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

TL;DR

This paper investigates how neutrino counting experiments can constrain non-unitarity in the neutrino mixing matrix, providing new bounds from gamma processes and exploring future collider sensitivities to invisible Z decays.

Contribution

It introduces new constraints on neutrino non-unitarity from gamma process measurements and assesses future collider capabilities for improved bounds via Z boson decay observations.

Findings

New bounds on non-unitarity from gamma process data

Projected improvements from future collider experiments

Enhanced sensitivity to neutrino mixing deviations

Abstract

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.

Tables

Table 1. Table 1: Summary from the ALEPH  Barate et al. ( 1998a , b ); Heister et al. ( 2003 ) , DELPHI  Abreu et al. ( 2000 ) , L3  Acciarri et al. ( 1997 , 1998 , 1999 ) , and OPAL  Ackerstaff et al. ( 1998 ); Abbiendi et al. ( 1999 , 2000 ) collaboration experimental data, collected above the W + ​ W − superscript 𝑊 superscript 𝑊 W^{+}W^{-} production threshold.

		$\sqrt{s}$ (GeV)	${\sigma }^{\mathrm{mes}}$ (pb)	${\sigma }^{\mathrm{MC}}$ (pb)	$N_{\mathrm{obs}}$	$ϵ(\%)$	$E_{\gamma }$ (GeV)	$\left|y\right|$
ALEPH	Barate et al. (1998a)	161	$5.3\pm 0.83$	$5.81\pm 0.03$	41	70	$x_T\ge 0.075$	$\le 0.95$
		172	$4.7\pm 0.83$	$4.85\pm 0.04$	36	72	$x_T\ge 0.075$	$\le 0.95$
	Barate et al. (1998b)	183	$4.32\pm 0.34$	$4.15\pm 0.03$	195	77	$x_T\ge 0.075$	$\le 0.95$
	Heister et al. (2003)	189	$3.43\pm 0.17$	$3.48\pm 0.05$	484	81.5	$x_T\ge 0.075$	$\le 0.95$
		192	$3.47\pm 0.40$	$3.23\pm 0.05$	81
		196	$3.03\pm 0.23$	$3.26\pm 0.05$	197
		200	$3.23\pm 0.22$	$3.12\pm 0.05$	231
		202	$2.99\pm 0.29$	$3.07\pm 0.05$	110
		205	$2.84\pm 0.22$	$2.93\pm 0.05$	182
		207	$2.67\pm 0.17$	$2.80\pm 0.05$	292
DELPHI	Abreu et al. (2000)	189	$1.80\pm 0.20$	$1.97$	146	51	$x\ge 0.06$	$\le 0.7$
		183	$2.33\pm 0.36$	$2.08$	65	54	$x\ge 0.02$	$\le 0.85$
		189	$1.89\pm 0.22$	$1.94$	155	50	$x\le 0.9$	$\le 0.98$
L3	Acciarri et al. (1997)	161	$6.75\pm 0.93$	$6.26\pm 0.12$	57	80.5	$\ge 10$	$\le 0.73$
							and	and
		172	$6.12\pm 0.90$	$5.61\pm 0.10$	49	80.7	$E_T\ge 6$	0.80-0.97
	Acciarri et al. (1998)	183	$5.36\pm 0.40$	$5.62\pm 0.10$	195	65.4	$\ge 5$	$\le 0.73$
							and	and
	Acciarri et al. (1999)	189	$5.25\pm 0.23$	$5.29\pm 0.06$	572	60.8	$E_T\ge 5$	0.80-0.97
OPAL	Ackerstaff et al. (1998)	130	$10.0\pm 2.34$	$13.48\pm 0.22$	19	81.6	$x_T>0.05$	$\le 0.82$
							and	and
		136	$16.3\pm 2.89$	$11.30\pm 0.20$	34	79.7	$x_T>0.1$	$\le 0.966$
	Abbiendi et al. (1999)	130	$11.6\pm 2.53$	$14.26\pm 0.06$	21	77	$x_T>0.05$	$\le 0.966$
		136	$14.9\pm 2.45$	$11.95\pm 0.07$	39	77.5
	Ackerstaff et al. (1998)	161	$5.30\pm 0.83$	$6.49\pm 0.08$	40	75.2	$x_T>0.05$	$\le 0.82$
							and	and
		172	$5.50\pm 0.83$	$5.53\pm 0.08$	45	77.9	$x_T>0.1$	$\le 0.966$
	Abbiendi et al. (1999)	183	$4.71\pm 0.38$	$4.98\pm 0.02$	191	74.2	$x_T>0.05$	$\le 0.966$
	Abbiendi et al. (2000)	189	$4.35\pm 0.19$	$4.66\pm 0.03$	643	82.1	$x_T>0.05$	$\le 0.966$

Table 2. Table 2: Expected uncertainties on N ν subscript 𝑁 𝜈 N_{\nu} and sin 2 ⁡ θ e ​ f ​ f superscript 2 subscript 𝜃 𝑒 𝑓 𝑓 \sin^{2}{\theta_{eff}} for different experimental proposals. Notice that for LEP we quote the present experimental values, whereas for CEPC, FCC-ee, and ILC we show future estimations.

	LEP Schael et al. (2006)	CEPC Ahmad et al. (2015)	FCC-ee Bicer et al. (2014)	ILC Baer et al. (2013)
$\sigma \left(N_{\nu }\right)\left[{10}^{-3}\right]$	$8.0$	$3.0$	$4.0$	$4.0$
$\sigma \left({\sin }^2{\theta }_{eff}\right)\left[{10}^{-4}\right]$	$1.6$	$0.23$	$0.01$	$0.1$

Table 3. Table 3: Test values for the invisible ratio R i ​ n ​ v 0 subscript superscript 𝑅 0 𝑖 𝑛 𝑣 R^{0}_{inv} used in the present work. We quote the expected uncertainty coming from future experiments.

		CEPC
$R_{inv}^0$	$\mathrm{\hspace{0.33em}5.9430}\pm 0.0065$	$5.9671\pm 0.0065$	$5.9801\pm 0.0065$
		FCC-ee / ILC
$R_{inv}^0$	$\mathrm{\hspace{0.33em}5.9430}\pm 0.0083$	$5.9671\pm 0.0083$	$5.9801\pm 0.0083$

Table 4. Table 4: Allowed values for α 11 subscript 𝛼 11 \alpha_{11} at 90 90 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 subscript 𝛼 11 \alpha_{11} is in the unitary limit and also when α 31 subscript 𝛼 31 \alpha_{31} and α 33 subscript 𝛼 33 \alpha_{33} are allowed to vary, fulfilling the Cauchy-Schwartz condition.

Experiment	$R_{inv}^0$	${\alpha }_{11}$	${\alpha }_{11}$ (${\alpha }_{31}$ and ${\alpha }_{33}$ free)
current	$5.9430$
CEPC	$5.9430$
FCC-ee/ILC	$5.9430$
CEPC	$5.9671$
FCC-ee/ILC	$5.9671$
CEPC	$5.9801$
FCC-ee/ILC	$5.9801$

Table 5. Table 5: Current bounds on non-unitary α 11 subscript 𝛼 11 \alpha_{11} parameters coming from  Fernandez-Martinez et al. ( 2016 ); Blennow et al. ( 2017 ); Escrihuela et al. ( 2017 ) .

Current bounds
${\alpha }_{11}\left(90\%\text{C.L.}\right)$ Escrihuela et al. (2017)		${\alpha }_{11}\left(2\sigma \right)$ Fernandez-Martinez et al. (2016); Blennow et al. (2017)
One parameter	All parameters
${\alpha }_{11}>0.98000$	${\alpha }_{11}>0.96000$	${\alpha }_{11}>0.99875$

Equations

$$U^{n\times n}=\left(\begin{array}[]{cc}N&S\\ V&T\end{array}\right),$$

$$N=N^{NP}U=\left(\begin{array}[]{ccc}\alpha_{11}&0&0\\ \alpha_{21}&\alpha_{22}&0\\ \alpha_{31}&\alpha_{32}&\alpha_{33}\end{array}\right)U\,,$$

$${\Gamma }_{inv}=N_{\nu }{\Gamma }_{\nu \bar{\nu }}$$

$${\Gamma }_{\nu \bar{\nu }}=\frac{G_F{M}_Z^3}{12\sqrt{2}\pi }.$$

$$R_{inv}^0\equiv \frac{{\Gamma }_{inv}}{{\Gamma }_{\ell \bar{\ell }}}=N_{\nu }{\left(\frac{{\Gamma }_{\nu \bar{\nu }}}{{\Gamma }_{\ell \bar{\ell }}}\right)}_{SM},$$

$${\Gamma }_{\ell \bar{\ell }}=\frac{G_F{M}_Z^3\left({g_V^{\ell }}^2+{g_A^{\ell }}^2\right)}{6\sqrt{2}\pi },$$

$$g_V^{\ell }=T_{\ell }-2Q_{\ell }{\sin }^2{\theta }_W,$$

$$g_A^{\ell }=T_{\ell }.$$

$${\Gamma }_{inv}=\frac{G_F{M}_Z^3{\sum }_{i,j}|(N^{†}N{)}_{ij}{|}^2}{12\sqrt{2}\pi },$$

$${\Gamma }_{inv}=\frac{G_F{M}_Z^3{\sum }_{\alpha ,\beta }|(N{N}^{†}{)}_{\alpha \beta }{|}^2}{12\sqrt{2}\pi }.$$

$$N_{\nu }=\sum _{\alpha ,\beta }|(N{N}^{†}{)}_{\alpha \beta }{|}^2.$$

$$G_F=\frac{G_{\mu }}{\sqrt{{\sum }_{ij}|N_{\mu i}{|}^2|N_{ej}{|}^2}}=\frac{G_{\mu }}{\sqrt{{\alpha }_{11}^2({\alpha }_{22}^2+|{\alpha }_{21}{|}^2)}}.$$

$${\sin }^2{\theta }_W=\frac{1}{2}\left(1-\sqrt{1-\frac{2\sqrt{2}\alpha \pi }{G_{\mu }{M}_Z^2}\sqrt{{\alpha }_{11}^2({\alpha }_{22}^2+|{\alpha }_{21}{|}^2)}}\right).$$

$$R_{inv}^0=\frac{{\sum }_{\alpha ,\beta }|(N{N}^{†}{)}_{\alpha \beta }{|}^2}{2({g_V^{\ell }}^2+{g_A^{\ell }}^2)}.$$

$$\sum _{\alpha ,\beta }|(N{N}^{†}{)}_{\alpha \beta }{|}^2$$

$$\sum _{\alpha ,\beta }|(N{N}^{†}{)}_{\alpha \beta }{|}^2={\alpha }_{11}^4+{\alpha }_{22}^4+{\alpha }_{33}^4+2{\alpha }_{11}^2(|{\alpha }_{21}{|}^2+|{\alpha }_{31}{|}^2)+2{\alpha }_{22}^2(|{\alpha }_{21}{|}^2+|{\alpha }_{32}{|}^2)+2{\alpha }_{33}^2(|{\alpha }_{31}{|}^2+|{\alpha }_{32}{|}^2).$$

$${\hat{s}}_Z=\frac{A_0}{M_W(1-\Delta {\hat{r}}_W{)}^{1/2}},$$

$$A_0={\left(\frac{\pi \alpha }{\sqrt{2}{G}_F}\right)}^{1/2}={\left(\frac{\pi \alpha \sqrt{{\alpha }_{11}^2({\alpha }_{22}^2+|{\alpha }_{21}{|}^2)}}{\sqrt{2}{G}_{\mu }}\right)}^{1/2},$$

$$M_W$$

$$\Delta {\hat{r}}_W$$

$$\alpha$$

$$G_{\mu }$$

$${{\hat{s}}_Z}^2$$

$$\frac{d^2\sigma }{dxdy}=H(x,y;s){\sigma }_0(s(1-x)).$$

$$H(x,y;s)=\frac{2\alpha }{\pi }\frac{\left[(1-\frac{1}{2}x{)}^2+\frac{1}{4}{x}^2{y}^2\right]}{x(1-y^2)},$$

$$x=2E_{\gamma }/\sqrt{s},y=\cos {\theta }_{\gamma },$$

$${\sigma }_0(s)$$

$${\sigma }_W(s)$$

$${\sigma }_{W-Z}(s)$$

$$F_W(z)$$

$$F_{W-Z}(z)$$

$${\sigma }_0(s)$$

$$\displaystyle+\frac{G_{F}^{2}}{\pi}M_{W}^{2}\Bigg{\{}\frac{s+2M_{W}^{2}}{2s}-\frac{M_{W}^{2}}{s}\left(\frac{s+M_{W}^{2}}{s}\right)\log\left(\frac{s+M_{W}^{2}}{M_{W}^{2}}\right)$$

Full text

I Introduction

Particle physics is currently in an era of great progress, with new experiments Baer et al. (2013); Bicer et al. (2014); Dong et al. (2018); Ahmad et al. (2015); Boland et al. (2016) 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 $SU(2)_{L}\otimes U(1)_{Y}$ electroweak symmetry Schechter and Valle (1980). The neutrino mixing in this case is described by an unitary $3\times 3$ 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 Nardi et al. (1994); Antusch et al. (2006); Smirnov and Zukanovich Funchal (2006); Dev and Mohapatra (2010); Ohlsson et al. (2010); Akhmedov et al. (2013); de Gouvêa and Kobach (2016); Antusch and Fischer (2016) and, more recently, a description in terms of a triangular parameterization has been discussed Escrihuela et al. (2015, 2017); Miranda and Valle (2016); Blennow et al. (2017).

In the presence of such a non-unitary (NU) mixing, neutrino counting experiments at high energies will differ from the Standard Model prediction Gonzalez-Garcia et al. (1990). This is the case of the invisible decay width of the $Z$ boson Schael et al. (2006, 2013) and also of the $\nu\bar{\nu}\gamma$ measurements Barate et al. (1998a, b); Heister et al. (2003); Abreu et al. (2000); Acciarri et al. (1997, 1998, 1999); Ackerstaff et al. (1998); Abbiendi et al. (1999, 2000); Hirsch et al. (2003). As far as we know, no constraints on non-unitarity have been reported from the $\nu\bar{\nu}\gamma$ 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 Carena et al. (2003). On the other hand, different proposals for the future generation of collider experiments are currently under development Fan et al. (2015), such as ILC Baak et al. (2013); Baer et al. (2013); Banerjee et al. (2015), FCC-ee Bicer et al. (2014); Blondel et al. (2016), and CEPC Dong et al. (2018); Ahmad et al. (2015); Liao and Wu (2018); Liang and Ruan (2018). 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 Schael et al. (2006, 2013).

In this work we study the constraints arising from the neutrino counting experiments around the $Z$ peak, specifically using data from $\nu\bar{\nu}\gamma$ 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 non-unitary 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 $\nu\bar{\nu}\gamma$

Non-unitarity has been subject to study for a long time Schechter and Valle (1980); Gronau et al. (1984); Nardi et al. (1994); Atre et al. (2009). Recent constraints can be found elsewhere Escrihuela et al. (2017); Fernandez-Martinez et al. (2016), 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 $U^{n\times n}$ matrix as the combination of four submatrices Hettmansperger et al. (2011)

[TABLE]

with $N$ a $3\times 3$ submatrix in the light neutrino sector, and $S$ the $3\times(n-3)$ 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 Escrihuela et al. (2015)

[TABLE]

where $U$ is the unitary PMNS mixing matrix for the standard $3\times 3$ case and $N^{NP}$ parameterizes the deviations from unitarity. In this way, we can encode all the parameters of the general description Schechter and Valle (1980); Rodejohann and Valle (2011), 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 $\alpha_{11},\alpha_{22}$, and $\alpha_{33}$ (all of them close to one) plus other three complex parameters $\alpha_{21},\alpha_{31},\alpha_{32}$ 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.

II.1 The invisible $Z$ decay

In the standard unitary limit, the branching for the invisible $Z$ decay into neutrinos will be given by Akhmedov (1999); Tanabashi et al. (2018),

[TABLE]

with $N_{\nu}$ the effective number of neutrino families and Tanabashi et al. (2018)

[TABLE]

Experimentally, the ratio $\Gamma_{inv}/\Gamma_{\ell\bar{\ell}}$ has been measured with greater experimental precision than $\Gamma_{inv}$ alone Tanabashi et al. (2018); Schael et al. (2006). Therefore, the number of light active neutrinos can be estimated from this relation, that in the Standard Model is given by Schael et al. (2006)

[TABLE]

with $N_{\nu}=3$. Here, the decay rate for the $Z$ boson into charged leptons is given by Tanabashi et al. (2018)

[TABLE]

where $g_{V}^{\ell}$ and $g_{A}^{\ell}$ are the vector and axial coupling for a charged lepton $\ell$:

[TABLE]

[TABLE]

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 Tanabashi et al. (2018)

[TABLE]

that can also be expressed as

[TABLE]

Comparing this expression with the unitary case discussed before, we can define for the non-unitary case

[TABLE]

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 Nardi et al. (1994); Langacker and London (1988); Atre et al. (2009)

[TABLE]

This correction cancels out in the ratio, $R_{inv}^{0}$, but can propagate to other observables, such as the weak mixing angle Fernandez-Martinez et al. (2016)

[TABLE]

From Eqs. (9) and (6), we can get an expression for the ratio in the non-unitary case:

[TABLE]

Let us notice that the deviation from unitarity, introduced by the parameters $\alpha_{ij}$, appears explicitly in the numerator through $|(NN^{\dagger})_{\alpha\beta}|^{2}$, but also implicitly in the denominator via $g_{V}^{\ell}$, 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

[TABLE]

If we neglect terms including third order or higher on off-diagonal parameters ($\alpha_{ij}\;i\neq j$), we obtain the following reduced expression:

[TABLE]

Different constraints for the $\alpha_{ij}$ parameters show that $N^{NP}$ is close to an identity matrix.

Besides, the precision of the measurements under consideration makes necessary to introduce radiative corrections. In the $\overline{MS}$ scheme, the weak mixing angle takes the form Tanabashi et al. (2018)

[TABLE]

where, in the non-unitary case, $A_{0}$ is given by

[TABLE]

$\Delta\hat{r}_{W}$ introduces the radiative corrections, and $M_{W}$ is the mass of the $W$ boson. According to PDG Tanabashi et al. (2018), the values of the relevant parameters are:

[TABLE]

For measurements at energies around the $Z$ peak it is common to use the effective weak mixing angle ${\bar{s}_{l}}^{2}$ instead of the $\overline{MS}$ scheme; both quantities are related through Tanabashi et al. (2018) ${\bar{s}_{l}}^{2}={\hat{s}_{Z}}^{2}+0.00032$.

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.

II.2 The process $e^{-}e^{+}\to\nu\bar{\nu}\gamma$

Another process that was also measured at LEP, and allows for a neutrino counting, is the single photon production with a neutrino-antineutrino pair Barate et al. (1998a, b); Heister et al. (2003); Abreu et al. (2000); Acciarri et al. (1997, 1998, 1999); Ackerstaff et al. (1998); Abbiendi et al. (1999, 2000); Hirsch et al. (2003). In this subsection we compute the expression for this observable in the NU case.

The differential cross section for the single photon production from electron-positron annihilation, $e^{+}e^{-}\to\nu\bar{\nu}\gamma$, can be written in terms of the radiator function $H(x,y;s)$ and the “reduced” cross section for the process $e^{+}e^{-}\to\nu\bar{\nu}$, $\sigma_{0}$, as Nicrosini and Trentadue (1989); Barranco et al. (2008); Berezhiani and Rossi (2002):

[TABLE]

The radiator function is defined by

[TABLE]

with

[TABLE]

and $\sigma_{0}$, the “reduced” cross section for the process $e^{+}e^{-}\to\nu\bar{\nu}$ is given by

[TABLE]

The three terms in Eq. (II.2) 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 Hirsch et al. (2003); Barranco et al. (2008); Berezhiani and Rossi (2002):

[TABLE]

where

[TABLE]

It is important to notice that the expression in Eq. (II.2), including the corrections from Eq. (II.2), is equivalent to the widely used expression:

[TABLE]

Nevertheless, we will continue using Eq. (II.2), since the introduction of the non-unitarity effects can be made in a more transparent way.

From Eq. (21), the total cross section is

[TABLE]

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:

[TABLE]

These corrections can be seen in Fig. 1: for the $W$ contribution (a), each neutrino line contributes with a term $U_{ei}$ in the scattering amplitude, while for the $Z$ contribution (b), the provided correction is of the form $U_{\alpha i}$. Since the mixing is non-unitary, flavor-changing neutral currents are allowed, hence the sum must be given over different flavors in the second term of Eq. (29).

Writing Eq. (29) explicitly, we will have

[TABLE]

Additionally, as discussed in the previous subsection, there will be NU corrections to $G_{F}$ and $\sin^{2}\theta_{W}$ 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, $\Gamma_{Z}$, 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 Akhmedov (1999); Tanabashi et al. (2018)

[TABLE]

and the non-unitary correction will appear through the $\Gamma_{\mathrm{inv}}$ contribution, as it had been computed in the previous subsection, and we will have:

[TABLE]

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

III.1 The process $e^{-}e^{+}\to\nu\bar{\nu}\gamma$

To obtain constraints for the NU case from the process $e^{-}e^{+}\to\nu\bar{\nu}\gamma$, we use the reported measurements from the ALEPH Heister et al. (2003), DELPHI Abreu et al. (2000), L3 Acciarri et al. (1997, 1998, 1999), and OPAL Ackerstaff et al. (1998); Abbiendi et al. (1999, 2000) collaborations. They are are listed in Table 1. 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, $x_{T}=x\sin\theta_{\gamma}$ (with $x=E_{\gamma}/E_{\mathrm{beam}}$), while $y=\cos\theta_{\gamma}$.

In order to make our analysis, we have computed the cross section from Eqs. (28) and (30), with the integration limits taken according to the last two columns of Table 1. Although we have a good agreement in our integration with many of the reported Monte Carlo simulations, there are some exceptions due, we believe, to our lack of knowledge of each experimental setup. Instead of excluding any experimental value, we have included a normalization error in our analysis, with a $10$ % uncertainty. Once we have obtained this expression, we have compared our theoretical expectation for the NU case with the experimental results of Table 1 through a $\chi^{2}$ analysis.

Our result for the non-unitary parameter $\alpha_{11}$ 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, $\alpha^{2}_{22}=\alpha^{2}_{33}=1$ and $\alpha^{2}_{21}=\alpha^{2}_{31}=\alpha^{2}_{32}=0$. We have chosen this parameter because diagonal parameters $\alpha_{ii}$ 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 $\alpha_{22}$ or $\alpha_{33}$. As it can be seen, it is possible to restrict the $\alpha_{11}$ NU parameter, and the constraint at $90$ % CL is given by

[TABLE]

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.

III.2 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 Schael et al. (2006, 2013) and future experiments Fan et al. (2015); Baak et al. (2013); Baer et al. (2013); Banerjee et al. (2015); Bicer et al. (2014); Blondel et al. (2016); Dong et al. (2018); Ahmad et al. (2015); Liao and Wu (2018); Liang and Ruan (2018) 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 Antusch and Fischer (2014, 2015); Fischer and Antusch (2018); Fernandez-Martinez et al. (2016). 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 $\sin^{2}\theta_{W}$ (Eqs. (10) and (11) respectively.) Perhaps the most important observable for our discussion is the value of the weak mixing angle that, at the 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 $\chi^{2}$ curve for this observable as a function of the $\alpha_{11}$ parameter. In this figure, besides considering the LEP Schael et al. (2006, 2013) 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 Abazov et al. (2011); Acosta et al. (2005); Aaltonen et al. (2013); TEW (2010), Atlas Aad et al. (2015), LHCb Aaij et al. (2015) and CMS Chatrchyan et al. (2011) result. It is possible to notice that the evaluation of this fundamental quantity of the Standard Model still can have an impact on the non-unitarity constraints. As in the previous subsection, for this plot we have only considered $\alpha_{11}$ as different from one and all other non-unitary parameters as equal to the standard case, that is, $\alpha^{2}_{22}=\alpha^{2}_{33}=1$ and $\alpha^{2}_{21}=\alpha^{2}_{31}=\alpha^{2}_{32}=0$.

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, $N_{\nu}$, or the effective value of the weak mixing angle, $\sin^{2}{\theta_{eff}}$. We show their sensitivity in Table 2.

In order to estimate the sensitivity of the future experiments we will consider again the ratio given by Eq. (5). In particular, the uncertainty of $R^{0}_{inv}$ is calculated from

[TABLE]

where $\sigma\left(\frac{\Gamma_{\nu\bar{\nu}}}{\Gamma_{l\bar{l}}}\right)_{SM}=\;0.00083$ Schael et al. (2006) and $\sigma(N_{\nu})$ is given in Table 2. With these hypothesis we obtain the results shown in Table 3.

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 Schael et al. (2006), $R^{0}_{inv}=5.9430$.

• •

The theoretical value calculated from the effective weak mixing angle including radiative corrections Patrignani et al. (2016), $R^{0}_{inv}=5.9671$.

• •

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 $\chi^{2}$ analysis in order to have a forecast of the future expected sensitivity, considering the following two scenarios:

• •

Firstly, we consider that $\alpha_{11}$ is the only parameter different from the standard case. The $\chi^{2}$ fit is made with the errors already discussed for each experiment. The results are compiled in Fig. 4.

• •

Secondly, we let $\alpha_{11}$, $\alpha_{13}$ and $\alpha_{33}$ to vary freely, while fulfilling the Cauchy-Schwarz condition:

[TABLE]

The other NU parameters are set to their SM value. The results obtained are shown in Fig.5. Notice that we have considered only $\alpha_{33}$ and $\alpha_{31}$ different from zero, since very similar results will be obtained with $\alpha_{22}$ and $\alpha_{21}$.

We summarize the expected accuracy for both cases in Table 4. 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 Fernandez-Martinez et al. (2016); Blennow et al. (2017); Escrihuela et al. (2017), that constraints $\alpha_{11}$ at the level of $0.999$ or below as we see in Table 5. 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 4 the future experiments under discussion will have the potential to show the evidence of new physics through a non-unitarity of the neutrino mixing-matrix.

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 $\nu\bar{\nu}\gamma$ 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 $\alpha_{11}$. 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 $\alpha_{11}$ 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. (II.1), the $Z$ decay measurement will mainly restrict the sum of the three diagonal parameters: $\sum_{i}\alpha_{ii}$ and, therefore, in a combined analysis, this measurement will help to restrict any of the diagonal parameters.

V Acknowledgments

This work was supported by the Conacyt grant A1-S-23238 and SNI (Mexico). LJF also thanks the Conacyt for the grant of Ayudante de investigador (EXP. AYTE. 16959) and a posdoctoral CONACYT grant.