# The 750 GeV di-photon LHC excess and extra \(Z^\prime \)s in heterotic-string derived models

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## Abstract

The ATLAS and CMS collaborations recently recorded possible di-photon excess at 750 GeV and a less significant di-boson excess around 1.9 TeV. Such excesses may be produced in heterotic string derived \(Z^\prime \) models, where the di-photon excess may be connected with the Standard Model singlet scalar responsible for the \(Z^\prime \) symmetry breaking, whereas the di-boson excess arises from production of the extra vector boson. Additional vector-like states in the string \(Z^\prime \) model are instrumental to explain the relatively large width of the di-photon events and mandated by anomaly cancellation to be in the vicinity of the \(Z^\prime \) breaking scale. Wilson line breaking of the non-Abelian gauge symmetries in the string models naturally gives rise to dark matter candidates. Future collider experiments will discriminate between the high-scale heterotic-string models, which preserve the perturbative unification paradigm indicated by the Standard Model data, versus the low scale string models. We also discuss the possibility for the production of the diphoton events with high scale \(U(1)_{Z^\prime }\) breaking.

## Keywords

Gauge Symmetry String Model Breaking Scale Chiral Spectrum Gauge Coupling Unification## 1 Introduction

The Standard Model matter spectrum strongly favours its embedding in \(\mathrm{SO}(10)\) chiral representations. This unification scenario is further supported by the perturbative logarithmic evolution of the Standard Model parameters; by proton longevity; and by the suppression of left-handed neutrino masses. This picture is reproduced in heterotic-string models [1, 2]. The free fermionic formulation [3, 4, 5] in particular has given rise to phenomenological three generation models that have been used to explore the unification of gravity and the gauge interactions. These models correspond to \(Z_2\times Z_2\) orbifold compactification at special points in the Narain moduli space and utilise discrete Wilson lines to break the non-Abelian gauge symmetry to an \(\mathrm{SO}(10)\) subgroup [6, 7, 8, 9]. The viable models constructed to date include the flipped \(\mathrm{SU}(5)\) (FSU5) [10]; the standard-like models (SLM) [11, 12, 13, 14, 15]; the Pati–Salam models (PS) [16, 17, 18] and the left–right symmetric models (LRS) [19, 20], whereas the \(\mathrm{SU}(4)\times \mathrm{SU}(2)\times U(1)\) subgroup did not produce viable models [21, 22, 23]. All these models give rise to extra observable gauge symmetries at the string scale. Flavour universal symmetries typically arise from the \(\mathrm{SO}(10)\) and \(E_6\) group factors. However, preserving an unbroken extra gauge symmetry down to low scales has proven to be elusive in the string models. The reasons being that suppression of left-handed neutrino masses favours the breaking of lepton number at a high scale, whereas the \(U(1)_\zeta \) gauge symmetry in the decomposition of \(E_6\rightarrow \mathrm{SO}(10)\times U(1)_\zeta \) tends to be anomalous in the string models and therefore cannot remain unbroken down to low scales. This extra *U*(1) in the string models is typically anomalous because of the projection of some states from the spectrum by the generalised GSO (GGSO) projections, i.e. anomaly cancellation requires that the chiral spectrum appears in complete \(E_6\) representations. However, the breaking of \(E_6\) at the string scale mandates that the chiral states exist in incomplete \(E_6\) multiplets.

Recently, however, we were able to construct a heterotic-string model in which the desired symmetry is anomaly free [24]. The derivation of this model utilises the spinor–vector duality that was discovered in free fermionic heterotic-string models, and was obtained by using the classification methodology developed in [25, 26, 27, 28, 29, 30]. The model of Ref. [24] is obtained from a self-dual \(\mathrm{SO}(10)\) model under the exchange of the total number of spinorial \(16\oplus \overline{16}\) and vectorial 10 representations of \(\mathrm{SO}(10)\). This is the condition that one has if the \(\mathrm{SO}(10)\times U(1)_\zeta \) symmetry is enhanced to \(E_6\). However, in the model of Ref. [24] the \(\mathrm{SO}(10)\) symmetry is not enhanced to \(E_6\). This is the case in the free fermionic model if the different 16 and \(10+1\) representations, which would form a complete 27 representation of \(E_6\), are obtained from different fixed points of the underlying \(Z_2\times Z_2\) orbifold [24]. Adding the basis vector that breaks the \(\mathrm{SO}(10)\) symmetry to the PS subgroup results in split multiplets, but the chiral spectrum still forms complete \(E_6\) multiplets, hence rendering \(U(1)_\zeta \) anomaly free. We remark that while complete \(E_6\) multiplets is sufficient for \(U(1)_\zeta \) to be anomaly free, it might not be necessary and alternative possibilities may exist.

The ATLAS and CMS collaborations [31, 32] reported recently an excess in the di-photon searches at di-photon invariant mass of 750 GeV. This excess can be attributed to a neutral scalar resonance of 750 GeV mass. Plausible candidates include the \(\mathrm{SO}(10)\) neutral singlet in the 27 of \(E_6\), and \(E_6\) singlets that arise in the string models, with the production and decay being produced via one-loop couplings to heavy vector-like matter states. Such vector-like matter states are precisely those required from anomaly cancellation of the extra \(Z^\prime \) gauge symmetry. Indications for an extra \(Z^\prime \) of order 2 TeV have been earlier suggested by the ATLAS and CMS experiments [33, 34, 35]. They have not been substantiated by the run II data, but the possibility of an extra \(Z^\prime \) at that scale nevertheless remains. The signature of the string model with a low scale \(Z^\prime \) in our model is characterised by the scalar resonance, by the extra \(Z^\prime \) and by the additional vector-like matter at the multi-TeV scale, required by anomaly cancellation.

Alternative string constructions that may account for such excesses have been recently suggested [36, 37, 38, 39, 40], based on low scale string models and *F*-theory scenarios. Low scale string models give rise to Kaluza–Klein and heavy string states. Therefore, future colliders will be able to discriminate between perturbative heterotic-string models and low scale string scenarios. Additionally, heterotic-string models give rise to states that do not satisfy the \(E_6\) quantisation of the \(Z^\prime \) charges. Such states arise in string models due to the breaking of the \(E_6\) symmetry by Wilson lines and can produce viable dark matter candidates. This would be the case if the \(Z^\prime \) symmetry is broken by a state that carries standard \(E_6\) charge. A residual discrete symmetry then forbids the decay of the exotic string state to states that carry the standard \(E_6\) charges. Such exotic states at the multi-TeV scale can provide viable thermal relics. On the other hand, if their exotic \(Z^\prime \) charges can be determined experimentally, they provide a distinct signatures of the heterotic-string models that utilise discrete Wilson lines to break the \(E_6\) symmetry.

## 2 The string model

*D*and \({\bar{D}}\) to \(\Delta \) and \({\bar{\Delta }}\) to avoid confusion with the notation for the vector-like quarks below. We note that the sextet states are in the vector representation of \({\mathrm{SO}(6)}\equiv {\mathrm{SU}(4)}\). They are vector-like with respect to the Standard Model subgroup, but they are chiral with respect to \(U(1)_{\zeta }\).

Observable sector field notation and associated states in [24]

Symbol | Fields in [24] | \(\mathrm{SU}(4)\times {\mathrm{SU}(2)}_L\times {\mathrm{SU}(2)}_R\) | \({U(1)}_{\zeta }\) |
---|---|---|---|

\({F}_L\) | \(F_{1L},F_{2L},F_{3L}\) | \(\left( \mathbf{4},\mathbf{2},\mathbf{1}\right) \) | \(+\frac{1}{2}\) |

\({F}_R\) | \(F_{1R}\) | \(\left( \mathbf{4},\mathbf{1},\mathbf{2}\right) \) | \(-\frac{1}{2}\) |

\({\bar{F}}_R\) | \({\bar{F}}_{1R},{\bar{F}}_{2R},{\bar{F}}_{3R},{\bar{F}}_{4R}\) | \(\left( \bar{\mathbf{4}},\mathbf{1},\mathbf{2}\right) \) | \(+\frac{1}{2}\) |

| \(h_1,h_2,h_3\) | \(\left( \mathbf{1},\mathbf{2},\mathbf{2}\right) \) | \(-1\) |

\(\Delta \) | \( D_1,\dots , D_7\) | \(\left( \mathbf{6},\mathbf{1},\mathbf{1}\right) \) | \(-1\) |

\(\bar{\Delta }\) | \({\bar{D}}_1,{\bar{D}}_2,{\bar{D}}_3,{\bar{D}}_6\) | \(\left( \mathbf{6},\mathbf{1},\mathbf{1}\right) \) | \(+1\) |

| \(\Phi _{12},\Phi _{13},\Phi _{23},\chi ^+_1,\chi ^+_2,\chi ^+_3,\chi ^+_5\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(+2\) |

\({\bar{S}}\) | \(\bar{\Phi }_{12},\bar{\Phi }_{13},\bar{\Phi }_{23},\bar{\chi }^+_4\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(-2\) |

\(\phi \) | \(\phi _1,\phi _2\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(+1\) |

\({\bar{\phi }}\) | \(\bar{\phi }_1,\bar{\phi }_2\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(-1\) |

\(\zeta \) | \(\Phi _{12}^-,\Phi _{13}^-,\Phi _{23}^-,\bar{\Phi }_{12}^-,\bar{\Phi }_{13}^-,\bar{\Phi }_{23}^-\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(+0\) |

\(\chi _1^-,\chi _2^-,\chi _3^-,\bar{\chi }_4^-,\chi _5^-\) | |||

\(\zeta _i,\bar{\zeta }_i,i=1,\dots ,9\) | |||

\(\Phi _i,i=1,\dots ,6\) |

Hidden sector field notation and associated states in [24]

Symbol | Fields in [24] | \({\mathrm{SU}(2)}^4\times \mathrm{SO}(8)\) | \({U(1)}_{\zeta }\) |
---|---|---|---|

\(H^+\) | \(H_{12}^3\) | \(\left( \mathbf{2},\mathbf{2},\mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(+1\) |

\(H_{34}^2\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{2},\mathbf{2},\mathbf{1}\right) \) | \(+1\) | |

\(H^-\) | \(H_{12}^2\) | \(\left( \mathbf{2},\mathbf{2},\mathbf{1},\mathbf{1},\mathbf{1}\right) \) | \(-1\) |

\(H_{34}^3\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{2},\mathbf{2},\mathbf{1}\right) \) | \(-1\) | |

| \(H_{12}^1\) | \(\left( \mathbf{2},\mathbf{2},\mathbf{1},\mathbf{1},\mathbf{1}\right) \) | 0 |

\(H_{13}^i,i=1,2,3\) | \(\left( \mathbf{2},\mathbf{1},\mathbf{2},\mathbf{1},\mathbf{1}\right) \) | 0 | |

\(H_{14}^i,i=1,2,3\) | \(\left( \mathbf{2},\mathbf{1},\mathbf{1},\mathbf{2},\mathbf{1}\right) \) | 0 | |

\(H_{23}^1\) | \(\left( \mathbf{1},\mathbf{2},\mathbf{2},\mathbf{1},\mathbf{1}\right) \) | 0 | |

\(H_{24}^1\) | \(\left( \mathbf{1},\mathbf{2},\mathbf{1},\mathbf{2},\mathbf{1}\right) \) | 0 | |

\(H_{34}^i,i=1,4,5\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{2},\mathbf{2},\mathbf{1}\right) \) | 0 | |

| \(Z_i,i=1,\dots ,\) | \(\left( \mathbf{1},\mathbf{1},\mathbf{8}\right) \) | 0 |

The model is derived by fishing a self-dual model under the spinor–vector duality at the \(\mathrm{SO}(10)\) level, i.e. prior to incorporation of the basis vector \(\alpha \). The \(\mathrm{SO}(10)\) model exhibits the self-duality property under the exchange of the total number of spinorial plus anti-spinorial, and the total number of vectorial \(\mathrm{SO}(10)\) representations. This is in fact a key ingredient in the construction of the model and in the possibility of having an anomaly free \(U(1)_\zeta \) as part of a low scale \(Z^\prime \). The spinor–vector duality was observed in the classification of free fermionic vacua with \(\mathrm{SO}(10)\) GUT group [41, 42, 43, 44]. The statement is that for every vacuum with a total number #1 of twisted \({16}\oplus \overline{16}\) spinorial representations and a total number #2 twisted 10 vectorial representations, there exist another vacuum in which the two are interchanged. To understand the origin of this duality it is instrumental to consider the case in which \(\mathrm{SO}(10)\times U(1)_\zeta \) is enhanced to \(E_6\). The chiral and anti-chiral representations of \(E_6\) are the 27 and \(\overline{27}\) representations that decompose under \(\mathrm{SO}(10)\times U(1)_\zeta \) as \(27=16_{1/2}+10_{-1}+1_2\) and \(\overline{27}=\overline{16}_{-1/2}+10_{1}+1_{-2}\). Thus, in the case of \(E_6\) the total number of \({16}\oplus \overline{16}\) is equal to the total number of 10 representations, i.e. this case is self-dual under the spinor–vector duality map. In the case of \(E_6\), \(U(1)_\zeta \) is anomaly free by virtue of its embedding in \(E_6\), whereas in vacua with broken \(E_6\), \(U(1)_\zeta \) is in general anomalous [45]. Furthermore, the case of \(E_6\) correspond to a string vacuum with (2, 2) worldsheet supersymmetry. The \(N=2\) worldsheet supersymmetry on the bosonic side of the heterotic string has a spectral flow operator that exchanges between the spinorial and vectorial components of the \(E_6\) representations [43, 44]. The vacua with broken \(E_6\) symmetry only possess (2, 0) worldsheet supersymmetry. In these vacua the would be spectral flow operator induces the map between the spinor–vector dual vacua. The string vacua, however, also admit a class of self-dual vacua under the spinor–vector duality map without enhancement of the gauge symmetry to \(E_6\). This is possible because the different spinorial and vectorial components that make up complete \(E_6\) representations are obtained from different fixed points of the underlying \(Z_2\times Z_2\) orbifold. In this case the chiral spectrum still resides in complete \(E_6\) representations, and \(U(1)_\zeta \) is anomaly free, but the gauge symmetry is not enhanced to \(E_6\).

*U*(1)s with

As seen from Table 2 the model also contains vector-like states that transform under the hidden sector \(\mathrm{SU}(2)^4\times \mathrm{SO}(8)\) gauge group. They comprise four bi-doublets denoted by \(H^\pm \), which carry \(Q_{\zeta }=\pm 1\) charges, 12 neutral bi-doublets, denoted by *H*, and five states that transform in the 8 representation of the hidden \(\mathrm{SO}(8)\) gauge group with \(Q_\zeta =0\).

Spectrum and \(\mathrm{SU}(3)_C\times \mathrm{SU}(2)_L\times U(1)_{Y}\times U(1)_{{Z}^\prime }\) quantum numbers, with \(i=1,2,3\) for the three light generations. The charges are displayed in the normalisation used in free fermionic heterotic-string models

Field | \({\times }\mathrm{SU}(3)_C\) | \(\times \mathrm{SU}(2)_L \) | \({U(1)}_{Y}\) | \({U(1)}_{Z^\prime }\) |
---|---|---|---|---|

\(Q_L^i\) | 3 | 2 | \(+\frac{1}{6}\) | \(-\frac{2}{5}\) |

\(u_L^i\) | \({\bar{3}}\) | 1 | \(-\frac{2}{3}\) | \(-\frac{2}{5}\) |

\(d_L^i\) | \({\bar{3}}\) | 1 | \(+\frac{1}{3}\) | \(-\frac{4}{5}\) |

\(e_L^i\) | 1 | 1 | \(+1 \) | \(-\frac{2}{5}\) |

\(L_L^i\) | 1 | 2 | \(-\frac{1}{2}\) | \(-\frac{4}{5}\) |

\(D^i\) | 3 | 1 | \(-\frac{1}{3}\) | \(+\frac{4}{5}\) |

\({\bar{D}}^i\) | \({\bar{3}}\) | 1 | \(+\frac{1}{3}\) | \(+\frac{6}{5}\) |

\(H^i\) | 1 | 2 | \(-\frac{1}{2}\) | \(+\frac{6}{5}\) |

\({\bar{H}}^i\) | 1 | 2 | \(+\frac{1}{2}\) | \(+\frac{4}{5}\) |

\(S^i\) | 1 | 1 | 0 | \(-2\) |

| 1 | 2 | \(-\frac{1}{2}\) | \(-\frac{4}{5}\) |

\({\bar{h}}\) | 1 | 2 | \(+\frac{1}{2}\) | \(+\frac{4}{5}\) |

\(\mathcal{D}\) | 3 | 1 | \(-\frac{1}{3}\) | \(+\frac{4}{5}\) |

\({\bar{\mathcal{D}}}\) | \({\bar{3}}\) | 1 | \(+\frac{1}{3}\) | \(-\frac{4}{5}\) |

\(\phi \) | 1 | 1 | 0 | \(-1\) |

\(\bar{\phi }\) | 1 | 1 | 0 | \(+1\) |

\(\zeta ^i\) | 1 | 1 | 0 | 0 |

A distinct property of the free fermionic heterotic-string model, as noted from Table 1, is the existence of the exotic states \(\phi _{1,2}\), and \(\bar{\phi }_{1,2}\). These states arise due to the breaking of \(E_6\) by a discrete Wilson line in the string model. Such states do not arise in pure field theory GUT models, and may be a distinct signature of the specific string vacuum of Eqs. (2.1) and (2.2), i.e. they may be a distinct signature of the particular Wilson line used in this model. These exotic states are \(\mathrm{SO}(10)\) singlets and therefore are neutral with respect to the Standard Model gauge group. The breaking of the \(U(1)_{Z^\prime }\) gauge symmetry may leave a discrete symmetry that forbids the decay of these exotic states to the lighter Standard Model states. This is the case if the \(U(1)_{Z^\prime }\) gauge symmetry is broken by one of the \(S_i\) states. The mass scale of the exotic states, relative to the \(Z^\prime \) breaking scale, then determines the type and whether they can provide a viable dark matter candidate [59, 60].

## 3 The di-photon events

*ZZ*,

*WW*, etc. A plausible explanation for the production and decay of such a resonance is via a Standard Model singlet scalar field that couples to heavy vector-like quark and lepton pairs. In is noted that indeed scenarios along this lines have been proposed for such a resonance [117, 118, 119].

*U*(1) symmetry \(\langle \mathcal{S}\rangle =v^\prime \). Provided that this VEV is of the order of the TeV scale, say \(v'=5\) TeV, then it ensures that the \(U(1)_{Z^\prime }\) remains unbroken down to the TeV scale. Furthermore, ensuring that the extra \(U(1)_{Z^\prime }\) is anomaly free mandates that the extra vector-like quarks and leptons obtain their mass of the order of \(O(v')\) from the couplings in Eq. (3.1). Thus, all the ingredients needed to generate the characteristics of the diphoton events naturally exist in the string model. The model then associates the diphoton events with the existence of additional \(U(1)_{Z^\prime }\) gauge symmetry at \(O(v')\). With \(M_{Z^\prime }\sim g^\prime v'\) and \(M_\mathcal{S}\sim \lambda ^\prime v^\prime \) being naively the masses of the heavy \(Z^\prime \) vector boson and the Higgs field \(\mathcal{S}\), respectively, we see that in this model the masses of the \(Z^\prime \) vector field and the \(\mathcal{S}\) scalar field are closely related. These characteristics fit well with both a di-boson excess at 1.9 TeV [33, 34, 35] as well as with the di-photon excess at 750 GeV [31, 32]. Furthermore, the model predicts the existence of the additional vector-like quarks and leptons in the same vicinity. The existence of the \(U(1)_{Z^\prime }\) symmetry at the TeV scale, and the associated anomaly cancellation requirement, naturally explain the existence of the vector-like quarks and leptons at the \(U(1)_{Z^\prime }\) breaking scale. However, as follows from Eqs. (3.2) and (3.4), the diphoton events can also be mediated in the string models by scalars with \(Q_{Z^\prime }=0\), and in this case the mass scale of the vector-like states is disassociated from the \(U(1)_{Z^\prime }\) breaking scale. This is a less appealing scenario, but one which is allowed in the string model. In this case \(U(1)_{Z^\prime }\) may be broken at a high scale along a flat direction by utilising the VEVs for, say \({\bar{\chi }}_4^+\) and \(\chi _5^+\).

## 4 Conclusions

Early indications from LHC run 1 and 2 provide evidence for excess in the di-photon channel with a resonance of the order of 750 GeV, and a less significant di-boson excess at 1.9 TeV. Such signals fit naturally in our heterotic-string derived model, with a high scale seesaw mechanism and a low scale \(U(1)_{Z^\prime }\) breaking. Furthermore, the model predicts the existence of additional vector-like quarks and leptons related to the \(U(1)_{Z^\prime }\) breaking scale.

Some of the proposals to explain the di-photon excess are in line with the explanation employed in our paper (see e.g. [117, 118, 119]), and alternative scenarios have been proposed as well (see e.g. [64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116]), Some of these alternative scenarios employ a composite scalar singlet. The string derived model [24] does contain hidden sector fields, charged under \(U(1)_\zeta \), which can form composites that can mimic the charges of the singlet field \(S_i\). Investigating whether this can provide an alternative scenario in the model at hand is left for future work.

The existence of vector-like quarks at the TeV scale poses a challenge when confronted with proton decay limits. Generically we anticipate that these states couple to the light quarks and may generate proton decay. How to avoid this conundrum remains a puzzle. Some plausible suggestions include the existence of local discrete symmetries that forbid the ominous couplings [120], and the special placement of matter fields in unified multiplets [121].

Alternative explanations have also been proposed in the case of models with a low string scale [36, 37, 38, 39, 40]. The low scale string scenarios give rise to additional Kaluza–Klein and heavy string modes and therefore will easily be discerned from the heterotic-string scenarios. Explorations into the multi-TeV regime will adjudicate between the competing scenarios. We await the return of the collider.

## Notes

### Acknowledgments

AEF thanks Shmuel Nussinov for discussions and the theoretical physics groups at Oxford University, the Weizmann Institute and Tel Aviv University for hospitality. AEF is supported in part by the STFC (ST/L000431/1).

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