet combination QCD is repulsive so that this state is more loosely bound, the effective becoming For example for Q in the fundamental of both TC and QCD and N TC = 3 the effective coupling of the octet at α TC ≈ α s is reduced by ≈ 60% compared to the singlet, leading to |R 15]. When this is smaller than the width of the bound states (Γ), Q-onium states will fill complete SU (2) multiplets with little mixing between them. Only the SU (2) singlet states can couple to gluons in this case and the rates to electro-weak final states are identical to the analogous TC pion scenario, see [4]. In the opposite regime ∆m Q /Γ > 1, the Q-onium mass eigenstates will be aligned with the fermion Q charge eigenstates, which must thus be summed incoherently. In this regime, the relative decay rates into electro-weak final states do not follow from SU (2) relation of the EFT [16]. In particular, the di-photon excess will be dominated by the bound state made by the SU (2) component of Q with the highest electric charge. As we will see, the first regime is relevant for the Q−onium system made of Q = (3, 2) 1/6 while the latter holds for Y = (3, 2) −5/6 when η 1 Q width is dominated by decays to gluons.

III. Q−ONIUM PRODUCTION AND DECAYS AT LHC
Since in the heavy Q limit 1 S 0 and 3 S 1 states are related by spin-symmetry, on can relate all their dominant interactions purely in terms of SM gauge group invariants and charges. We choose η 1 Q → gg decay as the reference width. One finds, where I 3 = 1/2 for the fundamental representation. For ∆m Q /Γ > 1 the formula above applies for each SU (2) component with d 2 = 1. The η Q decay to gluon or (only for η 1 Q ) photon pairs, while ψ Q decay to pairs of fermions or three gauge bosons. The prompt single production of η 1,8 Q states at the LHC then proceeds dominantly through gluon fusion, while ψ 1,8 Q are produced via qq annihilation. For ∆m Q /Γ > 1 (relevant for the models U , X and Y in table II) the decay rates into SM states for the various components of a Q SU (2) multiplet are predicted as, where combinations differ due to QCD effects so that |R 8 (0)| < |R 1 (0)| while we neglect the splitting between spin-0 and spin-1 states due to TC interactions. The ψ decay widths into 3 SM gauge bosons are also predicted. For example the width of ψ 1 Q into 3 gluons reads, which is extremely small. Many other relations can be found generalising the ones in [9]. The formulas above can be easily adapted to the degenerate SU (2) limit (∆m Q /Γ < 1). The ones with electroweak gauge bosons final states can be read from [4]. For η 8 Q → gγ only the hypercharge contributes in eq. (8).
where ψ refer to the SU (2) singlet and the neutral component of the triplet, (both color singlets) respectively. The other ratios are not modified. Considering η 1 Q → ZZ, Zγ decays and assuming a signal crosssection σ(pp → γγ) ≈ 5 fb [16] at LHC 13, the bounds from run 1 [18,28] and recently run 2 [19] translate into a constraint on the dimension of the SU (2) representation (d 2 ) and hypercharge (Y ) Finally, we note in passing that for Q in non-trivial SU (2) representations, charged η and ψ states will also be formed. However, they are singly produced only through weak interactions and thus less relevant for LHC phenomenology.
In the narrow width approximation, the resonant production cross-sections of η Q and ψ Q are given by, where D X is the dimension of the representaton, J X the spin and P is the parton producing the resonance at the LHC: gluons for X = η Q and quarks for X = ψ Q . The parton luminosity coefficients at LHC 13 (8) for the production of a 750 GeV resonance in the s-channel are C gg = 2137(174), C uū = 1054(158) and C dd = 627(89) [2]. In our phenomenological analysis we also include (approximately) known NLO QCD K−factors of K η 1 gg = 1.6 [20], K η 1 qq = 1.2 [21], K ψ 1 qq = 1.3 [22] and K ψ 8 qq = 1.3 [23]. On the other hand, QCD corrections to prompt production of a massive color octet scalar are presently not known. In Table II   consistent with results in [24] considering a somewhat similar scenario, but our results can easily be rescaled for different values.
The di-photon signal cross-section is reproduced for, implying that Γ(η 1 Q → γγ)/M ≥ 0.7 × 10 −6 with the equality saturated when the width is dominated by decays into gluons.
Let us consider models with an SU (2) singlet Q in detail. The width into photons reads Reproducing the di-photon signal assuming that the total with is dominated by decays to gluons then requires, or larger if extra decay channels exist. Given the irreducible QCD contribution to R 1 (0) [14], eq. (15) can only be satisfied for Q 0.5(3/N TC ) 1/4 in the α TC α s limit, suggesting the necessity of extra decay channels. These are naturally provided by TC glueballs and lighter TC pions. For example in QCD Br(η c → γγ) ∼ 10 −4 due to decays into hadrons. Given that η Q /ψ Q are almost degenerate, all other cross-sections are predicted in this model up to the difference between wave-function of singlets and octets. Using gluon and quark lumininosities at 13 TeV we find, where jets from ψ 8 Q include b-quarks. Consequently, for α TC α s , η 8 Q gives the dominant contribution to the resonant di-jet cross-section at LHC 13. At LHC 8 instead η 8 Q and ψ 8 Q give comparable di-jet signals. Estimates for various representations are given in Table  II assuming no extra decay channels and equal wavefunctions for singlet and octets. Note that in this regime σ(pp → η 8 Q → gγ) typically provides the strongest experimental constraint [26]. For all color octet rates however these estimates should be taken as a conservative upper bound given that the QCD effects make the octets more weakly bound. A comparison with bounds at the 8 TeV LHC [25,26] including contributions from η 1 Q , ψ 1 Q (and for the case |R 8 (0)| |R 1 (0)| (|R 8 (0)| |R 1 (0)|) , respectively. Thus Q with the SM quantum numbers of the right-handed down quarks (D) and up-quarks (U ) are disfavored only in the deeply bound regime where QCD effects are negligible. In the weakly bound regime where color octet effects are subleading, D is only marginally compatible with existing di-jet bounds.
In addition to single prompt production, the lowest lying η Q and ψ Q states will also be produced in fragmentation of high p T QCD produced QQ pairs (inclusive continuum production). In these processes, the confining TC dynamics forces formation of heavy highly excited QQ bound states, which in turn decay to the lowest lying states (η 1,8 Q and ψ 1,8 Q ) by radiating TC glueballs, gluons and photons (see for example [7]).
Depending on the TC dynamics, either prompt or inclusive continuum production of η Q may dominate. As an example we consider color triplet Q. Both prompt η Q and continuum inclusive QQ production are completely determined in terms of the η Q binding energy (∆E, or correspondingly m Q ) and the η Q radial wavefunction at the origin (|R 1 (0)|). We compare the two in Fig. 2, where we plot σ(pp → QQ) computed at NNLO in QCD [31] and normalized to N TC (in orange dashed contours) as well as the ratio σ(pp → η 1 Q )/σ(pp → QQ) (in blue full contours) both at 13 TeV. We observe that in the tightly bound regime (∆E/M ∼ O(1) and/or |R 1 (0)| 2 /M 3 ∼ O(1)) prompt η Q production can easily dominate over continuum QQ production. However even in the weakly bound Coulomb regime, when continuum production is bigger, it may not necessarily be the dominant source of η Q 's, since only a (small) fraction of QQ events will result in a specific η Q state.
On the same plot we also show the combinations of ∆E and |R 1 (0)| as predicted in the Coulomb limit (drawn with purple dotted line) and mark various values of α T C for the case N TC = 3 (in purple points). This can be compared with the experimentally determined values for the case of η c and η b (marked in black points), where their binding energies are approximated by the mass differences between the 1S and 2S states in Table I . Finally, we (shaded in green), Q = (3, 2) 1/6 (shaded in red) and overlapping X = (3, 1) 4/3 , Y = (3, 2) −5/6 (shaded in blue), taking for concreteness NTC = 3 and assuming predominantly prompt ηQ production and no additional significant decay modes (in full lines) or saturating their total decay width of ∼ 45 GeV with hidden decay channels (in dot-dashed lines). For illustration we also report the values predicted in the Coulomb regime for various choices of αT C for NTC = 3 (marked in purple points and connected with a dotted purple line), as well as an estimate [14] of the irreducible QCD contribution to |R1(0)| (upper edge of the black hashed region) .
overlay the values of |R 1 (0)| 2/3 reproducing the LHC diphoton excess at N TC = 3, for the cases of U = (3, 1) 2/3 (shaded in green), Q = (3, 2) 1/6 (shaded in red) and overlapping X = (3, 1) 4/3 , Y = (3, 2) −5/6 (shaded in blue), assuming predominantly prompt η Q production and no additional significant decay modes besides η Q → gg, γγ (in full lines) or saturating their total decay width of ∼ 45 GeV with hidden decay channels (in dot-dashed lines). We observe that without additional decay modes or production mechanisms the preferred region of |R 1 (0)| lies below the bottomonium regime for all four cases. As discussed above, color octet states are expected to be suppressed in this regime. Additonal η 1 Q decay modes, for example into lighter TC hadrons however can move the preferred region to higher values of |R 1 (0)|. Finally we note that increasing N TC , considering additonal production mechanisms such as continuum production, or considering multiple states (see Sec. IV), pushes the preferred values of |R 1 (0)| further down.

IV. EXCITED STATES AND THE DI-PHOTON WIDTH
One interesting aspect of the Q-onium system is the existence of excited states with equal quantum numbers that correspond to the radial excitations of the system. We estimate the binding energy as, In the Coulomb regime (3) the splitting between 1 S 0 and 2 1 S 0 for N TC = 3 is given by, When non-perturbative effects become important a larger splitting (and correspondingly a larger signal) is obtained. In Table I the mass splitting between (1S) and (2S) states in QCD are reported. For the bottomonium in QCD for example the splitting is 6%, much larger than indicated from the formula above (α s (m b ) ≈ 0.2). This invites to interpret the apparent large width of the diphoton excess as preferred by ATLAS (Γ/M = 0.06) as due to the presence of the di-photon excitation, 2 1 S 0 . The ratio of the prompt production cross-sections depends to leading order only on the values of the wavefunctions at the origin. In the Coulomb regime this is just 1/8 for 2 1 S 0 / 1 S 0 . Taking the charmonium and bottomonium cases in Table I as guidance, the 2 1 S 0 rate could be only around a factor 2-3 smaller contributing in significant way to the total cross-section. A bound state with binding properties similar to the bottomium η b would produce an apparent width of 6%. With the quantum numbers of U the value of the wave-function in QCD would indicate a width into photons Γ(η 1 Q → γγ)/M ∼ 3 × 10 −6 that could be allowed with decays into lighter TC hadrons. Given the experimental resolution into photons it will be possible with more data to distinguish this from the scenario of a single broad resonance. Detecting several peaks with decreasing strength will be a clean signature of this scenario.
In the regime where the splittings cannot be resolved the excited states contribute to the total cross-section enhancing it by an O(1) factor. In addition we also predict the existence of higher spin particles. Spin-1 resonances analogous to J/Ψ states of charmonium can be produced from qq initial states. The ψ 8 gives a significant cross-section into di-jets, see Table II. The splitting among these states is expected to be small (1% for bottomonium) so that they will not appear as separate resonances in the di-jet invariant mass distribution. The ψ 1 on the other hand decays into pairs of leptons with a cross-section that could be measured with future data. States with orbital angular momentum are more difficult to produce since their production cross-sections are proportional to derivatives of the wave-function at the origin and thus suppressed.

V. INVISIBLE DECAYS
So far we have assumed that the lightest new fermions are the ones that make the Q-onium. The only states lighter than the di-photon resonance are then TC glueballs. If kinematically accessible the di-photon resonance could also decay into TC glueballs. These would decay back to SM through higher dimensional operators G 2 TC F 2 SM generated by loops of heavy fermions. The final state with 4 SM gauge bosons is a generic feature also of models with fermions in the confined regime that could be searched for at the LHC.
The scenario can be simply generalized by adding fermions with mass smaller than M/2, either lighter or heavier than Λ TC . By far the safest possibility is that these are singlets. For m < Λ TC and n ≥ 2 the lightest particles are TC pions. The ones made of different species are stable and they constitute viable Dark Matter candidates whose relic abundance could also be thermally produced [4].
The di-photon resonance can decay to such TC pions. The decay into TC hadrons can be estimated in perturbation theory as the decay into TC gluons that will eventually hadronize, One also finds, For α TC > α s the decay into TC hadrons could be dominant. The final states are mostly TC pions. Including this invisible decay from eq. (13) the di-photon excess is reproduced for, Note that only η 1 Q can decay to TC hadrons. Such invisible decays of η 1 Q thus effectively imply a larger production cross-section of η 8 Q . Consequently it not possible to achieve a genuine large width in these models if color octet states are unsupressed, due to indirect constraints, particularly di-jets and photon-jet resonance searches.
The lightest TC baryons are stable so they are also good dark matter candidates [32]. Their cosmological stability is robustly guaranteed by the fact that the TC baryon number is broken by dimension 6 operators while stability of TC pions could be violated by dimension 5 operators. Interactions with the SM will be mediated by the Q-onium. Such interactions are however strongly suppressed. For example in QCD η c decays into pp with a branching ∼ 10 −3 . TC baryons couple strongly to TC pions so they will be in thermal equilibrium with them. As a consequence if these are in thermal equilibrium with the SM the thermal relic abundance will be too small. Dark Matter as thermal relic could be reproduced in region of parameters where the annihilation cross-section of TC baryons is suppressed, for example when fermion masses are above Λ TC .

VI. SUMMARY
To conclude we can compare the Q-onium system with other composite di-photon scenarios discussed in the literature. When the fermions are lighter than Λ TC the lightest states are TC pions. For an irreducible SM representation the quantum numbers of the TC pions are identical to the ones of η Q studied here and the singlet state (the η ) will couple to gluons and photons if the constituents carry color and electric charge, providing a perfect candidate for the di-photon excess with identical branching fractions into SM gauge bosons as η 1 Q . Moreover the heavier spin-1 resonances ρ will have the same quantum numbers as the ψ states. For the Q-onium spin-0 and spin-1 particles are almost degenerate leading to stronger constraints from di-jets.
Even without new strong interactions a bound state would form just because of QCD interactions [33][34][35]. The main difference in this case is that the value of the wave-function that controls the decay rates is set by α s without non-perturbative enhancements, leading to smaller cross-sections. Obtaining the required di-photon rate requires Q = 4/3 or larger. Since the color octet state is not bound under QCD, this scenario can avoid the strongest constraints from dijet and jet-photon resonance searches discussed here. Note however that such QCD effects can be relevant in certain region of parameters also in our setup, enhancing in particular the singlet signal and weakening or eliminating the color octets.
The main prediction of the Q−onium scenario is the presence of other resonances with the pattern sketched in Fig. 1. Color octet resonances with spin-0 and spin-1 can be copiously produced at the LHC and could be visible in the di-jet or jet-photon invariant mass distributions. Singlet spin-1 resonances could produce signals in di-leptons. Interestingly the large width suggested by ATLAS data can most easily be reproduced by the production of nearby radial excitations. Detecting such a pattern would be a clear smoking gun of Q-onium. On the other hand, due to poorer di-jet invariant mass resolution, η 1 Q , η 8 Q ψ 8 Q will likely not appear as individual resonances in di-jet searches.