Radiative effects of exotic fermions

Revista Mexicana de Física 43, No. 2 (1997) 219-223
Radiative effects of exotic fermions
M. ÁVILA
Facultad de Ciencias, Universidad Autónoma del Estado de More/os
Cuernavaca 62210, Mor., México
O.A. SAMPAYO'
Departamento de Física, CINVESTA V-IPN
Apartado Postal 14-740, C.P. 07000, Mexico D.F., Mexico
Recibido el 9 de agosto de 1996; aceptado el 7 de octubre de 1996
ABSTRAeT. \Ve evaluate the contributions oC exotic fermions, which transform as N -dimensional
vector-like multiplets under the S1\1group SU(2) x U(I), to the oblique parameters defined to
constrain ncw physics from high-precision measurements. It is found that present data implies
a lower bound 011 the mass of these fcernions ror arbitrary N. For instance, m 2: 360 GeV roe
N = 10.
RESUMEN.
Evaluamos
las contribuciones
de los fermiones exóticos que se transforman
en un
Itlultiplete vectorial N-dimensional bajo el grupo del modelo estándar (ME) SU(2) x U(I) a los
parámetros oblicuos definidos para constreñir nueva física de las medidas de alta precisión.
Se
halla que los datos actuales implican una cota inferior a la masa de estos fermiones para N
arbitrario. Por ejemplo, rn ~ 360 GeV for N = 10.
rAes:
12.60.-i; 12.15.Lk
The measurements of the gauge boson properties (~V,Z), have reached a level of precision
that allows to test thc standard model (SM) [1) at the level of radiative corrections at
LEP [21. /vIoreover, LEP results have al so been used to imposc contrains on the presence
of ncw physics.
\Vithin the SM group SU(2) x U (1), thc lcft-handed components of the known fermions
transform as doublcts, whereas the right-handcd components transform as singlets, which
has been tested with success [3]. Howcver, in extensions of thc SM of the grand unilied
type [4], the existcncc of new fermions with ditrerent transformation properties is predicted in general. Thc cases of ferlllions that transform as singlets, mirror or vector-like
dOllblets are the ones most commonly stlldied in the literature [5]; vector-like and chiral
triplets have also becn considered [6].
In this paper we study the illlplications of exotic ferlllions that transform as vectorlike multiplets of N-dimcnsion, lIndcr the SM group. \Ve are interested in evaluating the
.On leave on absence from: Departamento
de Física, Facultad de Ciencias Exactas
Universidad ::-';acional del ~tar de Plata, Funcs 3350; 7600 :\olardel Plata, Argentina
y Naturales,
220
M.
ÁVILA
AND
O.A.
SAMPAYO
contribution of these fermions to the obliqlle parameters, which defined to constrain the
contribution of new physics to observables measured with high-precision at LEP.
The left- and right-handed components of the N-dimensional multiplets, of hypercharge Y, are written as follows:
(1)
thus, the dimension of the mllltiplets is N = 2J + 1, with J being its spin.
The Lagrangian for this type of representation has the following form:
(2)
where the covariant derivatives are defined as
(3)
Equations (2) and (3) can be rewritten in terms of the physical gauge bosons
(W", Z, A). Then, the Lagrangian that describes the interactions of the fermions fJ
becomes
(4)
where B = ewA - 8WZ, W3 = 8WA + ewZ. The matrix elements of the operator T+ are
zero everywhere except for the elements below from the principal diagonal, with values
T(;+I,S) = '¡8(2J - 8 + 1), where 8 = 1,2, ... ,2J. The operator T3 is diagonal, with
elements T3(s,s) = J - 8. 8w(ew) denote the sine (cosine) of the weak mixing angle.
The resulting interactiolls between the fermions ami gauge bosons are
w f-J-s'Y ~fJ-s+I W-~,
C(J,s)
Z
with the corresponding
(5)
-
C(J,s) fJ-s 'Y~fJ-sZ~,
(6)
C(J,s) JJ-s 'Y~fJ-s A~,
(7)
coeffieients given by
CrlS)=-;
,
y28w
(8)
)8(2J-8+1),
Z
2]
[8\\'Y - (J - 8)ew
C(Js) = -- e
,
8WCW
C(J,s) = -e(Y
+J
- 8).
,
(9)
(10)
Beeause of gauge invariance, the fermion masses are degenerate (Mi = Mj). The
mass terms for these fermions does not require a Higgs mechanism.
In order to derive bounds on the mass of these exotie fermions, one can evaluate
theír radiative contribution to observables measured with high-precision. For the case
RADlATIVE
EFFECTS
OF EXOTIC
FERMIONS
221
presented here, it is assumed that the vector-like ferrnions do not mix with standard
fermions, thus, their (universal) contribution to radiative corrections can be evaluated
from the self-energies of the vector bosons, which define the oblique parameters [7,8]. In
particular, we shall use the set of V;-parameters introduced by Novikov et al. [9].1
According to Ref. 9, the contributions of new physics to the physical quantities
mw/mz,9A
and 9v/9A, can be expressed in terms of the parameters V;n, VA, VII, which
in turn are written as functions of the self-energies of the vector bosons (Ilvv', with
V, V' = J, Z, W), narnely,
_
",VA
==
1611 2 2 ( Ilzz(mz2)' - "Ezz(mz
2) - Ilw ( O)) ,
-3-swcw
_
1611 4
o-V
m = --sil'
3
[c~v(Ilzz (2)mz
(11)
- Ilw (2mil' )) .
-28Hl
2
2sw
+ Ilw(mw)
- Ilw(O)- Iln(mz)2 - -Il~z(O)
],
( 12)
C~ll
_
1611 2
2 [c~s~v
",VII = -3-(cw - sil')
2
cn' -
2
(
2
2Sw)
Ilzz(mz)
- Iln(mz)2 - Ilw(O) - -Il~z
StV
CW
(13)
- SWCwIl~z(m~)],
where ñ = o-(mz), and "Evv = Il(O)/m~.
The resuIts for the contribution of exotic fermions to the V -parameters
model are
(2YLsW+cw
4 4
16Nc
VA=-9-1_4u
N
c
Vm = 16N9 N
( 1 - 2sw
VII =
o.
1) (
2
N
-
12
2 ) [ (1 + 2u)Fu
-
1+2u-12uFu,
(2)1 + 2u/cw
2 )
Fu11'] ,
in the present
(14)
(15)
(16)
where u = m2 /m~, Fu = F(m~, m2), F~\' = F(m~v, m2), with m, mil', mz denoting
the fermion, W and Z masses, respectively. The function F is defined in Ref. lO. In
the large-mass limit (u -+ (0) all the V-parameters vanish, which shows the decoupling
nature of the fermions in this model.
The data for the physical quantities (mil' /mz, 9V, 9V /9A) used in the X2 fit, are taken
from Ref. 11. In our analysis, the value of the SM Higgs mass is kept as constant, whereas
the top and the fermion rnasses are varied lirst. Figure 1 shows the results of the lit for
mh = 300 Ge V, Y = Oand N = 2,3,4, ... , lO. The labels in front of the lines correspond
to N (number of fermions in each multiplet).
IOne could choose any of the parameterizations that have appeared in the literature, like the
S, T, U parameters of Peskin and Takeuchi [7),or the 'i parameters of Altarelli-Barbieri [81.
222
M.
ÁVILA
AND
O.A.
SAMPAYO
400
IGeV)
350
m
300
250
200
150
100
y=O
m¡f300
o
140
3
2
50
GeV
160
180
200
(GeV)
m,
FIGURE l. Result for the fit of m vs. m" for the model with vector-like fermions. The nllmber
in front of each curve corresponds to the dimension N of the multiplet.
The result of the lit shows that the values of the tal' mass are in agreement with
reeent experimental measurements [12]. For aH the previous cases it is found a lower
bound on m, whieh inereases for larger values of N. For instanee, we obtain m > 70 GeV
for N = 3, whereas for N = 10 the bound is m > 360 GeV.
In conc!usion, we have evaluated the radiative eorreetions from exotie fermions of
arbitrary dimension (N), using the oblique parameters that are defined to test the presenee of new physies from high-precision measurements. The interest of our result, is the
possibility to obtain a lower bound on the mass of the exotie fermions for arbitrary N.
ACKNOWLEDGEMENTS
\Ve aeknowledge to Dr. L. Díaz-Crnz for his partieipation
and the support from CONACyT and SNI (MEXICO).
in the early stages of this work
REFERENCES
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RADIATIVE
EFFECTS
OF EXOTIC
FERMIONS
223
9. V. Novikov et al., Nud. Phys. B397 (1994) 35.
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