On Decays of 𝑍^ into Unparticle Stuff

Abstract

We study the decay of a 𝑍^′-boson into 𝑈-unparticle and a photon. The extended Landau-Yang theorem is used. The clear photon signal would make the decay 𝑍^′→𝛾𝑈 as an additional contribution mode for study of unparticle physics.

1. Introduction

In 1982, Banks and Zaks [1] investigated gauge theories containing noninteger number of Dirac fermions where the two-loop 𝛽-function disappears. There is no chance to interpret theory at the nontrivial infrared (IR) fixed point where it possesses the scale-invariant nature in terms of particles with definite masses. The main idea is based on the following statement: at very high energies the theory contains both fields of the standard model (SM) and fields yielding the sector with the IR point. Both of these sectors interact with each other by means of exchange with the particles (fields) having a large mass 𝑀. The hidden conformal sector may flow to IR fixed point at some scale Λ<𝑀, where the interaction between fields has the form ~Λ^{𝑑BZ−𝑑}𝑂_SM𝑂_𝑈𝑀^−𝑎, where 𝑎>0, 𝑑_BZ and 𝑑 mean the scale dimensions of the Banks-Zaks (BZ) sector operator and the operator 𝑂_𝑈 of the 𝑈-unparticle, respectively; 𝑂_SM is the operator of the SM fields.

The unparticles (or the scale-invariant stuff) with continuous mass distribution, introduced by Georgi in 2007 [2, 3], obey the conformal (or scale) invariance. There is an extensive literature (see, e.g., the incomplete set of papers [4–13] and the references therein) concerning the phenomenology of unparticles.

The most interesting scale is Λ∼𝑂 (TeV), at which the dynamics of unparticles could be seen at CERN LHC through the different processes including the decays with the production of 𝑈-unparticles.

It has been emphasized [2, 3] that the renormalizable interactions between the SM fields and the fields of yet hidden conformal sector could be realized by means of exploring the hidden energy at high-energy collisions and/or associated with the registration of noninteger number of invisible particles. In this case, the conformal sector described in terms of “unparticles” does not possess those quantum numbers which are known in the SM.

Unparticle production at hadron colliders will be a signal that the scale where conformal invariance becomes important for particle physics is as low as a few TeV. At this scale, the unparticle stuff sector is strongly coupled. This requires that, somehow, a series of new reactions that involve unparticle stuff in an essential way turn on between the Tevatron and LHC energies. It will be important to understand this transition as precisely as possible. This can be done through the study of 𝑝𝑝,𝑝𝑝→𝛾+𝑈 and the identification of the effects from, for example, 𝑍-,𝑍^′-resonances [14] in 𝑝𝑝,𝑝𝑝→𝑓𝑒𝑟𝑚𝑖𝑜𝑛+𝑎𝑛𝑡𝑖𝑓𝑒𝑟𝑚𝑖𝑜𝑛.

The experimental channels of multigauge boson production ensure the unique possibility to investigate the anomalous triple effects of interaction between the bosons. We point out the study of nonabelian gauge structure of the SM, and, in addition, the search for new types of interaction which, as expected, can be evident at the energies above the electroweak scale. The triple couplings of neutral gauge bosons, for example, 𝑍𝑍^′𝛾, 𝑍𝑍𝑍^′, and so forth, can be studied in pair production at the hadron (lepton) colliders: 𝑝𝑝,𝑝𝑝(𝑒⁺𝑒⁻)→𝑍^′→𝑍𝛾,𝑍𝑍,…

In this paper, we study the production of unparticle 𝑈 in decays of 𝑍^′ with a single photon emission. There is a hidden sector where the main couplings to matter fields are through the gauge fields. Before using the concrete model, we have to make the following retreat. First of all, we go to the extension of the Landau-Yang theorem [15, 16] for the decay of a vector particle into two vector states. Within this theorem, the decay of particle with spin-1 into two photons is forbidden (because both outgoing particles are massless). The direct interaction between 𝑍^′-boson and a vector massive particle, for example, 𝑍 or 𝑈-vector unparticle, accompanied by a photon, does not exist. To the lowest order of the coupling constant 𝑔, the contribution given by 𝑔³ in the decay 𝑍^′→𝛾𝑈 is provided mainly by heavy quarks in the loop. What is the origin of this claim? First, it is worth to remember the known calculation of the anomaly triangle diagram 𝑍𝑍𝛾 [17], where the anomaly contribution result contains two parts, one of which has no dependence of the mass 𝑚_𝑓 of (intermediate) charged fermions in the loop while the second part is proportional to 𝑚²_𝑓. An anomaly term disappears in the case if all the fermions from the same generation are taking into account or the masses of the fermions of each of generation are equal to each other. The reason which explains the above-mentioned note is the equality to zero of the sum Σ_𝑓𝑁^𝑓_𝑐𝑔^𝑓_𝑉𝑔^𝑓_𝐴𝑄_𝑓, where 𝑔^𝑓_𝑉(𝑔^𝑓_𝐴) is the vector (axial-vector) coupling constant of massive gauge bosons to fermions, 𝑄_𝑓 is the fermion charge, 𝑁^𝑓_𝑐=3(1) for quarks (leptons). The anomaly contribution for the decay 𝑍^′→𝛾𝑈 does not disappear due to heavy quarks, and the amplitude of this decay is induced by the anomaly effect. The contribution from light quarks with the mass 𝑚_𝑞 is suppressed as 𝑚²_𝑞/𝑚²_𝑍′∼10⁻⁸-10⁻⁶, where 𝑚_𝑍′ is the mass of 𝑍^′-boson. Despite the decay 𝑍^′→𝛾𝑈 being the rare process, there is a special attention to the sensitivity of this decay to top-quark and even to quarks of fourth generation.

Since the photon has the only vector nature of interaction with the SM fields, the possible types of interaction 𝑍^′−𝑈−𝛾 would be either 𝑉−𝐴−𝑉 or 𝐴−𝑉−𝑉, where 𝑉(𝐴) means the vector (axial-vector) interaction.

2. Setup

Let us consider the following interaction Lagrangian density:−𝐿=𝑔_𝑍′𝑞𝑞𝑣^′_𝑞𝛾^𝜇−𝑎^′_𝑞𝛾^𝜇𝛾₅𝑞𝑍^′_𝜇+1Λ^𝑑−1𝑞𝑞𝑐_𝑣𝛾^𝜇−𝑎_𝑣𝛾^𝜇𝛾₅𝑞𝑂_𝜇𝑈, (2.1) where 𝑔_𝑍′=(5𝑏/3𝑠_𝑊𝑔_𝑍) is the gauge constant of 𝑈^′(1) group (the coupling constant of 𝑍^′ with a quark 𝑞) with the group factor 5/3, 𝑏∼𝑂(1), 𝑔_𝑍=𝑔/𝑐_𝑊; 𝑠_𝑊(𝑐_𝑊)=sin𝜃_𝑊(cos𝜃_𝑊), 𝜃_𝑊 is the angle of weak interactions (often called Weinberg angle); 𝑣^′_𝑞 and 𝑎^′_𝑞 are generalized vector and the axial-vector 𝑈^′(1)-charges, respectively. These latter charges are dependent on both (joint) gauge group and the Higgs representation which is responsible for the breaking of initial gauge group to the SM one; 𝑐_𝑣 and 𝑎_𝑣 are unknown vector and axial-vector couplings. Actually, the second term in (2.1) is identical to the first one up to the factor Λ^1−𝑑.

In conformal theory the unparticle does not have a fixed invariant mass, but instead has a continuous mass spectrum. In the paper we assume that 𝑂_𝜇𝑈 is a nonprimary operator derived by 𝑂_𝜇𝑈(𝑥)=𝜕_𝜇𝑆(𝑥) through the light pseudo-Goldstone field 𝑆(𝑥) which is the consequence of an approximate continuous symmetry. The scalar field 𝑆(𝑥) is a “grandfather” potential which serves as an approximate conformal compensator with continuous mass. The scale dimension of the gauge invariant nonprimary vector operator is 𝑑≥2 as opposed to 𝑑≥3 for primary gauge-invariant vector operators. In conformal theory, a primary operator defines the highest weight of representation of the conformal symmetry and this operator obeys the unitarity condition 𝑑≥𝑗₁+𝑗₂+2−𝛿_𝑗1𝑗2,0, where 𝑗₁ and 𝑗₂ are the operator Lorentz spins (primary means not a derivative of another operator). For the review of the constraints of gauge invariant primary operators in conformal theory and the violations of the unitarity, see [18] and the references therein. Furthermore, we consider 𝑑≥1 which does not contradict the unitarity condition because the operator 𝑂_𝜇𝑈(𝑥)=𝜕_𝜇𝑆(𝑥) is not gauge invariant. Because the conformal sector is strongly coupled, the mode 𝑆(𝑥) may be one of new states accessible at high energies. The operator 𝑂_𝜇𝑈 has both the vector and the axial-vector couplings to quarks in the loop.

We consider the model containing the 𝑍_𝜒-boson on the scale 𝑂(1TeV) in the frame of the symmetry based on the 𝐸₆ effective gauge group [19, 20]. The coupling constant of 𝑈(1)_𝑥 has the form 𝑔_𝜒=5/3𝑒/𝑐_𝑊.

The amplitude of the decay 𝑍^′→𝛾𝑈, where the coupling 𝑍^′𝑈𝛾 is supposed to be extended by the intermediate quark loop, has the following form: 𝐴𝑚𝑧_𝑢,𝑧_𝑞=𝑒²𝑐_𝑊533Λ^𝑑−1𝑞𝑒_𝑞𝑐_𝑣𝑎^′_𝑞+𝑎_𝑣𝑣^′_𝑞𝐼𝑧_𝑢,𝑧_𝑞, (2.2)

with 𝑧_𝑢=𝑃²_𝑈/𝑚²_𝑍′, 𝑧_𝑞=𝑚²_𝑞/𝑚²_𝑍′ for the momentum 𝑃_𝑈 of 𝑈-unparticle and the quarks 𝑞 (in the loop) with the mass 𝑚_𝑞. We deal with the following expression for 𝐼(𝑧_𝑢,𝑧_𝑞): 1𝐼=1−𝑧_𝑢12+𝑧_𝑞1−𝑧_𝑢𝐹𝑧_𝑞𝑧−𝐹^𝑞𝑧_𝑢−121−𝑧_𝑢𝐺𝑧_𝑞𝑧−𝐺^𝑞𝑧_𝑢, (2.3)

adopted for the decay 𝑍^′→𝛾𝑈 taking into account the results obtained in [20, 21]. For heavy quarks, 𝑚_𝑞>0.5𝑚_𝑍′, the functions 𝐹(𝑥) and 𝐺(𝑥) in (2.3) are 𝐹(𝑥)=−2sin⁻¹14𝑥²,𝐺(𝑥)=2√4𝑥−1sin⁻¹1,4𝑥 (2.4) while for light quarks (𝑚_𝑞<0.5𝑚_𝑍′), one has to use the following formulas: 𝐹(𝑥)=12𝑦ln⁺𝑦⁻²𝑦+𝑖𝜋ln⁺𝑦⁻−𝜋²2,𝐺(𝑥)=√𝑦1−4𝑥ln⁺𝑦⁻+𝑖𝜋, (2.5) where 𝑦^±=1±√1−4𝑥. The variable 𝑧_𝑢 is related to the photon energy 𝐸_𝛾 as 𝑧_𝑢=1−2𝐸_𝛾/𝑚_𝑍′. In the frame of the 𝑍_𝜒-model, we choose 𝑣^′_up=0,𝑎^′_up=√6𝑠_𝑊/3,𝑣^′_down=2√6𝑠_𝑊/3,𝑎^′_down=−√6𝑠_𝑊/3 for 𝑢𝑝 and 𝑑𝑜𝑤𝑛 quarks. Actually, the account of the only light quarks leads to the zeroth result for the amplitude (2.2). In the heavy quark sector, the contribution ∼𝑎_𝑣𝑣^′_𝑞 is nonzero for the only 𝑏-quarks and down quarks of fourth generation. We emphasize that the nonvanishing result for the amplitude 𝑍^′→𝛾𝑈 is the reflection of the anomaly contribution due to the presence of heavy quarks.

3. Decay Rate

In the decay 𝑍^′→𝛾𝑈, the unparticle cannot be identified with the definite invariant mass. 𝑈-stuff possesses by continuous mass spectrum, and can not be in the rest frame (there is the similarity to the massless particles). Since the unparticles are stable (and do not decay), the experimental signal of their identification could be looking through the hidden (missing) energy and/or the measurement of the momentum distributions when the 𝑈-unparticle is produced in 𝑍^′→𝛾𝑈 decay.

The differential distribution of the decay width—Γ(𝑍^′→𝛾𝑈) over the variable 𝑧_𝑢 looks like (see also [14]): 𝑑Γ𝑑𝑧_𝑢=12𝑚_𝑍′^𝑞𝑀_𝑞²𝐴_𝑑16𝜋²𝑚²_𝑍′^𝑑−1𝑧_𝑢^𝑑−21−𝑧_𝑢, (3.1) where _𝑞𝑀_𝑞²=16𝜋⁴𝑧_𝑢1−𝑧_𝑢²1+𝑧_𝑢𝐴𝑚𝑧_𝑢,𝑧_𝑞²𝑚²_𝑍′, (3.2)

and [2, 3] 𝐴_𝑑=16𝜋^5/2(2𝜋)^2𝑑Γ(𝑑+1/2)Γ()Γ().𝑑−12𝑑 (3.3)

One of the requirements applied to the amplitude in (3.2) is that it disappears in case of “massless” unparticle (Landau-Yang theorem), and when 𝑧_𝑢=1.

Some bound regimes in (3.1) may be both useful and instructive for further investigation. For this, we consider the quark-loop couplings in the amplitude (2.2) as the sum of the contributions given by light quarks 𝑞 and heavy ones 𝑄 (𝑄 may be referred to the quarks of fourth generation as well): _𝑞𝑒_𝑞𝑐_𝑣𝑎^′_𝑞+𝑎_𝑣𝑣^′_𝑞𝐼𝑧_𝑢,𝑧_𝑞+_𝑄𝑒_𝑄𝑐_𝑣𝑎^′_𝑄+𝑎_𝑣𝑣^′_𝑄𝐼𝑧_𝑢,𝑧_𝑄. (3.4) For the SM quarks with 𝑧_𝑞<1/4 and (𝑧_𝑞/𝑧_𝑢)>1/4 one gets the one-loop function 𝐼(𝑧_𝑢,𝑧_𝑞)𝐼𝑧_𝑢,𝑧_𝑞≃121−𝑧_𝑢13−𝑖𝜋1+𝑧_𝑢, (3.5) and the distribution 𝑑Γ/𝑑𝑧_𝑢 has the form 𝑑Γ𝑑𝑧_𝑢≃5𝐴_𝑑619+𝜋²𝑠_𝑊𝑐_𝑊²𝛼2𝜋²²𝑚²_𝑍′Λ^2𝑑−1𝑧_𝑢^𝑑−11−𝑧²_𝑢1+2𝑧_𝑢. (3.6) On the other hand, if the quarks inside the loop become heavy enough, 𝑧_𝑄>1/4, we estimate the following function 𝐼(𝑧_𝑢,𝑧_𝑄)𝐼𝑧_𝑢,𝑧_𝑄≃112𝑧_𝑄1−𝑧_𝑢121+𝑧_𝑢18𝑧_𝑄−1+1, (3.7) which is very small in the limit 𝑧_𝑄≫1.

The photon energy 𝐸_𝛾=𝑚_𝑍′(1−𝑧_𝑢)/2 in the limit 𝑧_𝑢→0 gets its finite value. The limit 𝑧_𝑢→1 is trivial and we do not consider this.

Within the fact of the combination 𝑐_𝑣𝑎^′_𝑞+𝑎_𝑣𝑣^′_𝑞 in (2.2) the decay amplitude of 𝑍^′→𝛾𝑈 does not disappear when the summation on all the quarks degree of freedom is performed.

Constraints on the unparticle parameter Λ can be obtained through, for example, limits on measurable collider phenomenology. In particular, it has been noted [22] that this bound never dips below 1 TeV. In Figure 1, the monophoton energy distribution 𝐸_𝛾⁻¹𝑑Γ/𝑑𝑧_𝑢 is presented for a range of photon energy 𝐸_𝛾 = 0–500 GeV and various choices of 𝑑. For simplicity, we use the 𝐸_𝜒-model assuming the flavor blind universality 𝑐_𝑣=𝑎_𝑣=1 for all three generation quarks; Λ and 𝑚_𝑍′ are set to be 1 TeV each.

Figure 1 

Energy distribution 𝐸𝛾−1𝑑Γ/𝑑𝑧𝑢×108 for 𝐸𝛾=0-500 GeV, depending on 𝑑=1.1,1.2,1.5,1.8; 𝑐𝑣=𝑎𝑣=1, Λ = 1 TeV, 𝑚𝑍′= 1 TeV.

The sensitivity of the scale dimension 𝑑 to the energy distribution is evident. As 𝑑 moves away from unity, the energy spectrum begins to flatten out gradually, excepting 𝑑=1.2 distribution which is above 𝑑=1.1. Such a behaviour is given by the factor (3.3).

The 𝑈-unparticle could behave as a very broad vector boson since its mass could be distributed over a large energy spectrum. The production cross-section into each energy bin could be much smaller than in the case where an SM vector boson has that particular mass. This may be the reason why we have not yet seen the 𝑈-unparticle trace in the experiment.

In the appropriate approximation when the relation between the total decay width Γ_𝑍′ of 𝑍^′-boson and 𝑚_𝑍′ is small, the contribution to the cross-section of the process 𝑝𝑝→𝑍^′→𝛾𝑈 can be separated into 𝑍^′ production cross-section 𝜎(𝑝𝑝→𝑍^′) and the branching ratio of the decay 𝑍^′→𝛾𝑈, 𝐵(𝑍^′→𝛾𝑈)=Γ(𝑍^′→𝛾𝑈)/Γ_𝑍′: 𝜎(𝑝𝑝→𝑍^′→𝛾𝑈)=𝜎(𝑝𝑝→𝑍^′)⋅𝐵(𝑍^′→𝛾𝑈). The 𝑍^′-boson can be directly produced at a hadron collider via the quark-antiquark annihilation subprocess 𝑞𝑞→𝑍^′, for which the cross-section in the case of infinitely narrow 𝑍^′ is given by 𝜎𝑞𝑞⟶𝑍^′=𝑘_QCD4𝜋²3Γ𝑍^′⟶𝑞𝑞𝑚_𝑍′𝛿𝑠̂−𝑚²_𝑍′, (3.8) where 𝑘_𝑄𝐶𝐷≃1.3 represents the enhancement from higher-order QCD processes. Conservation of the energy-momentum implies that the invariant mass of 𝑍^′ is equal to the parton center-of-mass energy √𝑠̂, with 𝑠̂=𝑥₁𝑥₂𝑠; 𝑥₁ and 𝑥₂ are the fractions of the momenta carried by partons in the process 𝑞𝑞→𝑍^′. The decay width Γ(𝑍^′→𝑞𝑞) is Γ𝑍^′⟶=𝐺𝑞𝑞^𝐹𝑚²_𝑍6𝜋√2𝑁_𝑐𝑚_𝑍′1−4𝑧_𝑞𝑣^′_𝑞²1+2𝑧_𝑞+𝑎^′_𝑞²1−4𝑧_𝑞, (3.9) where 𝐺_𝐹 is the Fermi coupling constant. In the narrow width approximation, the cross-section (3.8) reduces to (𝑧_𝑞≪1) 𝜎𝑞𝑞⟶𝑍^′≃𝑘_QCD2𝑎3𝐺_𝐹√2𝑚_𝑍𝑚_𝑍′²𝑣^′_𝑞²+𝑎^′_𝑞²𝑠/̂𝑚²_𝑍′−1²+𝑎², (3.10) where 𝑎=Γ_𝑍′/𝑚_𝑍′.

For the SM quarks, 𝑧_𝑞≤1/4, the branching ratio 𝐵(𝑍^′→𝛾𝑈) is 𝐵=5𝐴_𝑑13𝑎9+𝜋²𝑠_𝑊𝑐_𝑊²𝛼2𝜋²²𝑚²_𝑍′Λ^2𝑑−11𝑑()+2𝑑+2().𝑑+1)(𝑑+3 (3.11) In Figure 2, the branching ratio 𝐵(𝑍^′→𝛾𝑈) is presented with the assumption 𝑐_𝑣=𝑎_𝑣=1 for all three generation quarks; Λ is set to be 1 TeV, the range of 𝑑 is chosen as =1.1,1.2,1.5,1.8 for 𝑚_𝑍′= 0.5–3.0 TeV. The width Γ_𝑍′ is chosen in the framework of the sequential SM (𝑍^′_SSM), where the ratio Γ_𝑍′/𝑚_𝑍′ has the maximal value 𝑎=0.03 among the grand unification theories- (GUT-) inspired 𝑍^′ models [23].

Figure 2 

Branching ratio 𝐵(𝑍′→𝛾𝑈)×107 for 𝑐𝑣=𝑎𝑣=1, Λ=1 TeV depending on 𝑑=1.1,1.2,1.5,1.8 and 𝑚𝑍′= 0.5–3.0 TeV with Γ𝑍′ given by the 𝑍′SSM-model (𝑎=0.03).

We find the smooth increasing of 𝐵(𝑍^′→𝛾𝑈) with 𝑚_𝑍′ and its decreasing with the dimension 𝑑 excepting 𝑑=1.2 branching ratio.

In Figure 3, we plot the cross-section 𝜎(𝑞𝑞→𝑍^′→𝛾𝑈) in the case of up-quarks annihilation, where 𝑥₁∼𝑥₂∼√𝑥_min, 𝑥_min=𝑚²_𝑍′/𝑠; the range of 𝑑 is chosen as =1.1,1.2,1.5,1.8 for 𝑚_𝑍′= 0.5–3.0 TeV; 𝑎=0.03.

Figure 3 

Cross-section 𝜎(𝑞𝑞→𝑍′→𝛾𝑈)×102,𝑓𝑏 with the assumption of up quarks annihilation, where 𝑑=1.1,1.2,1.5,1.8,𝑚𝑍′=0.5−3.0 TeV, 𝑎=0.03.

For 100𝑓𝑏⁻¹ luminosity at the LHC, the detection of the process 𝑍^′→𝛾𝑈 can be achieved with about 10 signal events for 𝑚_𝑍′∼𝑂(1TeV) at 𝑑=1.1.

4. Constraints

The hidden sector can be strongly constrained by existing experimental data. One of the important and practical implications for unparticle phenomenon is the analysis of the operator form Λ^{𝑑BZ−𝑑}𝑀^𝑑BZ−2|H|²𝑂_𝑈, (4.1)

containing the SM Higgs field 𝐻. Within the Higgs vacuum expectation value (𝑣) requirement, the theory becomes nonconformal below the scale ΛΛ=^{𝑑BZ−𝑑}𝑀^𝑑BZ−2𝑣^{21/(4−𝑑)}<Λ, (4.2) where 𝑈-unparticle sector becomes a standard sector. For practical consistency we require Λ<√𝑠. It implies that unparticle physics phenomena can be seen at high-energy experiment with the energies Λ𝑠>^{𝑑BZ−𝑑}𝑀^𝑑BZ−2𝑣^{22/(4−𝑑)}, (4.3) even when 𝑑→𝑑_BZ. Note, that any observable involving operators 𝑂_SM and 𝑂_𝑈 in (4.1) will be given by the operator ̂Λ𝑜=^{𝑑BZ−𝑑}𝑀^{𝑑BZ+𝑛−42}𝑠^{𝑑+𝑛−4}, (4.4) where 𝑛 is the dimension of the SM operator. Then, the observation of the unparticle sector is bounded by the minimal energy 𝑠>𝑜̂^1/𝑛𝑀²𝑣𝑀^4/𝑛. (4.5)

In (4.5) we don't have dependence on both 𝑑- and 𝑑_BZ-dimensiions. The main model parameter is the mass 𝑀 of heavy messenger. If the experimental deviation from the SM is detected at the level of the order 𝑜̂∼1% at 𝑛=4, the lower bound on √𝑠 would be from 0.9 TeV to 2.8 TeV for 𝑀′𝑠 running from 10 TeV to 100 TeV, respectively. Thus, both the Tevatron and the LHC are the ideal colliders where the unparticle physics can be tested well.

5. Conclusion

We studied the decay of the extra neutral gauge boson 𝑍^′ into a vector 𝑈-unparticle and a photon. Both vector and axial-vector couplings to quarks play a significant role. The energy distribution for 𝑝𝑝→𝑍^′→𝛾𝑈 can discriminate 𝑑. The branching ratio 𝐵(𝑍^′→𝛾𝑈) is at best of the order of 10⁻⁷ for the scale dimension 𝑑=1.1. For larger 𝑑, the branching ratio is at least smaller by one order of the magnitude. Unless the LHC can collect a very large sample of 𝑍^′-bosons, detection of 𝑈 through the decay 𝑍^′→𝛾𝑈 would be challenging compared to the decay 𝑍→𝛾𝑈, where the branching ratio 𝐵(𝑍→𝛾𝑈)∼10⁻⁸ [14].

For 100𝑓𝑏⁻¹ integrated luminosity, the detection of 𝑍^′→𝛾𝑈 can be with about 10 signal events at 𝑑=1.1 for a 1.0 TeV 𝑍^′, while for larger values of 𝑑 there is the decreasing of the events number.

For the case when 𝑍^′-boson has continuously distributed mass [24], the branching ratio has an additional suppression factor due to nonzero internal decay width Γ_𝑍^int_′ in formulas: ^∞₀𝜌(𝑡)𝑑𝑡𝑝²≃1−𝑡−𝑖𝜖𝑝²−𝑚²_𝑍′+𝑖𝑚_𝑍′Γ_𝑍^int_′(𝑡)=1,𝜌𝜋Γ_𝑍^int_′𝑚_𝑍′𝑡−𝑚²_𝑍′²+Γ_𝑍^2int_′𝑚²_𝑍′. (5.1) The experimental estimation of 𝐵(𝑍^′→𝛾𝑈) could provide the quantity Γ_𝑍^int_′, and since the 𝛾-quantum energy has a continuous spectrum, by measuring the photon energy spectrum in the 𝑍^′-decay, one can discriminate the presence of the 𝑈-unparticle or not.

We have shown numerical results for 𝑍^′-bosons associated with the 𝑍_𝜒-model. The calculations are easily applicable to other extended gauge models, for example, little Higgs scenario models, left-right symmetry model, and sequential SM.