o-(ON p+ X) r o- (vN p- X) .
Instead of (150) one then finds
1 5
R,
=
sin2 + 5 (1r) sin4 , 1 5
-
sin2 + (1 + ) sin4 . (152) Some experimental values of
flu, R„,and r are shown in Table 4 (Conrad
et al.1998).
The relation between
R,and
R,as a function of sin2 B is plotted in Figure 12. This result has a couple of interesting features.
Experiment
R, fl, rCHARM 0.3091 ± 0.0031 0.390 ± 0.014 0.456 + 0.011 CDHS 0.3135 + 0.0033 0.376 + 0.016 0.409 ± 0.014 Average 0.3113 ± 0.0023 0.384 + 0.011 0.429 ± 0.011
Table 4. Neutrino neutral-current parameters.
The observed
R„is very close to its minimum possible value of less than 0.4. Initially this made the observation of neutral currents quite challenging. Note that
R„is even smaller. Its value provides the greatest sensitivity to sin2
O.It is also more precisely measured than
R,(in part, because neutrino beams are easier to achieve than antineutrino beams). The effect of
1'on the determination of sin2
0is relatively mild.
(148) (149)
(151)
R,
0.6
0.5
0.4
0.3 01 0.2 0.3 0.4 0.5 06
Rv
Figure 12. The Weinberg-Salam "nose" depicting the relation between 11,, and R,.
The solid line corresponds to r = 0.429, close to the actual situation; the dashed line corresponds to the idealised case r = 1/3 in which antiquarks in the nucleon are neglected.
The plotted point with error bars corresponds to the average of measured values.
A recent determination of sin2 8 (Zeller et al. 1999), based on a method proposed by Paschos and Wolfenstein (1973), makes use of the ratio
o- (v N -> v X ) - o-(0 N -> 1).A") R„ - r 1
R- sin B. (153)
o-(vNµX) - o-(0 N -> X) 1 - r 2
In these differences of neutrino and antineutrino cross sections, effects of virtual quark-antiquark pairs in the nucleon ("sea quarks," as opposed to "valence quarks") cancel one another, and an important systematic error associated with heavy quark production (as in vs c) is greatly reduced. The result is
sin2 0 (on—shell) = 0.2253 ± 0.0019(stat.) f 0.0010(syst.) , (154) which implies a W mass
May = Mz cos 0 (on—shell) = 80.21 + 0.11 GeV . (155) The "on-shell" designation for sin2 8 is necessary when discussing higher-order electroweak radiative corrections, which we shall do in the next section.
A more recent analysis by Zeller et al. (2001) finds
sin2 0 (on—shell) = 0.2277 + 0.0014(stat.) f 0.0008(syst.) , (156) equivalent to Mw = 80.136 ± 0.084GeV. Incorporation of this result into the electroweak fits described in the next section is likely to somewhat relax constraints on the Higgs boson mass: See Rosner (2001).
4.6 Z and top quark properties
We have already noted the prediction and measurement of the W mass and width. The
Z
mass and width are very precisely determined by studying the shape of the cross section for electron-positron annihilation as one varies the energy across theZ
pole. The results (LEP Electroweak Working Group [LEP EWWG] 2001) areMz = 91.1875 + 0.0021 GeV , Fz = 2.4952 ± 0.0023 GeV . (157) In much of the subsequent discussion we shall make use of the very precise value of
Mz
as one of our inputs to the electroweak theory; the two others, which will suffice to specify all parameters at lowest order of perturbation theory, will be the Fermi coupling constant Gp = 1.16637(1) x 10-5 GeV-2 and the electromagnetic fine-structure constant, evolved to a scaleM3:
a- (ms)(A,) = 128.933 + 0.021 (Davier and Hocker 1998). This last quantity depends for its determination upon a precise evaluation of hadronic contributions to vacuum polarisation, and is very much the subject of current discussion.The relative branching fractions of the
Z
to various final states may be calculated on the basis of (138). One may write this expression asr z ff
= (g 2 + g'2 )1 /2 / r -y5)a L + (1 + -y5)coiii •
(158)Table 5.
Contributions to rz predicted in lowest-order electroweak theory (including leading-order QCD corrections to hadronic channels). Here we have taken
sin2 9=
0.231and as(MD =
0.12.The partial width of
Z
intof f
is4G F
1
'(Z f f) =
371-‘7
A.d(a2L + a R)nc ,
where n, is the number of colours of fermions
f:
1 for leptons, 3 for quarks.The predicted partial width for each
Z vv
channel is independent of sin28:
GF MZ
F(Z
vv)
=127r = 165.9 MeV(159)
(160)
using the observed value of Mz. The partial decay rates to other channels are expected
or 1: 0.503: 1.721: 2.218 for sin2 9 = 0.231. A small kinematic correction for the non-zero b quark mass leads to a suppression factor
4m,,2 ) 112 9,,2 the precise determinations in the next section, is associated with loop graphs associated with top quark exchange (see the review by Chivukula 1995), and is of the same size, about 0.988. Taking a correction factor (1 + as/7r) with as(111) = 0.12 for the hadronic partial widths of the Z, we then predict the contributions to Fz listed in Table 5. (The ti channel is, of course, kinematically forbidden.)
The measured Z width (157) is in qualitative agreement with the prediction, but above it by about 0.7%. This effect is a signal of higher-order electroweak radiative corrections such as loop diagrams involving the top quark and the Higgs boson. Similarly, the observed value of F(Z —> e+ el, assuming lepton universality, is 83.984 + 0.086 MeV, again higher by 0.7% than the predicted value of 83.4MeV. We shall return to these effects in the next section.
The width of the Z is sensitive to additional vi) pairs. Clearly there is no room for an additional light pair coupling with full strength. Taking account of all precision data and electroweak corrections, the latest determination of the "invisible" width of the Z (see the compilations by the LEP EWWG 2001 and by Langacker 2001) fixes the number of
"light" neutrino species as = 2.984 ± 0.008.
The Z is produced copiously in e+ e- annihilation when the center-of-mass energy /7 is tuned to Mz. The Stanford Linear Collider (SLC) and the Large Electron-Positron Collider at CERN (LEP) exploited this feature. The cross section of production of a final state f near the resonance, ignoring the effect of the virtual photon in the direct channel, should be section should be apeak = 127rL3e+ e-/MZ 59.4 nb, corresponding to
o-(e+e- Z° -> all) = 913,+,-
5000. (164)
cr(e+e- it+ µ--) a2
This is a spectacular value of R, which is only a few units in the range of lower-energy e+e- colliders. Of course, not all of the cross section at the Z peak is visible: Nearly 12 nb goes into neutrinos! Another 6 nb goes into charged lepton pairs, leaving apeak, hadrons = 41.541 ± 0.037 nb.
(162)