Trinification can explain the di-photon and di-boson LHC anomalies

LHC data show a diphoton excess at 750 GeV and a less significant diboson excess around 1.9 TeV. We propose trinification as a common source of both anomalies. The 1.9 TeV excess can be produced by the lightest extra vector: a $W_R^\pm$ with a gauge coupling $g_R\approx 0.44$ that does not decay into leptons. Furthermore, trinification predicts extra scalars. One of them can reproduce the $\gamma\gamma$ excess while satisfying constraints from all other channels, given the specific set of extra fermions predicted by trinification.


Introduction
After the discovery of the Standard Model Higgs boson [1] at the Large Hadron Collider, data indicate the presence of other excesses that can be interpreted as resonant production of new bosons. The two most notable anomalies found in present LHC data are: 1. An excess of di-boson events peaked around 1.9 TeV, found in run 1 data [2,3]. This has been explained in terms of resonant production of extra vectors, such as Z or W R that decay into ZZ or W Z, respectively [4]. Run 2 data do not confirm this excess, but (as we will see) do not significantly contradict it.
2. An excess of di-gamma events peaked around 750 GeV, mostly seen at run 2 [3]. This has been explained in terms of resonant production of an extra scalar S [5,6]. In order to achieve the needed S → γγ decay width, extra vector-like fermions or scalars coupled to S with a relatively large coupling must be introduced [5,6].
Assuming that these excesses are not statistical fluctuations, we try to go beyond phenomenological considerations and explore which kind of bigger picture they can suggest.
The structure needed to explain both anomalies is present in models that extend the SM gauge group around the weak scale, thereby implying extra vectors, predicting a larger set of anomaly-free chiral fermions and needing extra Higgs scalars and Yukawa couplings. Among this class of models, trinification is the SU(3) L ⊗ SU(3) R ⊗ SU(3) c sub-group of E 6 that explains the observed quantised hypercharges, that can be broken to the Standard Model at a scale V just above the weak scale, that allows a totally asymptotically free extension of the SM [7,8], which is suggested by non conventional ideas about naturalness of the Higgs mass [9]. Trinification contains left-right models and 3-3-1 models as subgroups [10]. Focusing on trinification: • It predicts extra W ± R vectors with mass ∼ g R V with properties compatible with the di-boson anomaly: no decays into leptons, appropriate value of its g R gauge coupling.
• It predicts 27 chiral fermions per generation: the SM fermions, plus extra vector-like D and L fermions (and singlets), which receive masses M ∼ y V . Large Yukawa couplings y to fermions are needed in order to make M above present bounds, if V is not much above the weak scale.
• It predicts a larger set of Higgs scalars, which contain various singlets and doublets (with neutral components). One of them can be identified with the Higgs doublet at 125 GeV, and another with the scalar S at 750 GeV.
In section 2 we briefly summarise the main ingredients of trinification models needed to interpret the LHC anomalies. In section 3 we discuss how trinification can reproduce the di-boson anomaly. In section 4 we discuss how trinification can reproduce the di-gamma anomaly. Conclusions are given in section 5.
Field spin generations SU (3) Table 1: Field content of minimal weak-scale trinification. Q is the SM left-handed quark doublet, U is the SM right-handed up quark, E is the SM right-handed lepton. The D 1 and D 2 quarks mix making D (right-handed down quarks) and D (that gets a mass withD ) the L 1 and L 2 lepton doublets mix making L (left-handed lepton doublet) and L (that gets a mass with L ). N 1 and N 2 are singlets.

Trinification
The minimal model of weak-scale trinification is obtained including 3 families of chiral-free fermions under the gauge group G 333 = SU(3) L ⊗ SU(3) R ⊗ SU(3) c and 3 families of Higgses, as summarised in table 1. The three trinification gauge coupling constants (g 2 , g R , g 3 ) allow to reproduce those of the SM (g 2 , g Y , g 3 ) as The SM Yukawa couplings are obtained from the G 333 -invariant interactions The most generic vacuum expectation values that give the desired pattern of symmetry breaking are Defining V 2 ≡ n (V 2 n + V 2 Rn ) and the dimension-less ratios  Figure 1: Trinification heavy vector masses normalised to the W R mass. The next-to-lightest vector is a Z with mass ≈ 1.7 M W R . For concreteness, we used a basis where V R1 = 0.
• A left-handed weak doublet with mass M H L = g 2 V / √ 2, that contains two neutral components and two charged components.
• Two Z , Z vectors with masses: • A SU(2) R vector doublet A 2R splitted into two neutral components with mass M H 0 R = g R V / √ 2 and into 2 charged components with mass • A SU(2) R triplet A 3R that splits into a neutral Z and into 2 charged right-handed W ± R vectors with mass We see that W R is the lightest extra vector. The W ± R and H ± R mass eigenstates arise as in terms of the G 333 interaction eigenstates A ± R3 and A ± R2 . The mixing angle vanishes if β 1: Figure 2: Resonant production of a W ± R that decays into jj, W Z and W h.
process mass σ at √ s = 8 TeV σ at √ s = 13 TeV pp → S → γγ 750 GeV 0.5 ± 0.5 fb [3,11,12] 8 ± 2 fb [3] pp → W ± R → W ± Z 1.9 TeV 5.3 ± 2.0 fb [2] 1 ± 13 fb [3,13,14]   Concerning fermions, trinification adds to the chiral fermions under the SM group (Q, D, U , L and E in the usual notation), an extra vector-like D ⊕D and L ⊕L as well as some singlets N , N . In presence of three Higgs multiplets H, the theory has enough Yukawa couplings that the extra states can be naturally heavy, M D ,L ∼ y Q,L V thanks to large enough Yukawa couplings y Q,L [7]. The light and heavy fermion mass eigenstates can be parametrised as where θ D and θ L are (for each generation) mixing angles, that can be of order one, computable in terms of Yukawa couplings and vevs. Then, the W R coupling to quark mass eigenstates is An important feature predicted by trinification is that the W R does not couple to two light leptons. Rather, W R couples the SM doublet of left-handed leptons L to the heavy doublet of anti-leptonsL with coupling g R sin(θ R − θ L ), and right-handed leptons E to heavy neutrinos with coupling g R cos θ R , in the limit of negligible mixing angle between neutral fermions N 1 /N 2 . Furthermore, W R couples to pairs of heavy leptons.
3 The di-boson excess at 1.9 TeV LHC data show an excess of di-boson events around 1.9 TeV [2]. Trinification can explain this excess as pp → W ± R → W ± Z, and predicts an equal rate for pp → W ± R → W ± h, some amount Table 3: Cross section σ(pp → W ± R ) and partonic factors C ± , as function of the W R mass, assumed to be close to 1.9 TeV. We defined No excess is observed in di-leptons, while jj data show some excess. The measured rates are summarised in table 2. Fig. 2 shows the Feynman diagrams that describe such processes.

Trinification predictions
The cross section for producing a on-shell W ± R in narrow-width approximation is where the dimensionless partonic coefficients are to be integrated in the rapidity range |y| < ln √ s/M W R , summing over all up-type and downtype quarks. The factor K = 1 + 8πα 3 (M 2 W R )/9 ≈ 1.23 accounts for leading-order QCD corrections, with the partonic function distributions f q renormalised at Q = M W R . The numerical values of the C ± coefficients are given in table 3 using the MSTW2008NLO partonic distribution functions [24].
The cross sections into given final states X are We then need the decay widths and the branching ratios. Simple results hold into two extreme limits, described below.
1. All fermions and scalars are much lighter than M W R , so that the W R decay width into full G 333 multiplets is simply given by independently of the mixing angle θ R that defines W R . Taking into account that there are 3 generations of fermions and scalars, the total width is Γ W R /M W R = 15g 2 R /32π ≈ 0.029.
2. W R can only decay into the SM fermions and into the Higgs doublet, while all other states are so heavy that decays into them are kinematically forbidden. Interestingly, W R cannot decay into leptons. The decay width into a light quark generation is The W R decay width into the components of the SM Higgs doublet H = (χ ± , (h+iη)/ √ 2) (where, after SU(2) L breaking, χ ± and η become the longitudinal components of W ± and Z, respectively) is where θ H is a mixing angle that defines which components of H contain the light SM Higgs doublet H, given by cos θ H ≡ r ud cos θ R − r uL sin θ R with 2 In this limit, the total decay width is Γ W R /M W R = g 2 R (18 cos 2 (θ D + θ R ) + cos 2 θ H )/96π.

One loop QCD corrections enhance the decay widths into quarks by
If all the mixing factors are of order unity, in case 2 one expects BR(W R → W Z) ∼ 1/36. However this branching ratio can be much larger if cos 2 θ H cos 2 (θ D + θ R ). In case 1 one has BR(W R → W Z) = BR(W R → W h) ≤ 1/90.

Comparison with data
We are now ready to compare the trinification predictions with di-boson data.
The first issue that we address is the compatibility of run 1 data at √ s = 8 TeV (that showed an excess around 1.9 TeV) with run 2 data at √ s = 13 TeV, that show no excess so far. In view of the reduced luminosity so far accumulated at run 2, and of the larger backgrounds, run 2 data, as summarised in table 2, have much larger uncertainties. In our model the W R production cross section grows by a factor ≈ 7 when going from √ s = 8 to 13 TeV. 2 The decay width into SM vector bosons can also be computed taking into account that the small SM vacuum expectation values v mixes W R with the SM W by a small angle and that 1Σ,2Σ CL contour 2 dof Figure 3: Global fit of run 1 and run 2 data. Left: ±1σ experimental bands for the W R Branching Ratios into jj (blue), W Z (red), W h (yellow) as function of the couplingg R that controls the predicted W R production cross section. We also show the bands obtained using 8 TeV data (dashed contours with lighter colors), and the cross section at 8 TeV (upper axis). Right: global fit (green region) assuming only W R decays into jj, W Z, W h.
Taking into account all these factors, the incompatibility between the two data-sets is only mild, at the 1σ level, and the run 1 data still play the main role. This is shown in the left panel of fig. 3, where we extract the W R branching ratios into the various channels as function of the W R production cross section, plotted at 8 TeV on the upper horizontal axis, and parametrised in terms of the couplingg R on the lower axis. We next proceed in performing a global fit, assuming that W R only decays into jj, W Z and W h (case 2 in the discussion above) and using the theoretical prediction BR(W R → W Z) = BR(W R → W h). The result is shown in the right panel of fig. 3. We see that data prefer g R ∼ 0.25. While in a generic context this is a free parameter, trinification predictsg R = g R cos(θ D + θ R ), where g R = 0.444 and the mixing factor depends on the details of the model. This means that trinification is non-trivially compatible with data, that favour a substantial amount of mixing among right-handed down quarks, cos(θ D + θ R ) ∼ 0.5. The best fit has χ 2 best = χ 2 SM − 2.7 2 , which can be interpreted as a 2.7σ hint, up to look-elsewhere reductions. Similar results are obtained for a slightly lighter or heavier W R .
Allowing W R to decay into extra light states (up to the opposite extremum of case 1 in the discussion above) reduces the W R branching ratios into jj, W Z, W h, such that a higher production cross section becomes needed to fit the data. Trinification allows to raise the production cross section up tog R = g R . At the same time, the extra decay channels give rise to extra more complicated signals.
In order to keep the discussion simple, let us divide particles into three main categories: q (generic quarks), (generic leptons), H (Higgses: h, W , Z), all approximatively massless. Then q denotes the extra heavy quarks, the extra heavy leptons, and H denotes the extra heavy scalars (not eaten by the W, Z), with generic masses M . The possible W R decay modes are The extra primed particles, if produced, can decay through gauge interactions as (where the virtual W * R can, in many cases, only materialise into qq or HH light particles) and through Yukawa interactions as If W R → is kinematically allowed, one obtains final states with two leptons accompanied by a jj or W Z or W h pair or by a single W, Z, h. If is a heavy neutrino, Majorana masses can give rise to same-sign di-leptons, accompanied by the same extra particles listed above. In particular, for appropriate values of its parameters, trinification can also reproduce the eejj excess [25], as already discussed in [4] in the context of dedicated models. Needless to say, all decay widths can be computed in terms of unknown masses and of the mixing angles introduced above.

The di-photon excess at 750 GeV
The di-photon excess at 750 GeV [3] has been tentatively interpreted as a new 750 GeV scalar S with mass M ≈ 750 GeV that is produced as gg → S and decays as S → γγ through loops of extra fermions and/or scalars coupled to S by sizeable couplings [5,6]. ATLAS data [3] possibly indicate a large width Γ S /M ≈ 0.06, which would make this interpretation more difficult. In any case, the apparently large width can be faked by a multiplet of quasi-degenerate narrow resonances [5].
Trinification has built-in all the ingredients needed to provide such interpretation: extra scalars S as singlets and/or doublets in the Higgs H multiplets; extra fermions D and L that extend the SM chiral fermion content and receive masses only as yV ; Yukawa couplings y that must be sizeable enough in order to push the extra fermions above the experimental bounds.

Trinification predictions
In order to compute the relevant decay widths of S into γγ and gg, we parameterise its linear Lagrangian couplings to generic fermions Q f with Dirac mass M f , to generic scalarsQ s with mass M s and to generic vectors V with mass M V as The coupling g V is normalised such that it coincides with the SM coupling g 2 , if S is replaced by the SM Higgs h and V µ by the SM W ± µ . Then S acquires the following decay widths at one-loop level [5]: where multiplicities are implicitly included in sums. The color factors are C 1 = 0 for color singlets and C 3 = 1/2 for color triplets, and we have taken into account that trinification predicts no other coloured particles. The loop functions are [5] where f (x) = arctan 2 (1/ √ x − 1). The S decay width into gluons can be estimated as having assumed, for simplicity, N D ≤ 3 degenerate copies of D with mass M D M/2. This shows that the Γ(S → gg)/M > ∼ 10 −5 needed to fit the di-gamma anomaly [5] is easily achieved; furthermore trinification implies that some components of the H scalars have sizeable couplings to SM quarks, allowing for extra production channels of S. The S decay width into photons can be estimated as having assumed, for simplicity, N L ≤ 3 copies of leptons degenerate at M L > ∼ M/2, plus N s ≤ 9 copies of scalars with charge Q s = ±1 degenerate at the same mass and with the same cubic A s . We neglected the small contribution from heavy down-quarks and from vectors.

Comparison with data
In order to achieve the Γ(S → γγ)/M > ∼ 10 −6 needed to reproduce the γγ anomaly [5] one either needs Yukawa couplings of order unity or larger, or the presence of multiple charged states, or a large A s /M . In a context where the theory can be extrapolated up to infinite energy [8,9], some scalar quartic couplings can be predicted in terms of squared gauge couplings, times model-dependent order one factors. This implies that scalars typically have masses comparable to vectors, which must be heavier than about 2 TeV. 3 One can assume that one of the various scalar doublets contained in H 1,2,3 is accidentally lighter and can be identified with the 125 GeV Higgs boson, and that one singlet (or doublet) is accidentally light and can be identified with the 750 GeV resonance. In such a case, the accidental cancellation that makes its mass M smaller than V does not generically suppress, at the same time, its cubic couplings A s , such that A s /M can be large (phenomenologically limited by vacuum decay constraints) enhancing S → γγ. The mixing of S with the physical Higgs h must be small enough. The two neutral components of a doublet generically receive a small mass splitting ∼ v 2 /M ≈ 10 GeV and can mimic the large width favoured by ATLAS data [5]. The two components of all electroweak singlets in H generically receive mass splittings of order V .
If the S → γγ loop is dominated by a loop of either lepton doublets or scalar doublets (which have the same quantum numbers) one predicts the following extra electro-weak decays: These predictions satisfy the experimental bounds collected in [5] and quoted after the < symbols and correspond, in the language of effective operators of [5], to equal coefficients Λ B = Λ W , orΛ B =Λ W . The extra loop contribution from charged scalar singlets or from right-handed down quarks can reduce the ratios in eq. (30).

Conclusions
Models based on the trinification gauge group G 333 = SU(3) L ⊗ SU(3) R ⊗ SU(3) c explain the quantisation of electric charge [7]. Weak scale trinification models provide natural extensions of the Standard Model that can hold up to infinite energy, with all gauge and Yukawa and quartic couplings flowing to asymptotic freedom [8].
We have shown that these same models, without any extra ingredient, can simultaneously explain the γγ anomaly at 750 GeV (section 4) and the diboson anomaly at 1.9 TeV (section 3). Run 2 LHC data do not confirm the latter anomaly found in run 1 data, but do not significantly disfavour it (fig. 3a).
The lightest extra vector of trinification is a W ± R with a gauge coupling g R ≈ 0.444 compatible with the cross section needed to fit the 1.9 TeV anomaly ( fig. 3b). Furthermore, the group structure of trinification implies that it does not decay to SM leptons, in agreement with data.
At the same time, trinification predicts a set of extra fermions (chiral under G 333 but not under the SM) and of scalars which can be numerous and light enough to mediate the gg → S → γγ loops without giving, at the same time, excessive gg → S → γZ, ZZ, W + W − . This interplay between the 2 anomalies implies extra W R decay channels which give rise to the extra signals discussed in section 3.
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