Extra \(Z^{\prime }\)s and \(W^{\prime }\)s in heteroticstring derived models
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Abstract
The ATLAS and CMS collaborations recently recorded possible excess in the diboson production at the diboson invariant mass at around 2 TeV. Such an excess may be produced if there exist additional \(Z^{\prime }\) and/or \(W^{\prime }\) at that scale. We survey the extra \(Z^{\prime }\)s and \(W^{\prime }\)s that may arise from semirealistic heteroticstring vacua in the free fermionic formulation in the seven distinct cases: \(U(1)_{Z^{\prime }}\in SO(10)\); family universal \(U(1)_{Z^{\prime }}\notin SO(10)\); nonuniversal \(U(1)_{Z^{\prime }}\); hidden sector U(1) symmetries and kinetic mixing; left–right symmetric models; Pati–Salam models; leptophobic and custodial symmetries. Each case has a distinct signature associated with the extra symmetry breaking scale. In one of the cases we explore the discovery potential at the LHC using resonant leptoproduction. The existence of an extra vector boson with the reported properties will significantly constrain the space of allowed string vacua.
Keywords
Vector Boson Hide Sector Proton Decay Flavour Change Neutral Current Standard Model State1 Introduction
The Standard Model multiplet structure strongly favours its embedding in chiral 16 representations of SO(10). This can be most emphatically demonstrated by recalling that the Standard Model gauge charges are experimental observables. The Standard Model, including righthanded neutrinos, has three group factors, three generations, and six multiplets per family, and therefore heuristically the number of parameters required in the standard model is 54. Embedding the Standard Model states in SO(10) representations reduces this number to one, which is the number of 16 spinorial SO(10) representations required to accommodate the standard model states.
Gravitational interactions are not accounted for in the Standard Model. A contemporary selfconsistent framework that facilitates the exploration of the synthesis of the gravitational and gauge interactions is provided by string theories, which are conjectured to be effective limits of a more fundamental theory. Heteroticstring theory is the perturbative limit that allows for the embedding of the Standard Model states in chiral SO(10) representations as it gives rise the spinorial 16 representations in its perturbative spectrum. Three generation models with viable gauge group and Higgs states have been constructed using a variety of methods. Among those the free fermionic formulation [1, 2, 3] of the heterotic string [4] provided a particularly fertile ground. In these three generation models the SO(10) symmetry is broken at the string level to one of its maximal subgroups.
Recently, the ATLAS and CMS collaborations [5, 6, 7] reported an excess in fat jet production which is kinematically compatible with the decay of a heavy resonance into two vector bosons, generating a wide range of interest [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. A possible interpretation of the observed excess is as an extra \(Z^{\prime }\) or \(W^{\prime }\) with a mass of the order of few TeV [5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. The existence of an extra \(Z'\) inspired from heteroticstring theory attracted considerable interest in the particle physics literature [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. However, constructing string models that allow an \(Z'\) to remain unbroken down to low scales has proven to be very challenging. The reason being that the extra U(1) symmetries that are studied in the literature are either anomalous or have to be broken at the high scale to generate qualitatively realistic fermion mass spectrum. Furthermore, flavour changing neutral current (FCNC) constraints indicate that the extra \(Z^{\prime }\), below the DecaTeV scale, has to be family universal and imposes an additional strong constraint on the viable string vacuum. Extra vector bosons in the TeV region will exclude the majority of heteroticstring models constructed to date. Recently, a semirealistic string derived model that allows for a light \(Z^{\prime }\) model was constructed in Ref. [46].

Family universal U(1)s that admit the SO(10) and \(E_6\) embedding of the standard model charges.

Extra \(W^{\prime }\) and \(Z^{\prime }\) arising in left–right symmetric heteroticstring models.

Family nonuniversal U(1)s.

Hidden sector U(1) symmetries and kinetic mixing.

Extra vector bosons from extensions of the colour group.

Leptophobic and custodial SU(2) symmetries.
2 Additional U(1)s in heteroticstring models
In this section we elaborate on the type of extra gauge bosons that may arise from heteroticstring vacua. Our discussion is in the framework of the free fermionic formulation. Details of the construction and the models that we discuss are given in the references provided and will not be repeated here. In this paper we only mention the features that are relevant for the discussion of the light extra \(W^{\prime }\)s and \(Z^{\prime }\)s, which are obtained from the untwisted Neveu–Schwarz sector. The last category that we consider includes vector bosons from additional sectors.
Under parallel transport around the noncontractible loops of the worldsheet torus of the vacuum to vacuum amplitude, the worldsheet fermions pick up a phase. The allowed phase assignments are constrained by the requirement that the vacuum to vacuum amplitude is invariant under modular transformations. Models in the free fermionic formulation are obtained by specifying a set of boundary basis vectors and the associated oneloop GGSO phases [1, 2, 3], which both must satisfy a set of constraints derived by the requirement that the vacuum to vacuum amplitude is invariant under modular transformations. In this paper we will focus on the socalled NAHEbased models [56], which are typically produced by a set of eight (or nine) boundary condition basis vectors denoted by \(\{\mathbf{1}, S, b_1, b_2, b_3, \alpha , \beta , \gamma \}\), where the set \(\{\mathbf{1}, S, b_1, b_2, b_3\}\) is the socalled NAHE set [56]. The basis vectors of the NAHE set preserve the SO(10) symmetry. Basis vectors that extend the NAHE set may preserve the SO(10) symmetry in which case they are denoted \(b_{4, 5, \ldots }\), or they may break the SO(10) symmetry, in which case they are denoted \(\{\alpha , \beta , \gamma , \ldots \}\). At least one basis vector beyond the NAHE set must break the SO(10) symmetry.
Spacetime vector bosons in the free fermionic models arise from the untwisted Neveu–Schwarz sector and possibly from additional sectors that are obtained from combinations of the basis vectors. The vector bosons from these additional sectors enhance the gauge symmetry which is obtained from the untwisted NS sector. The generators of the SO(10) symmetry and of any additional U(1) symmetries are obtained from the untwisted NS sector. The vector bosons arising in the additional sectors do not play a role in the case of extra gauge symmetries from SO(10) subgroups, or from extra NS U(1) symmetries. They arise in the case of custodial symmetries [57]. The projection of the spacetime vector bosons arising from the untwisted NS sector depends only on the boundary condition basis vectors, and it does not depend on the GGSO phases [1, 2, 3]. The type of enhancement from the additional sectors does depend on the GGSO phases, but it will not play a role in our discussion here. The boundary condition basis vectors, and the GGSO phases, leading to the models that we discuss, are given in the references.
All of the three generation free fermionic models share a common structure due to the underlying SO(10) symmetry and the spectrum available to break the U(1) symmetry which is embedded in SO(10) and is orthogonal to the weak hypercharge. In all these models this extra U(1) symmetry is necessarily broken by a Higgs field with charges identical to those of the righthanded neutrino, i.e. the Standard Model singlet that resides in the 16 spinorial representation of SO(10). The reason is the absence of the adjoint and higher level representations in the massless spectrum of these string models. All the semirealistic models contain three chiral 16 representations of SO(10) decomposed under the final SO(10) subgroup and electroweak Higgs doublet representations that arise from the vectorial 10 representation of SO(10).
One distinction between the models is the scale at which the SO(10) extra U(1) has to be broken. For instance in the case of the FSU5 models it must be broken at the MSSM GUT scale, to generate masses for the \(SU(5)\times U(1)\) vector bosons which mediate proton decay via dimension six operators. In the three other cases it could in principle remain unbroken below that scale, because these models do not contain vector bosons that may mediate proton decay via dimension six operators.
2.1 Observable nonuniversal U(1)s
In addition to the family universal U(1) symmetries in the observable \(E_8\) gauge group, the string models contain two additional U(1) symmetries that are combinations of \(U(1)_{1,2,3}\) and are orthogonal to \(U(1)_\zeta \). These are family nonuniversal and therefore must be heavier than roughly 30 TeV due to flavour changing neutral current (FCNC) constraints [84]. Additional observable \(U(1)_{4,5,6}\) symmetries may arise from complexification of real fermions as discussed above. One combination of those may be family universal while the other two are not. In Ref. [85] it was proposed that the family universal anomaly free combination of \(U(1)_{1,2,3,4,5,6}\) in the model of Refs. [61, 62] plays a role in adequately suppressing proton decay mediating operators, as well as allowing for suppression of lefthanded neutrino masses via the seesaw mechanism. However, it was shown in Ref. [86] that the U(1) discussed in Ref. [85] must in fact be broken near the string scale. This is expected as this U(1) symmetry is a combination of \(U(1)_\zeta \), which is anomalous, with the family universal combination of \(U(1)_{4,5,6}\). In Ref. [87] it was shown that two of the anomaly free nonuniversal combinations may similarly, adequately suppress proton decay and generate small neutrinos via a seesaw mechanism. As discussed above they must be broken above the DecaTeV scale. The additional combinations of \(U(1)_{1,2,3,4,5,6}\), aside from \(U(1)_\zeta \), will not be considered further here.
2.2 Hidden sector U(1)s
In addition to the U(1) symmetries that arise in the observable sector, the string models may contain \(U(1)_h\) symmetries that arise from the hidden \(E_8\) gauge group. Such \(U(1)_h\) symmetries may mix with the weak hypercharge via kinetic mixing [88, 89, 90, 91] provided that there exist light states in the spectrum that are charged under both \(U(1)_Y\) and under the hidden sector \(U(1)_h\) factor. Depending on the details of the spectrum, kinetic mixing may then arise from oneloop radiative corrections [88, 89, 90, 91] and is proportional to \(\mathrm{Tr}Q_YQ_h\).
The existence of hidden \(U(1)_h\) symmetries in semirealistic heteroticstring models is highly model dependent, but there are some generic properties that may be highlighted. The PS class of models typically do not contain U(1) factors in the hidden sector. The reason being that the PS models utilise only periodic/antiperiodic boundary conditions, and that the set of basis vectors that generate a PS model typically contain a single SO(10) breaking vector.
The FSU5 models utilise rational boundary conditions, which break SO(2n) symmetries into \(SU(n)\times U(1)\). Provided that the hidden sector gauge symmetry is not enhanced, the hidden sector may contain unbroken U(1) factors. In the FSU5 model of Ref. [58] the hidden sector gauge group is enhanced and this model does not have any hidden sector U(1) factors. In the FSU5 models that were classified in Ref. [92] all the hidden sector gauge group enhancements are projected out and therefore these FSU5 models do contain two hidden U(1) symmetries.
The SLM [59, 60, 61, 62, 93, 94] and LRS [66, 67] models utilise two basis vectors that break the SO(10) symmetry. These models generically contain several hidden sector U(1) factors, irrespective of whether the hidden sector symmetry is enhanced or not.
We now turn to a discussion of the matter states appearing in the models and the feasibility of kinetic mixing. Before getting into specific SO(10) subgroups several broad observations can be made. All the models that we discuss have \(N=1\) spacetime supersymmetry, but the general properties that we extract are also applicable in tachyon free nonsupersymmetric vacua [95, 96, 97]. The first division of the matter sectors is into those that preserve \(N=4\), and those that preserve \(N=2\), spacetime supersymmetry. In the discussion of kinetic mixing it is sufficient to focus on the \(N=2\) sectors. These sectors are obtained from combinations of the basis vectors \(b_{1,2,3}\) with the other basis vectors. The basis vectors \(b_{1,2,3}\) in the NAHEbased models produce spinorial SO(10) representations that are neutral under the hidden sector. The sectors \(b_i+2\gamma \) produce states that transform as vector representations of the hidden sector gauge group, and they are singlets of the SO(10) subgroup. States that transform in the 10 vector representation of SO(10) are neutral under the hidden sector gauge group. All the sectors discussed thus far therefore cannot give rise to kinetic mixing with the weak hypercharge because they are not charged with respect to both \(U(1)_Y\) and \(U(1)_h\).
States that can induce kinetic mixing in free fermionic models can therefore only arise from sectors that break the SO(10) symmetry. These sectors arise in combinations of the basis vectors \(b_{1,2,3}\) with the SO(10) basis vectors \(\alpha , \beta , \gamma \). Here we can further divide into sectors that break the SO(10) symmetry to the PS or FSU5 subgroups. We will focus here on the examples of the FSU5 and SLM models. In the case of the FSU5 models all SO(10) breaking sectors contain states that carry fractional electric charge. The states may transform as singlets or fiveplets of SU(5) and both types of states will carry fractional electric charge. These states must therefore be decoupled from the massless spectrum [98, 99], or confined [58, 92], at a high scale and cannot generate sizeable kinetic mixing.
The SLM models contain a richer variety of SO(10) breaking sectors, which can be divided according to the surviving SO(10) subgroup, which can be \(SU(5)\times U(1)\), \(SO(6)\times SO(4)\) or \(SU(3)\times SU(2)\times U(1)^2\) [98, 99]. The first two cases produce states with fractional electric charge, which must be either decoupled or confined [98, 99]. The last category of states produces states that carry standard charges under the Standard Model gauge group but carry nonstandard SO(10) charges under \(U(1)_{Z^{\prime }}\). One type of states in these sectors are neutral under the weak hypercharge and therefore cannot generate kinetic mixing. The other type of states arising in these sectors are states that transform as 3, \({\bar{3}}\) and 2, \({\bar{2}}\) of the observable SU(3) and SU(2) groups, respectively, and carry the standard Standard Model charge under \(U(1)_Y\). These states interact via the strong and electroweak interactions, and therefore cannot remain light to the required scale to produce sizeable mixing [88, 89, 90, 91]. We conclude that kinetic mixing of a hidden sector \(U(1)_h\) with \(U(1)_Y\) is not viable in free fermionic models.
3 Light U(1)s
In this section we consider the possibility that an extra U(1) symmetry is leftunbroken in the heteroticstring vacuum; the phenomenological constraints; and the distinctions between the different models. The four cases that we discuss are: (i) the \(U(1)_{Z^{\prime }}\) in Eq. (2.3); (ii) the \(U(1)_{{\mathcal {Z}}^{\prime }}\) in Eq. (2.3); (iii) the nonAbelian left–right symmetric extension \(SU(2)_R\times U(1)_C\); (iv) the PS models. For completeness we also mention two additional cases: (v) the \(SU(4)\times SU(2)\times U(1)_C\) models; (vi) the leptophobic \(Z^{\prime }\) and custodial SU(2) models.
3.1 Case I: low \(BL\) breaking scale
Spectrum and \(SU(3)_C\times SU(2)_L\times U(1)_{Y}\times U(1)_{Z^{\prime }}\) quantum numbers, with \(i=1,2,3\) for the three light generations. The charges are displayed in the normalisation used in free fermionic heteroticstring models
Field  \(SU(3)_C\)  \(\times SU(2)_L \)  \({U(1)}_{Y}\)  \({U(1)}_{Z^{\prime }}\) 

\(Q_L^i\)  3  2  \(+\frac{1}{6}\)  \(+\frac{1}{2}\) 
\(u_L^i\)  \({\bar{3}}\)  1  \(\frac{2}{3}\)  \(+\frac{1}{2}\) 
\(d_L^i\)  \({\bar{3}}\)  1  \(+\frac{1}{3}\)  \(\frac{3}{2}\) 
\(e_L^i\)  1  1  \(+1 \)  \(+\frac{1}{2}\) 
\(L_L^i\)  1  2  \(\frac{1}{2}\)  \(\frac{3}{2}\) 
\(N_L^i\)  1  1  0  \(+\frac{5}{2}\) 
h  1  2  \(\frac{1}{2}\)  \(+1\) 
\({\bar{h}}\)  1  2  \(+\frac{1}{2}\)  \(1\) 
\(\phi ^i\)  1  1  0  0 
\({\mathcal {N}}\)  1  1  0  \(+\frac{5}{2}\) 
\({\bar{\mathcal {N}}}\)  1  1  0  \(\frac{5}{2}\) 
Taking \(m_t\sim 173\)GeV; \(k\sim 1/3\); \(\langle {\bar{\mathcal {N}}}\rangle \sim 3\) TeV we note that to accommodate a tau neutrino mass below 1eV we need \(\langle \phi \rangle \sim 1\)keV. While not impossible, it requires the introduction of a new scale, which may be ad hoc from the string model building perspective [103, 104].
3.2 Case II: high \(BL\) breaking scale
Spectrum and \(SU(3)_C\times SU(2)_L\times U(1)_{Y}\times U(1)_{{\mathcal {Z}}^{\prime }}\) quantum numbers, with \(i=1,2,3\) for the three light generations. The charges are displayed in the normalisation used in free fermionic heteroticstring models
Field  \(SU(3)_C\)  \(\times SU(2)_L \)  \({U(1)}_{Y}\)  \({U(1)}_{Z^{\prime }}\) 

\(Q_L^i\)  3  2  \(+\frac{1}{6}\)  \(\frac{2}{3}\) 
\(u_L^i\)  \({\bar{3}}\)  1  \(\frac{2}{3}\)  \(\frac{2}{3}\) 
\(d_L^i\)  \({\bar{3}}\)  1  \(+\frac{1}{3}\)  \(\frac{4}{3}\) 
\(e_L^i\)  1  1  \(+1 \)  \(\frac{2}{3}\) 
\(L_L^i\)  1  2  \(\frac{1}{2}\)  \(\frac{4}{3}\) 
\(N_L^i\)  1  1  0  0 
\(D^i\)  3  1  \(\frac{1}{3}\)  \(+\frac{4}{3}\) 
\({\bar{D}}^i\)  \({\bar{3}}\)  1  \(+\frac{1}{3}\)  2 
\(H^i\)  1  2  \(\frac{1}{2}\)  2 
\({\bar{H}}^i\)  1  2  \(+\frac{1}{2}\)  \(+\frac{4}{3}\) 
\(S^i\)  1  1  0  \(\frac{10}{3}\) 
h  1  2  \(\frac{1}{2}\)  \(\frac{4}{3}\) 
\({\bar{h}}\)  1  2  \(+\frac{1}{2}\)  \(+\frac{4}{3}\) 
\(\phi ^i\)  1  1  0  0 
3.3 Case III: Low scale left–right symmetric models
Spectrum and \(SU(3)_C\times U(1)_C\times SU(2)_L\times SU(2)_{R}\times U(1)_{\zeta }\) quantum numbers, with \(i=1,2,3\) for the three light generations. The charges are displayed in the normalisation used in free fermionic heteroticstring models
Field  \(SU(3)_C\)  \(\times SU(2)_L \)  \(SU(2)_R \)  \({U(1)}_{C}\)  \({U(1)}_{\zeta }\) 

\(Q_L^i\)  3  2  1  \(+\frac{1}{2}\)  \(\frac{1}{2}\) 
\(Q_R^i\)  \({\bar{3}}\)  1  2  \(\frac{1}{2}\)  \(+\frac{1}{2}\) 
\(L_L^i\)  1  2  1  \(\frac{3}{2}\)  \(\frac{1}{2}\) 
\(L_R^i\)  1  1  2  \(+\frac{3}{2}\)  \(+\frac{1}{2}\) 
\({\mathcal {L}}_R\)  1  1  2  \(+\frac{3}{2}\)  \(+\frac{1}{2}\) 
\({\bar{\mathcal {L}}}_R\)  1  1  2  \(\frac{3}{2}\)  \(\frac{1}{2}\) 
h  1  2  2  0  0 
\(\phi ^i\)  1  1  0  0  0 
3.4 Case IV: low scale Pati–Salam models
Spectrum and \(SU(4)_C\times SU(2)_L\times SU(2)_{R}\) quantum numbers, with \(i=1,2,3\) for the three light generations. The charges are displayed in the normalisation used in free fermionic heteroticstring models
Field  \(SU(4)_C\)  \(\times SU(2)_L \)  \(SU(2)_R \) 

\({\mathcal {Q}}_L^i\)  4  2  1 
\({\mathcal {Q}}_R^i\)  \({\bar{4}}\)  1  2 
\({\mathcal {H}}\)  \({\bar{4}}\)  1  2 
\({\bar{\mathcal {H}}}\)  4  1  2 
D  6  1  1 
h  1  2  2 
\(\phi ^i\)  1  1  0 
A low scale breaking of the PS symmetry may be obtained via the VEV of the neutral scalar component in a \(({\bar{4}},1,2)\) representation, whereas a high scale breaking requires an additional pair of heavy Higgs fields, \({\bar{\mathcal {H}}}\oplus {\mathcal {H}}= ({\bar{4}}, 1,2)_{\mathcal {H}}~\oplus ~({ 4}, 1,2)_{\mathcal {H}}\), to break the symmetry along supersymmetric flat directions. The dimension four operators are induced from the quartic order terms \({\mathcal {Q}}_L{\mathcal {Q}}_L{\mathcal {Q}}_R{\mathcal {Q}}_R\) and \({\mathcal {Q}}_R{\mathcal {Q}}_R{\mathcal {Q}}_R{\mathcal {Q}}_R\). With a low breaking of \(SU(2)_R\) these operators are sufficiently suppressed. The PS model with a high scale breaking include the (6, 1, 1) representation to generate mass to the coloured states of the heavy Higgs states, via the couplings \({\bar{\mathcal {H}}}{\bar{\mathcal {H}}}D+{\mathcal {H}}{\mathcal {H}}D\). With a low scale breaking these states are not required because an additional pair of heavy Higgs states is not required as the breaking can be implemented along a non flat direction. In this model suppression of lefthanded neutrino masses may be obtained by the generations of VEVs of the order of 1keV, similar to the discussion in Sect. 3.1, or may be generated from the quartic order coupling \({\mathcal {Q}}_R {\mathcal {Q}}_R {\mathcal {Q}}_R {\mathcal {Q}}_R\) as in Sect. 3.3. We note that the mass structure of the extra vector states in this PS scenario requires elaborate analysis, with the possibility that the charged \(W^{\prime }\)s are relatively light, whereas the neutral \(U(1)_{Z^{\prime }}\) is comparatively heavy, as is the case in the Standard Model. These considerations raise the prospect that there will be a need to probe the DecaTeV scale and above.
3.5 Case V: low scale \(SU(4)\times SU(2)\times U(1)_L\) models
For completeness we comment on the case with SO(10) broken to the \(SU(4)\times SU(2)\times U(1)_L\) model.^{2} This model was considered in Ref. [116] as a field theory extension of the Standard Model. The field theory model considered in Ref. [116] utilises the Higgs field in the (15, 2, 1) representation of \(SU(4)\times SU(2)\times U(1)_L\), to avoid the relation between the Dirac mass terms of the top quark and the tau neutrino. The string models do not contain such representations and therefore the only available route to satisfy the neutrino mass constraints is to assume \(\langle \phi \rangle \sim 1\) keV. The \(SU(4)\times SU(2)\times U(1)\) choice for the SO(10) subgroup of the string model is attractive because it admits both the doublet–triplet splitting mechanism [117] and the doublet–doublet splitting mechanism [68, 69, 70]. However, as discussed above, while a field theory model consistent with the phenomenological constraints can be constructed [68, 69, 70], it was shown in Refs. [68, 69, 70] that such string models are not viable because it is not possible to form complete families. This demonstrates that the string constructions are more restrictive than the field theory constructions. This is anticipated, as the string framework consistently incorporates gravity into the construction. An alternative method to produce \(SU(4)\times SU(2)\times U(1)\) three generation vacua is by enhancement of the NS gauge group from additional sectors [57, 118, 119].
3.6 Case VI: leptophobic \(Z^{\prime }\) and custodial SU(2)s
Finally, we comment briefly on the possibility of generating leptophobic \(Z^{\prime }\) [118, 119] and custodial SU(2) symmetries [57] in the free fermionic heteroticstring models. As mentioned in Sect. 3.5 the gauge group arising from the NS sector may be enhanced by spacetime vector bosons that are obtained from additional sectors in the additive group. Examples of such three generation string models were presented in Refs. [57, 66, 67, 118, 119]. In these models the three generations still arise from the sectors \(b_{1,2,3}\) and hence descend from the spinorial 16 representations of SO(10), but they transform in representations of the enhanced gauge symmetry. Leptophobic U(1)s are obtained when \(U(1)_{BL}\) combines with the a universal combination of the horizontal flavour symmetries to cancel out the lepton number and produce a gauged \(U(1)_B\) [118, 119]. We note that in the custodial SU(2) model only the lepton transforms as doublets of \(SU(2)_C\) [57]. Hence, the model will have distinct signature compared to the LRS models of Sect. 3.3. Namely, the additional \(W^{\prime }\) vector bosons couple to leptons but not to the hadrons, whereas a leptophobic \(Z^{\prime }\) [118, 119] couples to hadrons but not to the leptons.
4 Prospects at the LHC
In this section we illustrate LHC prospects for a hypothetical phenomenological scenario of a low scale heteroticstring derived \(Z^{\prime }\), based on the high \(BL\) breaking scale model of Sect. 3.2. In particular, we show the LHC 8 TeV Drell–Yan (DY) invariant mass distribution at the nexttonextto leading order (NNLO) in the QCD strong coupling constant (\({\mathcal {O}}(\alpha _s^2)\)) [120], for the production of a \(Z'\) with mass \(M_{Z'} = 3\) TeV.
Recently, the ATLAS [121] and CMS [122] collaborations have published measurements of the DY differential cross section \(\text {d}\sigma /\text {d}M\) in bins of dilepton invariant mass M at centerofmass energies \(\sqrt{S}\) of 7 and 8 TeV. In particular, \(\text {d}\sigma /\text {d}M\) has been measured as a function of the invariant mass of dielectron and dimuon pairs, up to 2 TeV. These measurements are very precise in the mass region around the \(Z_0\) peak, and no significant deviations from the SM prediction have been observed in the mass range explored. However, data at large invariant mass are still affected by large uncertainties due to systematical, statistical and luminosity errors.
The uncertainty associated to scale variation of the NNLO QCD theory prediction amounts to a few percent, while that associated to the parton distribution functions (PDFs) luminosity is larger than 15 %, especially in the large invariant mass region. In this kinematic region PDFs are probed at large x, where they are in general not well constrained. This has a significant impact on the parton luminosity uncertainties as they are the major source of uncertainty and represent a limiting factor to obtain precise predictions for the production of highmass dilepton resonances.
LHC runII will allow us to measure this and other differential observables with higher precision in the highmass region, and thus will confirm or rule out the existence of extra \(Z'\)s in the mass range of a few TeV.
4.1 Details of the calculation
In this section we briefly describe the details of the calculation and the choice of the parameter space. Electroweak corrections [123, 124, 125, 126, 127, 128] are not included here, a more thorough analysis exploiting other differential observables [129, 130] is left for future studies.
The theory is calculated by using an amended version of CandiaDY [131, 132], a program that calculates the DY invariant mass distribution up to NNLO in QCD for a large variety of \(Z'\) string derived models. The full spin correlations as well as the \(\gamma ^*/Z/Z'\) interference effects are included in this calculation. The charge assignment is that of the high \(BL\) breaking scale model described in Sect. 3.2 and is given in Table 2. Furthermore, we have chosen \(\tan \beta =10\), the \(Z'\) coupling constant \(g_z\) equal to the hypercharge \(g_Y\) and \(M_{Z'}=\) 3 TeV.
The main phenomenological results for the high \(BL\) breaking scale model are illustrated in Fig. 1. The left figure shows the \(Z'\) invariant mass distribution (blue) compared to the SM (black) while, in the right figure, the same prediction is normalised to that of the SM in the 1–2 TeV mass range. Bands with different hatching represent the sum in quadrature of the uncertainty relative to the CT14NNLO PDFs [133] rescaled to the 68 % C.L., plus the uncertainty associated to independent variations of the \(\mu _F\) and \(\mu _R\) scales. Different choices for the PDFs, obtained from recent analyses [134, 135, 136] including LHC runI measurements, give similar results. The heteroticstring prediction is almost indistinguishable from the SM in the 1 TeV mass region, and deviations start to be more evident around 2 TeV where the central value starts to rise. In the highmass region far from the resonance, the SM central value is larger than the heteroticstring prediction, but there is a substantial overlap between the two uncertainty bands. The decay width of the \(Z^{\prime }\) predicted by this model is \(\Gamma _{Z^{\prime }} = 8.76\) GeV that is more than three times larger than that of the SM \(Z_0\).
5 Conclusions
In this paper we surveyed the possibility of low scale \(Z^{\prime }\)s and \(W^{\prime }\)s in three generation heteroticstring vacua. The semirealistic free fermionic models produce the Standard Model spectrum and the necessary Higgs states for viable symmetry breaking and fermion mass generation. The models possess the SO(10) embedding of the Standard Model state and the SO(10) normalisation of the weak hypercharge. Hence, they can reproduce viable values of \(\sin ^2\theta _W(M_Z)\) and \(\alpha _s(M_Z)\). These are the first order criteria that a viable string vacuum should obey. The next major constraints on the string models are proton stability and suppression of lefthanded neutrino masses. These two constraints are in tension because on the one hand proton stability prefers a low scale \(U(1)_{Z^{\prime }}\) breaking, whereas suppression of neutrino masses works more naturally with a high scale \(U(1)_{Z^{\prime }}\) breaking. As we discussed in Sect. 3.1 low scale \(U(1)_{Z^{\prime }}\) breaking requires the introduction of the ad hoc VEV \(\langle \phi \rangle \sim 1\)keV. The alternative \({\mathcal {Z}}^{\prime }\) discussed in Sect. 3.2 uses a high scale \(U(1)_{Z^{\prime }}\) breaking but anomaly cancellation necessitates the augmentation of the spectrum into complete 27 multiplets, potentially generating new proton decay operators. We further remark that while field theory models allow much more model building freedom, the straitjacket imposed by synthesising the Standard Model with gravity in the framework of string theory is by far more restrictive.
Each of the cases discussed in Sect. 3 has a distinct signature. Case I in Sect. 3.1 has an additional \(Z^{\prime }\) but no additional states charged under the Standard Model, with the only additional particles being the three righthanded neutrinos. Case II of Sect. 3.2 requires the existence of additional colour triplets and electroweak doublets in the vicinity of the \({\mathcal {Z}}^{\prime }\) breaking scale. Case III of the left–right symmetric models of Sect. 3.3 contains \(W^{\prime }\)s in addition to \(Z^{\prime }\). Similarly, case IV of Sect. 3.4 gives rise to additional vector bosons from the SU(4), and \(SU(2)_R\) group factors. Case V in Sect. 3.5 produces the SU(4) vector bosons but not the \(SU(2)_R\). Finally, in the models of case IV with a leptophobic \(Z^{\prime }\) or custodial SU(2) the additional vector bosons couple to either the quarks or the leptons but not to both.
For case II of Sect. 3.2 we studied the NNLO Drell–Yan invariant mass distribution at the LHC 8 TeV for a \(Z'\) with mass \(M_{Z'} = 3\) TeV, and estimated the main sources of uncertainty in the QCD theory prediction. The uncertainty associated to the partonic content of the proton is the dominant one and is a limiting factor for precision at the present time.
Observations of one or more additional vector bosons at the LHC will choose the right model or eliminate all of the above, and in fact, the majority of semirealistic string models constructed to date. Furthermore, the observation of additional vector bosons at the LHC will restrict the exploration of string vacua, and it will elevate the utility of highenergy dilepton pair production at hadron colliders.
Footnotes
Notes
Acknowledgments
AEF thanks the theoretical physics groups at CERN and Oxford University for hospitality. This work was supported in part by the STFC (ST/L000431/1) and by the Lancaster–Manchester–Sheffield Consortium for Fundamental Physics under STFC Grant ST/L000520/1.
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