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arXiv:hep-ph/0702076v3 8 Apr 2007
Bounding the Number of Light Neutrinos Species in a Left-Right
Symmetric Model
A. Guti´ errez-Rodr´ ıguez,1M. A. Hern´ andez-Ru´ ız,2M. A. P´ erez,3and F. P´ erez-Vargas1
1Facultad de F´ ısica, Universidad Aut´ onoma de Zacatecas
Apartado Postal C-580, 98060 Zacatecas, Zacatecas M´ exico.
Cuerpo Acad´ emico de Part´ ıculas Campos y Astrof´ ısica.
2Facultad de Ciencias Qu´ ımicas, Universidad Aut´ onoma de Zacatecas
Apartado Postal 585, 98060 Zacatecas, Zacatecas M´ exico.
3Departamento de F´ ısica, CINVESTAV.
Apartado Postal 14-740, 07000, M´ exico D.F., M´ exico.
(Dated: February 2, 2008)
Abstract
Using the experimental values for the rates RLEP
exp
= Γinv/Γl¯l= 5.942 ± 0.016, RGiga−Z1=
Γinv/Γl¯l= 5.942 ± 0.012 (most conservative) and RGiga−Z1= Γinv/Γl¯l= 5.942 ± 0.006 (most
optimistic) we derive constraints on the number of neutrino light species (Nν)LRSM with the
invisible width method in the framework of a left-right symmetric model (LRSM) as a function
of the LR mixing angle φ. Using the LEP result for Nν we may place a bound on this angle,
−1.6 × 10−3≤ φ ≤ 1.1 × 10−3, which is stronger than those obtained in previous studies of the
LRSM.
PACS numbers: 14.60.Lm,12.15.Mm, 12.60.-i
Keywords: Ordinary neutrinos, neutral currents, models beyond the standard model.
E-mail:1alexgu@planck.reduaz.mx,2mahernan@uaz.edu.mx,3mperez@fis.cinvestav.mx
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I. INTRODUCTION
The number of fermion generations, which is associated to the number of light neutrinos, is
one of the most important predictions of the Standard Model of the electroweak interactions
(SM) [1]. In the SM the decay width of the Z1boson into each neutrino family is calculated
to be Γν¯ ν= 166.3 ± 1.5 MeV [2]. Additional generations, or other new weakly interacting
particles with masses below MZ1/2, would lead to a decay width of the Z1into invisible
channels larger than the SM prediction for three families while a smaller value could be
produced, for example, by the presence of one or more right-handed neutrinos mixed with
the left-handed ones [3]. Thus the number of light neutrino generations Nν, defined as the
ratio between the measured invisible decay width of the Z1, Γinv, and the SM expectation
Γν¯ νfor each neutrino family, need not be an integer number and has to be measured with
the highest possible accuracy.
The most precise measurement of the number of light (mν < 45 GeV ) active neutrino
types, and therefore the number of associated fermion families, comes from the invisible Z1
width Γinv, obtained by subtracting the observed width into quarks and charged leptons
from the total width obtained from the lineshape. The number of effective neutrinos Nνis
given by [4]
Nν=Γinv
Γl
(Γl¯l
Γν)SM,
where (Γl¯l
Γν)SM, the SM expression for the ratio of widths into a charged lepton and a single
active neutrino, is introduced to reduce the model dependence. The experimental value
from the four LEP experiments is Nν= 2.9841±0.0083 [2, 5], excluding the possibility of
a fourth family unless the neutrino is very heavy or sterile. Nν is the effective number of
light neutrino generations deduced from the Z1invisible width based on the expected partial
width for one light neutrino generation (Nν= Γinv/ΓSM
ν
). This result is in agreement with
cosmological constraints on the number of relativistic species around the time of Big Bang
nucleosynthesis, which seems to indicate the existence of three very light neutrino species [6].
On the other hand, the LEP result measures precisely the slight deviations of Nνfrom three.
In particular, the most precise LEP numbers can be translated into Nν= 2.9841±0.0083 [2],
about two sigma away from the SM expectation, Nν= 3. While not statistically significant,
this result suggests that the Zν¯ ν-couplings might be suppressed with respect to the SM
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value [4, 7].
Using the experimental value for RLEP
exp
=
Γinv
Γl¯l
= 5.942 ± 0.016 [5], we will determine
the allowed region for (Nν)LRSM as a function of the mixing angle φ and estimate bounds
for the number of light neutrinos species in the framework of a left-right symmetric model
(LRSM) [8, 9]. We will also use the LEP results to get a constraint on the LR mixing
angle φ. On the other hand, if one assumes that the results for Γinvand Γl¯lfor a future
TESLA-like Giga-Z1experiment agree with the central values obtained at LEP, one would
measure (Γinv
Γl¯l)Giga−Z1= 5.942 ± 0.012 (most conservative) or (Γinv
(most optimistic) [4], in this case we estimate also a limit for the number of light neutrinos
Γl¯l)Giga−Z1= 5.942 ± 0.006
species.
This paper is organized as follows: In Sec. II we present the expressions for the decay
widths of Z1 → l¯l and Z1 → ν¯ ν in the LRSM. In Sec.
computation and, finally, we summarize our results in Sec. IV.
III we present the numerical
II.WIDTHS OF Z1→ l¯l AND Z1→ ν¯ ν
In this section we calculate the partial widths for Z1 → l¯l and Z1 → ν¯ ν using the
transition amplitude given in Ref. [8] in the context of the LRSM. The expression for the
transition amplitude for the channel Z1→ l¯l is given by
M(Z1→ l¯l) =
g
cosθW[¯ u(l)γu1
2(agl
V− bgl
Aγ5)v(¯l)]ελ
µ(Z1), (1)
where u(v) is the lepton (antilepton) spinor and ελ
µis the Z1boson polarization vector and
the expressions for the couplings a and b in the LRSM are:
a = cosφ −
sinφ
√cos2θW
andb = cosφ +
?
cos2θWsinφ,(2)
where φ is the mixing angle of the LRSM [7, 11].
After applying some of the trace teorems of the Dirac matrices and of sum and average
over the initial and final spin the square of the matrix elements becomes
Σs|M|2=
g2M2
3cos2θW[a2(gl
Z1
V)2(1 +2m2
l
M2
Z1
) + b2(gl
A)2(1 −4m2
l
M2
Z1
)].(3)
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Our next step, now that we know the square of the Eq. (3) transition amplitude, is to
calculate the partial width of the reaction Z1→ l¯l:
Γl¯l=GFM3
Z1
6π√2
?
1 − 4ηl[a2(gl
V)2+ b2(gl
A)2+ 2ηl(a2(gl
V)2− 2b2(gl
A)2)], (4)
where ηl=
m2
M2
l
Z1.
For the Z1-decay width into ν¯ ν we obtain
Γν¯ ν=GFM3
Z1
12π√2
?
1 − 4ην[1
2(a2+ b2) + ην(a2− 2b2)],(5)
where ην=
m2
M2
ν
Z1.
The partial widths Eqs. (4) and (5) are applicable to all charged leptons and all neutrinos
respectively.
III. RESULTS
In order to compare the respective expressions Eqs. (4) and (5) with the experimental
result for the number of light neutrinos species Nν, we will use the definition for Nνin a SM
analysis [10],
Nν= Rexp(Γl¯l
Γν¯ ν)SM, (6)
where the quantity in parenthesis is the standard model prediction and the Rexpfactor is
the experimental value of the ratio between the widths Γinvand Γl¯l[2, 5],
RLEP
exp = (Γinv
Γl¯l
) = 5.942 ± 0.016. (7)
This definition replaces the expression Nν=
Γinv
Γν¯ νsince (7) reduces the influence of the
top quarks mass. To get information about what is the meaning of Nν in the LRSM we
should define the corresponding expression [7],
(Nν)LRSM= Rexp(Γl¯l
Γν¯ ν)LRSM.(8)
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This new expression is a function of the mixing angle φ, so in this case the quantity defined
as the number of light neutrinos species is not a constant and not necessarily an integer.
Also, (Nν)LRSMin formula (8) is independent from the Z2mass and therefore depends only
of the mixing angle φ of the LRSM. Experimental values for Γinvand for Γl¯lare reported
in literature which, in our case, can give a bound for the angle φ. However, we can look to
those experimental numbers in another way. The partial widths Γinv= 499.0 ± 1.5 MeV
and Γl¯l= 83.984±0.086 MeV were reported recently [2], but we use the value given by (8)
for the Rexprate of Ref. [5]. All these measurements are independent of any model and can
be fitted with the LRSM parameter (Nν)LRSMin terms of φ.
In order to estimate a limit for the number of light neutrinos species (Nν)LRSMin the
framework of a left-right symmetric model, we plot the expression (8) to see the general
behavior of the (Nν)LRSMfunction, Fig. 1. For the mixing angle φ between Z1and Z2, we
use the reported data of Maya et al. [7]:
− 9 × 10−3≤ φ ≤ 4 × 10−3, (9)
with a 90% C.L. Other limits on the mixing angle φ reported in the literature are given
in Refs. [11, 12]. In this figure we observed that for the mixing angle φ, around 0.65 rad,
(Nν)LRSMcan be as high as 5.9, and for values of φ around -0.95 rad, (Nν)LRSMis as low
as 0. This shows a strong dependence in φ for leptonic decays of the Z1boson. Therefore,
according to the above discussion, if we consider (Nν)LRSMas the number of neutrinos, the
restriction on the number of species can be “softened” if we consider a LRSM. In Fig. 2, we
show the allowed region for (Nν)LRSMas a function of φ with 90% C.L. The allowed region
is the inclined band that is a result of both factors in Eq. (8). In this figure (Γl¯l
Γν¯ ν)LRSMgives
the inclination while Rexpgives the broading. This analysis was done using the experimental
value given in Eq. (7) for Rexpreported by [5] with a 90% C.L. In the same figure we show
the SM (φ = 0) result at 90% C.L. with the dashed horizontal lines. The allowed region in
the LRSM (dotted line) for (Nν)LRSMis wider that the one for the SM, and is given by:
2.925 ≤ (Nν)LRSM≤ 3.02 or (Nν)LRSM= 2.987+0.033
−0.062, 90% C.L., (10)
whose center value is quite close to the standard model of three active neutrino species.
In the case of a future TESLA-like Giga-Z1experiment we obtain the limits
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2.926 ≤ (Nν)LRSM≤ 3.019 or (Nν)LRSM= 2.987+0.032
2.929 ≤ (Nν)LRSM≤ 3.016 or (Nν)LRSM= 2.987+0.029
−0.061, 90% C.L. (most conservative)(11)
−0.058, 90% C.L. (most optimistic), (12)
which are consistent with those reported in the literature [4].
Finally, and just for completeness, we reverse the arguments that is, we fix the number
of neutrinos in the LRSM to be three then the theoretical expression for R will be given by
RLRSM=3Γν¯ ν
Γl¯l
. (13)
The plot of this quantity as function of the mixing angle φ is shown in Fig. 3. The
horizontal lines give the experimental region at 90% C.L. From the figure we observed that
the constraint for the φ angle is:
− 1.6 × 10−3≤ φ ≤ 1.1 × 10−3,(14)
which is about one order of magnitude stronger than the one obtained in previous studies
of the LRSM [7, 11, 12].
In the case of a future TESLA-like Giga-Z1experiment we would obtain the following
bounds for the mixing angle φ:
− 1.1 × 10−3≤ φ ≤ 0.9 × 10−3, (most conservative),
−0.8 × 10−3≤ φ ≤ 0.33 × 10−3, (most optimistic).
(15)
(16)
IV. CONCLUSIONS
We have determined a bound on the number of light neutrinos species in the framework
of a left-right symmetric model as a function of the mixing angle φ, as shown in Eq. (10)
and Fig. 2. Using this result and the LEP values obtained for Nν, we were able to put a
limit on the LR mixing angle φ which is better than the one obtained in previous studies of
these models.
In summary, we conclude that in the LRSM it is possible to obtain from the experimental
results a value for Nνdifferent from 3 (not necessarily an integer number). In particular for
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the left-right symmetric model with Dirac neutrinos, (Nν)LRSM is in the neighborhood of
three. However, if new precision experiments find small deviations from three, this model
may explain very well these deviations with a small value of φ. We have shown that new
data of Rexp=Γinv
Γl¯l
can considerably shrink the allowed region of (Nν)LRSM. In the limit of
φ = 0, our bounds takes the value previously reported in the literature [2, 4, 5].
Acknowledgments
We would like to thank O. G. Miranda for useful discussions. This work was supported
by CONACyT and SNI (M´ exico).
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?
(N ) ?LRSM
FIG. 1: (Nν)LRSMas a function of the mixing angle φ.
?
(N ) ?LRSM
Allowed Region
FIG. 2: Allowed region for (Nν)LRSM as a function of the mixing angle φ with the experimental
value RLEP
exp. The dashed line shows the SM allowed region for Nνat 90% C.L., while the dotted
line shows the same result for the LRSM.
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RLRSM
?
FIG. 3: The curve shows the shape for RLRSM as a function of the mixing angle φ. The dashed
line shows the experimental region for RLEP
exp
at 90% C.L..
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