Neutrino counting experiments and non-unitarity from LEP and future experiments

Non-unitarity of the neutrino mixing matrix is expected in many scenarios with physics beyond the Standard Model. Motivated by the search for deviations from unitary, we study two neutrino counting observables: the neutrino-antineutrino gamma process and the invisible $Z$ boson decay into neutrinos. We report on new constraints for non-unitarity coming from the first of this observables. We study the potential constraints that future collider experiments will give from the invisible decay of the Z boson, that will be measured with improved precision.


I. INTRODUCTION
Particle physics is currently in an era of great progress, with new experiments [1][2][3][4][5] envisaged for the future. The existence of neutrino oscillations, as well as the discovery of the Higgs Boson are the main motivations for the development of new experiments that will measure the standard physics parameters with unprecedented precision, while also searching for new physics.
In the Standard Model picture, there are three active light neutrinos with an interaction governed by the SU (2) L ⊗ U (1) Y electroweak symmetry [6]. The neutrino mixing in this case is described by an unitary 3 × 3 matrix. If more (heavy) neutrino states exist, the corresponding mixing matrix will be bigger and it will have, at some level, a deviation from unitarity. Such picture has been studied since long time ago [7][8][9][10][11][12][13][14] and, more recently, a description in terms of a triangular parameterization has been discussed [15][16][17][18].
In the presence of such a non-unitary (NU) mixing, neutrino counting experiments at high energies will differ from the Standard Model prediction [19]. This is the case of the invisible decay width of the Z boson [20,21] and also of the ννγ measurements [22][23][24][25][26][27][28][29][30][31][32]. As far as we know, no constraints on non-unitarity have been reported from the ννγ process.
On the opposite side, the current measurement of the invisible decay of the Z boson lies two standard deviations below the Standard Model prediction, a measurement that has already been studied with detail [33]. On the other hand, different proposals for the future generation of collider experiments are currently under development [34], such as ILC [1,35,36], FCCee [2,37], and CEPC [3,4,38,39]. These proposals will be running at the very high energy regime, searching for new physics and measuring the Standard Model parameters in a different energy scale. They will also test physics at relatively lower energies, in order to improve the measurements on already known observables. In particular, it is expected that the invisible Z decay width will be measured with improved precision, if compared to the current reported measurement by LEP [20,21].
In this work we study the constraints arising from the neutrino counting experiments around the Z peak, specifically using data from ννγ measurement. We also analyze the invisible Z decay to have a complete scenario in the same framework and study the potential of future neutrino counting experiments in the same energy regime to constraint the nonunitary parameters, and compare these perspectives with the current constraints. We will show that the perspectives in these future experiments are very promising.
In section II, we will start the discussion by describing the non-unitarity formalism that we will use. Then, in section III we present the analysis used to obtain constraints on the non-unitary parameters, as well as the found results and perspectives for future experiments.
Finally, in section IV we present our conclusions.

II. NON-UNITARITY, INVISIBLE Z DECAY AND ννγ
Non-unitarity has been subject to study for a long time [6,7,40,41]. Recent constraints can be found elsewhere [16,42], either considering only the restrictions coming from neutrino experiments, or including the ones from charged leptons. In both cases it is useful to consider the mixing matrix as describing the transformation of three light neutrinos and n − 3 neutral heavy leptons. In this way, one can see the U n×n matrix as the combination of four submatrices [43] with N a 3 × 3 submatrix in the light neutrino sector, and S the 3 × (n − 3) submatrix that describes the mixing of the extra heavy isosinglet states.
One useful way to parameterize the non-unitarity of the mixing matrix N is the triangular parameterization [15] where U is the unitary PMNS mixing matrix for the standard 3 × 3 case and N N P parameterizes the deviations from unitarity. In this way, we can encode all the parameters of the general description [6,44], for an arbitrary number of additional neutrino states, in a compact notation. In this general framework, we can describe the non-unitary phenomenology by using the three real parameters α 11 , α 22 , and α 33 (all of them close to one) plus other three complex parameters α 21 , α 31 , α 32 that contains extra CP violating phases and whose magnitude is small.
In what follows we will discuss two neutrino counting observables in the context of this triangular parameterization.
A. The invisible Z decay In the standard unitary limit, the branching for the invisible Z decay into neutrinos will be given by [45,46], with N ν the effective number of neutrino families and [46] Experimentally, the ratio Γ inv /Γ ¯ has been measured with greater experimental precision than Γ inv alone [20,46]. Therefore, the number of light active neutrinos can be estimated from this relation, that in the Standard Model is given by [20] with N ν = 3. Here, the decay rate for the Z boson into charged leptons is given by [46] Γ ¯ = where g V and g A are the vector and axial coupling for a charged lepton : When we consider the non-unitarity formalism, applied to the invisible decay rate of the Z boson, we will find that the contribution of the three active neutrino flavors will be given by [46] that can also be expressed as Comparing this expression with the unitary case discussed before, we can define for the non-unitary case It is important to notice that the theoretical expression for the decay rate will be affected by non-unitarity with several corrections. However, we must notice that there is another correction due to the definition of G F . In order to introduce this correction, we can write the equivalent expression to Eq. (5) for the non-unitary case. For this purpose, we start by considering that, from muon decay, a non-unitary mixing will affect the value of the Fermi constant to be [7,41,47] This correction cancels out in the ratio, R 0 inv , but can propagate to other observables, such as the weak mixing angle [42] From Eqs. (9) and (6), we can get an expression for the ratio in the non-unitary case: Let us notice that the deviation from unitarity, introduced by the parameters α ij , appears explicitly in the numerator through |(N N † ) αβ | 2 , but also implicitly in the denominator via g V , because it contains the expression for the weak mixing given in Eq. (11). The explicit form for the numerator in the previous formula will be If we neglect terms including third order or higher on off-diagonal parameters (α ij i = j), we obtain the following reduced expression: Different constraints for the α ij parameters show that N N P is close to an identity matrix.
Besides, the precision of the measurements under consideration makes necessary to introduce radiative corrections. In the M S scheme, the weak mixing angle takes the form [46] where, in the non-unitary case, A 0 is given by ∆r W introduces the radiative corrections, and M W is the mass of the W boson. According to PDG [46], the values of the relevant parameters are: For measurements at energies around the Z peak it is common to use the effective weak mixing angles 2 l instead of the M S scheme; both quantities are related through [46]s 2 l = s 2 Z + 0.00032. Now we can turn now our attention to the comparison with the experimental results to obtain constraints and future perspectives for the NU parameters. However, before entering into this discussion we will also discuss another neutrino counting observable.
The differential cross section for the single photon production from electron-positron annihilation, e + e − → ννγ, can be written in terms of the radiator function H(x, y; s) and the "reduced" cross section for the process e + e − → νν, σ 0 , as [48][49][50]: The radiator function is defined by and σ 0 , the "reduced" cross section for the process e + e − → νν is given by The three terms in Eq. (24) come from the contribution of the W , the Z boson, and their interference, as can be seen in the Feynman diagrams in Fig. 1.
For energies above the Z resonance, finite distance effects on the W propagator need to be considered. These effects are taken into account by the following substitution [32,49,50]: where It is important to notice that the expression in Eq. (24), including the corrections from Eq. (25), is equivalent to the widely used expression: Nevertheless, we will continue using Eq. (24), since the introduction of the non-unitarity effects can be made in a more transparent way.
From Eq. (21), the total cross section is If we now examine this process in a non-unitary mixing framework, it is almost straightforward to obtain the non-unitarity effects in the reduced cross section: These corrections can be seen in Fig. 1 Writing Eq. (29) explicitly, we will have Additionally, as discussed in the previous subsection, there will be NU corrections to G F and sin 2 θ W as described in Eqs. (10) and (11) respectively. Finally, it should be noticed that in the last two terms of Eq. (29), the decay width, Γ Z , appears in the denominator.
Since we are considering the NU case, we must also introduce the corresponding corrections.
The total Z decay width can be calculated as [45,46] Γ Z = Γ inv + Γ + Γ had (31) and the non-unitary correction will appear through the Γ inv contribution, as it had been computed in the previous subsection, and we will have: Now that we have introduced the theoretical expressions for the two neutrino counting observables with the formalism for the non-unitary case, in the triangular parameterization, we will discuss the corresponding current constraints and future perspectives for these two cases.

III. EXPERIMENTAL TESTS
A. The process e − e + → ννγ To obtain constraints for the NU case from the process e − e + → ννγ, we use the reported measurements from the ALEPH [24], DELPHI [25], L3 [26][27][28], and OPAL [29][30][31] collaborations. They are are listed in Table I. The center of mass energy for each run is listed in the first column. The background subtracted measured and Monte Carlo cross sections are given in columns two and three, respectively. The number of observed events after background subtraction are given in column four, while the efficiency corresponds to column five. Lastly, the kinematical cuts for the outgoing photon energy and angle are reported in the last two columns. For these cuts,   Table I through a χ 2 analysis.
Our result for the non-unitary parameter α 11 is shown in Fig. 2, for each experiment separately, and for a combination of all of them. In this analysis, we have considered any other NU parameter as equal to the Standard case, that is, α 2 22 = α 2 33 = 1 and α 2 21 = α 2 31 = α 2 32 = 0. We have chosen this parameter because diagonal parameters α ii give the main contribution for deviations from unitarity. Besides, any diagonal parameter contributes on equal footing and, therefore, our constrain can be equally applied to α 22 or α 33 . As it can be seen, it is possible to restrict the α 11 NU parameter, and the constraint at 90 % CL is given by To our knowledge, this is the first time that a constraint for NU is reported using this observable and it is possible to see that the limits are competitive.

B. The invisible Z decay
We now turn our attention to the particular case of the Z decay into neutrinos. This process has already been measured by LEP [20,21] and future experiments [1][2][3][4][34][35][36][37][38][39] can improve the measurement of this important observable. Previous works have already reported constraints on NU parameters using this observable for a combined analysis from different measurements [42,[51][52][53]. Here we focus on this particular parameter using the specific triangular parameterization and making more emphasis in the perspectives from future experimental proposals.
Before analyzing the invisible decay constraints on NU, it is important to remember from the previous section that the NU case will affect the theoretical prediction of different parameters, such as G F and sin 2 θ W (Eqs. (10) and (11)   relevant energy, differs up to three standard deviations depending on the experiment that measures it. Its impact is illustrated in Fig. 3, where we show the χ 2 curve for this observable as a function of the α 11 parameter. In this figure, besides considering the LEP [20,21] measurement for the weak mixing angle, we also show how this constraint changes if we consider other measurements for the weak mixing angle. That is the case of the Tevatron [54][55][56][57], Atlas [58], LHCb [59] and CMS [60] result. It is possible to notice that the evaluation of this fundamental quantity of the Standard Model still can have an impact on the nonunitarity constraints. As in the previous subsection, for this plot we have only considered α 11 as different from one and all other non-unitary parameters as equal to the standard case, that is, α 2 22 = α 2 33 = 1 and α 2 21 = α 2 31 = α 2 32 = 0. Provided that we have a precise measurement of the weak mixing angle, we can return to the computation of constraints on NU from current and future experimental proposals that will improve the measurements of different observables, such as the number of neutrinos, N ν , or the effective value of the weak mixing angle, sin 2 θ ef f . We show their sensitivity in Table II.
In order to estimate the sensitivity of the future experiments we will consider again the LEP [20] CEPC [4] FCC-ee [2] ILC [1] σ(N ν ) [   ratio given by Eq. (5). In particular, the uncertainty of R 0 inv is calculated from where σ Γνν Γ ll SM = 0.00083 [20] and σ(N ν ) is given in Table II. With these hypothesis we obtain the results shown in Table III. Within this framework, it is possible to obtain an idea of the future sensitivity of these experiments on the NU parameters. A forecast for this sensitivity can be computed considering three different cases of a future measurement of the ratio R 0 inv : • The experimental value reported at [20], R 0 inv = 5.9430.
• The theoretical value calculated from the effective weak mixing angle including radiative corrections [61], R 0 inv = 5.9671.
• A value two standard deviations (of CEPC) above of the previous value, R 0 inv = 5.9801.
To consider these futuristic scenarios, it takes into account the possible non-standard result where the effective number of neutrinos is smaller than three. Besides, it also considers the less expected case where a future experiment might have a statistical fluctuation, and measures a value above the SM prediction. For these three cases, we perform a χ 2 analysis in order to have a forecast of the future expected sensitivity, considering the following two scenarios: • Firstly, we consider that α 11 is the only parameter different from the standard case.
The χ 2 fit is made with the errors already discussed for each experiment. The results are compiled in Fig. 4.
• Secondly, we let α 11 , α 13 and α 33 to vary freely, while fulfilling the Cauchy-Schwarz condition: The other NU parameters are set to their SM value. The results obtained are shown in Fig.5. Notice that we have considered only α 33 and α 31 different from zero, since very similar results will be obtained with α 22 and α 21 .
We summarize the expected accuracy for both cases in Table IV. We can see from these results that future collider experiments could give a constraint on the diagonal non-unitary parameter that will be stronger than the current global limits [16,18,42], that constraints α 11 at the level of 0.999 or below as we see in Table V. It is also interesting to notice what would be the constraint in the case of a measurement different from the SM prediction; as illustrated in the same  Table IV: Allowed values for α 11 at 90 % C.L., considering present experimental values and future proposals from CEPC, FCC-ee and ILC experiments. We consider either the case when any NU parameter other than α 11 is in the unitary limit and also when α 31 and α 33 are allowed to vary, fulfilling the Cauchy-Schwartz condition. potential to show the evidence of new physics through a non-unitarity of the neutrino mixingmatrix.

IV. CONCLUSIONS
We have reviewed the measurements for neutrino counting observables close to the Z peak and reported a new analysis for the non-unitary formalism for the case of the ννγ process.
The corresponding constraints have been introduced in this work and we have shown that they are competitive with other current constraints. As far as we know, this is the first time this analysis is done. We have used the triangular parameterization to perform this analysis.
We have also analyzed the invisible Z decay into neutrinos, in the same triangular parameterization. In this case we have focused in the importance of a precise determination of the weak mixing angle and in the perspectives to improve current constraints by using future collider experiments, that are expected to be constructed as a continuation of the precision program for particle physics. They will allow to obtain better restrictions to new physics from several processes at different energy regimes. For this purpose, we have focused in the invisible decay width in the Z peak, that will be measured in the first stages of the future collider experiments ILC, FCC-ee and CEPC.
We have shown that any of these experiments will have enough sensitivity to improve the current constraint on non-unitarity. We have focused especially in the diagonal parameter α 11 . To obtain this result we have used different test values. In particular, for a measurement as low as the current LEP central value, future experiments will give a positive signal for non-unitarity at 90 % C. L., while a future measurement in accordance with the Standard model prediction will restrict the limit for α 11 to be bigger that 0.999, that is, a precision at the level of 10 −3 . It is also important to notice that, as can be seen from Eq. (13), the Z decay measurement will mainly restrict the sum of the three diagonal parameters: i α ii and, therefore, in a combined analysis, this measurement will help to restrict any of the diagonal parameters.