V o l u m e 168B, n u m b e r 3
PHYSICS LETTERS
6 M a r c h 1986
LIMITS ON MASSES OF EXCITED ELECTRONS AND MUONS F R O M N E U T R I N O S C A T T E R I N G O F F E L E C T R O N S ''~ J.A. G R I F O L S and S. P E R I S Departarnent de Nsica Tebrica, Universitat A utbnoma de Barcelona, Bellaterra (Barcelona), Spain Received 15 October 1985
The d a t a on v.e ~ v~e and ~.e + hue from the C H A R M C o l l a b o r a t i o n are used to set new and interesting b o u n d s on the masses of hypothetical excited electrons and muons. In particular, it turns out that excited electrons c a n n o t exist with masses below a couple of h u n d r e d GeV.
In recent years the electroweak theory based on the SU(2) X U(1) gauge group has witnessed an impressive success. Indeed, it correlates in a quantitative manner a large amount of experimental data. In spite of this fact there is a general agreement among physicists that a threshold for new physics should be reached at about the 1 TeV energy scale. The two main scenarios contemplated are, on the one hand, supersymmetry [ 1] - predicting a plethora of new particles and interactions - and, on the other hand, compositeness [2] - where the elementary of quarks/leptons and/or gauge bosons is abandoned. In the second line of investigation, once we give up elementarity of quarks and leptons we are naturally led to the idea of excited quarks and leptons. There is already an important amount of research carried out in the literature surveying the properties of excited quarks and leptons and analyzing the consequences of their existence [3]. In particular, bounds for excited electron and muon masses have been derived from the anomalous magnetic moment of the electron and the muon [4]. In this paper we shall derive new limits on masses of excited leptons, using recent data on rue --> rue and ~ue ~ ~,e scattering [5]. More precisely, we shall use the error on the value for the Weinberg angle extracted from those data to put interesting limits on masses of excited electrons and muons. We assume as a starting point that the standard Weinberg-Salam model provides a basically correct ef* W o r k partially s u p p o r t e d by research project C A I C Y T .
264
fective lowest-order description of the low-energy electroweak phenomenology. The Weinberg angle, as obtained from neutrino and antineutrino scattering on electrons is given by sin20 w =¼ (1 + ~), "where g enters the ratio R,
o(v#e~vue ) ~2 ~j+ 1 R -
o@ue-+~,e)
-
(1)
1~2 + ~ + 1'
and is defined by, = -
a/b,
a and b being the parameters that appear in the effective current × current amplitude
T= ( - i G v / 2 V r~) vu( 1 -- 3'5) "/ovu" &~O(a- b75) e. (2) To lowest order, i.e. tree level Z0-exchange, the parameters a and b take the values
a=-l+4sin2Ow,
b=-l.
They get shifted when standard radiative corrections are taken into account. But they are also modified if, in addition to ordinary Z°-exchange, extra amplitudes involving excited leptons are present. As a consequence, a shift in sin20w is induced. The change in sin20w in terms of the changes in a and b is given by 8 sin20w
=¼a(Sa/a- 8b/b).
(3)
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 168B, number 3
PHYSICS LETTERS
6 March 1986
We shall compute 6a and 8b in the assumption that excited teptons do exist and have the following couplings to ordinary leptons [3] :
.6? = (g//2m *) L*ouu ' r £L W#V e
-
~I.
e
e
-
v~
(+ c r o s s e d )
+ ( g ' . f ' / Z m * ) -L* o u -¢' Y £L Buy
+
h.c..
(4)
Wuv and Buy are the field strengths of the SU(2) and U(1) fields, respectively. L* and £ denote isospin doublets for the excited and ordinary leptons, and m* is the mass of L*. The lagrangian density in eq. (4) respects weak isospin symmetry and the chiral nature o f the couplings avoids conflict with the experimental value for the muon and electron anomaly. There are a number of diagrams - which include vertex corrections, self-energies and boxes - involving the hypothesized excited leptons and their interactions. Using eq. (4) one can easily compute their effect on a and b. The loop integrals over momenta are to be performed up to a cut-off momentum A (the hypothesized new scale of substructure). For distances small on the scale of A - 1 , the form factors associated to the new interactions should kill the contributions o f higher momenta to the amplitudes. We shall consider only the leading behaviour in A. It turns out that our amplitudes are at most quadratic in A. The amplitudes which we take into account are depicted in figs. 1 - 4 . Any other diagram either does not contribute to 6 sin20w or its A-dependence is less than quadratic. Indeed, let us briefly comment on the diagrams which are not considered. Amplitudes with Z0-vacuum polarization insertions, being proportional to the tree amplitude do not change the value of sin20w . Also, amplitudes with mixed 7Z vacuum polarization insertions vanish in the low q2 limit. Another class o f diagrams which is absent is the one with self-energy corrections on the neutrino legs. No insertion can modify the handedness of the neutrino and therefore the corresponding amplitude does not correct the value of the Weinberg angle. Finally, note that no vertex corrections for the m,Z vertex nor the vv7 vertex do appear in figs. 1 - 4 . In the first case, it obviously gives no contribution to 6 sin20w . In the second case, the correction is only logarithmic in A. This is because the form factor for 7vv enters only through its derivative - the neutrino charge radius - which is clearly less dimensioned. Having stated these remarks we proceed now to give our results derived from figs. 1 - 4 .
-
e
•
Fig. 1. Diagrams contributing to 8 sin20w •
vl~
v~t
e
e
Z~Y
v._~
Vl~
e
ve~-
e
Fig. 2. Diagrams contributing to a sin20w .
v~
v~
v~
v.
z
z W
Z,Y ~
e
e
e ~e ~-e
F~. 3. Diagrams contributing to 6 sin20w.
Z,Y e*-
g
ZI e
'~
e-
W v~'-
e
Fig. 4. Diagrams contributing to 6 sin20w. All diagrams in figs. 1 - 4 give 6a = 8b and their explicit contributions are ~a 1
127 2 2 2 2 2 2 ) = _ x-~ (g /rr ) ( M ~ A / m u . me.
X 14f 4 + (f2 _ f , 2 tan20w)2]
(5)
from fig. 1,
6a 2
=
3 (g2/lr2) (A2/m2.)
X I ( f 2 + f , 2 tan20w) sinZ0w + 2f2 coS20wl
(6)
from fig. 2 and, 265
Volume 168B, number 3
PHYSICS LETTERS
me, (Te~/) 100-
sin20 w = 0.215 -+ 0.032.
.....i
10-
A=10 TeV I
I
I
A=5 TeV
Ill A='I TeV
IIII .1 i
I
.03 Ii
,...-,i
.....-I
.02 .0! , .01
.11 i .
.
.
.02 .03
.
.
.
.
.
i *
.1
.
.
.
.
.
.
.
.
I
I
.
.
.
.
.
.
.
.
I0 rnlt~[reVJ
Fig. 5. Allowed values for m e , and rntz, for different scales A. For each A, the permitted area lies above the two-branched curve.
~ a 3 + 4 = i 9~ [g2(cos20w_ sin20w)/lr 2] (A2/me.)
X ( 3 f 2 + f , 2 tan2Ow)
(7)
from figs. 3 and 4. We note in passing that, i f f = f ' , the photon exchange diagram in fig. 1 vanishes. This is because the magnetic coupling of the neutrino to the excited neutrino is proportional to (e/2m*) ( f - f') as follows from eq. (4). Since f , f ' should be O(1) we shall assume f = f ' = 1 for simplicity. We are ready now to compute 6 sin20w parametrized as a function of A and the masses of the excited electron and muon. The CHARM Collaboration gives for sin20w [5],
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6 March 1986
(8)
The central value for sin20 w in (8) is in good agreement with the world's average. We take it as input for the lowest-order calculation, and we shall saturate the above experimental error with/i sin20 w as derived above, i.e. using eqs. (3), ( 5 ) - ( 7 ) . We shall vary A, me. and mu, independently. Of course, both masses should be ultimately related to the scale A. However, since even ordinary fermions differ widely in mass, we feel allowed to vary arbitrarily me., mu. and A for they can easily differ from each other by more than an order of magnitude. Fig. 5 shows the permitted values of me, and mt~. for different compositeness scales (A = 1, 5 and 10 TeV). The vertical and horizontal straight lines are the limits from e+e - machines. For each A, the area above the two-branched curve is allowed by the neutrino scat. tering data [i.e. eq. (8)]. The dips in fig. 5 correspond to an accidental cancellation of the amplitudes. Both the position and the depth of dip cannot be trusted literally. The position depends essentially on the relative strength o f f and f ' and the dip will be partially filled by the non-leading contributions in A. The bounds obtained here are interesting. Indeed, even at A = 1 TeV (the alleged threshold for new physics) and barring fortuitous cancellations, the excited electron must be heavier than a couple of hundreds of GeV. This is fortunate since (g - 2)e gives no useful bound on me, (i.e. it is well below the direct limit set already by PEP and PETRA).
References [1] See e.g.H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 75. [2] See e.g.H. Terazawa, Proc. XXII Intern. Conf. on High energy physics, Vol. I (1984) p. 63. [3] J. Kiihn and P. Zerwas, Phys. Lett. 147B (1984) 189. [4] F.M. Renard, Phys. Lett. l16B (1982) 264. [5] CHARM Collab., F. Bergsma et al., Phys. Lett. 147B (1984) 481.