Volume 252, number 1 PHYSICS LETTERS B 6 December 1990

A lower bound on the "number o f neutrino species" measured at the Z ° peak

S.M. Bilenky Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Head Post Office, P.O. Box 79, SU-IOI O00 Moscow, USSR

W. Grimus and H. Neufeld Institut fiir Theoretische Physik, Universit?it Wien, Boltzmanngasse 5, A-1090 Vienna, Austria

Received 23 August 1990

We consider a model where the three standard neutrinos are linear combinations of three light and one heavy neutrino mass eigenstates. Using input from low-energy experiments we get a lower bound on the "effective number of neutrino species" mea- sured at the Z ° peak.

In the s tandard model [ 1 ] o fe lec t roweak interact ions the difference of the total width o f the Z ° and the width for the decay into all visible channels is a t t r ibu ted only to neutr inos and thus propor t ional to the number of families [ 2 ]. This number has now been de te rmined with high precision by LEP exper iments [ 3 ] at the Z ° peak. The exper imenta l results show that within the s tandard model the number o f families agrees with three.

In a recent paper [ 4 ] Jarlskog has considered the possible consequences of r ight-handed neutr ino singlets for neutr ino count ing via the invisible width of the Z °. Let us briefly recall the arguments for n lepton families: The n lef t -handed neutr ino fields 1/Li ( 1 ~ i ~ n ) occurring in the S U ( 2 ) doublets are related to N neutr ino mass eigenfields (N>~ n) by

N 17Li = E UiaO')La ' ( 1 )

a = l

where the lepton mixing matr ix Uia ( 1 <~ i ~ n, 1 <~ a <~ N) is an n × N submatr ix of an N × N uni tary matr ix Uab ( 1 ~< a, b ~ N) . The charged current lagrangian ~ c c expressed in terms of mass eigenfields is given by

g - . . + ffCC = - (.OLaUia7 lLiW • + h . c . , (2) , / 2 = 1 i=1

and the weak neutral current in teract ion reads

LPNC= g ~ ~ ~)LaU~aUib~lt(l)LbZg. (3 ) 2 cosOw a,b=~ i=~

F rom eq. (3) we obta in the par t ia l decay width o f the Z ° into all neutr ino types:

GyM3 ~ I ~, U~aUib 2 Rab (4) F ( Z ° - > n e u t r i n ° s ) - 12v/2 n ~,b=l i=l

with

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland ) 1 19

Volume 252, number 1 PHYSICS LETTERS B 6 December 1990

and

Rab=(l mE+m2 (m]-m2)2.]w(M2, m],m 2) 2M 2 2M~ ] g2z

Here, w is the kinematical function [ 5 ]

w(a, b, c) = [a2 +b2 +cZ-2(ab+ac+bc) ]1/2

(5)

O(Mz --ma -mb) • (6)

Note that the first factor in eq. (6) is missing in ref. [4]. Eq. (4) is valid for both the Dirac and the Majorana neutrino case.

With the definition

2

nexp:---- a,b= ~ I i=~l g~agib Rab (7)

and the inequality Rab ~< 1 one obtains the result [ 4 ]

nexp ~< n. (8)

This means that the "effective number of neutrino species" measured at the Z° peak nexp, defined by

F(Z °--, neutrinos ) = Fo n~xp, ( 9 )

where

ro= /1242 is always smaller than the number of left-handed doublets n for non-trivial mixing.

Let us now investigate the possible deviation of n.xp from its upper bound n if experimental constraints are taken into account. To simplify the discussion we will restrict ourseNes to the special case n = 3, N = 4 for the rest of this letter. We know from experiment [ 6 ] that three neutrinos must have very small masses and only the fourth one could have a large mass. In the following we will neglect the masses of the light neutrinos in eq. (6) and therefore we obtain

n~xv = 3-0"2( 1 -R44 ) - -2a( 1 - t r ) ( 1 - R , 4 ) (10)

with

3

a= ~ [Ui412=l-IU4412. i=l

The largest possible deviation n~p from 3 is attained by assuming m4/> Mz since in this case R44 = R14--0. To proceed further we will estimate tr from the following experimental and theoretical inputs: (i) F(~t---.e-9~v.); (ii) unitarity of the three-generation quark mixing matrix; (iii) F(n+ ~e+ve)/F(n+ ~ t +v.); (iv) F(x--- .~t-9.v.) or F ( x - - , e - % v 0 . In the further discussion we shall assume that the mass of the heavy neutrino is larger than the x mass. The decay width o f l a - ~ e - % v , including O ( a ) radiative corrections is given by [6]

2 5 [ ( 2 5 ) ] ( 8mZ~fl 3 m 2 ) G~,m~ a r ( ~ - - ~ e - 9 ~ v , ) = 192n 3 1+~-~ - - - n z 1-- m~/k, +-5M~ " (11)

120

Volume 252, number 1 PHYSICS LETTERS B 6 December 1990

It follows from eqs. (2), ( 5 ) and ( l 1 ) that G, and GF are related by

3 3 G2=G 2 ~" IU.,12 ~ IUejI2=G2(1-1U.4 z)(1-1Ue412 ) . (12)

i=1 j=l

In the next step we will use some input from the quark mixing matrix V. Nuclear beta decay, when compared to muon decay, gives [ 6 ]

IVual2G2(1-IU~4I 2 ) I__V~_a 12 G 2 - 1 - [U-'~.412 = (0"9744+0"0010)2" (13)

An analysis of K~3 decays yields [ 6 ]

IVusl 2 1 --lUg,412 -- (0"2196+0"0023)2" (14)

We will now combine (13) with (14) and assume I Vua 12 + I Vus 12 + I Vub 12 = 1. We have checked that the con- tribution of Vub is completely negligible. Thus we obtain an estimate for the possible size of the matrix element U~4,

1-1U~412=1.002+_0.003. (15)

In order to obtain also information about U~4 we will study the ratio Z " 2 / 2 2"2 ( )

r(n+--,e+v¢) [ m o i l i n g - m e ] I-IU~412 3or m~ R = F(n+__,lx+v, ) =\~-~,] \ ~ ] 1S[U-~,4~ 1 - --Inn ~ " (16)

In this formula the dominant radiative corrections [ 7 ] have been taken into account. Using the measured ratio R~xp = ( 1.228 +- 0.022) × 1 0 - 4 leads to

1 - lUg412=0.994+0.021 • (17)

Finally, the possible range of I Ux4 I can be determined from an analysis of F ( x - --, kt- 9.v.) or F(x - - . e - 9~ v~). The corresponding coupling constants G~. and G~ are related to GF by

G2, =G2(1 - [Ux412)(1- [U~4I 2) , (18)

and

2 2 G ~ = G v ( 1 - 1 U , 4 1 2 ) ( 1 - [Ue4l 2) , (19)

respectively. Extracting [ U,4 [ from eq. (18) yields

1 - [ U,412=0.961 +_0.077, (20)

whereas eq. (19) leads to

1 - [ U,412=0.937+_0.058. (21)

We want to remark that because of the smallness of the lepton mixing matrix elements I U~4 12 ( i= e, ~t, x) there are no interesting effects in neutrino oscillation experiments in this model.

From ( 15 ), ( 17 ), (20) and (21 ) the numerical value of the parameter tr can now be determined:

tr=0.043+_0.101 from eq. (20 ) ,

=0.067+-0.083 fromeq. (21). (22)

Consequently the minimum of n~xp is given by

121

Volume 252, number I PHYSICS LETTERS B 6 December i 990

min nex p = 3 - t r 2 - 2 t r ( 1 - a ) = 2 . 9 2 + 0 . 1 9 from eq. ( 2 0 ) ,

= 2 . 8 7 + 0 . 1 5 f r o m e q . ( 2 1 ) . (23)

Obviously, the largest contr ibut ion to a can come from I U~412. This quant i ty could be measured with high accuracy in a t a u - c h a r m factory.

Eq. (23) shows how much the "effective number o f neutr ino species" could still deviate from three i f the most stringent low-energy da ta are used. Now we want to compare our findings with nexp extracted from the newest ALEPH [3 ] results. With the model independent exper imental width Finv = 4 9 5 + 41 MeV we f ind

n~xp =2 .98 + 0 . 2 9 , (24)

thus yielding a margin for devia t ions from 3 comparable with that o f our low-energy analysis (23) . In this letter we have considered a model where the three s tandard neutr inos are l inear combina t ions of three

light and one heavy mass eigenstate. Denot ing the "effective number of neutr ino species" as it is measured at the Z ° peak by n¢~p, such a model predicts nexp ~< 3 [4 ]. Consider ing this result, one is immedia te ly led to the quest ion how far nexp can deviate f rom 3. To pursue this problem we have used low-energy da ta to constrain those elements o f the lepton mixing which connect the charged leptons with the heavy neutrino. These results allow to set a lower bound on n~xp. It is found that n¢xp > 2.7 is val id at the one s tandard devia t ion level. A s imilar range is obta ined ifn¢xp is der ived from the latest A L E P H data. Thus, we conclude that low-energy da ta analyzed within the abovement ioned extension o f the s tandard model allow a sizeable devia t ion of the "effective number o f neutr ino species" from 3. If, in the future, n~xp turned out to be defini tely smaller than three but still respecting the lower bound eq. ( 2 3 ) , a fourth, heavy neutr ino would provide a natural mechanism to explain such a result. It should be kept in mind, however, that the model discussed in this let ter is not the only one which can give a lower n~xp. The same effect can be obtained, e.g., by a new heavy neutral vector boson [ 8 ] but this mechanism differs in the semileptonic decay widths used in this letter.

References

[ 1 ] S.L. Glashow, Nucl. Phys. 22 ( 1961 ) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Proc. 8th Nobel Symp., ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968 ) p. 367.

[2] R. Bertlmann and H. Pietschmann, Phys. Rev. D15 (1977) 683; J. Ellis et al., Ann. Rev. Nucl. Part. Sci. 32 ( 1982 ) 443.

[ 3 ] ALEPH Collab., D. Decamp et al., Phys. Lett. B 231 (1989) 519; B 234 (1990) 339; B 235 (1990) 399; DELPHI Collab., P. Aarnio et al., Phys. Lett. B 231 (1989) 539; B 241 (1990) 425; DELPHI Collab., P. Abreu et al., Phys. Lett. B 241 (1990) 435; L3 Collab., B. Adeva et al., Phys. Lett. B 231 (1989) 509; B 237 (1990) 136; OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B 231 (1989) 530; B 235 (1990) 379.

[4 ] C. Jarlskog, Neutrino counting at the Z-peak and right-handed neutrinos, CERN preprint CERN-TH-5657/90 (1990). [ 5 ] H. Pietschmann, Weak interactions - formulae, results and derivations (Springer, Berlin, 1983 ). [ 6 ] Particle Data Group, J.J. Hernandez et al., Review of particle properties, Phys. Lett. B 239 (1990) 1. [7] W.J. Marciano and A. Sirlin, Phys. Rev. Lett. 36 (1976) 1425;

T. Goldman and W.J. Wilson, Phys. Rev. D 15 ( 1977 ) 709; D.A. Bryman, P. Depommier and C. Leroy, Phys. Rep. 88 (1982) 151.

[8] G. Altarelli et al., CERN preprint CERN-TH-5591/89 (1989).

122